microscopic structure of the c12 giant resonances

MICROSCOPIC STRUCTURE OF THE C 12 GIANT RESONANCES.
by
Claude B rassard
BSc. Universite* de M ontreal, 1964
A D issertation P resen ted to the Faculty o f the
Graduate School o f Y ale U niversity
in candidacy fo r the degree o f
D octor o f Philosophy.
1970 .
-2-
To Nicole.
I
\
-3-
ABSTRACT
A comprehensive study, involving the development
of a number of new techniques, both experimental and
theoretical, has been carried out on the proton radiative
capture reactions leading to the first four states in C
.
Excitation functions have been extended to 36 MeV, through
the previously inaccessible excitation energy range,
utilizing the MP tandem accelerator and the on-line data
acquisition system of this laboratory.
Microscopic ^ - M a t r i x
calculations, utilizing the one-particle one-hole wave
functions of Gillet have been compared not only with these
data, but also with the gamma radiation angular distributions
measured systematically throughout the energy range studied;
it has been necessary to evolve an internally consistent
technique
for carrying out these calculations, inasmuch
as detailed study has revealed that none had existed
previously.
Measurements reported herein strongly suggest
that for certain instrumental reasons, all the previously
reported absolute cross sections for high energy radiative
capture reactions (i.e. above 10 MeV excitation) have been
over-estimated by as much as 50%.
This has important
consequences concerning the isospin purity arguments.
Apart from this, while the theoretical predictions repro­
duce both the excitation functions and angular distributions of the gamma radiation to the C
12
ground state
remarkably well, it has been found that the Gillet wave
functions do not constitute a significant improvement
-
4-
over pure j-j configurations in this respect.
Gross
discrepancies between the absolute magnitude of the
predicted and experimental cross sections to the excited
12
states in C
(far exceeding the 50% mentioned above)
suggest that more complex configurations, primarily
two-particle, two-hole, in nature play a dominant role.
The data support.the view that capture to the 4.43 MeV
state involves a giant dipole based on that state in
analogy with the dipole resonance based on the ground
state*
The capture cross section to the 3” state at
9.63 MeV in C
12
shows that this state involves relatively
little one-particle one-hole structure.
As an integral
part of the instrumental development involved in these
measurements a new electronic system capable of counting
some forty times faster than any previously reported,
on-line
gain control and pile up monitoring, and on­
line data analysis capabilities have been demonstrated.
-
5-
ACKNOWLEDGEMENTS
It is a pleasure to express my most sincere
gratitude to the director of WNSL, and my thesis director
D.A. Bromley; the work reported herein could not have
been completed without his continued interest and
enthusiastic participation.
Working under Dr. Bromley's
direction has proved very pleasurable, as well as
profitable.
Many thanks are due to my collaborators:
H.D. Shay
and J.P. Coffin have participated in the experiment,
while
H. Duncan had considerable influence on the
theoretical aspect of the work.
W. Scholz was involved
in the early phases of the experiment and calculations
prior to his departure from Yale.
It is clearly impossible to mention all my
colleagues at WNSL whose collaboration will nevertheless
be gratefully remembered; among them Drs. M.W. Sachs,
M.J. Levine, J.P. Allen, W. Thompson, A.A. Aponick and .
J.C. Overly, as well Mr. T. Harrington, and Mrs. E.B.
Fehr have contributed to various phases of this work.
Many persons at Yale have provided advice at
various critical stages; I wish to take this opportunity
to thank Drs. F.W. Firk, D.C. Lu and J. Sandweiss, for
discussions, as well as Dr. W.E. Lamb who contributed
indirectly to the theoretical chapter of this thesis
-6-
through his much appreciated lectures on quantum
mechanics.
Finally, I wish to thank the U.S. Atomic Energy
Commission and the Universite de Montreal, for financial
assistance, and the Oak Ridge National Laboratories for
the loan of a detector with which some of the measure­
ments reported herein were obtained.
-7 TABLE OF CONTENTS
ABSTRACT:
3
ACKNOWLEDGEMENTS:
5
TABLE OF CONTENTS:
7
LIST OF FIGURES AND TABLES:
10
Chapter I: INTRODUCTION:
13
L ist o f R e feren ces fo r Chapter I:
Chapter II: THE EXPERIM ENTAL METHODS:
37
39
1 - The A c c e le r a to r .
41
2 - Beam Line and Beam O ptics.
43
3 - T arget M aking Technique.
45
4 - The 5"x6" Nal(Tl) C rystal.
46
5 - The 9 " x l2 " Nal(Tl) C rystal.
47
6 - Angular D istribution Apparatus.
57
7 - The E lectron ic C ircu its.
58
8 - The Fast Counting System .
65
9 - The Light P u lser P ile-u p M onitor and Gain Stabilizer.
68
1 0 - The Data A cqu isition Com puter P rogra m .
11- New D evelopm ents in Instrum entation.
Chapter III: THE EXPERIM ENTAL RESULTS:
72
- 73
75
1 - O bject o f the M easurem ents.
76
2 - T yp ica l Spectra and Angular D istributions.
85
3 - A D iscu ssion o f Experim ental A c c u r a c y ; interpretation
o f e r r o r bars.
89
Chapter IV: THE ANALYSIS OF THE DATA:
93
1 - G eneral D escription of the WNSL Computer System.
95
2 - G eneral D escrip tion of the Data A n alysis P rogra m .
96
3 - A lgorith m fo r the Fit.
99
4 - A lgorith m for a fir s t ord e r C orrection to P ile-u p .
106
-8Chapter V: THE THEORY OF NUCLEAR REACTIONS:
109
( A ) Introduction.
HO
1 - D irection.
HO
2 - T ypical P a rticle-G a m m a C ro s s Section Calculation.
112
3 - P ra ctica l D ifficu lties with the Standard R eaction T h eories.
123
( B ) D ifferential C ro s s Sections: The Blatt and Biedenharn f o r ­
mula fo r Gamma R ays.
127
1 - Incom ing and Outgoing Spherical W aves of unit Flux.
127
2 - Incom ing Wave Am plitudes.
135
3 - The T ran sfer M atrix and the Scattering State o f the
Compound System.
139
4 - The Blatt and Biedenharn Form ula fo r Gamma R ays.
142
( C ) T ransition Am plitudes and the R -M a trix.
148
1 - Introduction.
148
2 - G eom etry o f Configuration Space: Channels.
150
3 - The Wave Function in the Channel Region: The S-M atrix.
154
4 - Wave Function in the Compound Nucleus R egion in
T erm s o f the S-M atrix.
162
5 - The D isp ersion R elation.
166
6 - Calculation o f the S-M atrix: the R -M a trix.
171
7 - Some P ro p e rtie s o f the N uclear S-M atrix.
174
8 - A pproxim ations to the S-M atrix: the W igner M any-
L evel Form ula.
( D ) R educed E lectrom agn etic Transition M atrix Elem ents:
1 - P a rticle s and H oles, Angular Momentum and Isospin.
176
—_
180
.
180
2 - M a n y -P a rticle M atrix Elem ents in term s o f S ingleP a rticle M atrix Elem ents.
183
3 - Elim ination o f the Hole.
185
4 - The E lectrom agnetic M ultipole O perators.
186
5 - S in g le-P a rticle T ransitions.
189
L ist o f R e fe re n ce s for Chapter V.
191
Chapter VI: CALCULATIONS OF THE B 1 1 ( p , y ) C 12 CROSS SECTIONS:
194
( A ) Introduction:
195
( B ) Specialization o f Form ulae from the Theory o f N uclear R eactions
to the R eaction B 1 1 ( p , y ) C 12.
198
-9 1 - F ram ew ork fo r the Calculations.
199
2 - The States o f C 12 and the Use o f G illet's Wave Functions.
206
3 - The N uclear Potential.
208
4 - Radial wave Functions.
2 12
5 - Radial M atrix Elem ents.
216
6 - R educed Widths and Boundary Value P aram eters.
218
8 - E lectrom agnetic S in g le-P a rticle M atrix Elem ents.
235
9 - Computation o f the R -M a trix.
237
1 0 - Phase Shifts and P en etrabilities.
238
1 1 - Calculation o f the T ra n sfer M atrix T.
239
1 2 - R eaction Am plitudes and the S-M atrix.
240
1 3 - The Blatt and Biedenharn Form ula.
241
( C ) The R esults o f C alculations.
L ist o f R e feren ces fo r Chapter VI.
Chapter VII: DISCUSSION.
242
260
262
L ist o f R e feren ces fo r Chapter VII.
295
Chapter VIII: SUGGESTIONS FOR FUTURE EXPERIMENTS:
297
Chapter IX : CONCLUSION:
300
APPENDICES
n -A
The F ast Counting System.
305
II-B
The Photom ultiplier Control Unit.
320
II-C
Fast D C -C oupled A m p lifier.
324
IV -A
The Data A n alysis P rog ra m .
327
V -A
P ro p e rtie s o f States Under T im e R ev ersa l.
369
V -B
N orm alization of the Spherical Waves to Unit Flux.
372
V -C
D erivation o f the Blatt and Biedenharn Form ula for Gamma Rays.
374
V -D
P a rticle s and H oles.
379
V -E
M a n y -P a rticle M atrix Elem ents in term s o f S in g le-P artile M atrix
Elem ents.
397
-
10 -
V -F
Elim ination o f the Hole.
402
V -G
E le c tr ic S in g le-P a rticle M atrix E lem ents.
408
V-H
S in g le-P a rticle M agnetic T ran sitions.
412
V -i
S in g le -P a r tid e M agnetic T ran sition s, Spin Contribution.
419
VI-A
C on version from Isospin to P roton-N eutron F orm alism .
42i
V I-B
Standard Configurations used in the T h eory and in the C alculations.
437
VI-C
The States o f C1 2 ; the use o f G illet's Wave Functions.
441
V I-D
The R -M a trix P rog ra m fo r Calculations o f D ifferential C r o s s Sections.
451
LIST
OF
FIGURES
AND
TABLES
I-1
Energy L evel D iagram o f C 12.
24
H -l
Plan o f WNSL.
40
II-2
Beam Line.
42
II—3
P r o c e s s e s in Nal(Tl) D etector System .
50
II-4
Peak Shapes.
52
II—5
Sim plified B lock Diagram o f E lectron ics.
60
II - 6
Fast Counting System with Light P u lser L og ic.
67
II—7
Light P u lser Spectrum.
69
II-A -1
P erform a n ce o f Fast Counting System .
306
I I -A -2
P u lse Shapes.
n -A -3
R elative R esolution vs Gate Opening T im e.
314
n -A -4
B lock Diagram of Fast Counting System .
316
II -B -1 to I I -B -3 :
The P hotom ultiplier Control Unit.
/"
308
321
II -C -1
F ast A m p lifier.
325
H I-1
P relim in ary R esults with the 5 " x 6 " C rystal.
77
IH -2 to III—13: Experim ental R esults obtained with the 9 " x l 2 " C rystal.
n i-1 4
T ypical Spectrum .
78
84
IH -16 to III—18: T ypical Angular D istributions.
86
IV -A -1 and I V -A -2 :
329
K eyboard arrangem ents for the Data A n alysis P rog ra m .
-
11 -
I V -A -3 (Table)
L ist o f Phases fo r the Data A nalysis P rog ra m .
338
I V -A -4 (Table)
L ist o f Subroutines fo r the Data A n alysis P rogram .
339
V -l
C om parison o f theoretical techniques.
114
V -2
S in g le-P a rticle States and j - j Configurations.
116
V -3
T opology o f Configuration Space.
152
VI - 1
R e a listic M odified Gaussian Potential fo r C 12.
209
VI-2
Neutron S in g le-P a rticle States.
210
V I-3
P roton S in g le-P a rticle States.
2 11
VI- 4 and V I-5
C om parison o f T h eory with Experim ent for y q
243
VI - 6 and V I-7
C om parison o f T h eory with E xperim ent for y±
245
VI - 8 and V I-9
C om parison o f T h eory with E xperim ent fo r y g
247
VI-12 to VI-1 4
P artial Contributions to Total C ro s s Sections, Log.
251
V I-15 to VI-17
P artial Contributions to Total C ro s s Sections, Linear.
254
C om parison o f G ille t's wave functions with pure j - j config.
249
V I-10 and VI - 1 1
-
12 -
CHAPTER I
INTRODUCTION.
-1 3 -
I
Chapter I
Introduction
A substantial amount of information is already
available concerning the forces which act between two
nucleons.
Nucleons interact through all four of the basic
interactions clearly identified thus far:
of these four
only gravity plays a very minor role in the ordinary
nucleus although dominant in the neutron stars; the weak
interactions are responsible for beta decay; the electro­
magnetic interaction dominates at large distances; and
the strong
or nuclear interaction, which is attractive
dominates when the inter-nucleon distance decreases below
several fermis.
Since the transitions induced by the weak interactions
are extremely show compared with those induced by the
nuclear and electromagnetic interactions, beta-decay will
usually not occur unless all other possible modes of decay
are forbidden.
For this reason, the weak interaction is
generally not included in the study of nuclear reactions.
The electromagnetic interaction acts rapidly enough
to compute with the strong interaction and cannot be
neglected.
It plays a role in two different ways:
the
Coulomb repulsion between protons is one of these; the
other is in the emission and absorption of radiation by
nuclear systems.
-
14 -
I
It is the very strong, short-range nulcear force
which accounts for most of the properties of nuclei, how­
ever, and the electromagnetic interactions play the role
of a small perturbation.
The nuclear force is relatively
well known, at least between two free nucleons.
We know
that it has a short range, of the order of one fermi; and
that it is identical, or almost identical between any two
nucleons.
That is to say, it is closely charge independent.
We also know that the force depends quite strongly on the
orientation of the nucleon with respect to its direction
of motion (spin-orbit force).
These qualitative considera­
tions apply to the force between two nucleons, whether they
are free or imbedded in nuclear matter; the more quantitative
aspects of the force, however such as its dependence on
the inter-nucleon distance (this dependence has been
measured quite accurately in the case of free nucleons) may
well be affected by the presence of other nucleons.
Apart from the knowledge of the forces which act
between nucleons, there is also a many-body aspect of the
problem which consists in understanding the motion of
individual nucleons in a complex nucleus from a microscopic
point of view;
slow.
progress in this direction is relatively
In fact, present nuclear structure calculations make
little use of our detailed knowledge of the nuclear force;
approximations inherent in the models used lead to ad hoc
"residual interactions" which are largely phenomenological
and in many cases have little to do with the nucleon-nucleon
-
interactions.
15 -
Furthermore, the results of these calculations
are typically insensitive to the particular choice of the
"residual interactions".
Almost all the detailed information available on
the structure of the nucleus has been obtained through
nuclear reactions.
Information on the motion of nucleons
inside the system of two colliding nuclear fragments is
obtained by observing the emerging reaction products.
Various ions can be used as projectiles, their orientation
with respect to their direction of relative motion can be
changed, and their relative velocity can be varied.
Often
there is a great number of possible reaction products,
which can be in many different states of internal motion,
and it is possible to measure the probability of formation
of these reaction products at various angles, as a function
of their orientation.
Of all the nuclear reactions possible,
those involving the emission or absorption of electromagnetic
radiation play a very special role.
The electromagnetic
interaction is an ideal probe of the nucleus, for several
reasonst
its detailed nature is well known, and it is only
a small perturbation on the system, as a good probe should
be.
Furthermore, the relatively long wave length of the
electromagnetic radiation implies a rapidly converging
multipole expansion and the difficult problem of determining
the reaction mechanism, again involving the strong interaction,
does not occur for gamma rays.
the nuclear
From the point of view of
s-Matrix, reactions involving nuclear particles
-
16 -
can give information on the channel surfaces only, in
configuration space, whereas the electromagnetic inter­
actions can probe the inside of the nucleus.
because the
This is
s-Matrix theory treats the nucleus as a
black box, and this statement does not, of course, i:.iply
that the nuclear particles do not penetrate the nuclear
surface.
Unfortunately, it is experimentally difficult to
utilize the full power of the electromagnetic probe,
because gamma radiation cannot be accelerated or detected
directly.
Experiments which involve gamma rays both as
projectile (entrance channel) and as detected reaction
product (exit channel) are for practical purposes limited
to relatively low energies, where the number of open
particle channels is small; at higher energies, gamma
rays are used either in the entrance or in the exit channel.
Difficulties with utilizing gamma rays as projectiles
arise because of the poor energy definition of the gamma
ray beams currently available.
Bremsstrahlung produced
gamma rays, which have long been the only available source
at high energy have a continuous energy distribution, which
requires a subtraction of spectra in order to extract the
yield corresponding to a given energy.
The energy resolution
is poor, and the errors are difficult to estimate and
generally fairly l a r g e . M o r e
recently, "monoenergetic
gamma ray beams have been obtained by positron annihilation
in flight
(2 )
.
a
great improvement in the quality and
dependability of the
an<3
cross section measure-
-
17 -
ments has resulted.
The option of utilizing the electromagnetic probe
in the exit channel instead, by detecting the emitted
gamma radiation, did not really exist until recently; the
particle accelerators with high enough energy to explore
the regions studied in the photonuclear work had poor
(3)
beam current and very limited energy variability.
With
the advent of the high energy tandem electrostatic accelera­
tors, however, continuous beams of excellent resolution,
high intensity, and continuously variable energy have be­
come available.
The high resolution, high efficiency detection of
gamma rays is more difficult than is the detection of
nuclear particles; except at very low energies, where the
relatively new but physically small lithium drifted
germanium (GeLi) detectors perform very well, it is
necessary to use large Nal(T^) detectors.
These detectors
(for high energy radiation) are typically 8" in diameter
by 9” long or larger; their resolution for 20 MeV gamma
rays is of the order of 5% only, and they typically are
shielded from ambient radiation with tons of lead.
This is
to be compared with a semiconductor particle detector,
typically one square inch in area or less, with a 30 keV
resolution for 20 MeV charged particles.
It is interesting to compare the two types of
experiments:
the photonuclear experiments, such as
» {"$*(£) and the capture experiments such as ({> ff i )
and
-
18 -
First, it must be emphasized that these reactions
are not the inverse of each other, in general, and that for
this reason both types of experiments must be performed.
The reactions leading to the ground states of the corres­
ponding residual nuclei are the inverse of each other,
however, and may be compared through detailed balance.
In
terms of energy resolution, accuracy, and reliability, a
further comparison of the various techniques is interesting.
The ( p t f i )
experiments typically surpass the bremsstrahlung
</,/>) experiments by an order of magnitude in resolution,
accuracy and reliability (Except for the determination of
the absolute magnitude of the total cross-section, as
noted below).
The positron annihilation technique offers
a great improvement over the bremsstrahlung measurements,
but its typical energy resolution is far from matching that
of the tandems.
Ftirthermore, the bremsstrahlung tail
which accompanies the positron annihilation peak in the
incident gamma ray beam has the following two consequences;
it substantially increases the errors at very high energy,
and it makes the extraction of deexcitation branching ratios
very difficult.
Nevertheless, it is possible to obtain excellent
results for ground state transitions with the "monoenergetic"
photons, and a substantial amount of data has been published
already, concentrating mainly on medium and heavy nuclei.
Some polarization measurements have been reported, but the
angular distributions of the reaction products are generally
not available.
-
19 -
I
It remains that the photonuclear reactions benefit
from one great advantage which the capture reactions do
not share:
they have access to the
and { jf t W )
processes, whereas only i f 3* ^ ) can be performed conveniently
with a tandem accelerator.
comparison of the
It is well known that a
and (^f*Y I) absolute cross-sections
can, in principle, yield important information on the
energy dependence of the isospin mixing^^ in nuclei.
The advent of the tandem accelerators has already
been demonstrated to mark the opening of an entirely new
era in the study of these relatively old but still very
important radiative capture experiments.
It is important,
in planning for applications of the technique to nuclear
structure studies to focus on those situations wherein
maximum elucidation of the many body aspects of the
structure might be anticipated.
A very large number of theoreticians have attempted
to find an approximate solution to the many body problem
in closed shell nuclei; by computing their excited states
as linear combinations of low-lying one-particle one-hole
configurations. (®»"7»8»9»10)
^s expected that the electro­
magnetic transitions to the ground states of these closed
shell nuclei will emphasize the one-particle one-hole aspects
of the excited states,because the operators responsible for
electromagnetic transitions are one-body operators, and
therefore it is the gamma ray transitions which must be
investigated in order to test the relevance of such calcula­
-
tions.
20 -
The calculations yet available are very approximate,
however, and somewhat inconsistent (they use experimental
energies for the configurations, but harmonic oscillator
wave functions).
Having decided that a radiative capture experiment
is ideal for studying the one-particle one-hole aspect of
the excited states of closed shell nuclei, we must focus
first on those nuclei for which maximum information might
be gained on their detailed structure.
We are helped in
this task by the data already available on the photonuclear
reactions, which have established the trends in gamma ray
abosrption cross sections.
We have learned, from these
early experiments, that the gamma ray absorption is
dominated by a broad peak around 15 to 20 MeV of excitation,
in the plots of absorption cross section as a function of
energy.
This peak is generally referred to as the giant
resonance.
In heavy nuclei, where a great number of
states contribute to the giant resonance, the number of
degrees of freedom is so large that models of a statistical
nature yield excellent results; the giant resonance takes
the shape of one or two Lorentzian peaks.
In medium-heavy
nuclei, the more recent high-resolution studies have detected
an incredible amount of structure in the giant resonance,
which seems to originate in the fact that the contributing
states are numerous, but not in sufficient number to
average out smoothly as it is the case in the heavier nuclei.
In the light nuclei, especially near closed shells or semi-
closed shells the cross section typically presents much
less structure than in the case of the medium-heavy
nuclei, reflecting the smaller number of states contri­
buting substantially to the observed transitions.
The slowly varying excitation functions associated
with the heavy nuclei and the success of collective models
in describing them suggest that comparatively little
information on the detailed structure of these nuclei
will be gained by the study of their giant resonance.
The situation for the medium-heavy nuclei is very different
their excitation functions are so complex that there is no
hope at present of understanding them in a detailed scheme
(although it is possible to interpret the variations in
the cross section as fluctuations).
It is therefore in
the light nuclei, near closed shells, that a comparison
of theory and experiment is expected to yield maximum
insight in the details of the nuclear structure, and it is
not surprising that theoreticians have concentrated there
by
far the greatest attention; the relatively simple
structure of the giant resonance in these light nuclei
constitutes an ideal test for nuclear models.
The giant
resonance was early regarded as one of the most simple
collective motions in nuclei, namely a dipole oscillation
of the imagined neutron and proton "fluids" relative to
one another.
More detailed information concerning this
collective degree of nuclear freedom was surprisingly late
in coming particularly as compared to the situations in­
volving the more complicated but lower lying, and therefore
-
22-
more accessible, quadrupole and octupole collective
inodes.
The delay, in the dipole case, simply reflected
_ the paucity of
adequate, high quality experimental data.
One of the first nuclear models intended to describe
this giant resonance in terms of the motion of individual
nucleons was devised by W i l k i n s o n ^ ^ , who interpreted the
gamma ray absorption as causing a single-particle transition
of one major shell in the nucleus; this model was subse(7)
quently refined by Elliott and Flowers' ' and by Brown and
(8)
Bolsterli' ' ; the latter introduced the schematic model,
which demonstrates that under certain circumstances the
residual interactions tend to concentrate the dipole strength
into a single state which is pushed up in energy to appear
as the giant resonance.
More recently, an impressive
number of authors have contributed to the study of the
giant resonance in light nuclei; in particular some of the
most detailed theoretical calculations now available have
(1 0 ,1 1 )
12
been performed by Gillet and Vinh Mau
in C
and
16
0
, and the predicted one-particle one-hole wave functions
have been published.
In choosing a particular nucleus for the present
experiment, one of the most important criteria was
certainly the availability of such detailed theoretical
model wave functions, and for this reason, we have concentra12
16
ted the initial efforts on C
and 0 , where a detailed
comparison of theory with experiment seemed feasible.
I'
Preliminary measurements of the reactions B (}>»$)C
12
and
-
(^>,2^)0
23 -
indicated clearly that although slightly more
difficult, the C
12
reaction was more interesting because
of the possibility of measuring simultaneously the decay
to three states of this nucleus, whereas only the ground
16
state decay could be resolved in the case of 0
with the
12
instrumentation initially available. C
was therefore
the most interesting nucleus to study, and it is the
II
,
B ( f t Y )C
reaction which constitutes the object of the
16
present report. Work on 0
and on heavier nuclei
continues in this laboratory with the much improved
instrumentation which has been designed to reflect the
experience gained in the work reported herein on C
12
.
Some limited degree of comparison of Gillet's
calculations with experiment was possible
previously
12
I I
since excellent data for the B
>ff) C
reaction already
(12 )
existed
; however, these data were limited to 14 MeV
of incident proton energy, whereas protons of up to 22
MeV are routinely obtained with the WNSL tandem.
Some
relatively poor quality 90° data, obtained with a proton
linear accelerator, also existed in the 20 MeV region;
/3 \
these data showed a peak at 20.5 MeVv ' incident proton
energy, which corresponds to one of the four l”
(T =1)
states predicted by Gillet, but which had not been
observed in the photonuclear studies.
The presence of
this resonance is a question of great interest, and it
was recognized that a high precision measurement such as
is made possible by the availability of a tandem could
not fail to clarify the situation.
It was also felt that since Cl2 is the most
-
ENERGY
LEVEL
24 -
D IAG R AM
OF T H E
MASS-12 S Y S T E M
36 M e V
18.72
C " + T>
Bn+p
9.64
7.65
7.37
Be8 + a
4 .4 4
0.00
3“
0 +
2+
0
+
12
F ig. 1-1: This m a ss-1 2 energy level diagram shows the C * 2 ground state
and fir s t three excited states, and the neutron, proton and alpha p article tresholds.
The three transitions rep orted herein and the energy range o f the m easurem ents
a re indicated. The transition to the 7. 65 M eV state is v ery weak and an upper
lim it was set on it.
-
25-
extensively studied nucleus in the periodic table, information
on the yet unobserved reactions involving C
nucleus would be extremely valuable.
12
as a compound
Accordingly, cross
sections and angular distributions for the reactions
(/>»^)
have been obtained for the decay to the ground state and
first three excited states of C
12
, as indicated in Fig. 1-1,
for incident proton energies of 14 to 21.5 MeV, and the
results of these measurements are reported herein.
It should
be emphasized that in no other work has it been possible
to measure transitions other than the transitions to the
ground state and to the first excited state, in C ^ ,
following radiative capture; as we shall demonstrate herein,
the availability of radiative capture data in the higher
excited states makes possible a much more stringent and
informative test of the one-particle one-hole wave functions,
in C1 2 ,
Preliminary measurements
(14)
were obtained for the
90° cross section, with energy increments of 200 keV, and
on the basis of the lack of structure in these excitation
functions, complete angular distributions were subsequently
measured using 500 keV energy increments.
The preliminary
measurements utilized a 5" x 6 " Nal(T^) crystal, whereas
the angular distribution measurements benefited from a
considerable improvement in resolution, through the use
of a larger, 9" x 12" crystal.
These angular distributions
were obtained by moving the crystal and its 4,000 pound lead
shielding relative to the target.
The gamma rays from the
i p * # ' ) reaction were identified by their energy; competing
-
reactions, such as
26 -
and
(f
produce gamma rays
of much lower energies than the relevant capture gamma
rays.
The construction of the beam line, vacuum systems
and angular distribution carriage presented no unusual
problems and are not reported herein.
One of the serious
difficulties which is always present in high energy gamma
ray measurements using absorption spectrometers however
is that presented by the cosmic ray background.
Because
of the large size of the Nal detectors used in this work,
the counting rate from cosmic rays is frequently higher
than is the yield from the actual experiment.
This problem
is routinely solved by surrounding the Nal crystal with an
anticoincidence shield.
Unfortunately, the detector used
herein, since i t represented, an interim solution, on loan
from the Oak Ridge National Laboratory, was not fitted
with such a shield; in order to minimize cosmic ray
problems, it was therefore essential to maximize the fore­
ground counting rate.
Higher foreground rates were desirable
for another important reason, namely reducing the necessary
accelerator time to a minimum.
The only direct limit on the counting rate in such
capture experiments at relatively high energy is always
electronic pile-up from the low energy neutron and gamma ray
background radiation.
At the highest proton energies re-
ported herein only one photon in every 10
6
emerging from
the target itself was in anyway related to the phenomena
of
interest.
difficulty.
Electronic pile-up thus presented a very real
-
27-
This difficulty was:solved by the design and con­
struction of a new, fast-counting system, capable of in­
creasing the counting rates by a factor of forty over
that attainable with the conventional double-delay-line
amplifer system; the new system offers great advantages
over the pile-up rejection circuits currently in use else­
where, because of its superior performance, and because it
does not reject pulses, and in consequence does not require
a corresponding correction.
This fast counting system also
constitutes an improvement from the point of view of
circuit simplicity; it is described in some detail in
appendix II-A.
Other major experimental problems encountered during
the present experiment include the necessity of constantly
monitoring the severity of any remnant pile-up, and, most
important, the gain of the photomultiplier tube and system.
These two problems were solved simultaneously by the use of
a gallium phosphide light pulser, which was optically
coupled to the NaI(T^) crystal; the light pulses were shaped
to be identical with gamma ray scintillations of an appropriate
energy, and the accumulated spectrum of these light pulses
automatically extracted from the system output during
on­
line operation was used by the laboratory computer to
evaluate, on-line, the pile-up and photomultiplier gain
situations.
Reports were printed periodically for the
operator's attention, and in addition, the photomultiplier
gain was controlled automatically by the computer.
This
-
28 -
on line IBM 360/44 is not only an extremely powerful tool,
but was found to be virtually essential in the data
acquistion phases of an experiment such as the one
reported herein; it has also been found that the possibility
of
on-line data analysis dr .stically reduces the time
needed to complete an experiment, by permitting the
optimization of the various parameters which prevail during
the acquisition of the data.
Automation of the radiative capture instrumentation
initiated in this experiment has continued; in particular
such items as beam current intensity, beam energy and beam
optics adjustments and detector positioning are all being
placed under direct on-line computer control in our
16
continuing studies on 0
and on heavier nuclei utilizing
our anticoincidence shielded 12" x 12" Nal system.
Another important technical difficulty, which is
related to the background problem discussed above, is the
necessity of a high proton beam definition.
An extremely
clean ion-optical system was designed by positioning the
beam defining slits on the remote side of a concrete
shielding wall, and imaging these slits on target with a
magnetic quadrupole triplet.
The accurate positioning
of the beam on target was accomplished through a remotely
monitored flip aperture in front of the target; this aperture
was removed during the actual runs, so that the beam would
strike nothing but the target itself, in the target chamber
area.
These precautions were necessary because the beam
cound not be allowed to strik any collimators in the
-
29 -
vicinity of the target, without creating disasterous
background levels.
After perfecting the experimental method as described
above, it became possible to obtain angular distributions
12
for the y
and y
capture transitions in C
with
such an accuracy as to provide a meaningful test of the
theoretical predictions.
The transition to the second ex­
cited state was too weak to be observed, and the upper
limit on its cross-section has been established herein to
be a factor of five below the
same energy.
of C
12
y
cross section,at the
The transition to the 3“ third excited state
has been observed, however, and the accuracy on its
total cross section is typically 10 %; unfortunately, the
large experimental errors on its angular distribution do
not permit as accurate an appraisal of the theoretical
predictions in this case.
Since it was not possible to determine the target
thickness accurately, the total cross sections were
initially normalized to Allas*
section data.
-q 1
^)C*2
90° cross
Unfortunately, we have since become con­
vinced that the total cross sections reported from that
work (as well as from other (/»^) work) are too large by
a factor of approximately 1.5; the reasons for
assertion are detailed in chapters II
this
and VII, and they
have to do with considerations of Compton scattered gamma
radiation in the thick paraffin neutron shields which are
invariably used in front of the detector in such experiments.
-
30 -
For the present, the absolute magnitudes of the total
cross sections are therefore open to some question.
The
figures representing our results herein have been normalized
as indicated for comparison purposes only.
A gross
correction to all such data nevertheless appears in
order, and in C
12
it would considerably improve the
(13)
agreement with the measured ( ^ 4 ) cross section'
.
It is well known that this question of the normalization
of the total cross section has very deep consequences
with regards to isospin mixing in nuclei. Firk, Wu and
(4)
Thompson
had estimated the isospin mixing to be very
large in C
12
on the basis that the (^»/0) cross section
(obtained from (/ȣ) through detailed balance) appeared
much larger than the
cross section* ; the correction
on the (/»/) cross section proposed herein would consequently decrease the amount of isospin mixing in C
in agreement with the low value observed in 0
16
.
12
The pro­
posed correction would also reduce the estimated contri­
bution of the { fi/b o ) to tlie dipole sum rule.
The most immediate result of our measurements is
that the high energy peak in the ^
and
^
cross sections
reported by Reay, Hintz a n d L e J 3 ^,
as corresponding to
a high lying fragment of the giant resonance simply does
not exist.
The total cross sections decrease monotonically,
by a factor of three, between proton energies of 14 and
21.5 MeV, for both ^
and ^
.
Their angular distributions
*
We wish to thank Dr. Firk for discussing these matters
with us.
-
31 -
are fairly constant, within errors, and are typically
peaked at 60°, with an almost vanishing cross section
at 180°.
The situation is somewhat different for the pre­
viously unobserved
^
transition; the angular distribution
is more isotropic, but less constant with energy, and the
cross section shows variations by a factor of two in
the 14 MeV to 21.5 MeV range.
rapidly as do
the
^
and
It does not decrease as
^
cross sections with energy,
and for this reason it is easier to measure at high energy.
The cross section does not vanish at 13 MeV, as our pre­
liminary measurements had suggested; however, the ^
peak
in the gamma ray spectrum simply disappears in the low
energy background, and below 12 MeV the experimental errors
become extremely large.
In comparing theory with experiment for the photonuclear
reactions, or for the radiative capture reactions, the major
difficulty has always been that the typical one-particle
one-hole calculation (and some more detailed open-shell
calculations as well^24^) approximate the nuclear potential
by a harmonic oscillator well; this harmonic oscillator well,
having infinite walls, leads to discrete resonances and
transitions.
This situation hinders a quantitative comparison
of cross sections and angular distributions.
In order to
1 17 \
alleviate this problem Boeker and Jonker'
' have computed
cross sections and angular distributions in the framework
of
R
-Matrix theory; their results were only partly
successful, because of unresolved difficulties in calculating
-
32 -
reduced widths and because of some apparent residual sign
errors in their calculations.
Numerous calculations have
also been carried out in the coupled-channel formalism
and some of them had substantial success in C
(18)
12( 19)
Unfortunately, no theoretical predictions were
12
available in C
for excitation energies above 28 MeV,
and no calculations have been done for the excited state
transitions; furthermore, the only theoretical estimates of
E2 transitions (which lead to asymetrical angular distribu­
tions) were those of Boeker
and Jonker, whose results
disagreed in sign with the experimental results.
We there­
fore decided to perform complete calculations, including
El, Ml and E2 transitions, for the ground state decay and
the decay to excite states as well, on the basis of
Gillet’s "approximation I" wave functions.
We have chosen to perform these calculations within
the framework of TZ -Matrix theory, and after much unsuccess
ful.
effort to resolve phase differences and other dis­
crepancies between the five or six major references involved in carrying out the complete calculation, we
-7
realized
that the major difficulty with this otherwise extremely
powerful approach was the lack of a complete internally
consistent, and coherent calculational framework.
We
therefore reluctantly were required to start ab initio i.e.
from basic quantum mechanics, and chapter V of the present
work constitutes the result of this undertaking.
reader will find in this chapter a complete
The
-
coherent
33 -
scheme for calculating the cross sections of
nuclear reactions, based on Messiah's "Quantum
M e c h a n ic s "
(21).
Other problems which had plagued earlier calculations in­
cluded the approximations inherent in the many-level formula
and the difficulty of evaluating reduced widths.
The
many-level formula has been replaced herein by a more
accurate matrix inversion procedure, and a technique has
been devised for computing meaningful reduced widths; this
latter point is treated in detail in chapter VI.
A one-particle one-hole coupled channel calculation
12
in C
typically involves four states and six channels;
our calculations involve sixty channels and sixty-four states,
and the results are plotted next to the data for comparison,
at the end of chapter VI.
We wish to emphasize that the
calculation involve only one free parameter, which has
some influence in the region around 34 MeV excitation only;
except for this narrow region, the predictions are unique,
including the absolute magnitude of the total cross section.
We would emphasize, as above, that the determination of
the absolute experimental total cross section remains open
to some question.
We believe that the latter should be
reduced by a factor of approximately 1.5 (which factor we
are unable to determine precisely at present), so that the
( A / . ) calculated cross section, for example, does nbt
agree very well with experiment; the graphs at the end of
*
We have not yet tackled the problem of predicting the
polarization; however, we intend to do so in the near
future. We wish to thank Dr. Firk for this recent
suggestion.
-
34 -
chapter VI are therefore somewhat misleading.
The angular distributions for the ^
transition
are predicted very well, and the discrepancies in the
shape of the total cross section are attributed mostly
to the presence of many-particle many-hole states, as
discussed in chapter VII.
in the case of the
The situation is very different
transition, where gross discrepancies
in both the total cross section and angular distribution
lead us to believe that the
transition originates
mostly from a giant dipole resonance built directly on the
first excited state, namely two-particle two-hole states.
This first excited state is probably dominated by the
or in?
configuration, and the giant
resonance
can be built on top of this state by promoting one more
particle from the
, J h
jp
sub-shell, thus reaching two-
particle two-hole states.
been introduced by Greiner
Models of a similar nature have
(25)
et al. who unfortunately
limited their calculations to the ground state transition.
Examination of the ^
transition, which contrary to the ^
transition is predicted to be much more intense than it
has been measured, lead us to believe that this 9.64 state
of C
12
has a fairly small component of one-particle one-hole
configurations.
The figures at the end of chapter VI also include
graphs showing the contributions of the various channels
to the total cross section, and a comparison of results
obtained with Gillet's wave functions (of "approximation I")
-
35 -
with those obtained by replacing these wave functions by
pure j-j configurations.
The latter results are striking,
because the differences are very small, and the important
consequences of these results in evaluating calculations
of the type performed by Gillet will be mentioned below,
and discussed at greater length in Chapter VII.
One of the interesting and perhaps surprising results
of this work is the realization of the. fact that the gamma
decay to excited states is as interesting, and sometimes
even more interesting than the decay to the ground state,
and that its study will permit much more detailed testing
of models.
Unfortunately, such measurements of decay to
excited states are experimentally very difficult, and are
feasible only in a limited number of cases with the
techniques which are available at present.
There is clearly
a need for a higher resolution, high energy gamma ray
detector, also immune to the intense background produced
in such experiments.
The construction of such an apparatus
may well not be beyond the present day capabilities.
We conclude that one-particle one-hole type calculations
with the ad hoc residual interactions
have already been
pushed too far and we believe that at this point progress
will come from radically new techniques and...ideas, rather
than from improvements on the same theme.
There is ample
evidence that if the one-particle one-hole model works at
all, it is because the states which are being described are
indeed, for the most part, almost pure one-particle one-hole
-
36 -
)
in character.
In trying to account for the observed
experimental facts by mixing the one-particle one-hole
st&tes, the success has also been substantial, although
C
12
can be considered a failure from this point of view.
We have been led to believe strongly, that the impurities
of more complex configurations in these primarily oneparticle one-hole
states include a great number of
very small contributions, in particular from very highlying shells, and therefore that the particle-hole
expansion would converge very slowly.
From this point of
view, it is not clear whether an open-shell calculation for
C
12
for example would constitute a substantial improvement
over a lp-lh calculation, because of the truncation in
the single particle levels which is inevitable in such a
treatment.
We have thus demonstrated not only
that the
serious experimental difficulties inherent in radiative
capture studies at energies well above the giant resonance
can be surmounted but also that the data thus obtained do
indeed provide an interesting and a powerful probe for
the nuclear structure involved.
-
The techniques, both
experimental and theoretical, which have been evolved
herein are directly applicable to a broad range of
experimental situations and will be exploited in a continuing
experimental program in this laboratory.
-
37 -
LIST OF REFERENCES FOR CHAPTER I
(1)
B.C. Cook, J.E.E. Baglin, J.N. Bradford and J.E.
Griffin, Phys. Rev. 143, 724 (1966)
(2)
B.L. Berman et al., Phys. Revv 162, 1098 (1967)
R.L. Bramblett et al., Phys. Rev. 148, 1198 (1966)
(3)
N.W. Reay, N.M. Hintz and L.L. Lee, Nucl. Phys.
44, 338 (1963)
(4)
C.P. Wu, F.W.K. Firk and T.W. Phillips, Phys. Rev.
Letters 20, 1182 (1968)
(5)
M.A. Kelly, PhD Thesis; UCRL 50421 (1968)
(6 )
D.H. Wilkinson, Physica XXII, 1039 (1956)
Ann. Rev. Nucl. Sci. 9, 1 (59)
(7)
J.P. Elliott and B.H. Flowers, Proc. Roy. Soc.,
242 A, 57, (1957)
(8 )
G.E. Brown and M. Bolsterli, Phys. Rev. Letters,
3, 472 (1959)
(9)
N. Vinh MaU and G.E. Brown, Nucl. Phys. 29, 89 (1962)
(10)
V. Gillet and N. Vinh Mau, Nucl. Phys. 54, 321 (1964)
(11)
V. Gillet, PhD Thesis, Universite^ de Paris (unpublished)
-
38 -
(12) R.C. Allas, S.S. Hanna, L. Me^fer-Schutzmeister and
R.E. Segel, Nucl. Phys. 58, 122 (1964)
- (13)
R.C. Morrison, PhD Thesis, Yale University, (un­
published)
(14;
C. Brassard, W. Scholz, and D.A. Bromley, Proceedings
of the Conference on Nuclear Structure, Tokyo, Japan
(1967) p. 139
(17)
E. Boeker and C.C. Jonker, Physics Letters 6 , 80 (1963)
(18)
B. Buck and A.D. Hill, Nucl. Phys. A 9 5 , 271, (1967)
(19)
M. Marangoni and A.M. Saruis * Nucl. Phys. A132, 649,
(1969)
(20)
J.H. Fregeau, Phys. Rev. 104, 225 (1956)
(21)
Albert Messiah, Mecanique Quantique, DUNOD, Paris
(1962)
Quantum Mechanics, North Holland Pub., Amsterdam.
( 22 )
H.J. Rose and D.M. Brink, Rev. Mod., Phys. 39,
306 (1967)
(24)
D.J. Rowe and S.S.M. Wong, Physics Letters 30B, 147
(1969)
S.S.M. Wong and D.J. Rowe, Physics Letters 30B, 150
(1969)
(25)
D. Drechsel, J.B. Seaborn, and W. Greiner, Phys. Rev.
Letters 17, 488 (1966).
-
39 -
CHAPTER n
THE EXPERIMENTAL METHODS.
NUCLEAR STRUCTURE LABORATORY
F IR S T
FLOOR
PLAN
-4 0 -
F ig .I I-1 : Plan o f WNSL; the experim ent was p erform ed in target room 1, on the
45 d egree left beam line.
-
II-l
41 -
The Accelerator
The experimental work presented herein was performed
at the Wright Nuclear Structure Laboratory utilizing the
proton beam of the MPl - Tandem Accelerator.
Fig. (II-l)
is a plan of the building showing the accelerator, the
four target areas and the control room equipped with an
on-line IBM 360/44 computer.
The Tandem accelerates a beam of several micro­
amperes of protons to a maximum energy of approximately
22 MeV; heavier ions can be accelerated to much higher
energies.
It is basically a linear machine in which a
central electrode is electrostatically charged to a very
high positive potential (up to more than eleven million
volts) by the Van de Graaff belt charging principle.
Negative ions are injected at the low energy end, become
positive in the central electrode through collisions in a
"stripping canal" and emerge at the high-energy end,
having been accelerated twice by the same potential.
The
Tandem shares with other Van de Graaff accelerators an
excellent energy resolution (
/ £
- /0 ^ )»
And
stability, achieved through a feedback mechanism which
enables the momentum analyzed beam to precisely control and
maintain the voltage on the terminal.
The IBM 360 model 44 computer was extensively used
in this experiment, both on-line and off-line.
During the
experiment, the data were accumulated in the computer
B E A M L IN E FOR T H E
R A D IA TIV E CAPTURE E X P E R IM E N T S
3 0 ° LEFT
B E A M LINE
SHIELDED
BEAM
DUMP
TARGET
CHAMBER
FLIP
APERATURE
45° L E F T
bJ
BEAM
LINE
FOCUSING Lf
QUADRUPOLEi
MA G N E T I C
U SLITS
COIL S T E E R E R S
PARAFFIN
SWITCHING
MAGNET
9"xl2" N a K T f )
C R Y S T A L WITH 57 AV P
PHOTOMULTIPLIER
F ig. II-2 : Sim plified view o f the beam transport sy stem , and o f the target and d etector area.
-
43 -
memory and displayed; checks of the severity of pile-up
and of the stability of the detector, were automatically
performed and spectra obtained under unacceptable
conditions were rejected.
The light pulser system used
for monitoring pile-up and stability of the detector
will be described later this chapter.
The computer was
also used off-line for data analysis and for theoretical
calculations.
II-2
Beam Line and Beam Optics.
The experimental arrangement is shown in greater
details in fig. II-2.
After the beam emerges from the
accelerator, it is analyzed by a 90° magnet; at the image
slits of this analyzing magnet, the feedback signal which
controls the terminal voltage is produced.
Various focusing
and steering elements guide the beam to the switching
magnet, where it can be directed to any one of the various
experimental areas.
The switching magnet is shown on fig. (II-2).
The
beam is focused through a set of slits which is used to
define it.
The focusing quadrupole images these slits on
the target, after the earth*s magnetic field has been
corrected for with the magnetic steerers.
Finally, the
beam is dumped in an insulated and shielded Faraday Cup.
It is crucial, in an experiment of this type, to
minimize the neutron and gamma ray background produced by
the beam.
High energy background (which would lie in the
-
44 -
region of interest in the gamma spectrum) must be
eliminated as completely as possible, since it would
interfere directly with the measurements, but low energy
background is also harmful, because it increases electronic
pile-up,,and pile-up is almost always the limiting factor
in experiments of this type.
Therefore , clean beam optics
and a well shielded beam dump are important.
The first,
and perhaps most important criterion is that the beam
should never be allowed to strike a material with a high
@ -value for the (
y
) reaction.
Furthermore, as little
beam as possible should strike areas which are not shielded
from the detector, as in the immediate vicinity of the
target (target holder, etc.); for this reason, no beam
defining aperture can be placed in front of the target.
Instead, the beam is defined by a slit system which is
located on the remote side of a heavy concrete shield wall.
In order to permit' positioning of the image of these
slits accurately on target, a remotely controlled aperture
is flipped in for beam tuning, while during the experiment
itself it is replaced automatically by a slightly larger
aperture to provide a continuous monitor on the beam
position.
Typically 0.02% of the beam or less
strikes this fixed aperture, which is constructed of
0.050" tantalum to minimize background.
The size of the
beam spot was approximately 0 .1 " x 0 .2 " and could be
repeatedly positioned with an accuracy much better than
0 .0 1 ".
-
45 -
The beam dump was simply shielded with paraffin
and lead until it produced no appreciable background
compared with foreground counts from a B // target of
thickness.
A suppressor ring was provided to prevent the
secondary electrons emitted by the target from being
collected in the Faraday cupj it was operated at a potential
of 5 kV, so that even the highest energy electrons could
not traverse it.
II-3
Target Making Technique.
Relatively thick boron targets can be produced rather
easily, provided carbon impurities can be tolerated, and
when local homogeneity is not too important.
The technique
consists in mixing the boron powder with some "alcodag" (a
graphite dispersion in alcohol similar to aquadag and
commercially available from Acheson Chemical).
The mixture
is then dropped on a glass slide and allowed to dry.
When
the slide is submerged in water (preferably in hot water)
the film will separate spontaneously, provided it has not
been allowed to dry too completely.
The target thickness can be determined by weighing,
but because of unevenness
the accuracy is very poor and
it becomes necessary to normalize the total cross-section
on a different way.
It is also necessary to insure that
the beam always strikes exactly the same area on the
target surface.
Targets were produced successfully in this manner,
in thickness
ranging from 1 mg/cm
2
to approximately 5
-
2
mg/cra ; Enriched B
II
46 -
(99%) was used and the carbon
binder impurities were typically around 25%.
The carbon impurities could be tolerated in this
experiment because of the very low { D -Value of the gamma
ray producing reactions associated with C
12
as a target.
The performance of the targets manufactured with the
technique described above was excellent; no deterioration
was ever noticed, even under the most severe conditions of
high beam current density.
II-4
The 5” x 6 ” Nal(T^) Crystal
Lithium drifted Germanium detectors (GeLi) show
outstanding performances for the detection of low energy
gamma rays, and are used extensively forthis purpose.
Be­
cause of their small physical size, however, they are
totally inadequate in the detection of high energy gamma
radiation encountered in experiments of the type reported
here; neither their resolution, nor their detection
efficiency can compare with those of a large NaI(T<f)
detector,
above 20 MeV.
A complete excitation function for the reaction
B ( / ; ^ f ) a t 90° was obtained with a 5" x 6 ” NaI(T^)
crystal detector, in a preliminary study.
This detector
was small for the high energy gamma rays involved and it
was very difficult to obtain reliable data on the decay
to the third excited state.
14 MeV to 22 MeV was covered
The proton energy region from
in steps of 200 keV, with a
-
47 -
target of comparable thickness.
No structure was observed
in the total cross-section which appeared to vary smoothly
with energy, within errors, and these results justify the
0.5 MeV energy increments used in subsequent measurements.
This detector is fitted with an Amperex XP1040
photomultiplier tube, and did not include a built-in light
pulser; this was a very serious disadvantage.
The anode
is D.C. coupled to avoid baseline shifts at high counting
rates.
The detector's performances include a risetime of
10 ns (due to the crystal itself) and a resolution of 9%
137
for a Cs
source, which is rather good for a crystal of
137
Cs
. size with an XP1040 photomultiplier. We would
stress that the photomultiplier was run at relatively low
gain (1 0 ^) and that the linear signal was taken at its
anode, and not at one of the dynodes as it is customary.
The reasons for this are mentioned in the discussion of
the electronic instrumentation used.
The divider string
was zener-diode stabilized, and required several hours to
stabilize.
II-5
The 9" x 12" Nal(T-l) crystal.
The cross-section and angular distributions for the
which are reported in detail herein
were obtained with a 9" x 12M NaI(T^) crystal, viewed by
an Amperex 57 AVP 9" photomultiplier tube.
The crystal
was surrounded with a cylinder of paraffin and shielded
with 4000 lbs of lead.
Most of the measurements were taken
-
48 -
at a distance of 35" between the target and front
crystal face, with a 6 M diameter aperture, and with 16"
of paraffin in front of the crystal to prevent
fast
neutrons from reaching it.
The risetime
of the detector was of the order of
40 nanoseconds, partly due to the crystal and partly
due to the photomultiplier•
It is worth noting that the
photomultiplier risetimes quoted in specification sheets
do not apply to their use with Nal crystals.
This follows
because these risetime measurements are performed for in­
finitely
short light pulses, whereas the Nal crystal is
essentially an "infinitely long" light source.
The risetime observed for a N a I ( T ^ ) pulse is related
instead to the output pulse length in response to
an infinitely short light pulse; some manufacturers are
beginning to quote this as well as the "risetime".
The resolution of the detector, for Cs
extremely bad, of the order of 40%..
137
, was
This was entirely
due to the characteristics of the photomultiplier. It was
137
possible to obtain a resolution of 10% for C s
by using
6 three-inch photomultipliers, but additional difficulties
with gain stabilization and other circuit problems prompted
us to keep the 57 AVP; the resolution at high energy does
not depend critically on the photomultiplier and was
12
adequate in the case of C
. Future experiments in this
laboratory will use the more sophisticated combination of
smaller photomultipliers, with a new 1 1 V
x 1 2 " crystal
-
49 -
I
which has been designed on the basis of experience gained
in the present studies and is now assembled.
A crucial
problem which arises in the analysis of all high energy
gamma ray spectra is the determination of the peak shapes.
We shall treat this problem here, since it is really part
of the detector's performance.
Large crystals have
essentially 100 % efficiency especially at very high
gamma ray energies.
This is because the pair production
cross-section is high enough that the probability of a
gamma ray traversing 12" of Sodium Iodide without an inter­
action is extremely small.
On the other hand, the gamma
ray will not necessarily deposit all its energy in the
crystal; in fact, it is quite probable that some-^energy
will escape under the form of secondary gamma radiation.
This effect produces a tail on the low energy side of the
peak.
Peak shapes are difficult to study, for several
reasons.
It is impossible to find a radioactive source
with such high energy gamma rays, so that an accelerator
must be used to produce them.
Most reactions do not produce
monoenergetic gamma rays, and are therefore useless for
studying peak shapes.
It seemed possible to obtain some
results with the reaction H
3
L
4
)He , but the counting
rates are fairly low for a reasonable target thickness, and
not having an anticoincidence shield in the present studies
\ we were unable to see any peak above the cosmic ray back­
ground; if we had had an anticoincidence shield in this
LEAD SHIELDING
G E O M E T R Y OF A T Y P I C A L D E T E C T I O N S Y S T E M
F O R H I G H - E N E R G Y G A M M A RAYS
F ig . II-3 : This diagram shows three types o f gamm a ra y events counted in the cry sta l: no. 1 is a n orm al
count; no. 2 was included in the d e te cto r 's so lid angle , but was Com pton sca ttered by the paraffin; no. 3 was not
included in the d e te cto r's so lid angle in itially, but was Compton sca ttered into it by the paraffin. Events o f type
2 and 3 produce a low energy tail on each peak o f the gamma ray spectrum .
-
51 -
intermediate system we would not, in any case, have had
sufficient beam time to perform the measurement.
The
situation is made much worse by the fact that peak shapes
depend critically on the geometry, in particular on the
size of the lead collimator used in front of the crystal,
and on the thickness and nature of the absorbers used in
front of it (walls of target chamber, paraffin etc.).
This
latter has not been adequately recognized in previous
work.
Fortunately, the response of N aI(T^) crystals to
high-energy gamma radiation was measured accurately at
Lweimore, by B.L. Berman.*
The measurements were made
relatively easy, at least in principle, through use of
monochromatic gamma rays from positron annihilation in
flight.
The experiment consisted in sending gamma rays
of known energy parallel and very close to the axis of
the test crystal,and measuring its response.
Consistently
it was found that the low energy tail could be fitted very
well with an exponential.
We have only learned of those
measurements very recently, and it is in part pure good_
fortune that all our data were analyzed with a Gaussian,
matched to just such an exponential low energy tailI
The exact peak shapes used for the fits were deter­
mined from the N ^ ( y 6,^) 0^
and B /7( ) C ^
experiments; we
have found an empirical parametrization which suffices
over a 10 MeV interval with the same set of parameters,
and gives excellent fits.
the N^(y 4 ^ ) 0^
The
data, and especially
data, revealed that the exponential low
energy tail levels off and becomes fairly constant at
*B.L. Berman - private communication.
(RELATIVE)
COUNTS
OF
NUMBER
(RELATIVE)
COUNTS
OF
NUMBER
ADC C H A N N E L N U M B E R
F ig II-4 : R esp on se of the detector system o f fig .I I -3 . The area under the heavy
line corresp on d s to the total number of counts; the area between the faint and heavy lines
rep resen ts the counts originating from Compton sca ttered gamma radiation.
-
53 -
about 10 % of peak height (for our own detector geometry)
down to at least six MeV below the ground state transition
peak, where we loose it in the intense low energy background.
This behavior initially appeared absolutely inconsistent
with our understanding of the various processes in the
crystal.
It is only when calculating the probability
of interaction of the gamma ray with the paraffin absorber
that the nature of the low energy tail became clear; the
probability for Compton scattering in 16" of paraffin is
very high, about 40%.
Many of the Compton scattered gamma
rays are detected by the crystal, but at lower energies.
At the time of recognition we had no way of proving this;
however, the experiments of Berman confirm that when the
gamma rays are well collimated and when no absorber is
used, the constant low-energy tail does not exist.
The following computation of the total cross-sections
suggested itself as most nearly correct:
we took into
account the correction due to the paraffin absorber and
the target chamber walls, using the tabulated total
absorbtion coefficients; to take into account the fact
that some of the Compton scattered gamma rays will be
counted by the crystal and constitute the low energy tail
of the peak, all the counts which lie above the exponential
tail were considered spurious and were not added with the
peak.
Fig. II-4 shows the peak shapes for 35 MeV and 20
MeV gamma rays.
The heavy line represents the peak shapes
used for fitting, and the thin line defines the area under
which we integrated.
-
54 -
Fig. II-3 shows the three basic processes which
may occur, from the point of view of determining the
peak shapes.
Gamma ray # 1 is detected in the crystal
without any interaction in the paraffin, and contributes
to the area under the faint line in the peak shapes of
fig. II-4.
Gamma ray #2 , which was headed towards the
crystal, interacted in the paraffin but was counted in
the crystal as a lower energy gamma ray; assuming that a
correction is performed for the total absorbtion in the
paraffin, this gamma ray should not be counted in the
integration under the peak.
Gamma ray #3 was Compton
scattered in the crystal and will also appear in the tail;
it should not be counted because it was headed outside the
solid angle defined by the lead collimator of the crystal.
Although the exact values of the parameters used
to produce the peak shapes is of no intrinsic interest,
(this particular detector assembly has since been dis­
mantled) we believe that the algorithm used to produce
the peak shapes would probably perform as well with other
crystals, and that it may be useful to describe it; we
shall not try to give it in closed form, but rather to
make it suitable for computer use.
A Gaussian is computed first, and its width is
assumed to be proportional to the square root of the
number of primary photoelectrons collected in the photo­
tube, that is to say proportional to the energy of the
gamma ray.
For a detector with a better photomultiplier
-
55-
system, it would probably be better to assume that there
is an intrinsic resolution which should be added (in
quadrature) with this energy dependent part of statistical
origin.
The Gaussian is replaced at low energies by an
exponential tail, to take into account the possibility
of some energy escaping the boundaries of the crystal.
The criterion used for matching is that of equal values
and first derivatives at the junction.
Therefore, only
one free parameter is introduced by the tail; the relative
height at which the connection is made.
The first step is to generate the following function
O < X C X0 <r /
X > X,
where
d
X
A
is inversely proportional to the energy, and <20 and
are adjusted so that the junction X0 occurs at a height
-d/a
given by:
is the energy of the gamma ray, and
which depends on the crystal.
peak at
X
) 1
if is a parameter
The function
= 1 , and the height of the peak is unity; it
can be easily scaled
for peak-fitting purposes.
The contributions of the first escape and second
escape peaks are then folded in in the following way:
where K
&
is a parameter which depends on the crystal, and
represents the rest mass of the electron:
S = 0 .5/1 J £ ( M z V )
-
56 -
In the energy region we have explored, and with our
fairly poor resolution, the peak shape was very insensitive
to o( , but we have kept it for completeness.
The second
escape peak was not added, because we think that in such
a large crystal, at least one of the 511 keV gamma rays
resulting from positron annihilation is detected; this is
because when one of the annihilation gamma rays escapes,
it is usually because it is coming back to the front of
the crystal, in which case the other one sinks deeply
in the crystal, and will amost surely be detected.
This
argument depends on having a very large crystal and tight
collimation; we do not really expect it to hold for a
9" diameter crystal with a 6 " aperture, since the
photoelectric absorbtion coefficient for Nal is 0.3 cm”^.
For ther.high energies involved in this experiment, however
the peaks are not resolved and it does not really matter.
We integrated under the peak shape
Cx)
in order
to determine the total number of counts under a peak in
the spectrum.
This peak, however, was not used for fitting,
because the tail of Compton scattered gamma rays has not
yet been added.
The number of ways in which this can
be done is essentially infinite; the only merit of the
following scheme is that it fitted our spectra very well.
Let yo be the maximum value of the function ^
let j G
tail:
(x) , and
be the value of the desired low energy constant
the function
defined below was added to ^ , ( X )
-
from
57 -
X = 0 to the top of the peak of
This function goes to yC* at X = - «>o and to zero at
the peak of ^
ends.
, with a vanishing derivative at both
Thus, the final peak
shapes have a continuous
first derivative everywhere.
We should stress at this
point that we do not expect the constant low energy tail
to extend all the way to zero energy, but to be cut off
at a point which depends on the geometry of the collimator,
the thickness and size of the absorber in front of the
crystal, and other less important factors.
A satisfactory
measurement of these effects would be very difficult to
perform, but the results of a Monte-Carlo calculation
could be quite interesting, and relatively easy to
obtain,
II- 6
Angular Distribution Apparatus.
In order to measure the angular distribution of
the gamma rays, it was necessary to move and precisely
position the crystal and its lead shielding at various
angles around the target.
One method consists in taking
excitation functions at various angles; in this case,
the detector assembly is moved only once for each angle,
and an overhead crane would have been perfectly suitable
as a source of crystal motion.
-
58 -
It is much more desirable, however, to take a
complete angular distribution at a given energy before
changing the beam energy, because this procedure minimizes
systematic errors in the angular distributions, and it
substantially decreases the time spent at tuning the
accelerator.
(In our case, the average counting time
spent on a run was 20 minutes, and optimizing
the beam
transmission required approximately 15 minutes per energy
step, since the steps were fairly large (0.5 MeV).
These
considerations prompted us to build a motorized, remotely
controlled carriage capable of accurately positioning
the five thousand pound detector without human intervention.
The accuracy is better than 0.5 degree and the change of
angle is completed within less than a minute.
II-7
The Electronic Circuits.
A considerable amount of effort went into the
design of new electronic circuits for this experiment.
All efforts were centered on solving the problems of pile
up, and therefore increasing permissable counting rates.
Two major problems, which did not plague other groups
involved in similar experiments, were the cosmic ray
background and the
paucityQf beam time.
Cosmic rays were
a major problem since we did not have an anti-coincidence
shield at the time; and we could only count on an average
of two days of beam time per month, because of the many
-
59 -
very active research groups at WNSL.
The extreme severity of the pile up problem for
high energy capture experiments is essentially a conse­
quence of the comparatively weak strength of the electro­
magnetic interaction; at the very high energies of excitation,
there are thousands of widely open particle channels, and
most of them lead to excited residual nuclei which subse­
quently de-excite emitting several gamma rays of low energy
which contribute to the background radiation.
The neutron
channels, in particular are very harmful because the
neutrons themselves are counted efficiently in the crystal.
Typically, above the giant resonance, one count in a hundred
millions originate in a
) reaction, all the other
constituting the low energy background.
These difficulties prompted us to develop more power­
ful counting techniques for high energy gamma rays, and
we think that these methods will remain the ultimate in
detection of the high energy gamma rays with Nal crystals.
These techniques are not important for the understanding
of the physics involved, and are of interest mainly to
people engaged in similar experiments.
The detailed
description of the circuits has therefore been put in
appendix II-A where they are discussed in some details.
Diagrams of the various circuits that it was found necessary
to design and to build locally will also be found in
Appendix II-B.
S I M P L I F I E D F L O W D I A G R A M OF E L E C T R O N I C S
AND E L E C T R I C A L C O N T R O L S Y S T E M S
I
Ci
0
1
F ig. II—5: A ll the con trol system s and e le ctro n ic equipment shown in this diagram , other than the IBM 360
com puter and its peripheral hardware w ere designed and built as part o f this experim ent; the fast counting system
is detailed in fig . II - 6 and appendix IIA.
-
61 -
Fig. 115 is a simplified diagram of the electronics
and electrical systems.
It shows all the most important
logical connections in terms of an information flow
diagram.
The fast counting system will be treated in
greater details below; in this section we shall attempt
to give some indication of the nature of the information
which is exchanged between the various systems of the
diagram.
Basically, all the information originates in the
Nal crystal, and after considerable treatment it is
stored on the computer's magnetic tape.
It will be
retrieved from the tape by the data analysis program,
which will be described in the chapter on data analysis.
All the systems mentioned on Fig. II-5 serve the purpose
of treating this information, or they establish certain
conditions which must prevail during the measurement.
The photomultiplier detects light pulses from two
different sources; the Nal(Ti’) crystal and the light
pulser.
The light pulser is a fast gallium phosphide
visible light source which is driven by a suitably shaped
pulse, so that its light output closely resembles that of
a gamma ray scintillation of comparable intensity.
Its
purpose is twofold; we wish to obtain a quantitative
measurement of the effects of pile up, and to stabilize
the gain of the photomultiplier tube.
A more complete
description of these important aspects of the experiment
will be the object of a separate section.
The risetime
of the pulses at the output of the photomultiplier tube
-
62 -
is approximately 35 nanoseconds, and the fall time is the
characteristic decay time of Nal(T^) scintillators, i.e.
280 nanoseconds.
The pulses of the photomultiplier are fed to the
fast counting system, which rejects all pulses of low
intensity and accepts only the relatively rare pulses
of very high intensity.
The low energy pulses come from
the gamma rays of reactions such as ( f t , « y ) ,
(ft
( ft» f t]f)
( f t > d ]f) etc., as well as from the direct detection of
slow neutrons in the crystal either through a activation
or inelastic scattering, and are of no interest in this
experiment.
The high intensity pulses originate in the
(ft* ^ f) reaction which we are interested in, and from cosmic
radiation.
For each of these large pulses, the fast counting
system produces a 3-microsecond pulse with a height propor­
tional to the charge output of the photomultiplier during
the first 100 nanoseconds of the duration of the fast
pulse, and this information is fed to an analog-to-digital
converter (ADC) of the computer interface, and used a as
gamma ray energy measurement.
The total counting rate
(above 2 MeV) is usually of the order of 200,000/sec or
more, and the counting rate of accepted (i.e. "large")
pulses is kept around several hundred per second.
None
of the interesting pulses is missed or rejected, except
for the very few and negligible cases involving pile up o f
two of the rare large pulses themselves.
At the beginning of each run, preliminary information
such as the run number, the beam energy and the detector
-
63 -
angle are entered in the computer memory through the
I
typewriter.
The run is started manually at the
computer interface, and the data acquisition program
proceeds to store the information supplied by the ADC.
The gamma ray events are stored temporarily in a 1000word array, and the light pulser events are stored in
a separate part of the computer memory.
When a certain
pre-determined number of light pulses events has been
accumulated, the severeness of pile-up and the stability
of the photomultiplier gain are automatically checked; the
results of this test are printed by the typewriter;
if they are favorable, the counts are transferred to
permanent storage.
At the end of the run, the permanent
storage is dumped on the magnetic tape
with the values
of the various parameters indicating the conditions under
which the run was performed.
The checks on pile-up and
stability occur concurrently with the accumulation of data;
in addition, the various spectra which are accumulated are
displayed on the computer's display unit and updated a t regular intervals.
When the data acquisition program
finds that the gain of the photomultiplier has shifted,
the correction to the photomultiplier's high voltage is
computed and the computer interface issues an analog signal
which is amplified by the photomultiplier control unit
and applied to the photomultiplier circuit.
It is crucial
to keep the photomultiplier gain constant to within 0 .1 %
during a run, because the broadening of the peaks resulting
-
64 -
from a shifting gain would ruin the accuracy of the data
analysis.
On the other hand, since DC coupling has been
used throughout the electronics (to eliminate baseline
shift at the extreme counting rates involved) the photo­
multiplier has to be used at relatively high (> 10 micro­
amperes) anode current to minimize zero shift in the electronics,
and the experiment would not be feasible without constant
checks and occasional readjustments of the gain.
The pulsed light source is powered by a mercury
relay which is triggered by the beam current integrator,
so that the shape of its pulse height spectrum is a faithful
indication of the severeness of pile up.
This precaution
was taken because the beam current is not always exactly
constant, although it is typically stable to within + 10 %
during a 20 minute run.
The tuning of the proton beam has been facilitated
by the creation of three beam control systems:
the beam
steerers, the flip aperture and the slit monitoring system.
The first two have already been mentioned in the section"
on beam handling, and their purpose is to permit a very
accurate and reproducible positioning of the beam.
The
slit monitoring system consists in a set of remotely con­
trolled reed relays which enable us to read, in the control
room, the current on any combination of slits and apertures
in> the beam line.
Focusing can thus be achieved with great
precision, and this is crucial, in view of the importance
of a clean ion-optical system.
-
II - 8
65 -
The Fast Counting System.
The fast counting system is the object of a separate
appendix, where a detailed discussion of the various theoreti­
cal and technical aspects involved can be found.
We simply
mention that its use can improve counting rates by a factor
of 40 (over a double-delay-line amplifier) for high energy
gamma rays detected with a Nal(T^) scintillator, and that
no pile-up rejection is involved; the number of counts does
not have to be corrected.
So far we have stated that the light pulser pulses
are essentially undistinguishable from the gamma ray
pulses; it may therefore not be clear how they can be
separated and stored in different arrays in the computer.
Discrimination between the two types of events is achieved
through timing.
A pick-up coil on the light source triggers
a discriminator which produces a light pulser event pulse,
and suppresses the gamma ray event pulse which is present
whenever a linear signal has been accepted to the linear
gate.
The computer can recognize these logical signals and
use them to properly route the information in the ADC.
It
is important that no cross-talk exist between the two types
of events; the light pulser peak is intentionally set at
the same peak height as the most interesting gamma ray
pulses.
We have carefully checked that no light pulser
counts ever go into the gamma ray spectrum and vice versa.
The pulses from the photomultiplier anode are first
amplified by a factor of five in a direct-coupled pre-
-66-
amplifier.
The purpose of this amplifier is two fold*
it reduces the load on the photomultiplier and provides
the high output impedance which is needed at this point.
(The photomultiplier anode is back-terminated with 50
ohms of course).
This high impedance drives a shaping
line which is terminated with a suitable resistance, the
criterion being that the scintillation pulses be brought
back to zero when the reflection is added to the original
pulse.
This is not equivalent to a clipping line, which
would be terminated by a short circuit and would therefore
cause overshoot.
The pulse is then split in two parts;
one is used as a logic signal, and the other consititutes
the linear signal.
The logic signal is amplified and
triggers a discriminator which opens the gate for the linear
signal.
The discriminator level is adjusted so that only
the largest pulses will open the gate; the small pulses
are of no interest in this experiment.
At this point it
is worth mentioning that the criterion for the gate opening
/
could be a coincidence requirement, in the case of a (^»^") >
or (/,°^) experiment, for example.
A delay line is
used to increase the gate dead time so that it will not open
a second time if the discriminator triggers more than once.
Upon opening, the gate produces a logic signal which is
used to externally trigger the ADC.
Because the system is direct-coupled, sixty cycle
line noise which would be eliminated in an AC-coupled
system will be superimposed on the linear signal at the
input of the ADC.
Zero shifts which do not exist in AC
THE
W IT H
FAST
THE
C O U N T IN G
L IG H T
SYSTEM
PULSER
LO G IC
FROM
PHOTOMULTIPLIER
ANODE
50&
PA S S I V E
LINEAR
FANOUT
D E L A Y LINE
A V (SUICIDE)
L G 102
INTEGRATING
LINEAR GATE
DELAY
LINE
V
~ 50 n s
V V
LINEAR
ADC
ADC
TRIGGER
GAMMA
DELAY
LINE
(ADJUST)
TO
RAY E V E N T
LIGHT P U L S E R
EVENT
A
SP E C I A L
TERMINATION
~I 3 O H M
GATE
OPEN
FAST
PREAMP
DC
T 101
DISCR.
-TRI GG
n
DELAY
LINE ^
~ 150 ns
A
F R O M LIGHT
SOURCE'S
T 101
DISCR.
-TRIGG
=T=
<
PICK-UP COIL
DELAY
CAPACITOR
~ 5/iS
F ig. II-6 : The fast counting system estim ates the energy o f the gamm a ra y s a ccord in g to the leading edge
o f the corresp on d in g p u lses, and it is capable o f counting up to forty tim es fa ster than a d ou b le-d ela y -lin e am pli­
fie r . The gamm a ray event and light pulser event pulses p erm it the routing o f the gamma ray and light pulser counts
in different sp ectra . The T101 and LG102 m odules a re m anufactured by EGandG.
i
C5
-3
I
-68-
coupled amplifiers may also become very important in DCcoupled electronics.
It is unfortunate that at the
highest counting rates, the best performance can only be
obtained with completely direct-coupled electronics.
For this reason, we decided to attempt a direct solution
of the noise and zero shift problems associated with
the much more sdphisticated direct-coupled electronics.
Solving the noise problems involved such things
as twisting delay cables into a knot to eliminate in­
ductive coupling (even doubly shielded
cables pick up
noise) and insulating the special termination (it must
float)•
The zero shift problem was more difficult to
solve, but careful design of the fast preamplifier re­
duced it to less than 1 channel / ° C
or 1 channel per day.
The LG-102 linear gate and stretcher and the T101 dis­
criminators are commercially available from EG & G.
The
remainder of the equipment described in fig. H -6 was
designed and built as part of this experiment; the circuit
diagram of the fast DC-coupled preamplifiers is reproduced
in appendix II-C.
II-9
The Light Pulser Pile-Up Monitor and Gain Stabilizer.
The light pulser is a gallium phosphide visible
light source purchased from Electro Nuclear Labs.
These
light sources have been extensively tested for stability,
and while it was found that some of them are unstable over
-69-
O
o
O
—
“I 3 N N V H D U 3 d S l N f l O O
F ig. II-7 : Typical light pulser spectrum used to m onitor pile-u p and to perm it
on -lin e com puter con trol o f the photom ultiplier gain. A s seen by the shape of the peak,
pile-u p in a D C -cou pled system always leads to an in crea sed pulse s iz e , (counts a ccu ­
m ulated around channel 140)
-
70 -
long periods of time, others behave extremely well.
In
addition, individual diodes differ in light output by as
much as a factor of 100.
We have been able to secure
a small number of selected specimens.
More recently,
another company has advertized GaP light sources (Mallory),
at a fraction of the cost of those used here; however,
we have had no experience with them.
There can' be no question that the direct coupled
light source is an extremely powerful and convenient way
of measuring pile-up and gain shifts in the photomultiplierbased counting system.
Its great advantage is that the
calibrated light pulses are injected directly into the Nal
crystal, and thereafter follow the same route as the
scintillation pulses.
No other method permits direct
detection or control of problems originating in the jlhotomultiplier, and it is usually the photomultiplier which
is by far the weakest link in the system.
Moreover the
pulses from the light source can be made almost identical
to the scintillation pulses; they show, for example, the
same fluctuations reflecting statistical collection of
primary electrons.
Their spectrum is of Gaussian shape
and its width is a good measure of the number of primary
electrons involved in a pulse.
This information can be
used to evaluate accurately the efficiency of light collection
and the focusing characteristics of the photomultiplier.
Fig. II-7 shows a typical light pulser spectrum taken
during a run.
Channel zero corresponds to zero pulse
height, and channel 130 to about 30 MeV.
Note that there
are no pulses at all below channel 115, except in channel
-
zero.
71 -
The sharp cut-off on the left of the peak is
characteristic of a DC coupled system; the pulses always
have the same polarity, there is no overshoot as in
double-delay-line amplifiers, and therefore pile-up always
results in an increased pulse height.
The effects of
pile-up can be seen on the right hand side of the peak,
where they h a v e •produced a tail.
To know the fraction
of the pulses which were displaced because of pile up,
the peak can be fitted by a Gaussian with a tail.
The
situation is not satisfactory here, because the light
pulser peak is very widfe ; this reflects the very poor
photomultiplier used here, and the width should normally
be only one fifth of what is shown in fig. II-7.
The
effects of pile-up would then be much more obvious as
indeed they are in the new improved system which we have
assembled in this laboratory for continuing work in this
field.
There are 14 counts in channel zero and one count
in channel 191 which arise as follows:
the counts in
channel zero are generated when a light pulse is emitted
by the pulser when the ADC is busy analyzing a gamma ray
event, and their number is indicative of the effective
dead time(this dead time represents our only rejection of
pulses).
0.1%.
In this case, the dead time was approximately
The count in channel 191 arose because of pile-up
between two large pulses; this type of event was extremely
rare.
-72-
In fig. II-7 we see that only one percent of the
counts were substantially shifted by pile up and this is
typical of the situation which prevailed throughout the
experiment.
11-10
The Data Acquisition Computer Program.
Data acquisition is controlled by a computer program.
Each time a given "event"
occurs , a certain
sectionof
the program is executed.
An "event" is defined as the
occurence of a logical pulse at the computer interface.
This pulse can be produced manually, for example to start
and stop a run, or electronically, indicating the detection
of a gamma ray or the detection of a pulse from the lightn
pulser.
The list of possible types of events, as pro­
grammed for our experiment, is given
EVENTS
1, 13, 15 and
16t
below.
Enter various
sets of
parameters, such as the run number, the
angle of the detector, etc. (Manual).
EVENT 3
Reset the photomultiplier high voltage
to mid-scale.
EVENT 7
Update display.
EVENT 8
Beam current integrator - Also stops the
run automatically.
EVENT 9
Analyze the gamma ray pulse.
EVENT 10
Analyze the light pulser spectrum, and
perform
the pile up and stability checks at regular
intervals.
When successful, the temporary
-73-
spectra are added to the permanent storage.
EVENT 11
Start - Stop (Manual)
Control is returned
to the typewriter upon termination of a run,
and the spectra can be dumped on tape for
off-line analysis.
11-11
New Developments in the Instrumentation.
I
After the data reported here were accumulated, in
September of 1968, work has begun on a new, 11
x 12"
Nal(T^) crystal, under the responsibility of H.D. Shay.
After testing some 60AVP
photomultiplier tubes, it became
apparent that optimum performance would only be achieved
with smaller tubes, and the crystal has been fitted with
a set of 3" XP1031.
Tests have indicated that the new
detector has an excellent resolution for high energy
gamma rays, almost twice as good as the one used in the
work reported here.
The new detector is fitted with the
same type of light sources as we used previously.
A new type of anticoincidence shield was designed
as a cosmic ray rejection device.
It consists of a set of
six flat, very thin plastic scintillator plates surrounding
the detector; cosmic rays are rejected on the basis of a
fast coincidence between any two sides of the box.
Clearly,
those muons which stop in the crystal will be counted as
background radiation, and this limits the maximum efficiency
of the anticoincidence shield; however, we believe that the
thinner plastic and the fast coincidence requirement between
£wo of the plastic detectors will reduce the dead time
-74-
associated with the detection of background radiation in
the anticoincidence shield.
The assembly of the shield
has been completed, and it is currently undergoing a series
of tests.
A new, somewhat larger and more isotropic target
chamber has been built.
Long range plane include the
automated reading by the co iputer of the various conditions
prevailing during the experiment, such as the beam energy
and the exact position of the detectors.
These parameters
are currently entered on the computer typewriter by hand,
when it would be easy to convert them into a voltage and
have one of the front end’s DAC's read it.
We also
eventually intend to have the computer control the beam
energy and the detector position, so that it would proceed
to measure an excitation function with angular distributions
with a simple occasional operator intervention for purposes
of beam optimization.
-75-
CHAPTER III
THE EXPERIM ENTAL RESULTS.
-76*
III-l:
Object of the Measurements.
The differential cross-section for the reaction
Jj
B
T O
(/’jJ')C
has been measured herein at bombarding
energies (in the laboratory frame) of 14.0 to 21.5 MeV,
for laboratory angles of 30°, 45°, 60°, 90°, 120°, 145°
and 155°.
The energy steps were 0.5 MeV, and the energy
resolution limit reflecting target thickness was some­
what smaller than this.
This energy increment of 0.5 MeV
was selected on the basis of our earlier high resolution
survey.
A complete angular distribution was also taken at
7 MeV, and is in excellent agreement with previously
published r e s u l t s e x c e p t for the total cross section
as discussed below.
The yield was measured at 90° for
energies of 11.0 MeV to 14.0 MeV in step of 0.5 MeV, in
order to normalize our total cross-section to that of
Allas et al.(-*-)
This was necessary because of the
imprecision in determining our target thickness; the work
of Allas et a l .,^ ^
which represents the most accurate and
most detailed study of the grant resonance in C
12
, had
been done with substantially thinner B ^ targets for which
the thickness can be measured more accurately.
Fig.I-1 shows the decay to the 0
the 2 ^
ground state and
first excited state of C-1-2 , as well as to the 3
excited state which have been measured.
The preliminary
90° results obtained with the 5"x6" detector are shown
third
-77-
COMPOUND
to
NUCLEUS
28
EXCITATIO N
30
IN
MeV
34
32
36
e>
to
tD
cr>
%
X©
0
to
o’
2
-
k
A‘
9
•
ro
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9
*3*
©
©
©I €
T® © % ©
_
T ° ©o®
9
©
%
_9‘
<>
0
©0,
S
w 3
1 <
CO
CO
o
^
O
©
o ©
o
=L
2
8
©
2^
o
,©
e© cP1
°o®
— i
1
14
15
16
17
18
19
20
21
22
CD
PROTON
ENERGY
IN
MeV
F ig. I ll—1: P relim in a ry resu lts obtained with the 5"x6" detector; the
c r o s s section is shown together with the branching ra tios to the fir s t and third
excited states. The e r r o r s shown here have only been estim ated roughly, and the
absolute magnitude of the total c r o s s section is probably too high by as much as
a fa ctor of two, as d iscu ssed in the text.
-7 8 -
i
r
" B ( p , yo) l2 C
TOTAL CRO SS-SECTIO N
O*
22
20
18 9
m
i
16
14
12
10
1
I
8
M IC R O B A R N S
i
6
4
2
14
15
16
17
18
19
PROTON ENERGY,LAB.
20
22
MeV
F ig. I ll—2: The total c r o s s section for the ground state transition shown here has been
n orm alized to the low er energy resu lts o f A llas et a l, fo r display p u rposes. A s d iscu ssed in
the text, the absolute magnitude is thus o v e r-estim a ted by approxim ately a fa ctor o f 1. 5 .
A|
0.6
0.4
0.2
o
-0.2
16
17
18
19
PROTON E N E R G Y , LAB.
F ig. I l l —3: T his angular distribution c o efficien t has been obtained by fitting with Pq to
term s.
-7 9 -
a
2
" B ( p , y o )'2 C
0.4
ANISOTROPY
COEFFICIENT
A2
0.2
0
0.2
-
-0.4
I
I
i
I
I
T
1
J
14
15
I
1
I
I
1
I
I I
1
I
-
0.6
-
0.8
-
1.0
-
1.2
L
16
17
18
19
P R O T O N E N E R G Y , LAB.
20
21 MeV
F ig. I l l -4 : T his angular distribution c o efficien t has been obtained by fitting the differential
c r o s s section s with the Pq to
I
1
Legendre polynom ials.
1
"T
I
I
1
1
a3
" B ( p , y o )l2C
0.2
ANISOTROPY
COEFFICIENT
A3
0
-
- I
1
I
1
1
I
1
1
T
T
I
1
1
— -0.4
1
-
1
1
14
15
1
1
1
0.2
T
1
1
16
17
18
19
P R O T O N EN ER GY , LAB.
20
21
MeV
F ig. I l l -5 : T his angular distribution c o efficien t has been obtained by fitting the differential
c r o s s section with the P to
L egendre polynom ials.
0.6
-8 0 -
A4
" B ( p , y 0 ) l2C
ANISOTROPY
COEFFICIENT
0.6
A4
0.4
0.2
I
I
I
I
1
1
I
-
14
15
16
17
18
19
PROTON E N E R G Y , LA B .
20
21
F ig. IH-G: This angular distribution c o efficien t has been obtained by fitting with
0.2
MeV
to P^ term s.
40
11 B ( p , y , ) l z C *
TO TA L CRO SS-SECTIO N
a
30
X
x
X
I
10
14
15
_L
_L
16
17
18
19
PROTON EN ER G Y,LA B .
20
21
MeV
F ig. I l l -7 : The total c r o s s section for the transition to the fir s t excited 4. 44 M eV state
shown here has been norm alized to the low er energy resu lts o f A lla s ct al. for display purposes.
A s d iscu ssed in the text, the absolute magnitude o f the c r o s s section is thus ov er-estim a ted by
approxim ately a fa ctor o f 1 .5 .
M IC R O B A R N S
20
z
-8 1 -
0.6
0.4
X
i
I
i
i
i
i
1
i
1
1
I
I
I
0.2
"B(p,y,)'
2c *
I
A NISO TROPY
_l
14
15
I
-
CO EFFICIENT
I
16
17
18
19
PROTON E N E R G Y ,L A B .
I
20
0.2
A,
I____
21
F ig. I ll—8 : T his angular distribution co e fficie n t was obtained by fitting the m easured differential
c r o s s section s with the Pq to P4 Legendre polyn om ials.
t
1-------1-------1-------1------ r
a
2
" B ( p , y | )l2C *
0.2
A N ISO TR O P Y
C O E FFIC IE N T
A2
-
0.2
-0.4
I
-0.6
I
14
15
I
16
17
18
19
PROTON E N E R G Y ,L A B .
L
20
21
MeV
F ig. III-9: T his angular distribution c o e ffic ie n t was obtained by fitting the m easured differential
c r o s s section s with the 1^ to 1^ Legendre polynom ials.
-8 2 -
T
A3
n B (p ,y | ) l2C *
0.4
A N IS O T R O P Y CO EFFICIENT A 3
0.2
0
1
1
I
1
1
I
I
I
I
1
I
I
I
-
0.2
1
-0.4
14
15
16
17
18
19
PROTON E N E R G Y ,L A B .
20
2!
Me
F ig. in -1 0 : T his angular distribution co e fficie n t was obtained by fitting the m easured
d ifferential c r o s s -s e c t io n s with Pq to P4 Legendre polyn om ials.
7
1
1
1 ""
1
..... . I ......
1 ' B ( p f X|
'"
A4
) I2C *
CO E F F I C I E N T
ANISO TR O PY
1
0.4
A4
0.2
T
T
j
1[
1
0
'
-
I
-
0.2
-0.4
i
14
i
15
i
i
i
16
17
18
19
PROTON E N E R G Y , L A B .
1
20
1
21
MeV
F ig. ID -1 1: T his angular distribution c o efficien t was obtained by fitting the m easured
d ifferential c r o s s section s wile the Pq to 1^ L egendre polyn om ials.
-8 3 -
" B ( p , y } >l 2 C *
12
TO TA L C R O S S -S E C TIO N
<7
10 q
-
i
I
i
1
I
I
I
14
15
I
_L
16
17
18
19
PROTON ENERGY,LAB.
I
1
M IC R O B A R N S
8
i
I
20
MeV
F ig. 111-12: The total c r o s s section fo r the decay to the third excited 9 .6 4 M eV
state shown here has been n orm alized to the low er energy y resu lts o f A lla s et a l, for display
pu rposes. A s d iscu ssed in the text, the absolute magnitude o f the c r o s s section is thus o v e r ­
estim ated by approxim ately a fa ctor o f 1 .5 .
- -0.5
-
14
15
16
17
18
19
PROTON E N E R G Y ,L A B .
20
MeV
F ig. I l l —13: T his angular distribution c o e fficie n t has been obtained by fitting the differential
c r o s s section s with only the P^ and P2 L egendre polyn om ials.
-
1.0
PER CHANNEL
OF COUNTS
NUMBER
CHANNEL NUMBER
F ig. i n -1 4 : T ypical spectrum taken with the 9 " x l2 " Nal(Tl) cry sta l and the fast counting system , (see fig . I I - C - 1 )
-85-
in fig. Ill-l, whereas the more accurate results with
angular distributions obtained with the larger crystal
may be seen in fig. III-2 to 111-13.
bution coefficients
The angular distri­
are defined by the relation
transition the angular distributions coefficients were
obtained by fitting the differential cross-sectios
with only two terms;
The result presented herein
have obtained through
fitting the experimental spectra with the standard peak
shapes discussed in chapter II and a low energy exponential
background; the details of the data analysis techniques
are presented in chapter IV.
III-2
Typical Spectra and Angular Distributions.
Typical spectra are plotted in fig. 111-14 andJT/4-l
incident proton energies of 15 MeV and 19.5 MeV.
and
y
The
transitions are clearly resolved, whereas the ^
only gives a shoulder; there is, however, very little
doubt that this gamma ray is indeed the transition to the
third excited state of C
12
, for several reasons.
exactly where one would expect the ^
It falls
in the spectrum; the
competing transitions involving B 11 or the important
for
-8 6 -
n B(pty0 ),2C
FITTED
ANGULAR
WITH
UNITS
c;73
u>
to
DISTRIBUTIONS
= I A n Pn (cos0c m )
au
n*0
ARE ARBITRARY
CJ N
%
■o
Fig. m - 1 6
(0
to
CSI
—
-87-
"B(p,y,)l:iC
FITTED
ANGULAR
H
4
Oil
n« 0
W ITH ^f=Z
ARE
A n Pn ( c o s 0 )
ARBITRARY
0c.m.W eg)
UNITS
DISTRIBUTIONS
G
35
10
IO
b
b
T>
Sc.m.Weg)
T3
UP/-0
b
TJ
Fig. IH-17
-8 8 -
M B(pty 3 )l2c *
FIT WITH ~
alt
ANGULAR
DISTRIBUTIONS
= A o Po ( c o s 0 C m l + A 2 P2 k o s
UNITS
ARE
ARBITRARY
Fig. in-18
m )
-89-
contaminants in the target would be at lower energies, or
at least too far from the observed line.
Moreover the
energy of this gamma radiation tracks that of the incident
protons as characteristic of a radiative capture transition.
Some typical angular distributions are shown in
figs. 111-16 to 111-18.
For ^
and
^
they change very
little over the range from 14.0 to 21.5 MeV incident
proton energy; they are forward peaked around 60° and
the predicted yield goes essentially zero at 180°.
It
would be interesting to have a measurement near 180° ,
but this is precluded, except in the case of an annular
detector, because of the beam line.
For y 3 , the situation
is not so clear, because the errors are substantially
larger; therefore we have fitted
and
7^
.
with only two terms,
Fitting with two terms gives values of chi
squared which are usually lower than for the 5 term fit,
indicating that the 7~f ,
crucial.
Values of A z
5 * and
^
terms are not
obtained in this way are consistent­
ly negative.
III-3
A Discussion of Experimental Accuracy:
Interpretation of Error Bars.
All the error bars quoted with the data points have
been precisely calculated and are meaningful.
However,
they sometimes correspond to the statistical errors only,
and sometimes include the systematic errors.
A detailed
discussion of the experimental accuracy and the various
sources of errors is given below.
-90-
There are two types of errors; statistical and
systematic.
The statistical errors are those which
originate in the finite counting time and would go to
zero at the limit of infinitely long measurements.
They
can be calculated exactly; their evaluation is merely a
matter of solving a problem in statistics.
Actually, we
use some approximations in evaluating them, but these
approximations are extremely good when the errors are
small.
Also, these errors cannot, strictly speaking, be
interpreted as the probable error on the measurement,
because the true value is not known; more details on
these conceptual difficulties are included in the chapter
on data analysis.
The systematic errors are many and
their evaluation relies mostly on the experimenter's
physical sense.
It is important, in order to minimize counting
time, to limit the number of counts at a level which
yields statistical errors of the same order of magnitude
as the systematic errors.
Our counting time has been
such as to give very small statistical errors on the
total cross-section, but the accuracy was needed for the
angular distributions, where most of the systematic errors
cancel out.
In the total cross section of figs. III-2,
III-7 and 111-12, the error bars shown are the statistical
errors only.
They include the fact that the peaks are
not well resolved, but they assume that the spectrum is
truly the sum of the ^ , y
low energy background.
and
y
peaks and an exponential
Any errors generated by this
-91-
possibly inaccurate assumption is considered as systematic.
The most important systematic error comes from the
normalization.
This is a very important matter and it is
discussed in chapters I and VII.
Because of the method
used in making our target, the accuracy on the total
thickness is fairly good (about 10 %) but the additional
uncertainty in composition makes this substantially worse.
Since the dropper we use to transfer the mixture of boron
powder and graphite suspension is somewhat more efficient
at picking up the graphite we expect to get a slight
increase in the concentration of carbon as compared with
the nominal value; this means an estimated accuracy of
(-30%, +10%).
In other words, we expected the B ^
thick­
ness of the target to lie between 70% and 110% of its
nominal value.
To alleviate the difficulty, we hoped to
normalize our cross-section to that of Allas^^et al.
Unfortunately, if we believe their results, it would
mean that our target thickness is only 56% of its
-nominal value.
This is quite impossible.
We have
nevertheless, for display purposes, normalized all our
total cross-sections to their 90° results, pending a
more detailed appraisal of the situation.
The probable
cause for this discrepancy and the various reasons why
we believe that the cross-section is really only 60%, of
the previously reported values have very important
implications for the (particle, ^
) experiments in general,
as well as some repercussions on the estimated isospin
-92-
mixing in light nuclei.
We therefore consider this
problem again in great detail in the discussion.
In the angular distribution coefficients, the
systematic errors introduced by a carefully measured
substantial correction for the target chamber anisotropy
has been included with the statistical errors.
It is
the only important systematic error in the case of y
y
.
In the case of y
and
, the most significant
systematic error comes from the assumption that the
spectrum is a sum of three peaks and an exponential.
This type of error is also the most difficult to evaluate
and we believe that it would increase the error bars by
about a factor If 1.5 at 15 MeV, to a factor of 1.2 at
21 MeV (where the y
peak is cleaner) on the average.
This estimate comes partly from intuition, and partly
from the values of chi squares obtained in the angular
distribution fits.
An improved resolution such as that
we have achieved with our new system would greatly help
in reducing this particular type of systematic error.
It
would be worth stressing, however, that as we shall
demonstrate no matter what the exact values of the probable
errors are, the calculations will not come close to
fitting the data.
There i_s something fundamentally wrong
with the theory or the calculations.
-93-
CHAPTER IV
THE ANALYSIS OF THE DATA.
-94-
The analysis of high-energy gamma ray spectra
requires the use of a computer, because the peaks are,
in general, far from being completely resolved.
It
would take many hours of work to analyse one spectrum by
hand, and the accuracy of the results would certainly
be questionable; to analyse several hundred spectra by
hand is practically impossible.
The data reduction involves two basic steps:
the first is the extraction of different cross-sections
from the observed gamma radiation, and the second is
the angular distribution fit.
Since the angular distri­
bution parameters enter linearly in the usual expression
for the differential cross-section,
c d tr '
d fl
a standard least-squares fit suffices for the latter.
In consequence, we shall only discuss the energy spectrum
fitting program, which is referred to hereafter as the
"data analysis program".
Only the general philosophy is
discussed in this chapter, while the more technical
programming details and a complete listing of the computer
program are the object of the appendix IVa.
-95-
IV—1
General Description of the WNSL Computer System.
The computer system of WNSL consists of an IBM 360
/44 central processing unit with various input-output
devices.
Generally speaking, these can be classified into
two categories;
some devices are standard.IBM components,
whereas others were fabricated by IBM to joint Yale-IBM
specifications specifically for use in the Yale
system.
Standard input-output devices include a typewriter-printer,
two single-disk drives, two magnetic tape units, a line
printer, a card reader-punch and a Calcomp plotter.
The other more specialized input-output devices
will be described in greater details.
A display unit with
1 i gth pen permits efficient interaction of the computer
with the operator, and a convenient keyboard commands the
loading and execution of the various phases of the program.
This keyboard also includes a set of parameter keys which
the operator may use to communicate with the program and so
modify its execution.
The computer interface consisting of a set of ADC'S
and scalars, and a coincidence-anticoincidence matrix, all
properly interfaced with the CPU, has been briefly described
in the chapter on experimental methods.
It was used
extensively in the data acquisition phase of this experi­
ment, but plays no part in its analysis.
In particular, we should mention that considerable
work has been performed at WNSL, under the direction of
Mr. C£L.Gingell, to interface the computer with the
-96-
nuclear instrumentation; for example, the DAC’S which
permit the computer control of the photomultiplier gain
were designed and built at Yale by the WNSL staff.
With
the WNSL display system, it is possible to manipulate the
buffers in storage, using the integer functions ASSIGN,
UNASGN, AWAKE, RELEAS, ACTIVE, ERASE, and PUTPT.
function, TRACK, controls the light pen.
Another
Since these
functions have been used in the data analysis program
which is listed in appendix IV-A, their use will be
descri?oed briefly in that appendix.
The numerical
analysis problems are discussed in this chapter.
IV-2
General Description of the Data Analysis Program.
Perhaps the most important factor conditioning
the writing of a complex peak fitting program is whether
or not the program will be used on line.
By on
line, we
imply that the operator is waiting for the results, and
is available to intervene in the flow of operations; we
also imply that the program will he used during an
experiment, and that the final results are needed
essentially immediately to permit decision concerning
the further course of this experiment.
Some important considerations in writing such an
on line program are the followings
the program must be
relatively fast since the operator (whose time is valuable,
especially during a run) is waiting for the result.
The
program must be essentially "foolproof", otherwise in­
-97-
evitable operator; errors will result in a considerable loss
of time.
It must also be able to fit very small, barely
discernable peaks, and return meaningful errors, so that
the operator can decide how many peaks there are in his
spectrum, and what level of confidence to put on this
decision.
Two major steps have been taken to guard against
the possibility of operator error:
almost all entries
of data are made through a special subroutine which
eliminates the need for proper formats, and simply returns
an error code when a mistake is detected.
The absence of
format (the numbers are entered separated by commas) makes
entering data on the typewriter much easier and eliminates
the possibility of an abnormal termination which would
require complete reloading of the program.
cause of operator error
The other major
involves activation of an
incorrect keyboard contact; this could, for example, result
in execution of program phases in an improper order.
This
has been avoided by logically interlocking subroutines so
that they can only be performed in a certain number of
well defined sequences.
A determined
effort has been
made to retain as much flexibility as possible, but gross
errors (e.g. forgetting to call the cosmic ray subtraction
routine) are hopefully completely excluded.
Achieving stability of the fit on extremely small
peaks has been the major problem encountered in writing
this program.
It has been completely solved, however, so
that the program will find the location of even the smallest
-98-
local minimum which happens to lie near the starting value.
Information as to the statistical significance of such
minima also are available from the program.
The point is
that the program does the mathematics very well; it finds
all the local minima in the ^
function.
Whether the
particular minimum in question Corresponds to a physical
situation or not is a problem which lies outside of the
mathematics, and one which is left to the discretion of
the user.
Considerations of whether or not the peak
also appears at other bombarding energies, or whether the
peak corresponds to an existing state in the nucleus, as
well as the statistical significance of the minimum,play
an important role in evaluating the significance of each
minimum.
In addition, provision has been made for change of
input-output units during execution.
This is not usually
possible since input-output assignments are made at the
beginning of the job through "ACCESS" cards which are
read by the supervisor.
When a change of input-output unit
is desired, one must normally reload completely
this is
quite acceptable as a general procedure, but a more power­
ful system is desirable for an on line data analysis program.
The method used, as well as a list of about sixty inputoutput operations involved are given in appendix
with a listing of the program.
IV-A,
The program also subtracts
cosmic ray background, performs a first order correction
to pile-up, computes the statistical errors on peak
positions and heights, and corrects for various absorbers
-99-
which may be in use between the targfet and the crystal.
In principle, it can include up to nine
peaks
simultaneously as well as an exponential low energy
background,
peaks,
We have currently used it with only three
We have found it more convenient to use para­
metrized peak shapes, rather than stored standards;
however, the parametrization is rather complex, and we
had to minimize the number of times the shapes are
computed.
Only the peak height and position are used
as parameters; permitting the width to vary is inconsistent
with the hope of extracting small, barely resolved peaks.
Some details concerning the peak shapes utilized herein
are given in the section above which discusses the 9" x 12"
crystal.
Fitting one spectrum with three peaks requires an
average of two minutes, with about forty seconds of
computer time involved; during the remainder of the time
the computer is waiting for the operator.
Initialization
of the program (for the first fit) requires five to six
minutes.
IV-3
Algorithm for the Fit:
The parameters used for the peaks are the energy
position and the peak count (height). For the exponential
background, the position and the height are essentially
the same parameter, so that a further parameter must
be selected.
The choice of the parametrization of the
-100-
slope of the exponential background is very important;
it is desirable that near the minimum, the variation of
the
(p
with this parameter be approximately quadratic
over as wide a range as possible*
Satisfying this criterion
leads to easy and rapid convergence.
After trying many
different parametrizations, it was found that
where -jb
is the parameter, gives excellent results, if S
is adjusted so that
= 0 . 5 corresponds to the starting
value for the exponential.
„
The minimum of p
2
with respect to the linear
parameters is very easy to find, no iteration is involved,
a single matrix inversion immediately yields the position
of the minimum for any value of the non-linear parameters.
2
This is precisely because (p iLs a quadratic function of
the linear parameters.
The reader is referred to any
of the standard descriptions of the methods of linear
least-squares fitting.
Finding the values of the non-linear parameters
which minimize
is a problem of a different order
of magnitude, because the choice of the method depends
essentially on the fitting functions.
A non-linear
least-squares fit program has been in use for some time
at WNSL; it was adapted by J. Birnbaum to fit semi­
conductor particle detector
Gaussian line shapes.
spectra with a set of
The problem herein was more
demanding because of the more complex nature of our
fitting functions.
In particular, the low energy back­
ground and the low energy tails of each peak have caused
-101-
difficulties.
Fortunately, we have found a method
which converges well for typical high energy gamma ray
spectra; it may be considered basically as a refinement
of the linearization method currently used for Gaussian
peak shapes.
To illustrate the convergence problems, we
represent graphically the behavior of
of the position
"B
as a function
of one peak, assuming that all the
other parameters are relatively close to their optimum
values.
(Otherwise,
p)
will in general have a complex
behavior with the non-linear parameters, including in
general several minima, and the lowest minimum will not
necessarily have any significance)
The value of
£p2 will clearly show a minimum when the
peak position is near the true value, but the position
of the minimum will not correspond to the best fit and
the value of
will be somewhat larger than X
,
since the other parameters are not at their optimum
values.
Also, and most important, the
<P
will become
-102-
constant on either side of the minimum when the fitting
peak position no longer overlaps with the fitted peak.
This implies that on each side of the minimum, there are
inflexion
points, which we label as I
and
T .
The presence of these inflexion points is the
source of the complete breakdown of the commonly used
linearization method; convergence by this method will occur
only if the starting value is well inside the limits set
by
X
and
1
.
To find the position of the minimum, the
linearization method essentially approximates the function
by the first three terms of its Taylor expansion around the
starting value.
The estimated position of the minimum
becomes
where
/s
is the starting value and
position of the minimum.
is the estimated
At the inflexion points
/ and I
the second derivative vanishes and the estimated correction
becomes infinite; past the inflexion point, the correction
is finite, but has the wrong sign.
For a complex spectrum involving small peaks, it
is not realistic to assume that the operator will necessarily
be able to specify a starting value which lies well inside
the inflexion points, and in consequence convergence in
this region must be improved.
When relatively far from
the minimum, the most convenient method is probably a
simple search; some programs currently used for gamma ray
-103-
peak fitting use it exclusively.
Its main disadvantage
lies in its extremely slow convergence.
The method
described below is a suitable combination of the
linearization and search methods, but with the advantage
of extremely good convergence properties over a wide range
of starting values.
We wish to fit a spectrum with a
function
%An
Y)
where the
are linear and the
parameters.
The residual at
are non-linear
X = X{
is defined by
7?; (A „, a > » %(*>) -
i, fa & J
and depends on the values of the parameters A
^ ( X i ) is the number of counts in channel l
experimental spectrum to be fitted.
and B y,
j
, i.e. the
One of the f ' s
is
the exponential background, the others are the peak
shapes.
!
The function to minimize is defined as
<P
( /) * >
2 7
&£
i
and its lowest minimum is referred to as
(chi-squared).
The sum is to be performed over all channels in the
arbitrarily defined fitting region.
The
COj
are
statistical weights which are usually given by
= 27
m
fin
£> f c , V * )
they are modified slightly (increased) herein when the
light pulser pile-up correction and cosmic ray background
correction are performed, however.
-104-
Since it is easy to fit for linear parameters,
it is interesting to consider the following function
of the non-linear parameters*
Q
lC 8 „ )
Q
=
z ( / ) „ ( 8 n ,) ,' 8 „ )
where the set of
s
given set of
minimize
^
value of
.
Clearly, the set of
2 also minimize
<£P
(&>2
Q
the partial derivatives of
'B >nS which
X
.
Furthermore,
are equal to the partial
Q 2
derivatives of the function
for the
, and the minimum
is also equal to
same argument.
Q
minimizes
with respect to the
Indeed, by the chain rule it follows
that
-— z
Z>/)„
„
V
I
38*
2
because the first derivatives of ^
the linear parameters
(£ )2
definition of
with respect to
vanish at that point, by
.
These partial derivatives are
easily computed
fc
t
’‘
t ' z j ?
2
1
*< < *> & ) J r
*
where the partdal derivatives of the peak shapes and
exponential background with respect to the non-linear
-105-
parameter are trivial; for example, if
is the
peak position in channels,
^
4
[ } ( * « >J
and the approximation is good provided that the peak
covers a large number of channels*
The matrix of second derivatives is computed by
finite differences, symmetrized and diagonalized.
The
diagonalization of the matrix of second derivative
locally transforms the many-dimensional problem into
many one-dimensional problems, one along each of the
eigendirections.
The above considerations on the
value of the second derivative then applies separately
in each of these eigendirections.
The distance to the
minimum is estimated to be the ratio of the first
derivative to the second derivative.
If the step taken
is unreasonably large, or in the wrong direction (i.e.
in the the direction of the gradient) it follows that
we are close to the inflexion point or past it.
~
In such a case,we move a fixed amount in the direction
opposite to the gradient (i.e. downhill).
The fixed amount
should be set to a fraction of a peak width, in the program;
for the exponential background, it should be determined
empirically.
Whenever the step is
limited by this method,
the program actually performs a search, and we consider
the operation as non-converging.
As soon as the minimum
is approached, the estimated displacements become very
-106-
accurate and convergence is achieved in three or four
interactions at the most.
In practice, small peaks which are barely resolved
will always start with the search, whereas if there is no
such small peak in the spectrum, the displacements will
be accurate from the start and complete convergence will
be achieved in a few iterations.
This method absolutely
guarantees that' a local minimum has been found, when
convergence occurs; the user must determine whether this
is the local minimum which has physical significance or
not.
The matrix of second derivatives computed at the
minimum is used to compute the error matrix.
The reference which proved to be most useful in
dealing with the statistics of the present problem is
by F.T. Solmitzs
"Analysis of Experiments in Particle
Physics", published in the Annual Review of Nuclear
Science. .
IV-4
Algorithm for a First-Order Correction to Pile-Up.
When pile-up occurs in a direct-coupled system, its
effect is always to displace counts from a peak to the
region above it.
This is
demonstrated in Fig. II-7,
which shows the effects of pile-up on the peak shape of a
light pulser spectrum.
Provided one can measure the effects
of pile-up on a narrow, well isolated peak, such as the
light-pulser peak in our case, it is possible to fit the
pile-up tail which can then be removed from other peaks
-107-
which are not resolved.
Counts which, because of pile-up,
were accumulated into a higher peak can therefore be
relocated in their proper spectral region.
We emphasize at this point that it is not possible
to reconstruct the spectrum as it would have been without
pile-up; information lost because of pile-up is forever
lost.
What we can do however, is to remove, in first order
the transfer of counts from a lower peak into a higher peak
The technique consists in estimating the probable
spectrum of piled-up counts, using the gamma ray spectrum
and the shape of the light pulser peak.
We assume that
the pile-up tail is independent of the energy, which turns
out to be exactly the case in practice.
An iteration is
necessary, because the shape of the spectrum before pile-up
is not known.
The first step consists in fitting an ex­
ponential to the pile-up tail of the light pulser peak.
The response of the system is then known.
The gamma ray
spectrum (with pile-up) is used as a first step in the
iteration; we compute the probable pile-up spectrum which
would have resulted from its accumulation.
For this pur­
pose, exponential tails with heights proportional to the
number of counts in the channel are added together for each
channel of the gamma ray spectrum.
The pile-up spectrum
thus calculated is subtracted from the gamma ray spectrum,
which is renormalized to the same total number of counts
to produce the first estimate of the corrected spectrum.
We then simply iterate, taking the first estimate of the
corrected spectrum to evaluate the second estimate of the
-108-
probable pile-up spectrum.
It has been found empirically
that two iterations are usually sufficient to ensure
convergence at 5% pile-up.
The correction thus performed is statistical in
nature, and although it will remove the bias, it will
also increase the statistical errors slightly ; this can
easily be taken into account by defining the weights
properly.
The calculated statistical errors then become
meaningful, since they include the effect of pile-up.
The algorithm has been tested and found to work
satisfactorily except perhaps in regions where the number
of counts varies very rapidly with the channel number.
In such regions, the convergence is obviously poor and it
is not practical to increase the number of iterations,
because the process is very time-consuming.
We believe
however, that the algorithm can be improved to handle
these regions more efficiently.
In practice, we have
experimentally kept the pile-up at a level so low that
the corrections involved were less than one half of the
~
probable error » the correction is only in first order, and
would certainly not suffice in cases of severe pile-up.
-109-
CHAPTER
V
THE THEORY OF NUCLEAR REACTIONS.
-110-
At
V-A-l
Introduction
Direction:
At the onset of the analysis of the data to be
presented herein it was, perhaps naively, assumed that
simple recourse to the long established reaction formalism
12
and more recent detailed model wave functions for Cx
would suffice to permit stringent testing of these wavefunctions.
Such has not been the case!
Indeed, only
after lenghty and repeated attempts to resolve major phase
and other discrepancies in the usually quoted references,
was the rather unwelcome conclusion that a complete,
internally consistent, new development of the formalism
was required.
In this chapter all the formalism needed to calculate
cross-sections and angular distributions for particlegamma reactions is developed.
A working knowledge of the basic principles of
quantum mechanics, including angular momentum theory, is
assumed.
Everything needed, except for the electromagnetic
interaction operators, is derived in this chapter.
The electromagnetic operators constitute the dynamics
of the problem and therefore must be obtained from experi­
ment.
They have been known for a long time and can be
found with their proper phases in a recent review article
by Rose and Brink^3-^.
This article and Albert Messiah's
-111-
masterpiece ‘Mecanique Quantique'(2 ) are the only references
needed to understand the chapter.
In the present intro­
duction various papers are quoted, and although none of
their results were used directly in the following sections
we are greatly indebted to their authors.
-112-
V-A-2
Typical Particle Gamma Cross Section Calculation.
A calculation of the cross-section for a reaction
must start with the knowledge of the nature of the system.
More specifically, we need a Hamiltonian for the projectiletarget system; such a Hamiltonian is provided by the ShellModel with residual interactions.
this Hamiltonian'for C
12
Gillet^3 ^ diagonalized
, in the restricted subspace of
one-particle one-hole states of two major shell excitation,
or less.
He parametrized the residual interaction and
fitted the energies of various experimentally identified
levels.
The important assumptions, in this particular
model ares
1) Shell Model basic assumption that we can
replace part of the nucleon-nucleon force by a one body
fixed average potential.
2) Restriction to one-particle one-hole states only.
3) Restriction to two major shell excitation, or less.
4) Charge independence of the nuclear interaction.
5) Specific assumptions about the ground state of
C
12
and the nature of the residual interaction.
These approximations, and the conclusions we can
draw from the experimental data,are discussed in details
in Chapter VII.
In comparing theory with experiment, it
is important to realize that it is differential cross-sections
which are measured, and therefore that it is differential
cross-sections which must be calculated, not an irrelevant
-113-
set of states.
Atomic systems usually have an infinite
number of bound states, reflecting the infinite range of
the force:
spectra exhibit a great number of essentially
discrete resonances over a wide range of energies.
Of
course these states are not really bound,because of the
coupling of the atomic systems with the electromagnetic
field, and therefore are not really discrete; however,
their widths are very small, and except for some possible
cases of degeneracy the resonances can be considered
isolated.
The nuclear case of course shows very different
behavior.
Indeed, only a very few states are bound in
the atomic physics sense.
"particle stable".
These states are also called
They include the several stable
states of that number of nucleons, which have infinite
lifetime, and the excited states which decay through the
coupling to the electromagnetic field (gamma decay) or
the weak field (beta decay).
Because of the Coulomb and centrifugal barriers, however,
for several MeV above the limit of particle stability we
still find very narrow, isolated resonances (especially
in the heavier nuclei).
This is simply because the barrier
increases the lifetime of the state, which can then be
treated as a bound state.
In the early days of Nuclear
Physics the accelerators did not permit, in most cases,
exploration beyond this region.
The situation is very
different now, since both experiment and theory have pro-
CO
z
or
<
GO
EXCITATION
ENERG Y (MeV)
(a)
i.
i
EQ U A TIO N S
30
E
20
0
*
~\—
\
/
/
1
1
|
r
OF
i
M O TIO N
'
F totW
-
1
.1 -
0
20
30
EXCITATION
ENERGY
40
(MeV)
(b)
Fig. V - l : ( a ) shows the resu lts o f a coupled-channel calculation o f M arangoni
and Saruis, and (b ) has been obtained by the equations o f m otion method; note how the
coupled-chan nel method approach allow s a meaningful com parison of calculations with
experim ent, w hereas the other approach, which p redicts d iscre te transitions preclu des
such a d ire ct com parison.
-115-
gressed to considerably higher energies.
Instead of introducing a realistic potential and a
set of boundary conditions to define the stationary
states of the system, many theoreticians prefer to use
the harmonic oscillator potential; they obtain, in this
manner, a set of bound states which become difficult to
interpret.
Such a model, taken literally, predicts no
scattering or reaction cross-section.
At low energy the
situation is not catastrophic, because the spins and
parities of levels can be both measured and predicted,
and some degree of comparison can therefore be achieved.
When the level widths become larger than the level
spacing it is no longer possible to measure spins and
parities of individual levels, and interference terms
play a dominant role in the angular distributions, and
in the total cross-sections as well.
What the experimentalist needs in such cases is
a set of calculated cross-sections to compare with his
data, not a list of state energies and wave functions.
Boeker and J o n k e r ^ ^ , Buck and Hill^4 ^ , and Greiner
(5 )
et al
have exploited different techniques to solve
the problem, all of which yield a calculated cross-section.
Fig. V—1 illustrates the two methods of coinparing the
theory with experiment:
fig. V-la shows the result of
a calculation based on the Equations of Motion^®)
approach, while fig. V-lb is the result of a recent
Coupled-Channel calculation^®^,
The superiority of the
-116(a )
(b )
1 5 .4 5 ---------------- I f 5";
2p / 2
8.65 ------------------ 2 p 3/2
7.02 ------------------ | f 7/2
3.39 ------------------- |d 3/2
-I. I
------- 1
Id 5/2
-1.86 ------------------ 2S '/2
- 4 . 9 5 -------------------------
| p 172
FERMI
SURFACE
-18.72
|p 3/2
POSITIVE PARITY
CONFIGURATIONS
(Ip'3/2, l p l/2) 1,2
(1 p'3/2,2 p l/2) 1,2
(1 p"3/2,2 P3/2) 0,1,2,3
( lp - 3/2, l f 5/2) 1,2,3,4
( l p 3/2, l f 7/2) 2,3,4,5
(ls -|/2,2 s ,/2) 0,1
(1s 'l/2, 1d3/2) 1,2
(ls-|/2, l d 5/2) 2,3
NEGATIVE PARITY
CONFIGURATIONS
/.(Ip -3/2 ,2s
~ 1/2,) 1,2
(lp '3/2, I d3/2) 0,1,2,3
-35 MeV
---------------- Is 1/2
(lp-3/2, ld5/2)
1,2 , 3 , 4
(I s_l/2,1 p,/2) 0,1
12
Fig. V -2 : The s in g le -p a rticle energy le v e ls for neutrons in C
are shown m ( a ) ,
with the neutron en ergies obtained from experim ental data on neighboring nuclei by G illet;
( b ) is a lis t o f o n e -p a rticle on e-h ole configurations o f two m ajor shell excitation or le s s ,
with the total angular m om enta shown on the right-hand side.
-
117 -
Coupled-Channel calculation, in comparing theory with
experiment, is immediately obvious.
When dealing with
angular distributions this superiority is even more
apparent.
In all cases, the theories which predict discrete
transitions will at best :permit a qualitative comparison
of theory with experiment.
In evaluating a method to calculate differential
cross-sections, the main criterion is that it should, if
possible, introduce no new assumption or parameter.
Given
a set of states for the system, we want to calculate
cross-sections in the same framework, to be consistent.
As a case in point, in testing Gillet*s wave functions we
are allowed the use of all five assumptions listed above.
The object of this chapter is to indicate one way in
which this can be accomplished.
( 3)
Gillet*s wave functionsv ' for C
12
are given as
linear combinations of one-particle one-^hole configurations
in j-j coupling.
The unperturbed energies are taken from
experiments on neighboring nuclei; this is a standard
procedure in this kind of treatment.
Only configurations
involving an excitation of two major shells or less are
retained, resulting in the set of configurations of Fig. V-2b.
Fig. V-2-a shows the available single-particle states for the
particle and the hole.
Because each single-particle state
can be filled by a neutron or a proton, each of the
configurations of fig. V-2-b represents in fact two
configurations, one with T=0 and the other with T=l,
where T is the total isospin quantum number.
-118-
For the single-particle wave functions we have not
used the eigenfunctions of the three-dimensional harmonic
oscillator, because they do not permit a reasonable
estimate of the particle reduced widths; instead, we have
used the results of the appropriate numerical integration
of a realistic potential well for C
.
The wavefunctions
tabulated by Gillet depend on the bulk of the nucleus
rather than on the details of the surface, and therefore
we expect that they will be relevant for our single-particle
wave functions, which agree fairly well with the harmonic
oscillator eigenfunctions inside the nucleus.
In the usual case of unpolarized beam and target,
the first step in the calculation of a differential crosssection is represented by the familiar Blatt and Biedenharn
/Q \
formula'
'.
It may also be found, expressed in j-j couling,
(9 \
in a Chalk-River report by Sharp and Kennedy'
'.
The
differential cross-section for scattering or reaction is
expressed as a bilinear form in the S -Matrix elements
or "transition amplitudes"
o r _ _ _ l ____ y
dfi,
Ts kl f
fC t t
:
, /<*> ~
it'
„
Kl (£°s 9)
( v - 1>
The "square hat" notation in the denominator is defined
below. I
projectile,
and
5
are the spins of the target and the
k Q is the reduced wave number of relative
motion of the target-projectile system.
0
is the angle
between the direction of the incident beam and of the
are a set of easily calculated "geometrical" coefficients;
in j-j coupling, for incident charged particles and
outgoing gamma-ray we have:
J L + Z L '+ K - K r + r '
(V-2)
The various quantum numbers are defined below; the £
signifies that the sign depends on the quantum numbers
involved in the formula, in a way which is determined
by a set of phase conventions.
The index
t (and
■£') i s
called a "transition" index and contains the following
quantum numbers:
the total angular momentum (and parity);
all other quantum numbers needed to describe the outgoing
channel; all other quantum numbers needed to describe the
incoming channel; and finally, quantum numbers describing
the intermediate state of the compound system through
\tfhich the reaction proceeds.
formalism for (/>, y )
For example in a j-j couping
reactions,
t - ( j ’* '; i s j l ;
XL <r/o j A J
(V-3)
Tt*
parity;
are the system’s total angular momentum and
X \J & , £
are the orbital angular momentum, spin
-120-
and total angular momentum of the projectile;
the spin of the target.
X
and
L
O' -
is
represent the spin
of the residual nucleus and the electric
magnetic
ftb
O ' = 0) or
1 ) multipole order of the gamma-decay
X
transition radiation respectively.
state of the compound system, and y u *
identifies the
the final state
of the residual nucleus.
The "square hat" notation introduced above is
defined as follows:
a ft
/ ■ - .-»
cud
sT Z d T T T
=
*
/— »
rp
<Z b
a * = (fa ,)rL*
WJ
(v-4)
This simplifying notation is extremely useful, and it
is used extensively throughout the following derivations.
In the simple case of an isolated resonance and
pure j-j coupling, only one transition contributes strongly
to the cross-section, if one can assume that the lowest
allowed multiple dominate strongly.
Under these conditions
it becomes extremely simple to predict the angular distri­
bution.
Even in the case of E2-M1
mixing , for example,
only one parameter (the mixing ratio) is needed to define
the angular distribution, since only two transitions are
involved.
-121-
None of these various simplifying conditions apply
to our case!
We do not have pure configurations - The
resonances are far from isolated.
The spins of the
projectile, of the target and the residual nucleus are
not zero.
And finally E1-M1-E2 mixing is important.
It is worth pointing out that although total angular
momentum J and its projection M, as well as the parity
, are conserved, they are not good quantum numbers.
That is, the scattering system is not in a state of
definite angular momentum or parity.
produces wave packets
wave.
(The accelerator
which locally resemble a plane
They are therefore states of good linear momentum
and since the momentum operator does not commute with
CT
, the system cannot be in a state of good angular
momentum.)
The coefficients
W . a r e
Cw
purely geometrical
in character and it is the reaction amplitudes
which
contain all the information on the dyamics of the system.
cnr1
p j n 1
S
-matrix elements
vC
and t^le reaction
amplitudes are related by
(V - 5 )
t
where the sum is extended to all transitions corresponding
to the incoming channel
C
and the outgoing channel c '.
This sum is therefore a sum over intermediate states of
the same spin and parity.
-122-
The reaction amplitudes can be calculated with the
( 10)
Wigner many-level formula
:
3c =
5cc'.
+
<v-6)
is related to the partial widths
in channel
C
state
in channel
AO
c
factors and phase shifts.
X /•
/*; the
X
are complex
fll 12)
numbers obtained from the reduced widths
'
for
X
, by
of state
, and appropriate penetrability
E is the energy of the system
in the center of mass reference frame,
of state
state
X
X
t
and
- £
P*
A*
P
.
AO
E x the energy
the total width of the
For particle channels, the
c
reduced widths are obtained directly from the value of
the radial part of the wave function of relative motion
of the two fragments in that channel; for gamma ray channels,
the partial widths of state
X
for a given multipole is
proportional to the square of the reduced matrix element
of that multipole operator between the state X
(13,14)
the final state of the residual nucleus.
and
-123-
V-A-3 Practical Difficulties with the Standard Reaction
Theories t
Following the simplified description above, the
calculation of a differential cross-section from the
one-particle one-hole wave functions might appear
trivially simple.
Unfortunately, there are subtle
problems which make it necessary to go into a much more
detailed theoretical analysis.
None of these problems
arise on the case of an isolated resonance in pure j-j
coupling (or pure L-S coupling), and for this reason
little attention has been given to them in the literature.
The difficulties can be separated naturally into
three categories.
The first kind simply reflects the
fact that many authors do not give all the information
needed to define their results:
phase conventions, which
are arbitrary but nonetheless crucial, are sometimes
omitted; results are often derived in terms of relevant
quantum numbers and the wave functions are incompletely
specified; an example of this occurs when the order of
coupling is not explicitly stated.
The second kind re­
sults from the all too frequent errors of sign which we
have found to be present in the literature.
Wigner
(15)
himself, in his classic paper with Eisenbud
, intro­
duced a phase error which propagated through many publica­
tions before it was discovered b y H u b y ^ 6 ^.
Ironically
enough, Wigner and Eisenbud had been very careful in
-124-
defining their phases, which is more than many authors
can claim.
It is perhaps not surprising that we have
discovered several mistakes in the relevant literature,
usually involving signs.
Sign errors are particularly
probable, because they usually are self-cancelling in
most simple cases normally considered as checks on the
results of calculations, and also because physical
intuition is frequently of little help in checking for
them.
Rose and Brink
( 1)
, in a recent review article,
pointed out the necessity, in most detailed calculations,
of starting directly from fundamental premises, and of
carefully keeping track of all the phases.
Their paper
is certainly one of the first to pay special attention
to phase conventions, and as such it must be highly
recommended.
It is, however, not very well suited to
our particular case, and we branch off early from their
derivation (at the consideration of electromagnetic
multipole operators).
To justify the need to start from
basic quantum mechanics, instead of using formulae already
in the literature, we refer the reader to the conclusion
of Rose and Brink’s article.
The last major difficulty has to do with the Wigner
many-level formula.
Such a formula cannot be derived
for overlapping resonances without recourse to grossly
unjustified assumptions, as shown with the R-Matrix theory
part of the present chapter.
The derivation of the many-
-125-
level formula in some review articles
rests mainly
on a set of algebraic errors in the various steps leading
to it; other authors
clearly state the conditions of
validity of the formula, which are seen to hold only
in special circumstances.
The obvious solution to the
problem is to numerically invert the channel matrix, a
procedure made relatively easy in our case by the
availability of modern high-speed computers.
The plan of the chapter is as follows:
First the
Blatt and Biedenhamformula, for the case of particle-in
gamma-ray out, is derived in a j-j coupling formalism.
This takes care of the geometry and we obtain the crosssections and angular distributions in terms of the S matrix elements.
How this
S
-matrix can be obtained
from an intermediate quantity, the
in the following section.
-matrixes shown
The 7 £ -matrix itself is calcu­
lated from particle reduced widths and from the reduced
matrix elements of the electromagnetic multipole operator?,
both of which are obtained directly from the wave functions
of the stationary states of the compound nucleus.
The
last section is devoted to the calculation of the reduced
matrix elements, but the particle reduced widths are
discussed for a special case only , in Chapter VI.
Each
section is intentionally self-contained and can be used
independently of the others; however, some of the important
but lenghty details are included in corresponding
-126-
appendices.
In Chapter VI these various sections are used
together and provide a complete scheme for the calculation
of cross-sections for
) reactions, from a set of
one-particle one<*hole states of the appropriate nucleus.
-127-
B:
Differential Cross-Sections:
The Blatt and Biedenharn
Formula for Gamma Rays.
V-B-l
Incoming and Outgoing Spherical Waves of Unit Flux.
Time Reversal and Phase Conventions.
The first step in treating the problem of scattering
of two particles is the center of mass transformation.
Given two particles of mass ?n t
and
/>
at
7€
and
7T
and
77}^
and momenta
j»t
(lab. coordinates) we can
introduce the new set of coordinates:
g
777,/bf +
__
(V-7)
7 * ^/1 ^
7*7, 4- W fc
/€
=
r
.
^
and
P
-
m t +
'
P
h
are the momentum and position of the center of
mass motion, 71
is the relative position and p
relative momentum.
is the'
This transformation has some remarkable
properties (Messiah IX-11) which can be summarized by
saying that the new coordinates introduced behave like
those of particles of masses
A/»
r l e 777f 4- 777
rr?t 7T7.
,
and 771 = — -— 777, 4 7*z,
respectively, in equivalent one .body problems.
The
equations of motion for the center of mass position are
readily solved.
Classically, we have
P»
constant, and
quantum mechanically, the wave function for the center of
-128-
mass motion is a plane wave.
It will remain such for all
times, reflecting the conservation of total linear
momentum.
The problem of a nuclear collision is complicated
by the fact that each of the particles is composite and
the collision can lead to rearrangements.
This will
be considered in some details in the next section, but
for the moment we simply assume that there exists a pair
of well defined clusters of masses T/n and
•
2>
.
The target is assumed at rest, so that p ^ s o
and
the projectile is assumed to have well defined momentum
p,
.
In the absence of interaction we can write the
wave function of the system as the direct product of the
wave functions of internal motion of the two fragments
and the wave function of relative motion.
The wave
function of relative motion will necessarily be an
eigenfunction of the momentum operator, hence a plane
wave.
^
-
The
£ < 4
^
«***’
< v - 8 )
^ -axis is here chosen in the direction of the
relative momentum
The relative speed
j> = p ,~ p ^
\J-C
, for convenience.
has been used to normalize the
plane wave to unit flux, and <pt and
(f>z
, the internal
wave functions of the interacting systems, are assumed
normalized.
k c is the relative wave number and is
-129-
given by the equationi
f>
-
fik o
«
^ 7 then represents the system of two particles, a projectile
in an intrinsic state
and a target in an intrinsic
* travelling with well defined relative
state
momentum j> j the use of the normalization to unit flux
permits us to avoid the construction of a finite wave
packet and should only be considered a mathematical
artifice.
Both the target and projectile, in general, have
intrinsic angular momentum and if we assume, as is always
the case in practice, that they are in an eigenstate of
total angular momentum, we have:
r
0
-
27
if
<
m
i »
>
.
<z„
u/trH J
S' t>r ls<r>
The expansion coefficients
orientation of the
spins.
c r'/5 0 -) -
|
/*.274JZ--/
*
CL^ and o
/
*
determine the
Because of 'the degeneracy in
the projection of their spins, the projectile and the tar­
get are not in an eigenstate of the ^
angular momentum operator L
.
component
tl-ie
In a beam of many
particles, each particle is in a pure state characterized
by a set of complex numbers
i>
, and the beam is thus a
9*
statistical mixture of these.
Statistical mixtures in quantum mechanics are best
treated with a density matrix 2*
•
Restricting ourselves
to the spin part of the intrinsic state of the projectile,
-130-
^
is defined as
< ror•
or
(V-10)
2
=
"jj
^ b(i) b (i)
w
where
b (i) is a column matrix with elements
x is the particle index and
fs{
particles.
^
b^(i)
<T
;
then describes the set of
When performing a measurement on the
system, the result can be calculated from the density
matrix, so that the density matrix in fact plays, for a
statistical mixture, the role that the wave function
plays for a single system.
More specifically, the
expectation value of an observable A . is given by
KA "> -
Tr a c e
For a very large number
N
C^A)
of particles, with randomly
oriented spins, the density matrix becomes a constant
times the unit matrix.
To show this, we perform a rotation
The effect of the rotation is to transform the
coefficients £
according to their irreducible representa­
tion
4-
where the
=
/>,,
Cr
(v-iD
are the new
, for the rotated system.
The new density matrix, for the rotated system, then becomes
%
=
£>s h )
t
& S( R )
( v _ 12)
-131-
The density matrix must be invariant under rotations,
since we assumed a statistical mixture with no preferred
orientation.
However, Schur's lemma states that the
only matrix which commutes with all the matrices of an
irreducible representation is a constant times the unit
matrix.
From the normalization assumed above, we
therefore have
(V-13)
where 1 is simply the unit matrix.
In the case of a partially polarized beam or target
the situation becomes very complicated, since one has
to know, through appropriate measurement, the density
matrix of the spins and use it in calculating crosssections.
However, a completely polarized beam or target,
or a completely unpolarized beam or target are much
easier to deal with.
For unpolarized beam and target, it
is sufficient to assume that all the particles are in
one of their eigenstates
(or / Z jO* )» with equal
probability for each eigenstate; this is completely
unrealistic, but will lead to the correct results, simply
because the density matrix is correct.
_.
T
i
=
Y'
, defined as
1^ y
/s< r>
( V-14)
is the state of the system corresponding to the two definite
-132-
spin orientations
respectively.
V
and
<r
for the target and projectile
We have introduced a factor
so that the total beam is normalized to unit flux, and
not each spin state separately.
For convenience in dealing with the interaction,
which is localized in the immediate vicinity of the
center of mass, the
Ip
can be expanded as linear
combinations of eigenstates of angular momentum around
the center of mass; introducing incoming and outgoing
spherical waves of unit amplitude,
J2.
ie f
(M
H if> (s<r>
c c
(V— 15)
p /f i
k
c
Y
^
(Ii>> ! s r >
outgoing
is the relative wave number as before,
c
the distance
between the center of masses of the two fragments and S I *
stands for the angular coordinates of relative motion, a
*£ CA a )
polar angle and an azimuthal angle.
£jL_f_c is a solution
to the radial equation, which involves Coulomb functions
in the case of two charged particles, and spherical Bessel
functions in the case of a neutron.
^ (k n )
is otherwise defined by its asymptotic
expansion (valid past screening only, in the case of
Coulomb functions)
-133-
£ca*)~ e
<v“16>
_ i ( kjL+ < s O
The phases
4
are arbitrary and we shall make a choice
at this point and retain it throughout our analysis.
It will be convenient to choose the same phase as that
of the incoming part of the regular spherical Bessel
functions:
<£ =
The
- (e n ) § ”
(v-17>
are Condon and shortley's spherical harmonics,
as defined in Messiah's appendix B.
We shall never use
spherical harmonics alone, because their properties
under time reversal are riot convenient.
We always
use phases such that under the action of the "time reversal"
operator
Q
any state of good angular momentum
transform as
0 /J~M>
=
( -/J
(v-xs)
We assume for example that for the intrinsic states already
introduced for the projectile and the target,
Q ls<r> = C-0S~<rIS-<T>
and
The functions
I* '
6
I
=
c - o r ' 1' h
obey this rule:
-» >
indeed
<v' 19)
-134-
8 ic LY
- V(r')L f>i LYL -M
t t .fi
W
(v-20)
An account of the use of the time reversal operator to
ensure real scalar products will be found in appendix
V-A.
The proof that the spherical waves (V-15) correspond
to unit flux is the object of appendix V-B.
-135-
V-B-2
Incoming Wave Amplitudes.
In the case where there is no interaction between
the target and the projectile, we have seen that the
wave function of relative motion is a plane wave.
Expanding it in spherical waves,
(V-21)
where
^
order 4
Messiah B-ll and B-10
is the regular spherical Bessel function of
, as defined in Messiah’s appendix B- 6 .
The unit flux spherical waves defined in the last
section had good projections of spins and orbital angular
momentum.
By using suitable linear combinations of them,
waves of good total angular momentum
can be obtained.
and projection M
One other intermediate quantum
number is needed; this can be the "channel spin" if we
couple the two spins first, or the total angular momentum
i
of the projectile, if we couple its spin to the
orbital angular momentum.
Choosing the last case for
convenience,
(V - 2 2 )
-136-
Although Clebsch-Gordan coefficients are much
more convenient here than the 3-j coefficients used, the
3 -j
and 6 -j
coefficients (and 9-j
as well) will prove
much superior later than the Clebsch Gordan and Racah
coefficients.
3-j
, 6- j
Therefore, to unify the notation only
and 9-j
coefficients are used.
The
superiority of these coefficients comes from their higher
symmetry properties; because they have higher symmetry
properties, they are easier to use and also tabulate.
They should be used both for recoupling angular momenta
and for numerical applications.
They are defined by
Messiah (appendix C) and defined and tabulated by
Rothenberg et al.
(18)
The transformation above is
unitary, and its inverse is:
k
PF
i * y . (j i )
™
£
x
(h i) n » > is v >
k/s,
(l l l ) t
=
Z )
and similarly for the outgoing waves.
The asymptotic expansion for the plane wave,
from the form of ^
is:
.(V-24)
(k /l)
( k / l - 2 ■ iV )
,
r
iU A -W
V
-137-
C
~ J Z n 1 T. T z k ^ (-ft * & ) i *
jL
t k*II»>h<r> ~/?>T Z ■? zfc, (■&** ft ) i* Y
to(JU lI»> lsr>
£
4
k
s
T H ) ^ / ' Z^
_
V>v* '~ ?
P
I f f
^
x
Z
X
l ^
i ^
+
</ r»* y*/
A
*
_
m
(v. 25)
) /r ^ J
jg+j+s*!-# / £ s * )fi T J )
'prp
£
1JH>)
-i
L K M fr ™ > „ *
jrn; »<r
A
~
^
V
;
( ^
W
( W
(V - 2 6 )
What we have done essentially, is match boundary conditions
at infinity for the spherical waves, so that in the absence
of all interaction we have the expansion for a plane wave
with the assumed normalization and phase; it is valid
-138-
everywhere.
We now introduce the basic assumption of
scattering theory.
We assume that outside the range of
interaction (outside screening range of Coulomb Forces)
the incoming spherical waves are not changed.
is so seems obvious from causality.
That this'
The incoming part
of the spherical waves is then obtained everywhere up to
the nucleus through continuity arguments.
The boundary
conditions at the nuclear surface determine the amplitudes
and phases of the outgoing waves.
More precisely, we write a formula valid even when
there are interactions:
W A xes
The incoming wave amplitudes
(v-27)
are still given
by the same formula above; solving the scattering problem
consists in finding the outgoing wave amplitude.
How
this is done is discussed in detail in the next section
of this chapter.
V-B-3
The Transfer Matrix and the Scattering State of
The Compound System.
Let the set of compound nucleus states
satisfying a suitable set of boundary conditions at the
channel surface in configuration space be a complete,
orthonormal set.
Then any state of the system may be
expanded as
/
(V-28)
>
The state of the system f I
> , of course is not normalized,
since the incoming plane wave was normalized to unit flux.
That there exists a linear relationship
wave amplitude and the coefficients
demonstrated in the next section.
/
between the
o(^ will be
We have:
(V-29)
where the index
wave.
C completely identifies the incoming
The matrix
/ will be called the "Transfer matrix".
The scattering matrix for gamma ray channels does not come
naturally in a treatment such as this one, where the
electromagnetic interactions are only treated as a
perturbation, and in practice, they are always added on the
nuclear problem as an afterthought.
That is to say, the
scattering matrix (or S -matrix, or collision matrix) for
strong interactions is first computed with a Hamiltonian
free from electromagnetic interactions, the scattering
-140-
state of the system is then determined, and the electro­
magnetic transition probabilities are computed from it
by perturbation theory.
For this reason it seems more
natural to intorduce the transfer matrix above than the
S*-matrix
, for gamma ray channels.
This transfer matrix
will be computed in the next section, directly from the
S - matrix of strong interactions.
A dimensionless
quantity could also be introduced, which corresponds
to an extension of the S -matrix for gamma ray channels;
this will be dealt with in section V-C.
In a scheme where both the states
/ A.> of the
complete set and the incoming waves have good total
angular momentum
*1*
matrix
vT and projection
\T
is diagonal in
and
M t the transfer
M
.
This follows
because total angular momentum is conserved.
If the
same representation has been used for angular momentum
in both cases, then
/ > =
T
r
is also independent of M
P
and
!> *» >
-o
X 7M
where C
is now an index which completely identifies an
a
incoming spherical wave
vT
and projection
<t m
S^c
of total angular momentum
M . There are
(Z I+ i)(Z S + i)
possible
states of the system corresponding to different initial
orientations of the target and projectile spins.
Writing
for the state of the system corresponding to
polarization v> of the target and
o'
of the projectile,
-141-
we note that the above considerations apply to each of
these states separately:
O ’M j u < r
l» (r > =
z
/ ^ )
<TH j U r
X
o (.
(V-31)
;
,y r f j i/* *
Z
TX
c «
x ’ ceJ
c t /
A
cet.
CV
C now identifies the incoming spherical wave of total
, and orbital angular
angular momentum \T , projection M
momentum
, total projectile angular momentum ^
.
These quantum numbers are sufficient to specify the incoming
wave completely so that
C
is not needed in our case
and is only included for generality.
upper and low indices is important:
The meaning of
For each set of
upper indices there exists a matrix in the lower indices
«7V*; V P
(column matrix in the case of
d .
and rectangular in the case of
T
A J-M 'iK r
and
c€J
cr
and we could
X > c€j
write in matrix notation:
- / A
Finally, considering that
V + O' -
(V-32)
M
f the index
M
can be dropped and we write
(V-33)
-142-
V-B-4
The Blatt and Biedenharn Formula for Gamma Rays.
Let
O C f
be the yield in d S L for gamma rays
0 s
of polarization ( y leaving the residual nucleus in a
X
state of total angular momentum
and projection J f
,
corresponding to the 2^ o ' polarized part of the unit flux
incoming plane wave.
The differential cross-section for
an unpolarized incoming plane wave is
( 4 T )
=
» k
{. d J 2 > J
—
l L
(v-34)
^
The reason why the yields are simply added here is that
one could in principle measure the beam and target
polarization, the gamma ray polarization and the spin
projection of the residual nucleus.
to observables that commute).
(They correspond
The rule, as stated by
Feynman for example, is that one sums over distinguishable
The sum over v' and O ' comes from the form
final states.
of the spin density matrix as explained above.
From Messiah (XXI-12),
<=>>
< V - 35>
f r
And from Rose and B r i nk^^
6
o
_
l>0' - * 5 ( k , £ y )
«
ZS* !< )(f
£
1
(2.2)
1
V/ ^ ^ > l Zp (£ )
r
( v - 36)
-143-
V
is the interaction which causes
the transition, and
the density of states available for
the transition. The
transition is from an initial state
of the compound
nucleus labelled
to
a ^amma ray of wave number
Co>
a final state labelled
k>
)
{ * r } .
and polarization vector
would be the transition rate per unit solid angle
available for the gamma ray, if the initial state was
normalized to unity,
with our present normalization of
unit flux incoming plane wave,
CU
is simply the
differential cross-section as can be seen in the above
equation.
It may be worth mentioning at this point the
compatibility of notations and conventions between the
work of Messiah^), and of Rose and B r i n k ^ L
V
Introducing
explicitly,
a;
A
, =
P ^ f C k e J
Z rr*
/ <
1
^
i H
a
j i v i x ^
r
<v -37>
.
9
Eq.(2.31)Rose and Brink
Here the operator
depends on the nuclear coordinates
only; the electromagnetic field part of \ /
The one-body operator
H S fy "
has been removed:
/■/ is given explicitly by:
& & ***]}
(
v
'
3
8
>
Eq. (2.13) Rose and Brink
where
2^ / t c
t^ie nuclear magneton, ^
and ^
orbital and spin gyromagnetic ratios, and ^
and
are the
6
are
the momentum and polarization operators respectively.
is the wave number of the emitted gamma ray, and
the
k>
-144-
position vector of the particle.
particle's spin operator.
Finally S
is the
The one-body many particle
operator for electromagnetic transitions is simply
obtained by summing over all particles of the nucleus.
The exact form of / 4 is derived by Rose and Brink^3-^
from
1/ , which itself was obtained in Messiah ^2 ^( X X I - 3 1 ) .
It comes from experiment primarily on atomic systems.
What is needed in the present derivation is the
multipole expansion of
, also obtained in the article
of Rose and Brink:
-
-
r
*■
LM u
V
(V?39)
Eq.(3.21) Rose and Brink
As before,
is the polarization index and can take the
values +1 and -1.
The
f ) *
are the rotation matrices
t
(Messiah appendix C) and &
^
is the rotation taking the
axis (direction of incident beam) in the direction
k
of the emitted gamma ray.
The
^
are called
the interaction multipole operators, and will be given
explicitly in the last section of this chapter, when
their reduced matrix elements are calculated.
w
is
defined as zero for electric transitions, one for
magnetic transitions and can only take these two values.
From
V,
-145-
C 'x V J f -
/£
»
(V-41)
where formula (V-39) and (V-33) have been used.
The Blatt and Biedenharn formula results from the
algebraic transformation of the above expression, and all
the appropriate steps are detailed in appendix V-C.
We
have,
jctt
a h . * I f f * ?
'E
% (c o :6 )
)x
, n - W * * * 7 '* x*J--------------- r-7
Wt t , = (-/)
.
r
,
e t ' f f j - zJ'/<:‘ \ o o o J K
U L ' + k .+
l U
J ' * )[■ }}'K I
J(-t / oJ{ tri x J i f f s Jlcr'crlJ
{['+(->)
Caution:
(V-42)
These functions are not identical with those
(q \
of Sharp and Kennedy'
T } S
The notation is as follows:
Spin of target and projectile
X
t
.
Spin of final state of residual nucleus
-
transition index =
{crv, t s j r i
Sum runs over values of
xf
compatible with given SfTfX.
is the relative orbital angular momentum,
j
the total projectile angular momentum (result
of coupling £
and S
.)
-146-
\ T j 77*
Total angular momentum and parity of compound
state ■
A,
Index of compound state of
&
Multipolarity of emitted gamma ray; (w/=l magnetic,
k; = 0 electric transition)
are the complex transition amplitudes defined
in the text.
The coupling scheme and coupling order are as follows:
X v
•
S
/( £ s ) f T J >
J
J
0£FOR€ 5
eeF O R e
I
Two remarks seen appropriate in concluding this section.
For practical computations it is worth noting that
\ /
.
vv
, =
^
This can be used to reduce the number of
terms in the sum to include those with
£ only.
the relationship between transition amplitudes
an extended S
the sum over
. ,Q<>
Also
and
-matrix for gamma rays can be obtained if
A
, which is implied by the sum over
is performed first.
Since
tt
W . i s
independent of
t ,
A
we have
cC o*
d /L
* .a /
5* S
S XiCcose)
>? r
(V-43)
s v
1
=
T J t
X
-147-
stands for all the transition quantum numbers except
the state index, and will occasionally be refered to as
the "contracted transition" index.
introduced and with
c» { 4 S f l}
Channels can be
C '= { X C w j
we can write in a more familiar way:
(V-44)
Finally, formulae derived elsewhere for
\ / Ck)
Vv
tt
will not
necessarily agree with the formula derived here, because
of different coupling schemes or definitions of reduced
matrix elements, for example.
The difference, if any,
presumably is compensated for by a different definition
of the transition
amplitudes.
-148-
C - Transition Amplitudes and the ^ - M a t r i x
V-C-l
Introduction.
A nucleus of mass
is accelerated to a velocity
and strikes a target nucleus of mass y n
i
rest in the laboratory.
assumed at
If the momentum of the projectile
is well defined, the system must be in an eigenstate of
the momentum operator j> .
After separation of the center
of mass motion and relative motion, the wave function of
relative motion can be written:
y -
&
(v- 45>
&
We have chosen the y
axis in the direction of 1 A.
relative wave number k
the reduced mass
is given by
77?=: m i 7n% /Y7W,i«
The
t k o = m 14- with
^and
and^are the
internal wave functions of the projectile and target
respectively.
We assume
<jt and
<j>2 normalized to unity,
in which case lj) is normalized to unit flux.
If the two particles did not interact the wave
function \ p
would describe the system exactly.
Since
there is an interaction, however, there is a probability
that the projectile will be scattered by the target, or
that one or more other particles will emerge from the
collision.
The state of the system can then be described
as the superposition of the plane wave
suitable outgoing waves.
\J> and various
In practice these outgoing waves
are always taken to be spherical waves (eigenstates of
angular momentum), since these form the most convenient
-149-
complete set for this problem.
It should be emphasized,
as noted above, that the state of the system is not an
eigenstate of angular momentum, and it does not have
well defined parity.
It has a well defined energy however,
given, in the center of mass system, by
£ =
■£
+ fa ? +
) c z
This formula is valid in the non-relativistic limit only,
t
but includes the rest energy of the target and projectile.
-150-
V-C-2
Geometry of Configuration Space:
Channels
The first basic problem in the description of
nuclear reactions is the diversity of possible reaction
products.
Following Wigner^3-^ ^ 2^
let us neglect the
possibility of outright capture of the projectile and
assume that after the collision there are at least two
distinct nuclear fragments.
The possibility of capture
followed by gamma ray emission is not included because
the electromagnetic interactions are treated as a small
perturbation.
After the nuclear problem is solved and
we know the wave function of the scattering state, the
perturbation is turned on and the cross-section for
gamma ray emission is computed.
Justification for the
approach is that in most cases of interest in nuclear
reactions, the cross-sections involving gamma rays are
much smaller
than those involving nuclear particles.
Thus the wave functions can be obtained quite accurately
without the introduction of electromagnetic interactions.
This fact simply reflects the greater strength of the
nuclear forces.
Breakdown of the system into more than two nuclear
fragments usually becomes energetically possible at
relatively high energies only and will be completely
neglected.
Given A nucleons, configuration space is defined
as the space of their relative position coordinates and
their spin (and isospin, if isospin formalism is used)
-151-
coordinates.
The space part itself (position coordinates)
is 3(/4-l) dimensional.
Pb2(3® for example has a
configuration space consisting of 621 dimensions for the
position coordinates alone and 208 spins,
a little difficult to visualize.
taken to be the center of mass.
which make it
The origin is always
The elimination cf the
center of mass coordinates removes three degrees of
freedom, which explains why we have 3(A-1) spatial degrees
of freedom and not 3A.
Configuration space is divided naturally into 3
regions^^^:
the rest.
the compound nucleus, the channels and
Before we define these three regions we have
to give a more accurate definition of what is meant by
"fragments'*.
A nuclear fragment is a set of nucleons
which is not divided into two subsets far enough from each
other to have negligible nuclear interaction.
Then we
can define the compound nucleus as the region of configura­
tion space corresponding to only one fragment, and the
channels as the regions of configuration space corresponding
to exactly two fragments,
In the rest of configuration period
space three fragments or more are involved and the wave
function will vanish or decrease exponentially.
Obviously
the space part of the compound nucleus region is a simply
connected region around the origin.
Each channel is a
simply connected region and the different channels are not
-152-
TOPOLOGY
OF
C O N FIG U R A TIO N
SPACE
Fig. V -3 : This figure shows how the various regions o f configuration
space a re interconnected. Channels 1 , 2 , and 3 should be m ore p roperly
labeled as channel regions 1 , 2 , and 3; two channels belonging to the sam e
channel region a re disconn ected, w hereas the various channel regions are
disconnected from each other and connected to the compound nucleus region.
-153-
connected to each other, but interfaced with the
compound nucleus region.
Fig. V-3 may help to visualize the topology of
configuration space (how the regions are connected to
each other).
It should no'; be taken too literally
because it crudely over-simplifies the geometry.
The
qualitative definition of the three regions given above
will suffice for most purposes.
One improvement needed
is a more accurate definition of the very important
boundary between each channel and the compound nucleus,
called the "channel surface".
surface
in channel
c
We define the channel
as the locus of points of
configuration space such that the distance
between
the center of masses of the two fragments in
c is equal
to a constant
.
PEF/NtrtovOF
5C :
(V - 4 6 )
-154-
V-C-3
The Wave Function in the Channel Region; The5 -Matrix.
Knowledge of the nature of the two fragments in a
channel does not completely specify the state of the
system.
Clearly, the state of each of the two fragments
must also be known.
In what follows we assume that there
is only a discrete set of states available for each
fragment.
The reason for this simplification is that
if one of the fragments is in a continuum, then it is
particle unstable, which means three-body breakup is
energetically possible.
We have already assumed the
energy of the system does not permit three or more
nuclear fragments.
Assuming in the channel region that the interaction
between the two fragments does not disturb the internal
state of each of them, the wave function can be written
IjJ -
where <f> and
<f>
<j>
(p
( f i( J i)
(V-47)
are the wave functions of fragment 1
and 2 respectively, involving their internal coordinates
only, and
cp (Ji) is the wave function of relative motion
of their respective center of masses.
The wave function of
center of mass motion has been omitted since it is trivial.
The proof that p
has the form above is simply the assumed
breakdown of the Hamiltonian into the sum of three parts.
Clearly this is not always valid.
For example, Coulomb
-155-
excitation proceeds through interaction between the two
fragments when they are in the channel region.
In view
of our interest in high-energy low-mass projectiles
however, such effects are expected to be exceedingly small.
The wave function of relative motion cfCA)is a
solution to the problem of a free particle or a particle
in a spherically symmetric Coulomb potential.*
The
energy available for the relative motion is well defined
and equal to the total energy defined earlier minus the
rest mass of the two fragments in their respective state
of excitation.
Because the potential is spherically
symmetric, we can separate angular and radial variables.
The angular solutions will be the same in both cases
(usually spherical harmonics are used) and the radial
wave functions will be spherical Bessel functions for
the free particle (neutrons) or Coulomb functions for
charged particle channels.
This procedure is standard
and it is expected that the reader is familiar with it.
Let us introduce a complete set of angular wave functions
&
Although in practice spherical harmonics are used univer­
sally, it is preferable to keep full generality and only
assume the set to be complete and orthonormal.
*.
The assumption of spherical symmetry of the Coulomb
potential is not rigorous, but is adequate for practical
purposes.
-156-
For each
^
we have two linearly independent solutions
to the radial equation.
The combinations of these
which are the most useful here are (C
p ~ ' •'eg
f ' 0% )
complex conjugate
(>c
)
and its
corresponding to incoming
and outgoing waves respectively.
We finish defining them
by their asymptotic form:
€
(V-48)
where the real parameter
Pc =
and
is an arbitrary phase and
t*ie ra<^ial variable in channel
the wave number of relative motion.
C
For charged
particle channels, this asymptotic expansion is only valid
if we assume a "screening radius" outside of which the
long range Coulomb force is cut-off.
The exact form of J ^
in terms of spherical Bessel functions or Coulomb functions,
and more elaborate explanations on the process of intro­
ducing a screening radius will be given in the next
chapter; a definite choice for
6^
has already been made
in (V-17).
Various recursion relations can be derived from
the differential equation satisfied by the
, and
they are of the utmost importance for practical computa­
tions.
One can also derive the so-called Wronskian relation
Given two solutions ^
for
and
f
of the differential equation
times the radial wave function,
-157-
(V—49)
/ ,' fa
■ / re } / £ f a
*
c o A /s r /iA jT
This constant is easily evaluated in the asymptotic region
by
/'
/ * - / ' * /
<:£ ce
-ifo + % )
-i
€
=
ce Jct
{(& + &
€
-iC p c i&
e
-
i (& *&
C i) £
(v-50)
= -2 i
Therefore everywhere in the channel region, including
the channel surface,
Ce ~ c
<=t
,*
(v" 51)
^ 7
^c€ i e
* .
£
*
C
In each channel region characterized by two given fragments,
there is a complete set of incoming waves:
i t
u
(j i j
4 *
( v - 5;y
and outgoing waves
£
With
(p^
u ( J ij 4- 4<
the relative speed of the two fragments, the waves
are normalized to unit flux.
This is easily proved by
-158-
integrating the flux over a
sphere, a
out in detail in appendix V-B .
procedure carried
The direction of the flux
is also easily seen to he inwards for the incoming waves
and outwards
the outgoing waves.
are
The
(f> .
and
'--e. set of wave function for the first and
second fragments involved in channel region c
.
The
sets are not complete but contain these discrete eigen­
states which can be reached at that energy by the reaction.
They are assumed orthonormal.
So far we have defined the channel regions - (part
of configuration space corresponding to two well defined
fragments)
but we have not introduced the concept of
channels.
Each of the linearly independent incoming
(outgoing) waves defined above in a channel region is
called an incoming (outgoing) channel.
This is the
definition used in the remaining part of this chapter.
When angular momentum is introduced, a channel sometimes
labels all the
( u + / ) spherical waves which transform
into each other by rotations, so that to completely
specify the wave the channel C and the projection
M of
total angular momentum must be given*Which definition is
used is usually obvious from the context, and the ambiguity
should in practice cause no difficulty.
If
C
is now the channel index, instead of merely
labelling the channel region, we can write incoming waves
in a more convenient way:
-159-
i s t l
Ac tf&i
y ( s i c )< t> l< tl
e
(v- 53)
and similarly for the outgoing channels.
It is very convenient to introduce^3
Ic)
channel states
-
0
the
corresponding to the channel wave
/
£
<PC <j>c
functions
.
We use the round bracket
Dirac notation to denote channel states which are
incomplete (they depend on all coordinates except radial
coordinate of relative motion) and the regular Dirac
notion for states of the system which depend on all
the coordinates.
Thus
f c * } would be the state corresponding
to the wave function
(V-54)
/*»-*
£
ft)
r
u c s i
) < $ '4 ?
=■
£
(? *> f C )
7e
Also interesting is the incomplete scalar product which
involves integration over all internal coordinates of
the two fragments, and
A.
, the angular variables of
relative motion, but not over the radial coordinate /Lc .
Thus
(e tc ) - /
simply implies
/ / %<K €
where d ? ‘ and
= /
v
.
5
5
)
d/^represent all the internal coordinates
of the first and the second fragments respectively.
(c lc + >
(
is the radial state of the wave function
Also,
-160-
It is important to understand this scalar product,
because it is used to define the reduced widths.
the radial wave function of
(c l
When
is evaluated at the
previously defined channel surface, the scalar product
becomes "integration over the channel surface" as used
b y L ane(13)(22).
Although the channel states
10
do not form a
complete set in the restricted space in which their
wave functions are defined, the scattering state
I y
of the system can be expanded in terms of the IC) S in
(because the sets { 4 !
the channel regions.
and
con-
J
tain all the states of the fragments that are accessible)
i.t.
Here the ( 0
*c
/> = Z
(V - 5 6 )
Cfc fc)
are kets of radial motion, obviously not
normalized to unity or to unit flux.
This expansion is
valid everywhere in the channel region, including on
the channel surface.
nucleus region.
It is not valid in the compound
The wave function of the ket
is
obviously a linear combination of the incoming and out­
going radial functions.
.
fc (Pc)
X c& Z
fc W
C
(v-57)
-161Ac
is called the incoming amplitude.
The basic assumption of scattering theory is that
the incoming wave amplitudes are not changed by the
interaction.
They are assumed to be the same as if
there was no interaction between the target and projectile.
Thus they are fully known.
Everything we can measure depends on the outgoing
waves, and they must be calculated from our knowledge of
what occurs.'in the nuclear interior.
We shall prove
below the existence of a linear relationship between
the outgoing and the incoming wave amplitudes; namely,
(V-58)
.The matrix thus defined is the
S -matrix,
also called
scattering matrix or collision matrix; because so far it
does not take into account "gamma ray channels" , we
call it the "nuclear S -matrix".
The nuclear
S -matrix
is a complex matrix, and its elements are functions of '
the energy of the system.
It is a unitary matrix, and
under certain conditions of phase conventions it can also
be shown to be symmetric.
A substantial part of what
follows is devoted to its explicit calculation.
Transi­
tion" amplitudes for gamma rays will also be derived.
-162-
V-C-4
Wave Function in the Compound Nucleus Region
In Terms of the S - M a t r i x .
Our next task involves expansion of the wave functions
inside the compound nucleus region
in a complete set.
In principle any complete set could serve the purpose, but
it turns out in practice that if these basis wave functions
are cleverly chosen, we may, under some circumstances, ob­
tain fairiLy
accurate results by keeping only one term.
The idea is that in the case of an isolated resonance, only
one mode of motion is excited strongly; if we have its
wave function in our set, then the state of the system
around resonance will be well described
that one term.
by keeping only
Therefore we would like all the states
of the set to satisfy
/ / M >
=
Ex !*->
<v-59>
This does not suffice however to define the ^ 5 and the f M s
The reason is that it is an open-ended problem, since the
boundary conditions at infinity are not defined.
introduce such conditions later.
We shall
We simply assume, for
the moment, that the /A>Vare orthogonal and normalized
inside the compound nucleus region:
<A//0
-
(V-60)
This scalar product, different from the one introduced
earlier, corresponds to integration over all coordinates,
-163-
but inside the compound nucleus region only.
/A>
The states
are nevertheless continued in the channel region, so
that they are defined everywhere.
channel
C
In particular, in any
, we have the radial ket
(c/ A >
radial ket of IX ) in channel c ,
=
M >
OR
fc )C c (A >
e
(V-6 1 )
which represents an expansion of the state / A ) in channel
wave functions, valid in the channel region, up to and
including the channel surface.
The wave function
corresponding to the radial ket ( c / X ) of
depends on only one variable:
derivatives
in channel c
We define the
by
Ke
Similarly by
.
(A )
Cf>c
■
(c !
V 'C V L
f-c
c
-
< f< V
w© mean the value of
<v -'6 2 >
at the
*c
channel radius £C
Cc/A>
= q>(Rc)
A
c^
The quantities (
<=
J?c 1 2 7 ric ) *
(V-63)
are extremely important.
C
They are called reduced widths— more precisely— the reduced
width of state A
in channel C
.
A priori they are
complex numbers and we have
( c h )> -
< M c f
(V-64)
-164-
This is a case similar to that of the angular momentum
coupling coefficients;
(Clebsch Gordan, etc.) all of
them are defined as scalar products between complex
functions and therefore they are a priori complex.
It
is however possible to choose the arbitrary phases of the
complex functions so that all the scalar products are
real.
Transformation properties under time reversal
play an important role, here.
The properties of the time
reversal operator which are of importance for this purpose
are not its physical properties but its antiunitarity.
Any antiunitary operator which leaves the Hamiltonian
invariant would be just as good.
The procedure to obtain
all scalar products real is outlined in Messiah for the
general case.
(Messiah^2 ) XV-19).
It was dealt with in some detail for the case of
states of good angular momentum in section B of this
chapter.
We assume in the following that such a procedure
has been carried out: i.e.
(c!K> - Cc,A>*
(v_65)
One more condition is necessary to ensure that at
resonance the scattering state of the system /> be well
described
by only one term in the expansion.
(V-66 )
-165-
The resonance must occur near ^
this is that
small
be small.
(c /h y ^
.
The condition for
This is because a
means a large value of the wave function
inside the compound nucleus region, as compared with the
channel region, which is just the resonance condition.
To finish defining the states
//O
, the special boundary
condition
Cel A >
(V-67)
could be used, but for more generality we use the
boundary conditions
c
~ 4
c
~ ^
(V-68)
remembering to keep the arbitrary boundary condition
parameters bc
"quite small'?: .
Equation (V-59) together
with equation (V-68 ) and the phase conventions needed for
real scalar products define the states
energies ^
//))> and the
-166-
V-C-5
The Dispersion Relation.
We now come to the most difficult part of the
theory, namely, obtaining the state of the system in
terms of the incoming wave amplitudes.
Some conceptual
difficulties with the convergence of the expansion
/ > •=• 2 T
A
IA )> S A / >
(V-69)
(23)
are pointed out by Wigner, in a footnote'
famous paper.
' to his
Other difficulties are usually not
treated in deriving the formula, which is assumed to be
almost o b v i o u s ^ .. The reader will easily convince
himself that it is far from obvious and that each step
of the following derivation is indeed necessary.
First let us introduce the wave function in
/ >
configuration space, for
/* >
and
I
:
7 *
-»
(V-70)
/ >
^
From the definitions of ~X^ and Ip
H ip
,
*
tJ y
ZT* V
(V—71)
■H X j ~ t J X/i
and therefore
X l H ip - p H X l
f X *n p -
ty H x l
=
-
(£ -£ ,)
ip x ;
( £ - £ a) < I * > °
(v_ 72)
Xa ( € - £ a >
-167-
where the integral is carried out on all coordinates
but inside the compound nucleus region only.
The
Hamiltonian defined in that subspace only is not
Hermitian, since the condition of Hermiticity is
<MH!
>
* <
IHIX>* o s
<v - 7 3 >
This is to be expected because flux comes in and out of
the boundaries and the probability is not necessarily
conserved with time.
Now, assume the following Hamiltonian
( v '
where
7 4 )
f>- 7TJ^
and
are the momentum and mass of the ith
relative particle, and
V a real potential which depends
only on the space and spin (and isospin) coordinates.
Using the explicit form of
H
in the equation above
yields:
(V-75)
it
And transforming to a surface integral by Green's theorem
in 3(A-1) dimensions:
i
-168-
The integral is now over the channel surface ;
Sj
normal to
is
and therefore along the radial coordinate
in each channel.
The difficult point is that to evaluate
this integral, we must use a different set of coordinates
in each channel region.
Suppose for example that near
channel surface 6T we have two fragments T and
7Z
:
to
evaluate the surface integral above, we use the following
set of coordinates: 3 (w x - / )
for fragment I } 3 ( ti^ ~ i )
Tl
internal motion coordinates
internal coordinates for fragment
and the relative position of the center of masses
I
of
y?-
and
and
I
S
in spherical coordinates,
and
jQ.
^
.
are the number of nucleons in each
to
3t
fragment.
Cr
The crucial point is now the following:
o tS j is always
terms of the sum above vanish, because
zero; except when ^
all
corresponds to the relative motion
of the two centers of mass.
Thus in that channel we get
a contribution
with
s
t
cC
=
2
-
d
Expressing the gradient in spherical coordinates permits
us to evaluate this easily:
P n (cilc)(cl \ - <Mc\ (cl,c)M \)
=
_
{ < M C ) (€/>'- <A\c)
2 7*1^ I
_
“
/?e
_ *f£ fcm
2 Me
{(cl
e
(cl > )
*c
**
Kc
>; - 4 ft/ >, J
c
c
(V-77)
-169-
Let (c ! > be the value at j i
of the wave function
\
c
associated with the radial ket C d > .
Then we define
(C / \
CC/ \
H
4*
(V-78)
By summing over all channels the integral over the
channel surface is obtained and
( « ' i - i
r
v
^
-
^
o
(v- 79)
Each radial ket is a linear combination of the incoming
and outgoing unit flux radial kets defined above,
(c / > * A c
(C lc -y
+ Bc ( c l C 4 >
(v_80)
and $ c are complex numbers and are called incoming
and outgoing amplitudes in channel C .
For brevity a
new notation is introduced:
f f i /z
t,
•
-t m *
T
=
/(A 4 /
**
‘
(A.*)'
T F T )*
f
■
_
'
F*
(c I r - ' i
(c l X
By taking the complex conjugate on each side, we get
similar equations with
L.
c
and
Tc
(c /
C c /C + y „
and
are dimensionless quantities.
Above equation
becomes:
r
-
&
-
£
)
«
*
( v - 8 2 )
-170-
With the following definition of the reduced widths
-
( W * (C/K\
:
( v - 8 3 )
Writing in matrix notation^
(£)' fa*
L,
(V-84)
is a diagonal matrix with elements
ft »A
and
(LA 4 L*B) =
*
££
and
3
and
T denotes
I^ .” 6C TC
are column matrices with elements
transpose.
J
the diagonal matrices _/* and
TC and TC
L cc -
for later use.
Yxc •
We also introduce
with diagonal elements
With the
S
-matrix defined above
we have:
@ ) * r / a + M A
-
A
We could calculate the state of the system (the °(^s)
if we knew S .
(V-85)
-171-
V-C-6
Calculation of the
From
.S^ -Matrix:
S -matrix,
in terms of the
<*, * (i)
£a.£
The ^-Matrix.
Ctrl'S) / I
(V-86)
ft Re
also
(c/ ^
= ZCd*>g < M > =
Ccn\
(V- 8S)
So that in matrix notation again,
T«*rx
•
( i f f r s V /i
(v-89)
Another expression may be obtained from the calculated
above, namely:
<T
°(»ii
=
(i)
(T
zZe ) (i'+L*s )d
(v-90)
The very important ^ - m a t r i x is defined as
r
■ -»
A
^
/ a must be valid for
Because the two equations for
A
any incoming amplitude A
-r Cu Cs )
5
' — CL
CL
=
» we have the matrix equations:
r+r*s
U//7H
(v-92>
CC = LCL~L
-172-
We also use
5
=
"L
(fa d )
O z + d *)L
U fH Z tt
c X *- - J L
'
(V-93)
Some properties of 5 will be discussed in the next few
pages.
To determine the state of the system our main
interest lies with the
of S
•
Putting the calculated value
back in the equation above for
, one obtains
where
(V-94)
is a row matrix in channel space, which we have defined
as the transfer matrix.
All the calculations described
in the next chapter were performed using this formula.
i Z d
intuitive meaning is the following:
Its
reflects the
efficiency with which the channel C "feeds" the state
i
)
, or
alternatively, the degree of coupling between
the channel C and the state
A
.
The phase of
Tj^also
gives very important information, telling us if two
channels will contribute constructively or destructively
to build up the mode of motion
X
.
determines the lifetime of the state.
Clearly, it
For example, a
state which is loosely coupled to all channels will have
a long lifetime.
The transfer matrix also permits calcu­
lation of gamma ray cross-sections, something which cannot
be done directly with the nuclear
S -matrix;
this is
because the transfer matrix gives direct knowledge of the
-173-
scattering state of the system, which is needed for
gamma ray transitions.
How gamma ray cross-sections
are calculated from ' T is examined in detail in
section B of this chapter.
-174-
V-C-7
Some Properties of the Nuclear
S -Matrix.
#-/
Matrices of the form a
properties!
a
have very interesting
Before discussing them, it is useful to
recall the following property of complex matrices: the
operations
Qf taking the adjoint, the inverse, the complex
conjugate and the transpose of a matrix, all commute
with each other.
Also, a necessary and sufficient condition
to be unitary (i.e. A A * /) is that it
for a matrix
conserves length.
Another important remark is that a
matrix and its complex conjugate do not in general commute.
Thus
(in general)
S - - Cc CL
*
-
&>&
The first interesting property of
S
, is that
• I
SS
Indeed,
s s *
-
-
d
"
a
( - a
' ' a
) *
-
(V-96)
/
A more important property is its unitarity.
Unitarity of the S
-matrix is closely related to the
conservation of probability, and replaces, in a scattering
problem, the Hermitiaty condition on the Hamiltonian for
a bound state problem.
Is a a
for all row matrices f ; defining
y = f CL , the
condition becomes Jy CL f - ! y & I for all ij
are unitary.
The condition is indeed that
This is equivalent to
CO
Not all matrices of the form
Y j a a 'Y
or again to the condition that
=
a a
/ --
0t>
1 st
for all
be a real matrix.
y
-175-
Any matrix of the form
not necessarily real.
act
is Hermitian, but
However:
a a f - ( K L - f J & ’Z - l V - X U 'K + T I * - R L I* - riS e
tr‘
-
(f-li)l'
(V-97)
aa. =
rm +ieirr'+irK-Kri*- t*i/z
a a f- z u ‘e*n*+iz&n*+6’r >
iz+ fz -iT s-x feti*) +
-(xc f f/ ) n
^yn
a a ?
=■ o
The last equalities are from the Wronskian relation (V-51).
, and as shown
We have thus proved the reality of
above this implies the unitarity of
remembered that the reality of the
and the
S
.
It should be
^ -matrix was used,
-matrix is real because of our phase conventions
which give real reduced widths.
Thus the proof only holds
when the reduced widths are real.
Another property of the
S
-matrix which is useful
in reducing the number of required parameters, when it is
to be experimentally determined, is its symmetry:
o = c r
J
(V-98)
9
Indeed we have, using the property
s
T -
s
f<h
*
( s 'J "
-
s
ss - 1 ■
.
(V-99)
-176-
V-C-8:
Approximations to the S -Matrix:
The Wigner Many-
Level Formula.
The Wigner many-level formula was not used' in our
calculations, but its derivation is given here for
several reasons.
Its use being a standard procedure for
such a calculation, some justification is needed for
avoiding it.
The following derivation shows well the
conditions of validity of the approximation, and the
need for a more accurate procedure; several derivations
in the literature which do not involve the restrictive
approximations used here, obtained the desired results
through a series of algebraic e r r o r s ^ ^ .
Other articles(lO)
merely state the result with a set of correct, restrictive
assumptions.
Finally, the many-level formula having the
advantage of a clear physical interpretation and its
form being well established, its derivation from our
earlier results will provide a test of the accuracy of
these results.
The many-level formula is based on some very
interesting properties of matrices.
These are not widely
known, and a mathematical preliminary is in order.
first remark concerns matrices of the form
y
is a column matrix.
J y r
The
, where
Such matrices are singular
because one can obviously find a similarity transformation
which brings all the elements to zero except one on the
diagonal:
it is simply a transformation which sends
V
-177-
into a basis vector.
r r 7
Y Y r+ e
The matrix
has an inverse.
therefore has no inverse.
, where
€
is a diagonal matrix,
As will be seen later, the many level
formula comes from the properties of this inverse, in
particular that it can be calculated without actually
performing the inversion.
Indeed,
~
( y y r+£)
X ~
where
6
^
/+ x e ' YY e
Y , (c diagonal).
This curiousperperty of
by multiplying
(v-ioo)
matrices can be proved
C y Y r+ € )
very simply
by its stated inverse:
few simple algebraic steps yield the unit matrix!
a resonance, one level on the ~R.
A
Near
matrix will dominate
and we hope to approximate
YA f t
A
JOL
+
&
*
y isthe reduced widths col.
where
at £ o
^
where the contribution from
will be assumed diagonal.
matrix
of thestate
distant levels,
There is fairly good justification
for this approximation because the off-diagonal elements
of
come from many contributions of different signs,
and the diagonal elements are a sum of positive terms.
The slow energy dependence of
Although
72
^ ^ w i l l be neglected.
can be approximated in this way, it does
not mean that a good approximation to
will yield a
-178-
good approximation to S' :This
is in fact one
V VT
major difficulty.
£
Even if
5
approaches £ o
becausey y
,
does not exist in the
y
does not
is singular.
-
7? approaches
go to
The pole
in the^
when
j.
)
matrix
S -matrix, which stays unitary,
well behaved and varies slowly even at the limit when
the
^
-matrix, from which it is calculated, diverges.
Let us compute
-
S
in this approximation:
i* " 0 e * d ) " t e + d * )
=
-
1?
'( y
=
- I f ' ' [ a
( v _ 93)
jL
f r* e ) ' ' ( y ? T- t e * ) L
'
-
ft x
£ *
£ ' if} r r £ " ] ( W
(£ a - £ )
T+ £ ) L
X *
/ V ' /
(V-101)
i+x
But
.
.
d -d *
Cg*"- s~') - ( a - a )
/ ^ o ^ ^ / i s a diagonal matrix, which commutes with also
oL so that the notation with a fraction bar
diagonal
is unambiguous.
d - d
-
-
I *
L.
t
1
L
-
I L ' - L I *
/ u t
'
m
2 £
*
where the Wronskian relation (V-51) was used.
( V ' 102)
Also
-179-
U X =
/ + $ r£ ' ' Z
/+ ^
=
K ^ + oL
..
-
" /
»
l/i^F
)
JW
T/ _ U )
~ 'i l w * J r f e '
(e-E + A + I r ) ( ~ f )
<v-103>
, > ? „ * 7?& d
with the level shift
and the level width
^
Z7-
-h c ip
_.
9rf
J
)
*
(V-104)
^ &ili"
and
9
1
* * * 1’
r
(V-105)
Putting all these results together yields the familiar
many-level formula:
5 =
-
e
■iJ2{n. i H - — ~
i'
A - E + A - i r J -*
6,
£-£>£!-j/V
with the above definitions of ^
The phase shift f l
, A
(v-106 )
and P
.
is given by:
JL(72+cC*) ~ M-! (% + d *)
(v-io7)
In the form that we quoted in the introduction, (V-6),
the level shift was ignored, the phase
shifts 1 2 , were
assumed part of the definition of
and the sum over
^
states which is implicit in (V-106) (it comes from the
matrix multiplication of
9
by
) was not included; this
is why the many-level formula (V-106) yields
(V-6) was a formula for
^
.
S
whereas
-180-
D:
Reduced Electromagnetic Transition Matrix
Elements:
V-D-l
Particles and Holes, Angular Momentum and Isospin.
In dealing with the antisymmetrized wave functions
of many fermions, it is convenient to use the formalism
of second quantization.
The state of the system is
expressed as the direct product (suitably antisymmetrized)
of single-particle states; the latter are divided into
two classes, usually identified as "above the Fermi
surface", and "below the Fermi surface",
respectively.
It is convenient to keep track of the ;partides above
the Fermi surface, and of the unfilled single-particle
states below; the latter are referred to as "the holes".
These ideas are simple in principle, and the only
difficulty which arises is to obtain the proper phases.
In appendix V-D, the antisymmetric states, and the particle
and hole creation and annihilation operators are defined,
and the various phase conventions which enter in the
definitions are clearly stated.
No previous knowledge
of the techniques of second quantization is assumed,
mainly because we do not know of a suitable reference to
quote; Brown's
we
(25)
treatment parallels ours quite closely, but
favor a slightly different approach.
The behavior under rotations of the particle-hole
states may be inferred from the transformation properties
of the particle and hole creation and annihilation operators;
-180-
D;
Reduced Electromagnetic Transition Matrix
Elements:
V-D-l
Particles and Holes, Angular Momentum and Isospin.
In dealing with the antisymmetrized wave functions
of many fermions, it is convenient to use the formalism
of second quantization.
The state of the system is
expressed as the direct product (suitably antisymmetrized)
of single-particle states; the latter are divided into
two classes, usually identified as "above the Fermi
surface", and "below the Fermi surface",
respectively.
It is convenient to keep track of the :particles above
the Fermi surface, and of the unfilled single-particle
states below; the latter are referred to as "the holes".
These ideas are simple in principle, and the only
difficulty which arises is to obtain the proper phases.
In appendix V-D, the antisymmetric states, and the particle
and hole creation and annihilation operators are defined,
and the various phase conventions which enter in the
definitions are clearly stated.
No previous knowledge
of the techniques of second quantization is assumed,
mainly because we do not know of a suitable reference to
quote; Brown's
we
(25)
treatment parallels ours quite
closely,
but
favor a slightly different approach.
The behavior under rotations of the particle-hole
states may be inferred from the transformation properties
of the particle and hole creation and annihilation operators;
-181-
the latter are as follows:
P
a r 72"
=
X £>*, M
(v-io7)
,
g*
a-/ « T5’"
- 2T' 3D*
'o
o
tvnrmi,(%’) a- ■,
7
*
J”
«
(v—io8)
J *
W
% i ” p« ' ' 5 *
? R
^
?«
"
'
I T
(
%
"
•
&
"
>
v
'
i
o
9
)
((v' 110>
^
represents a rotation of the system, and
is the
operator which transforms the initial state into the
rotated state.
the particle creation operator
for single-particle state
(°f angular momentum
and projection m ) , and
operator for that state;
is the annihilation
/■
&-
and
A ■
are respectively
J™
the hole creation and hole annihilation operators for
state J f r n ' > •
The exact definition of these various
operators is the object of appendix V-D.
The rotation
matrices are those of Messiah'( 2 )J appendix C-IV, except
that he uses the latter
"R"
for them.
We also have to distinguish between the proton
creation operators and the neutron creation operators,
unless we treat the proton and neutron as two states of
the same particle (isospin formalism); since we only need
-182-
deal with one particle and one hole, it is simpler to
choose to distinguish between protons and neutrons, and
we shall use hats to identify the neutron operators.
The one-particle one-hole states will have the form:
> W crMf>'
T d ' %
(v-111).
a g S M j .
K
-
£
£ , / * >
Tnm
-
7Vrr?‘
(V - 1 1 2 )
*
.
where
Jtp
0
«
is the particle-hole vacuum.
The + sign
identifies the symmetric combination (corresponding to
T = 0 in isospin formalism) and the - sign corresponds
to the antisymmetric combination.'
A many-particle one-body operator $
, in this
description, becomes
s
.
£
5
%
)
r
+
;^r f> A d& n
5
%;
116 )
jJ~ ru u & en
and
/*•*
S *
where v S ^ a n d
^
S
(V-114)
are the proton and neutron single
particle operators corresponding to S> > the operators
c jC p
are particle creation and annihilation
operators corresponding to the situation where no Fermi
surface has been introduced, and are defined in appendix
-183V-D.
V-D-2
Many-Particle Matrix Elements in Terms of SingleParticle Matrix Elements.
In this section, we give the formulas for the
I ^ f ^
evaluation of matrix elements of the form
and S y ft/Z J , M , t }$*’/
Mz i >
.
The first case, fig.
a, corresponds to a transition to the ground state, and
the second ,b,
represents a transition to an excited one-
particle one-hole state.
FERMI
LEVEL
(a)
(b)
(c)
The transition to an excited configuration can occur through
a particle jump (fig. b) or a hole jump (fig. c); only one
jump is permitted, when the operator £ *
is a one-body
operator.
After some tedious manipulations of the creation
and annihilation operators, which are reproduced in
-184-
appendix V-E, we obtain
K
j f ' J t l t ifflC )
«
x c->/(it's,) *
<v- ii5)
7»r»'
S~
with
defined as
t
c - -- S '-S ”
S P* S "
2 * ± .
S
(V-116)
also
* 2
*
m ,™ /W z 7r?(
£
}
-
3
£
i-V (<**'&/>*•>
J>+Jl f™l+™l+/ (~
+
C -0
i
£*>,;/>. < £ ' * > '
l
s
+
1
'
(V - 1 1 7 )
The first term of the sum corresponds to the particle
jumps, and the second term to the hole jumps; S
should
be used when there is no change in symmetry, (-f—** -f or
transition, or X S T = 0 in isospin formalism) and
>—
S
(H
is for the transitions which change the symmetry
> —
and — — » +• ).
The term
&■
.J
should be
-185-
interpreted as follows:
it is equal to one if the two
single-particle states involved are the same (including
principal quantum numbers which are not included here), and
: i.'n ) '
^as a similar interpretaVThe above formula is not necessarily valid if the
zero otherwise.
tion.
initial and final states are the same.
V-D-3
Elimination of the Hole.
The purpose of this section is to establish the
formula for the reduced matrix element of a tensor operator
between two one-particle one-hole states, in terms of the
reduced matrix elements of the corresponding single-particle
operator.
We have
£
< p s /n p
L
(V-118)
and
< jkjtnsjifk‘xty -
'xhjL i f ’ i o ] * fll5'L ^
+
(V - 1 1 9 )
here again,
is equal to one if the holes are the
same, zero otherwise', and similarly for
S-
.
The same
-186-
rule applies to the use of S' or
S
as in section V-D-2.
Formula (V-119) applies even if the initial and final
L ^ O
states are the same, provided
.
if
/, =• O
,
however, the diagonal matrix elements include another
term due to the closed shell.
The derivation of these
formulas is in appendix V-F.
V-D-4
The Electromagnetic Multipole Operators.
The operator for electromagnetic transitions
introduced in section B of this chapter:
was
it is
(Rose and Brink 2.13)
•jjr ~
H J k j) =
i
t~
+ g s
V x (e e
*)}
(v-1 2 0 )
The various quantities appearing in this formula are
defined in section B.
This operator is a one-body, one-
particle operator, and the many-particle operator of electro­
magnetic transitions for the nucleus is obtained simply by
summing over all nucleons in the nucleus. • The amplitude
for emission of a plane wave
(A ^ Z )
is proportional to
the matrix elements of this operator; the amplitudes for
emission of spherical waves may be obtained by expanding
/■/
in multipoles«
(Rose and Brink 3.21)
W y The sum runs over all integers
(
/, and M
, and I t *
v
‘
1
2
takes
the values zero (electric transition) or one (magnetic
1
)
-187-
transitions).
From Rose and Brink
-r< T r>
/
(1)
we obtain the multipole operators
with their correct phases :
LM
Electric:
r <e>
C
a. ru*.LiLY ]
(
~ OL-OU KZl(2iH/J fjiAMc LH’ n'
v
-
1
2
2
)
Magnetic:
-J- Cm )
«
'
/ S,
U h
LL t i
*
„ (s fa a ) -
(
^
V(a li l ',K) •/
(zifau/tU-n))
£ % « ; - d m
V *
(V-123)
(1)
These formulae are obtained from Rose and Brink'
(3.17
final) (3.20 final) and (3.4).
The small spin dependent
term on the electric transitions has been neglected.
The "coupling constant" of the interaction is given by
the proton charge
where
C
or the nuclear magneton
is the velocity of light,
of the nucleon.
The symbol
of the system, and
H
and
/ i Li l
'L»r1
H
and m
is the mass
represents the Hamiltonian
is the commutator of
•
The spherical harmonics
are used here as operators.
Y
£~ft
Their action is to multiply
-188-
the wave function of the nucleus (in position coordinate
representation) by a function
position f l
of.the
angular
of the nucleon under consideration.
Before using expression # (V-122)
multipole operator
/
of the electric
for actual computation of
matrix elements, it is preferable to transform it slightly,
If f i y
and
ff)>
are respectively the initial and final
states of the nucleus, we have
= (Er £f)< il/iLYLHlf>
The last result, where
C
t l x and
C
(v-124)
are the initial
<ZY)ct
fc n o A
energies of the system, is only valid assuming that
the initial and final states have well defined energies:
/ J / i > = £ { /£>
and
/-///A- £ / / / >
The energy of the gamma ray emitted,
(V-125)
will be equal
to the energy difference between the initial and final states of the nucleus so that:
This formula is valid for the calculation of matrix
elements between eigenstates of the nuclear Hamiltonian
only if the initial state is on the left and the finai
-189-
state on the right.
equal"
^
We have used the sign "approximately
as a reminder that the small spin dependent
electric term was neglected, but no other approximation
has been introduced.
V-D-5
The Single-Particle Transitions
The electric single-particle matrix elements are
derived in appendix V-Gj we have
'f'tX/bkl f..
1
..
(V-127)
The magnetic transitions involve the sum of two parts;
the "space" part is derived in'appendix V-H, whereas
the "spin" part has been reserved for appendix V-I; the
result
is
summarized below.
and
(V-129)
-190-
/
with
and
S ^ S '=
'fa
.
ft
number of the emitted gamma rays;
k
is the wave
is an arbitrary
constant with the dimension of length, introduced for
convenience.
-& and
fa
are the proton charge and
nuclear magneton, respectively.
The
~ r« > i
/
, and other
t
such operators, are defined as in (V-116); the
has
nothing to do with parity, but simply identifies the
proton-neutron symmetry, as it does everywhere else, in
this chapter.
The
[s l
-type symbols are defined below
(14)
and include the center of mass corrections.
:
A'11(aS +C-if(z?-/)] fin. ry*
& S [ & - / ) '- C-Ol]
=
'A [ / -
■
a
+
&[/ ■-£]
r
=
$
1
A
fit. T ™ '
f f s f L )
a
rL%acc)
0.$ 8 0
(
^ 2 ,
v
_
i
3
0
)
7^" c s f i t r , )
f
where
_
-
.
fa r t *
“
A
and
H
C sp **)
are respectively the mass number and
the atomic number of the radiating nucleus.
in self-conjugate nuclei,
forbidden because
7 -Or>>
\ <7 j
7 ~ -0
vanishes.
In particular,
8.1 transitions are
-191-
List of References for Chapter V
(1)
M.J. Rose, D.M. Brink, Rev. Mod. Phys. 39, 306(1967)
(2)
Albert Messiah, Mecanique Quantique, DUNOD, Paris
(1962) ; Quantum Mechanics, North Holland, Amster­
dam
(3)
V. Gillet, PhD Thesis, Universite de Paris (unpublised)
V. Gillet and N. Vinh Mau, Nucl. Phys. 5 4 , 321 (1964)
(4)
B. Buck and A.D. Hill, Nucl. Phys. A95, 271 (1967)
(5)
M. Danos and W. Greiner, Phys. Rev. 146, 708 (1966)
(6)
E. Boeker and C.C. Jonker, Physics Letters 6, 80 (1963)
(8)
J.M. Blatt and L.C. Biedenharn, Rev. Mod. Phys. 24,
258 (1952)
(9)
W.T. Sharp, J.M. Kennedy, B.J. Sears and M.G. Hoyle,
Atomic Energy of Canada Ltd. (Chalk River) Report
CRT-556
(10)
Claude Bloch, Nucl. Phys. 4, 503 (1957)
(11)
E. Vogt, Rev. Mod. Phys. 34, 723; Nuclear Reactions,
Volume I, North Holland, Amsterdam.
(12)
J.M. Blatt and V.F. Weiskopf, Theoretical Nuclear
Physics, Wiley, New York 1952
(13)
A.M. Lane, Rev. Mod. Phys. 32, 519
-192-
(14)
J.M. Kennedy and W.T. Sharp, Atomic Energy of
Canada Limited (Chalk River) Report CRT-580
(15)
E.P. Wigner and L. Eisenbud, Phys. Rev. 72,
29 (1947)
(16)
R. Huby, Proc. Phys. Soc. A67, 1103 (1954)
(17)
E.P. Wigner, Group Theory and its Application to the
Quantum Mechanics of Atomic Spectra, Academic Press,
(1959)
(18)
M. Rothenberg et al., The 3-j and 6-j symbols, The
Technology Press (M.I.T.)
(19)
Biedenharn, Blatt and Rose, Rev. Mod. Phys. 24,
248 (1952)
( 20)
E.P. Wigner, Phys. Rev. 70, 606 (1946)
Phys. Rev. 70,; 15 (1946)
( 21)
I.E. McCarthy, Introduction to Nuclear Theory,
Wiley, New York (1968)
(2 2 )
A.M. Lane and R.G. Thomas, Rev. Mod. Phys. 30,
257 (1558)
(23)
Ref. 15, p. 34, footnote #10
(24)
J.S. Bell, Nucl. Phys. 1^2, 117 (1959)
-193(25)
G.E. Brown, Unified Theory of Nuclear Models and
Forces, North Holland, Amsterdam (1967)
(26)
A De Shalit and I. Talmi, Nuclear Shell Theory,
Academic Press, New York (1963)
(28)
M. Marangoni and A.M. Saruis , Nucl. Phys. A132,
649 (1969)
(29)
DoJ• Rowe and S.S.M. Wong, Physics Letters 3OB,
147 (1969) and 303 150 (1969)
-194-
CHAPTER
VI
CALCULATIONS OF THE
B 1 1 ( p , y ) C 12 CROSS SECTIONS.
-195-
A:
Introduction
The formalism developed in the preceeding chapter
constitutes a complete scheme for calculating differential
cross-sections of reactions from the knowledge of the
stationary states of the system.
It therefore provides
an extremely powerful tool in examining the results of
model calculations which predict such stationary states.
The theory has been developed in considerable details,
omitting only material already available in standard
reference books or in reliable articles; in this chapter,
we pull together the various expressions and component
calculations developed in the last in order to carry out
the actual calculation of cross section.
It is, of course,
obvious that this approach is not reversible i.e. given
the experimental cross-sections it is not possible to
derive from these a unique description of the stationary
states involved; rather we take such descriptions from
model calculations and attempt to validate them through
comparison of the cross-sections which correspond to them
with the experimental data.
The purpose of this chapter is then twofold.
First­
ly, it is intended to provide a concrete example of a
calculation carried out by these methods.
This example
will demonstrate that chapter V is indeed complete, from
a practical point of view, and that its results are easy
to use.
Secondly, it is intended to provide the missing
-196-
link which will enable us to directly and meaningfully
compare theory with experiment in C
predicts a set of states for
12
17
.
The theory
C based on some fundamental
assumptions about the nuclear structure, but a direct
comparison between the electromagnetic strengths of these
states and a (/,^) differential cross-section is irrelevant
at best, a qualitative comparison with the total crosssection may be useful.
One of the reasons is that the
resonances are wide and overlapping, and interferences
between states play an important role.
That the resonances
are wide and overlapping is clear from a comparison of
the density of stationary states and the corresponding
smoothness of the cross-section with energy.
The
importance of interference terms is proved experimentally
by the fairly large odd- f? terms in the angular distribu­
tion :
d er
a n
t ’ f
(vi-l)
In such a situation, where states and resonances (or
group of states and resonances) do not correspond one-toone, the importance of stationary states is greatly
decreased; however, it is still possible to predict
differential cross-sections from fundamental assumptions,
measure them, and compare.
This way, the fundamental
assumptions about the structure of a particular group of
nuclei can be tested; and it is these assumptions we are
-197-
really interested in.
It matters not that the stationary
states of the system have lost much of their relevance
as intermediate quantities.
It still remains, in all
cases where resonances and stationary states can be
matched one-to-one, that the technique of using states
as intermediate quantities is extremely powerful; this
applies for example to low-energy nuclear reactions,
(isolated resonances) analog states (sharp resonances
on slowly varying background) and perhaps to cases where
one single state, although not isolated, is presumed to
contribute dominantly to the cross-section (the peak of
the C
12
giant resonance, for example).
The calculation carried out in this chapter should
be thought of as only constituting an example.
The
method is by no means restricted to (/£>/) reactions, nor
is it limited to one-particle one-hole states.
However,
some minor additional calculations which have not been
covered in chapter V would probably have to be carried
out in these other cases.
In part B of this chapter, the theory of nuclear
reactions developed in the preceding chapter is specialized
to El, E2 and Ml transitions in C
12
following the capture
of protons; only one-particle one-hole states are
considered.
A brief description of the numerical
methods used in the computations is also given, and
complete listings of the programs will be found in
chapter VI-D.
-198-
Complete sets of calculated angular distributions
and cross-sections are compiled and plotted in part C,
together with experimental results, for comparison.
Gillet’s wave functions are also compared with pure
configurations, to see if they constitute a substantial
improvement over unmixed configurations.
B:
Specialization of Formulae from the Theory of Nuclear
It . . 1 2
Reactions to the Reaction B (p,y)C
In this section, the theory developed in chapter V
is specialized to produce a calculational scheme for
il ,
the reaction B (p,y)C
12
El, Ml and E2 transitions to
the ground state or any excited state of C
12
are considered.
All stationary states of C^-2 are assumed pure one-particle
one-hole states, with wave function given by Gillet^3-^.
All channels other than nucleon channels leading to a
one-hole state in the residual nucleus are neglected.
The dimensions of the various quantities introduced
in the theory chapter will be discussed in some details,
using the following convenient system of units:
Time
Second
Energy
MeV
Length
Fermi
Cross-Section Microbarn
In this system we have
-199-
r fc
=
197.3
MeV fermi
4
=
1.44
MeV fermi
777C2
=
938
MeV
„2
=
/ /m U to fa /in .
and
VI-B-1
—4
10 fermi
2
C is the speed of light, yS the nuclear magneton;
where
771
=
0.126 .MeV^fermi^2
4
are the proton mass and charge respectively.
Framework for the Calculations
The theory of nuclear reactions developed in the
preceeding chapter is very general and essentially exact.
In practice however, it is impossible to carry out
numerical calculations involving an infinite number of
states and channels.
Our first approximation will
consist in truncating the space of states and channels;
this will be done in a way which is consistent with
Gillet's work.
Essentially,
we keep all states Gillet
has included in his calculations, and all channels to
which these states are strongly connected.
Not only do we propose to test Gillet's wave
functions, but, more important, we want to test the
fundamental assumptions which enter into such a calcula­
tion.
For this reason, all the assumptions made in
calculating a set of states can also be used in calculating
the cross-section, and no additional approximation in
principle should be made.
Our intention is to stress the
-200-
importance of such a completely coherent scheme, from
first principles to calculated cross-section.
The truncation of the space of compound states
involves two separate approximations.
The first one
consists in neglecting all states which are not oneparticle one-hole (the vacuum is also included).
We are
not implying that there are no other states; it is well
known in fact that a
0
state exists at 7.65 MeV which
is clearly not one-particle one-hole.
It is simply
assumed that states of a more complicated nature can be
neglected with reference to the
) results.
A
discussion of the conditions of validity of the various
assumptions and the conclusions we can draw from the
results of experiment will be the object of another
chapter.
The second truncation consists in restricting
ourselves to states of two major-shell excitations or less.
These are states which in the extreme single-particle
model would have energies o k c o ,
/£ c v
and
This
leaves a finite-dimensional subspace, and a suitable basis
for this subspace has been given in the introduction of
chpater ~3~ (theory) .
This approximation is expected to
be fairly good, inside the nucleus, whereas neglecting
many-particle many-hole states may turn out to be
completely unjustified.
It is not sufficient to define the subspace of
states; one must also choose the eigenvectors in that
space, which will be the stationary states of the system.
We use the states of Gillet's "approximation I" whenever
-201-
possible, and pure configurations for the states which
Gillet did not tabulate in his thesis.
These states
were obtained by taking the unperturbed energies from
♦
neighboring ..nuclei and diagonalizing a parametrized
residual interaction; the parameters were obtained by
fitting level positions in G1^ and 0^6 .
procedure will be found in Gillet's
Details of this
thesis^).
Since Gillet assumed charge independence of the
nuclear force, the states are eigenstates of the total
isospin operator; they correspond to 7”= 0 or T = 1 .
12
Also, the ground state of C
^ taken to be the pure
.
particle-hole vacuum.
Knowledge of the states of C
12
is not sufficient
to calculate cross-sections of a reaction wherein it
constitutes the compound system, even in the case of
gamma-ray scattering.
Something must also be known of
the open channels.- In the case of a ( P » y )
experiment,
the knowledge of the ground state of the target nucleus
is very important.
The same is true to a certain extent-,
in a (y ,f> ) experiment.
Even this, however, does not suffice; in principle,
we must know the wave functions of all the states of the
fragments into which the compound system is allowed to
break.
This includes many possibilities; however, the
one-particle one-hole states are expected to decay mostly
by nucleon emission, where the excited particle is emitted,
leaving the residual nucleus in a one-hole state.
-202-
This process may be graphically represented as follows;
FERMI
SURFACE
(a)
' (b)
It is also possible for another nucleon to be emitted,
leaving the residual nucleus in a one-particle two-hole
state; for example, the diagrams
(c)
one
of the seven particles in the
emitted.
(c)and (d) in which
(d)
f p ^ subshell has been
Emission of an alpha particle from C
12
is
energetically possible from 7.4 MeV and would correspond
to the diagrams:
-203-
->a
-oxxxx
(e)
(f)
It is not possible to treat all the open channels, not
only because there are too many of them, but also and
mainly because the states of the residual nuclei are not
known.
It will prove sufficient, however, to retain
only the most important channels, the other (unobserved)
channels providing just a slight broadening and additional
shift of the resonances.
That this is so in first
approximation can be seen in the Wigner many-level
formula (V-106).
Until we reach the two-nucleon emission threshold,
at approximately thirty MeV, (above the C
12
ground state)
the most important channels for decay are those of
diagrams a and b.
These channels certainly remain most
important for one-particle one-hole states even above the
two-nucleon emission threshold, the two-nucleon emission
coming mostly from two-particle two-hole states.
It is worth considering separately the one-particle
one-hole states having a hole in
and those having
-204-
a hole in
/S^x .
In the first case, the excited particle
has a very substantial amount of energy and therefore
will leave the nucleus quite readily.
.3/2
nucleus will'consist of a hole in
state.
Ip
The residual
, which is a 3/2“
Looking at the 3/2“ states in C '*
and B "
see that there are two low-lying 3/2“ states;
state and an excited state at about 5 MeV.
expect a pure hole in
of these two states;
we
the ground
We therefore
to be a linear Combination
fp>^Z
we have assumed in the calculations
that the ground state of B 11 and c'* consist of a hole
, 3/2
in
ip
. It is an unreasonable assumption; however,
in first approximation, it will not change the results.
The one-particle two-hole part of the B 11
ground state
(target nucleus) will mainly contribute the formation of
two-particle two-hole states in C
neglecting anyway.
12
, which we are
In the outgoing channels, when the
energy is high enough to reach the first excited state,
the particle-hole state of C
12
will decay to both the
ground state and to the 3/2” excited state of B 11
and C tl
The decay can also reach one-particle two-hole states of
B /;
and C 11
by diagram c, but the probability for this
is fairly small because considerable energy is concentrated
in the excited particle.
The situation is somewhat different
when the hole is in the 1
state. Since the hole energy
12
is about 35 MeV in C , all these states lie at fairly
high energy, typically 30-50 MeV.
In the region of interest,
most of the energy is concentrated in the hole and
-205-
relatively little in the excited particles.
For this
reason, these states will tend to de-excite by processes
of diagram at almost as readily as by those of diagram 1- .
' l
In fact, because there are 8 nucleons in
3/2
Ip
and
only one excited nucleon, the process of diagram d
easily become more important than that of
could
.
When the nucleus deexcites through channels of
type
, the residual nucleus is left with a hole in !S ft
which is a
for / S %
^ ^
channel.
Although the main strength
holes lies at about 35 MeV for B n
there is a
*
and C f/ ,
level in both nuclei between 6 and 7 MeV
which should contain a significant portion of the /S^1 hole
configuration.
Therefore we always have this channel
open, above 7 MeV of proton bombarding energy, and it
has been taken into account.
configuration in the low-lying
The fraction of
^ ^
(S'/*
excited states
has been included in a reduced width parameter called
" /s'/* hole damping” .to be discussed below.
-206-
VI—B-2:
The States of C
12
and the Use of Gillet*s Wave
Functions.
12
As a basis for the states of C
Gillet has
fone-particle
chosen to use j-J coupled]
configurations.
( one-hole
These configurations are simply
o
=
0
Z ' O)
(L C / ) &
? » tv J
* L
x
^ . ) K>
(VI-2)
where the proton creation operator
C/jn
L?.>
creates a
proton in single-particle state
» *2?
*
v?s h )
f
! s ™ s>
(VI-3)
n>4,rrjs
«Af
and the neutron creation operator
CL. •
tjr *
neutron in the same single particle state.
are the corresponding hole operators.
to the
T=
0 states and the
—
The
creates a
The
O 's
sign refers
sign to the 7~=1 states.
A more complete definition of these configurations,
including phase conventions, will be found in appendix
VI—B .
The best way of visualizing these configurations
is to simply think of
them as being antisymmetrized
direct products of single-particle states; the creation
-207-
operators constitue a shorthand notation for Slater
determinants.
It is somewhat difficult to understand'Gillet*s
notations and phase conventions, and all the detailed
work involved will be found m
eigenstates of C
appendix VI-C.' The
as calculated by Gillet can be’ expressed
in terms of the configurations defined above, and Gillet's
tabulated particle-hole amplitudes. (See tables 6.1 - 6.6
in ref. 1 and tables 10 - 13 in ref. 10; ref. 1 is more
complete).
We have:
(VI-4)
where
J
/ A 'J f f 'f '
is state number A of total angular momentum
* , projection M •
It has parity
7*(Tf=
0 for even parity
IT * ~ 1 for odd parity) and proton-neutron exchange symmetry
V-
( 7"*= 0)*
Similarly,
/ X
J
is
antisymmetric under' .
proton-neutron exchange ( 7”= 1).
by Gillet.
are the amplitudes tabulated
. , - 3/2
For example, the amplitude of the i p
configuration in the second / ( T -
0) state, predicted by
Gillet at 12.3 MeV, is (table 6.2, ref. 1, or ref. 10 table
11)
-208-
Establishing formula VI-4 with the correct phases
required a careful analysis of Gillet's phase conventions,
and the detailed work involved is the object of appendix
*
VI-C.
Appendices VI-A and VI-B are prerequisites for
Appendix VI-C; appendix VI-A establishes the general
formula for conversion from the proton-neutron scheme
into isospin formalism, and appendix VI-B is an exact
definition of our standard single-particle states.
VI-B-3
The Nuclear Potential
In order to compute the nucleon reduced widths, a
12
shell-model approach similar to the method of Lane
was
used; this involves the concept of an average nuclear
potential.
The reduced widths are related to the value
of the radial wave functions at a certain distance from
the center of mass, and therefore are relatively
sensitive to the details of the surface of the nuclear
potential.
For this reason, a realistic nuclear potential
was chosen, and the radial wave function obtained by
numerical methods.
A realistic nuclear potential for C
12
is the
so-called "modified Gaussian potential" which fits very
well Fregeau’s
(13)
electron scattering data.
Assuming
the density of nuclear matter to be proportional to the
density of charge, we would expect a nuclear potential
of the same form,
-209-
F ig. V H -1: R ea listic m odified gaussian potential fo r C ^ :
( 2 ) fo r proton s. (3 ) rep resen ts the Coulom b repulsion potential.
( 1 ) fo r neutrons;
-210-
F ig. VI-2 : r tim es the radial wave functions o f the s in g le -p a rticle states
in the neutron potential o f fig. V I-1 ; these radial wave functions (for the unbound
sta tes) have been defined by requ irin g a log. derivative of -1 at r=4. 5 ferm i.
The s in g le -p a rticle en ergies a re also given.
-211-
0.4
0.3
0.2
0.1
-
0.2
-0.3
-0.4
Fig. VI-2 : r tim es
o f the proton potential o f fig.
sta tes) have been defined by
The s in g le -p a rticle en ergies
the radial wave functions o f the s in g le -p a rticle states
V I - 1 ; these radial wave functions (fo r the unbound
requ irin g a log. derivative o f -1 ar r = 4. 5 ferm i.
a re a lso given.
< J
^
J
(*+*)
(24 )
Fig : 2
-212-
(VI-5)
The constant..!^
has been determined to be 65 MeV by
requiring that the
if 1
state be bound, and have a binding
energy of approximately 10 MeV for neutrons.
This was
done for consistency with Gillet's experimentally determined
hole energies.
(Gillet^1 ) fig. 4.2)
Protons were assumed to move in the same nuclear
potential as the neutrons, and the Coulomb repulsion
originating in the modified Gaussian charge distribution
was added.Fig. VI-7 shows
the total proton and neutron
potentials.
VI-B-4
Radial Wave Functions
If
is a radial wave function corresponding
to orbital angular momentum quantum number
, and
fo C S l) 5 S I U ^ C A ) we have
(VI-6)
A 2
£
=
0. 6 ¥ 4
/ m m i 2 P e l/"
(VI - 7)
is the energy of the particle in the potential.
The
radial wave functions are further defined by two boundary
conditions:
finiteness at / l -
0 is one of them, and the
other condition depends on whether the particle is bound
-213-
or not.
If the particle is bound, the condition is that
goes to zero when yj,
increases to infinity.
If
the particle is not bound, the condition is that at an
arbitrary value of
derivative of £ ^ (4 .)
j i,
, say yz =• 7 ?
the logarithmic
have a definite value
7%
i
/U'flg)
=
4'
or equivalently
3 ? K 'c r a
a
' <VI-®>
, ,
‘
7^.
was chosen immediately outside the nuclear potential,
at 4.5 fermi; for larger values of / I , neutrons can be
considered free, and protons interact with the residual
nucleus through electrostatic forces only.
of
^
This value
= 4.5 fermi has therefore been chosen as the
channel radius in all the channels.
No other value of
has been tried with the modified Gaussian potential, and
7^
is thus not a fitting parameter.
The constant
bc
is called the boundary condition
parameter, and bc - -1 corresponds very closely to a
vanishing level shift; this means that if the states
satisfy the boundary condition with
6=
-1, the center
of a resonance will occur near the eigenvalue £
of the
corresponding eigenstate.
Since the energies for the
12
states of C
have been determined by Gillet from empirical
one-particle and one-hole energies, they correspond to the
resonance energies; this means that for consistency, we
-214-
should minimize level shifts.
For this reason, we have
set in all channels
= V .-5
and we already had
tTru
(VI-9)
For each value of-/ we thus obtain an infinite
number of radial wave functions, corresponding to 0,1,2-nodes.
when
When
72 =
0 we label them / s ,
/ =
1, we use
write
(ft,
the notation
where
72
2s
, 3s,
Z j> ,
3 f> ,.
etc.;
— - and
is the principal quantum number,
and is equal to the number of nodes plus one.
The differential equation has been solved numerically
by performing the change of variables
- A * * ' CiSg
(VI-10)
which yields
with the boundary condition at zero (starting values)
__
CufaCo) - o
Cis€
Co)
The differential equation for
= /
(VI-1 2 )
was integrated numerically
and we searched for the values of 2 :
boundary condition at
when
£2o
which satisfied the
and at ^
when £~ )> o
,
Fig. VI-2 and VI-3 show the radial wave functions obtained,
and the corresponding single-particle energies.
These
single-particle energies agree well with the empirical data
-215-
used by Gillet, except that we do not have the experimentally
observed -E 'S
splitting, since no spin-orbit force was
included in the potential.
The 2 /> state is also sub♦
stantially higher in Gillet*s work, but these slight
differences are not very important.
Some degree of
inconsistency is unavoidable, because Gillet used harmonic
oscillator wave functions ana empirical single-particle
energies, instead of obtaining both wave functions and
energies from a realistic potential; however, by using a
realistic potential well (which agrees closely with the
empirical energies) to calculate the reduced widths, the
inconsistencies are reduced to a minimum.
situation can be summarized as follows :
In fact, the
Gillet used poor
radial wave function (harmonic oscillator) to evaluate
the residual interaction, and we may expect his wave
functions to be fairly inaccurate.
It turns out, however,
that the calculated cross sections are much more
sensitive
to a change in reduced widths than they are to a change
in the wave functions; in fact, at the end of this chapter,
results obtained with Gillet*s wave functions are compared
with results corresponding to pure
tions.
coupled configura­
Except for details, the predicted cross-sections
and angular distributions are essentially the same.
This has very important consequences for calcula­
tions of the type of B r o w n a n d of G i l l e t ^ ^ 3-0^ and
they will be discussed in details in another chapter.
-216-
VI-3-5:
Radial Matrix Elements
The normalization of the radial wave functions
was performed numerically; we have
(VI-13)
The wave functions
therefore have dimension
-3/2
(length)
. For the electromagnetic transition matrix
elements we need to evaluate the integrals
-a
where
b
of length.
~ ™
(VI-14)
is an arbitrary parameter having dimensions
For electric dipole transitions we have
(VI-15)
and for electric quadrupole radiation,
£ 2 : 1
1
L ’ Z
i - t - 4 ' / * O o* Z
(vi-16)
The other integrals do not vanish, but they are not
needed in the calculations.
These radial integrals
have been obtained numerically and they are given in
table VI—1 for a value of
& = /£/
fermi.
It is not a good approximation to use harmonic
oscillator wave functions to compute the electromagnetic
transition matrix elements; Gillet hoped that these matrix
elements would depend on the whole volume, of the nucleus
and not be affected too much by the fact that harmonic
-217-
L= 1
1s
If
2
s
2
1.05
1s
Ip
Id
IP
0.18
1.41
1.05
Id
p
1.41
-0 .9 5
2 .0 4
•
-1.32
2 .04
If
2
s
2
p
1.76
-0 .9 5
0 .1 8
1.76
-1.32
(a)
CVJ
it
_J
Is
1S
i.n
1.37
If
s
2
p
s
2
- 1.25
-4.10
-4 .2 7
5 .1 0
4 .4 0
-4 .1 0
-4 .2 7
-1.25
4 .4 0
(b)
a) R adial integrals fo r E l transition s.
b) R adial integrals fo r E2 transitions.
They a re defined as:
>.
*
Ju » i un<eM(i)
with >P = 4. 5 fe r m i,
v
T A B L E VI-1
p
-0 .8 6
4 . 17
-0 .8 6
2
2.51
2.51
2
If
1.37
2.13
IP
Id
Id
IP
A ’a
^ = 1 .6 1 ferm i.
oscillator wave functions are a bad approximation near
This is true for M l
the surface.
but for £ /
transitions, of course,
transitions, discrepancies are typically
*
20% and for
£2,
transitions they can be as high as 60%.
This can make a difference of more than a factor of two
in the cross-section, and can completely change the angular
distribution.
The radial integrals given in table VI-1 are those
for the proton states.
Since the neutron results do not
differ appreciably from the proton results, and because
almost all the electromagnetic radiation comes from protons,
we have used the proton radial integrals exclusively.
Introducing different proton and neutron radial wave
functions would be inconsistent with our treatment of the
isospin as a good quantum number; we have neglected all
Coulomb forces so far, inside the nucleus.
The fact
that the radial integrals for protons and neutrons are
nearly equal confirms that the approximation is good.
VI-B-6:
Reduced Widths and Boundary Value Parameters.
The calculation of the reduced widths is in principle
extremely difficult.
The reduced widths are defined in
-219-
and the scalar product
C c ,A \
is an overlap integral
between the channel wave function of
function of
/
/C )
and the wave
, performed over the channel surface
*
defined by A c Consider C
nucleon channel,
12
in some shell-model state; for a
/lc
is just the distance between that
nucleon and the center of mass of the rest of the nucleus.
/c)
is the direct product of the state of 11 nucleons
corresponding to an eigenstate of the residual nucleus
<B" or C n ) which defines the channel region, and a spinangular wave function of relative motion for the pair of
fragments.
The channel states ! c ) have little to do with
the shell model, except perhaps when we represent the
state of the residual nucleus by a shell-model state.
We can always assume that
M^>
is given as a
linear combination of configurations (direct products of
single-particle states); this is perfectly general.
If
is a shell model state, these single-particle states are
eigenstates.of a particle in an average potential and
only contains a finite number
number) of configurations.
/A>
(usually a~very restricted
It is important to understand
that the shell model assumptions are really the truncation
of the basis of configurations and the introduction of
phenomenological "residual" interactions, and not the
introduction of an average potential.
There lies its
\ real weakness, and we shall see that the shell model breaks
down completely in the calculation of some reduced widths.
-220-
Let the shell model state //> be given as
/a > =
where
represents pure configurations, * that is to say,
simple direct- products of single-particle states coupled
to good angular momentum.
of a state //>
To evaluate the reduced width
into a channel
/c )
it is sufficient to
know the reduced widths of the configurations
into
channel j c ) ; indeed,
f a c
?
=
f a
^
(VI-18)
Therefore, we will restrict the discussion to the scalar
products
Cc/fi)>
.
Also, in order to see more clearly the
essential features, we will assume the configurations
to be simply the direct product of an 11-nucleon configura­
tion
! f in )
coupled to good angular momentum, with a one-
nucleon state in
coupling,
/f i )
~
j
f f iX
.
Thus
f i n ) } fit)
Here, the nucleon involved in ( f i P
( vi - i 9 )
is the one which consti­
tutes one of the fragments in channel /C)
;
thought of as constituting the core, and
particle.
In practice, the states ( f i „ ) and
could be
the extra
jf iP
would
be coupled to good angular momentum, but this will be
avoided here, to keep the discussion as simple as possible.
In a similar way, the channel state / )
can be written
-221-
as the direct product of the internal states of both
fragments with the state of orbital
relative motion.
In
this case, one of the fragments is a nucleon, and its wave
*
function reduces to just a spin state (and i?ospin state
if isospin formalism was used).
It is convenient to
couple the spin of that nucleon with the orbital angular
momentum of relative motion, but as previously, we shall
momentarily refrain from coupling the result with the
angular momentum of the residual nucleus.
Therefore we
write
/C )
-
H >
H jm )
(VI-20)
where
//^
is the 11-nucleon state of the residual
nucleus in channel
/c) ,
a n d t h e
result of coupling the
spin state of the nucleon fragment with the relative motion's
orbital angular momentum.
<?//> =
The scalar product-becomes
C * / i» / a
The first term vanishes if
(v 1 _ 21)
'> < i / A . >
and
U jj r n ') do not involve-
the same particle, or if the single-particle state !/& ,) does
not correspond to angular momenta
state /y ,y
fy * * 7
•
Otherwise, the
is nothing but the direct product of
/■ P jm )
with a radial wave function.
//O
=
U p (A )
U
p *>
- U ^C a )
(VI-22)
If we have been careful to define all the phases consistent­
ly, the radial wave function
will be identical with
A
-222-
the radial part of
/A ^
, otherwise, it may differ by
a phase.
The second term in the equation above, < Z T /A n A ,
simply expresses the probability amplitude that the
state /A ff A
made by pulling a nucleon out of the 12-
nucleon state / f t )
of C 11
or B /;
, will look like the eigenstate / I )
; Some knowledge of the wave functions of
and B n
the states of
will have to be assumed in order
to calculate the reduced widths.
The only other quantity
needed is the value of the radial wave function
at
, which is simply U g C^c X •
-3/2
The dimension . of
(1 < /) is (length)
as can
the channel radius
^
be seen easily from our normalization
/P
f
‘
and therefore
U > ) '
lc U =
I
(VI-23 )
has dimension (Energy)^.
The
constants in equation (V-83) can be evaluated;
^
=
A c e (2 7 > te c ? z y f‘ < r / A > ? ? U s ( z )
-
2 .2 * 7
< r /A >
X
(vi-24)
where we have taken the reduced mass
7^-
4.5 fermi and
c
{/■ *(% .) is in fermi
A
v
7?7^
— 3 /2
.
= 857 MeV,
The reduced
it
widths are then obtained in MeV^.
This equation cannot be used as it stands, because
we have not expanded our states /A )
configurations
Ifi)
in terms of the
just discussed, but in terms of
configurations of good total angular momentum, and we
-223-
have taken the symmetric and antisymmetric combinations
of proton and neutron particle-hole states.
When the
formula is modified to take these facts into consideration,
we obtain
"
the -f
HX-2S)
7 F " ?
sign always applies, unless JS is an antisymmetric
configuration and
/ X y and
C
is a neutron channel.
involve the same particles and the
same projection of angular momentum.
given by the above formula only if
Finally,
y c is
C
have
and
the same total angular momentum; otherwise,
Y
vanishes.
Having obtained these very important preliminary
results, we discuss the validity of using such an approach
to compute reduced widths.
We hope to demonstrate that
the shell-model breaks down in at least two different ways
when we try to use it to evaluate reduced widths, as we
have done above.
This discussion will be important for
understanding the limits of validity of our calculations", and although at the moment we can offer no simple solution
to alleviate the problem involved, we hope that these will
be solved in the near future.
It is relatively easy to
deal with an example, and one-particle one-hole states in
12
C
will be used to illustrate the difficulties. Let us
examine one of the low-lying
T = 0 states below the proton
and neutron tresholds, for instance the first excited
-224-
state, which is a 2^" state.
This state is predicted by
a particle-hole calculation to be mostly a
if *
/P
state, with small admixtures of higher configurations.
These small admixtures of other configurations are ob­
tained by diagonalizing a Hamiltonian which is the sum of
an unperturbed Hamiltonian and a so-called "residual
interaction".
The effect of the residual interaction is
to pull down the energy of some states from their unper­
turbed values, and in this way, one hopes to reproduce
the experimental energy spectra.
The way in which this
is accomplished involves mixing the higher configurations
with the low lying configurations.
It is quite probable that the procedure substantially
improves the wave function inside the nucleus, but the
end result is to make the wave function near the surface
completely wrong.
This occurs even in the case where- a
realistic finite-walled potential has been used; we shall
not bother to talk about the harmonic oscillator potential,
which is totally inadequate for dealing with the unbound
states.
(In C
12
only a few states are bound, anyway).
That the wave function near the surface is completely wrong
is quite obvious.
Since the state we are now dealing with
is a low lying bound state, all the nucleon channels are
closed and the radial wave function of relative motion is
exponentially decreasing with X tQ
vanish.
.
All the reduced widths
But if we try to compute the reduced widths, using
-225-
the approach outlined above, we find that the radial wave
function in the nucleon channels is far from decreasing
exponentially.
It is true that the part of the wave function
corresponding to the
/f a
2 / f a z configuration leads to
exponential decrease in the nucleon channels, because the
/f a
radial wave function is bound, but all the higher
configurations mixed in,
/ f a *^*
/
for example, lead
to oscillatory behavior in the nucleon channels, and they
all lead to non-vanishing reduced widths.
reach a contradiction:
Therefore we
those same parts of the wave
function which were used to pull down the energy of the
state below the proton and neutron thresholds also enable
it to decay, when in fact it has become energetically
impossible; the admixtures which improve the wave function
inside the nucleus serve to make it completely wrong, even
qualitatively, near the surface.
We would like to stress the fundamental reasons
why this contradiction occurs, and try to show both a
physical picture of the situation and the mathematical
mechanisms which come into play.
Dealing with the
physical picture first, we see that when one of the
nucleons reaches the surface of the nucleus, the
representative point in configuration space approaches a
set of channel regions, each channel region corresponding
to an eigenstate of the residual nucleus.
As the nucleon
N leaves the residual nucleus, we penetrate into the various
channel regions, and in each region the residual nucleus
is in an eigenstate of energy.
What will determine whether
-226-
a given region is closed or open is just the difference
between the total energy of the system, and the sum of
the nucleon rest mass and the energy of the eigenstate
of the residual nucleus corresponding to the region in
question; this difference is the energy available for the
relative motion of the two fragments.
If this energy is
positive, the corresponding channels are open; if this
energy is negative, the radial wave functions of relative
motion decay exponentially in these channels.
This means
that what happens near the channel surface in configuration
space really depends on the properties of the states of
the residual nucleus, more than on the properties of the
states of the' compound system.
As we would have expected,
as soon as one nucleon moves away from the nucleus, the
average field to which it was contributing simply starts
to collapse.
Mathematically speaking, since the expansion
in terms of shell-model states is complete, we could still
treat the nuclear surface with particle-hole formalism;
however, in order to obtain a correctly closed channel ~
(for bound states) when higher energy configurations are
included, it will obviously take an infinite number of
terms with the correct relative phases.
This is analogous
to the expansion of the function
a
u
-*
“
/
l" X +
-g j
-
X3
-jy
(VI-26)
\ which goes to zero exponentially as x increases, although
-227-
each of the terms in the expansion goes to infinity in
absolute value.
:finite number
Obviously, if we were to keep only a
of terms in the expansion, the function
*
would go to
t
00 when
X
increases.
Consider the shell
model's one-particle - one-hole expansion of two major-shell
excitation or less, as used by Gillet; In the channels
where we know from simple energy considerations that the
wave function must decrease exponentially when / l „
goes to
infinity, we have only one or two terms in the expansion,
and they have oscillating behavior.
For the same reason
-X
that we need an infinity of terms in -6
, we also, strictly
speaking would require an infinity of terms in the shell
model.
In practice, of course, we would be quite happy with
enough terms to provide a reasonable approximation near
the surface.
But these higher terms could only be obtained
by studying the properties of the surface, not by fitting
level positions, which are mainly dependent on the bulk of
the nucleus.
When discussing the "surface" we have made
no great distinction between the physical surface of the -7
nucleus and the channel surface in configuration space;
it is hoped that we have not created confusion, since we
have been talking mostly about nucleon channels, and the
channel surface in configuration space then corresponds
to one of the nucleons being at the physical nuclear surface
Thus the surface region corresponds to the nuclear surface,
in a sense.
The fundamental distinction nevertheless exists
-228-
There is another important case which arises, and
which will prove to be very important in calculating the
B
//
,
(Jt>j y f )C
12
cross-section from Gillet's wave functions:
it is the case of states above the nucleon'emission
threshold but close to it.
Essentially, the same type of
problem arises; the particle-hole state is a poor approxi­
mation near the channel surface.
the channel regions,
Again, when we go near
we find that the wave function
splits into the various channel regions corresponding to
the eigenstates of the residual nucleus; the residual
nucleus rearranges itself, and the wave function will
oscillate in some channels while it will exponentially
decay in most of them.
We choose the case which is the
most important for these calculations, as an illustration;
it is the ( / S ~ ^
)1
( 7~-l) state, predicted by
Gillet to be mixed mainly with the ( /£> ^
configuration.
)
The admixture of other configurations is
very small and will be neglected in this discussion, al­
though the calculations take it into account.
According
to our formula for the reduced widths, the conf. ( / S ' ^ 1
is bound.
This is because particles in the / s
and / p
shell are bound, in a realistic average potential for C
12
This means that the state Gillet predicted at 33.8 MeV
,3/z
would have a very small reduced width , coming from its
/ -^ / 2
(
/p
f< Z
)1
admixture.
)/
If calculations are
performed according to these reduced widths, a narrow
resonance (of the order of several hundred keV wide) with
a peak cross-section well exceeding the observed giant
.
-229-
resonance is produced.
The situation here is the same as
for analog states; the state is formed through a small
impurity; it can decay by particle emission only through
*
this small impurity, but it will strongly gamma-decay
12
to the ground state of C . Thus a very narrow and intense
resonance is produced, a compound nucleus type resonance
over the direct reaction background.
This phenomenon was
not observed experimentally , and there may be.several
reasons for this, but in the light of our preceeding dis­
cussion it would be advisable to reconsider the calculation
of the reduced widths.
In particular, the reduced widths
of the configuration (
no sense.
)1” certainly make
There are some 20 MeV available above the
particle emission threshold, and it is clear that not
only can the
/l A
/p
particles as well.
particle escape, but so can the
tip /'1
Ip
This is because as soon as one of the
particles gets near the surface, the highly strained
residual nucleus will collapse; or in quantum-mechanical
terms, the wave function in that region of configuration'
space will become that of a linear combination of
residual nucleus states with a nucleon orbiting around,
and many of the corresponding channels will be open.
What this proves again is that for a particle-hole
expansion to be useful near the surface we must include a
very great number of terms.
\ an almost pure (
/S ~ ^ Z
It is not possible to have
/ />
)1~ state standing around
-230-
20 MeV above the threshold.
However, we may assume that this
approximation is fairly accurate inside the nucleus, and
try to estimate the reduced widths from the time the
nucleons spend at the periphery of the nucleus.
When the
/p
/p
nucleon or any of the
nucleons
arrives near the nuclear surface, we have seen that the
rest of the nucleons may
rearrange themselves and this
nucleon may then leave through a
ij/2.
p
/j%
/>
or a
channel.
The probability that this will occur depends oh the scalar
/ S 'A
product between the one-hole in
or one-particle
two-hole state left behind, and the eigenstates of the
residual nuclei corresponding to the open channels.
In
principle, all these open channels should be considered
individually in the
7 ft -matrix; the channels which
/ 3 /2
/fa
correspond to the emission of a
as important as the
/t'/z
/p
nucleon may be
nucleon channel, and since
//V2
there
are 8 nucleons in the
enhancement by a factor of 8.
/P
shell, we get an
Since the outgoing protons
and neutrons are not observed in a ( f a i t f )
experiment, the
main effect of these open channels will be to increase
/ 'A
the total widths of the states with a hole in the IS
state,
if we assume the many-level formula to hold approximately.
For this reason, instead of including separately all the
channels leading to the excited states of C
we have included only two (for each spin):
leading to the lowest-lying
respectively.
//
and B
it
,
the channels
Zz ^ states of C 7 and B n
For the value of the radial wave function
-231-
CB: )
which enters the formula (Jan. 21 p &
we have used the value of the / f i
wave function at ^
^
= 4.5 fermis.
)
bound state radial
The justification
*
for this is..that while the nucleon is still'within the
residual nucleus, we may hope that its radial wave
function will be fairly accurate; at 4.5 fermis, when it
is just leaving the influence of the nuclear potential,
it will go into wide oscillations corresponding to the
energy available for the relative motion in the various
channels.
-232-
Two additional factors must be taken into account;
, L 3/2.
one is the enhancement due to the eight ! p
particles,
which we are not including explicitly, and the other is the
overlaps
<?T
(ft, }
between the states of,the residual
nuclei and the state formed by pulling a
nucleon out of the given C
12
//> ^ z
or
f
one-particle one-hole state.
Grossly speaking, the enhancement is expected to bring a
factor of three.
(nine nucleons instead of one) in the
reduced width; the effect of the various scalar products
JJ
is more difficult to determine, and the effect
of artificially giving all the width to the
!/>
nucleon
is also very difficult to estimate, except that we can be
certain that the effect will be to reduce the effective
//5 ^ width.
Following this discussion, we have used for the
reduced widths of all the states which involve a hole in
the /S
shell:
Ypc
=
"
^
(VI-27)
When the parameter CO is varied within reasonable limits,
i.e. from 0 to 3, the only important effect is to change
the contribution 6f the 1~ state at 33.8 MeV.
At
C 0=0
the resonance is very narrow and the peak cross-section
is of the order of 300 y t f - barns; for CO = 3, the resonance
disappears completely.
for C O - 1 and
The calculations have been plotted
CO = 2.8; it is felt that a reasonable value
of CO should fall between these two extremes.
-233-
The calculation of the other widths, however, is
very sound; the
2s ,
/d ,
/ / ’ and.
Z p reduced widths are
expected to be quite accurately given by formula (VI-25).
♦
The reduced widths for proton channels are somewhat
larger than those of neutron channels; this is to be
expected, because of the additional Coulomb repulsion for
the protons.
Thus we would expect a slightly higher cross-
section for ty ,p ) than for { y n ); this is in qualitiative
agreement with experiment.
There is also the slightly
higher energy available for proton decay, which favors
the protons.
Since Gillet has neglected the Coulomb
forces and assumed isospin independence of nuclear forces,
we have neglected the slight difference (5% or less)
between the calculated proton and neutron widths; the
proton channels being much more important for us than
the neutron channels, we have used the calculated proton
widths for all the nucleon widths.
These values of the reduced widths correspond to
a boundary value
Ip
and
Z .S .
parameter
b, -
-1, in channels
fd , / / \ .
The boundary value parameters have been
chosen at -1, because it corresponds to the situation
where the energy of the eigenstate is equal to the ob­
served resonance energy;
since Gillet took the unper­
turbed energies from experiment and fitted position of
resonances in C
12 , in order to use bis level energies
we must use the boundary condition
bound states,
js
and
//>
/) -
-1.
For the
, the situation is somewhat
-234-
different.
There is no configuration with a f s particle,
since the whole I s
single-particle level lies below the
and all the / S
Fermi surface,
particles have been
treated as truly bound; therefore no problem arises.
/p
The
single-particle states, however, have a non-vanishing
width, as discussed above; however,
to estimate because the radial
/ jb
this width
wave function is not
known accurately near the channel surface
7? - matrix theory assumes that all states
/ tA
is difficult
.
Because
which feed
channel will have the same logarithmic derivative
in that channel (i.e. satisfy the boundary condition) we
need a logarithmic derivative of -1
for £/■
zero logarithmic derivative for
CC(Ae )
that the
//>
^
or a
•
we assume
bound state in the realistic potential well
of fig. VI-2 and of fig. VI-3 is fairly accurate inside the
nucleus, then the radial wave function satisfying the
boundary condition will have a value at
than the bound
/p
different energy.
slightly higher
radial wave function, and a slightly
Therefore, all the relevant reduced
widths have been obtained accurately, except for the
Ip
widths which we have under-estimated by using the value of
the bound state.
This will have an effect on the low-lying states
which contain the ( /
) configuration; these
states lie below or just above the nucleon threshold, and
. /
the above approximation should be very bad.
'/*
The lf> widths
also influence the states containing the ( / S
^
)
-235-
configuration, but for these states the parameter CO intro­
duced above will also include the uncertainty in the value
of the radial wave function near 7?
C
♦
The reduced widths used in the calculations are
as follows:
SINGLE
PARTICLE
STATE
/
(MeV
1/2
)
IP
0 . 4 2 7 (or 0.427cu)
Id
1. 45 0
If
1.85 0
2s
1.638
2p
1.8 8 3
VI-B-8: Electromagnetic Single-Particle Matrix Elements.
Following (V-127) the electric dipole and quadrupole
operators are
-
i '**’*
( H o ) { $ 4 s } (ki)
'
{oTo}
*
“»■? d ) * 1*-
& '% } T 777
(
^ Ja»*
'
(VI-29)
-236-
<5 = 1/2,
Here
b
of length; we used
used for
for
b
-
1.61 fermi; the upper constant is
(no change in 7~~) and the lower constant
*
“
7 [ < e > ~ •( A T =
dimension
is an arbitrary constant with dimensions
±
/ ).
The matrix elements have
MeV*5 fermi*5 , since everything is dimensionless
in the formula (V-127),except the electric charge of the
proton JL = 1 . 2
(MeV. )^ .
The only energy dependent term is ( & b ) . .
Therefore,
it is convenient to evaluate the energy independent terms
once and store them; the matrix elements are easily obtained
at any energy by multiplication with
)•
The space and spin parts of the magnetic transition
operators are' obtained from (V-128) and (V-129) respective­
ly.
&
j l
& /)p s + e
l°p %
}
<xp/TL<m>t(s^)/i^'y° :* 7 I r
■(vi-3o)
( U
V
k *<*• £ * •
The notations are the same as above.
(VI-31)
The sign factor in
front of the space and spin parts are not the same.
The
u
matrix elements will have units (MeV fermi)'2 , provided
the wave number k ,
of the gamma ray is expressed in fermi-3-,
-237-
Computation of the "R -Matrix.
VI-B-9
The 7? -Matrix is easily computed from its definition
(V-91)
_ *5” ' Efac Yac '
nc c ' ~ £
£ a-£
~rp
In a representation where the channels have good angular
momentum and parity, the
-Matrix elements corresponding
to two channels of different angular momentum or parity
will vanish.
For this reason the 7 2 -Matrix can be con­
sidered as a set of non-zero sub-matrices along the
diagonal.
Each sub-matrix will be called a block; a block
thus corresponds to given angular momentum J" , its
projection
M and parity
and I f but different M
If .
Two blocks with the same (7
are identical and we need to keep
track of the M = 0 blocks only; the Blatt and Biedenharn
formula will take care of the sum over magnetic quantum
numbers.
It will be necessary in practice to avoid computing
the 7P -Matrix at the poles.
We have found however that
it is possible to carry out calculations very near the
poles, even if the matrix to invert becomes nearly singular.
We have performed all this part of the calculations using
double precision, expecting some difficulties, but we feel
that single precision would probably have been adequate.
Since the reduced widths
the
Y
0AC
^ -Matrix is dimensionless.
u
have dimension (Energy) ,
As everywhere else in
the calculations, we have used 1 MeV as the energy unit.
-238-
VI-B-10:
Phase Shifts and Penetrabilities.'
The phase shifts and penetrabilities which enter
the many-level formula are taken into accobnt in the
channel vector of defined as (V-93)
where
=
L
d ,I L , b
b
and
I
I-hJ
are diagonal matrices in channel space;
is the boundary condition parameter in channel C ,
*
introduced in (V-68)^./and I
were defined in (V-81):
they are respectively the value and the derivative of
the incoming radial wave function in channel
C f
evaluated at the channel radius and multiplied by a
constant.
We obtain for charged particle channels:
( £ * * '< * )
r
4
*
~ w
- 4
”
%
5
-
4c-
4cc
i ( W
€
( V I - 32)
iG j)
(vi-33)
—
S e '->
£
.
b
T*
- & ( £
t ± £
)
(vi-34,
<4 + 1 Ft
The prime means derivative with respect to the argument
which is defined by
fc 3
, where
number of relative motion in channel C .
k
,
is the wave
and Gg are
the regular and irregular Coulomb Functions of order -jf
where J !
is the orbital angular momentum quantum number of
relative motion in channel C ; they are evaluated at the
-239-
following value of their arguments:
£
?/
2
A ir
^
^
where Azp
__
Q .7 P 9
yzp
(VI-35)
JT P
^
is the energy of the protons in the laboratory,
(Only valid for channels consisting of B 7/
in MeV.
and
proton fragments, of course).
The Coulomb phase shifts G l
are defined by
(&j>i)(-e-i-nn).... f/s
°
/(lfiy )(l-!+ iy ) —
( u i y ) l
(VI-36)
For the neutron channels, the formulae are similar:
4
Tcc *
-
( t t y *
(vi- 37)
bc c - t - e c ( ? i ± ± ^ )
<tc c
r
/
The
regular and irregular spherical Besselfunctions^ and
?2g
(Messiah* s^8 ^ definition) are evaluated at Q
kc
where
s
is the wave number of relative motion and ^
*
the
channel radius.
VI-B-11:
Calculation of Transfer Matrix
T
.
The transfer matrix is computed from the expressioni
(V-94)
^
EE e
fe + n c r-ta *)) L
-240-
where y j
is a row matrix in channel space;
are diagonal matrices.
1 , c i and £
A
, for each value of
Thus
is a column matrix in channel space, with dimensions of
t
(time)35.
All the quantities involved have already been intro­
duced, and it is just a matter of numerically inverting a
matrix and multiplying a few matrices together.
behavior of
7^
near a pole is rather interesting.
though the diagonal elements of L
the poles and the matrix
the unit matrix
cause
The
1
)
do not go to zero near
{ ‘K + d
)
, the denominator (
VJ* to have a pole near
£
Al­
- £
a
does not approach
- Zz
) does not
In fact, 7-is
very well behaved near the poles, even though it is not
defined at the poles themselves.
The transfer matrix is computed by blocks like the
- R -Matrix, one block corresponding to a given value of
total angular momentum C7" and parity 'ftJ (and M = 0).
V-B-12t
Reaction Amplitudes and the S - M atrix.
The reaction amplitude
transition t
corresponding to a
was defined (V-C3) as
The matrix elements between pure configurations were com­
puted from the single-particle matrix elements using
(V-118) or (V-119).
From the knowledge of the wave
functions of the initial and final states the matrix ele-
ments between states were deduced.
Jt
is dimensionless, since the matrix element has
dimensions (Energy •
length)^
dimensions (time) 2.
Here,
k.
and the transfer matrix has
is the wave number of the
gamma ray.
After performing the -Jhove matrix multiplication, the
transition amplitudes may be summed over the intermediate
state A , yielding elements of the augmented
S -Matrix
(not the nuclear S -Matrix, but those elements for which
one channel involves two nuclear fragments and the other
involves a gamma ray) which we have also called "contracted
transition" amplitudes.
The only step left is the sum
over transition pairs of the Blatt and Biedenharn formula.
When performing the sum over states A to obtain the
amplitudes for contracted transitions, we calculate the
relative importance of each state in contributing to the
amplitude, and compute the phase (amplitudes are complex
numbers) with which it contributes.
These are printed
when requested, and they permit to determine the states
which contribute at any given energy, and their relative
importance.
VI-B-13;
The Blatt and Biedenharn Formula.
The last step, which yields the cross-section and
angular distribution is accomplished with the Blatt and
Biedenharn Formula:
-242rCF)
t
where the
coefficients are given in (V-42).
In
Cw
this formula,
kc
is the wave number of relative motion
in the incoming channels.
♦
The computer program computes the angular distri­
bution coefficients
, also called anisotropy coefficients,
Sir.
d
•
fl
c
and the total cross-section
'
^ to t ~
The
relative contributions of all the transition pairs to
A
the coefficients
and the total cross-section G ^-a r
are also computed when requested.
Thus, at any given energy,
the contribution of any incoming wave and outgoing multipole
can be clearly determined.
C:
The Results of the Calculations.
Figs. VI-4 to VI-9 show a comparison of the calculated
cross-sections and angular distribution coefficients with
the experimental results.
Eelow 28.8 MeV excitation
(14 MeV incident proton energy) the data are those of
Allas et a l . ^ ^ .
Their work is the most recent and most
/
detailed study of the reaction B (/^y )c
12
in this energy
region; however, as discussed in other chapters of the
present thesis, the absolute magnitude of the total cross
section has probably been overestimated by as much as 50%.
Above 28.8 MeV, the experimental results reported are ours .
In the total cross-sections, the error bars have been
omitted, because they are approximately of the size of the
data points shown.
For the angular distribution coefficients,
EXCITATION
ENERGY
, MeV
A ll the resu lts of calculations presented in figu res V I-4 to V I-9 are based on
G ille t's " approxim ation 1 " o n e -p a rticle on e-h ole wave functions.
20
22
24
26
28
EXCITATION
30
ENERGY
32
34
36
38
, MeV
Fig. V I-4: The data below 28. 8 M eV a re from A lla s et al; those above
28. 8 M eV are ou rs. The heavy line corresp on d s to o5= 2 . 8 and the faint line c o r ­
responds to 0 5 = 1 . 0 , where oi is the damping param eter defined in the text. No
other param eter is involved in the calculations. The experim ental c r o s s section
is probably low er than is shown h ere, by a fa ctor o f approxim ately 1 . 5, as d is ­
cu ssed in the text, so that the calculated c r o s s section does not agree with exp e­
rim ent as w ell as the figure suggests.
-244-
EXCITATION
ENERGY
, MeV
Fig. V I-5: These figures show the angular distribution coefficients A 1?
A and A fo r the y ^ transition. The heavy line and faint line rep resen t w =2. 8
and co= l. 0 as in fig V I-4. The data below 28. 8 M eV a re from A lla s et a l, and
those above 28. 8 M eV a re ou rs.
-245-
EXCITATION
EXCITATION
E N E R G Y , MeV
E N E R G Y , MeV
F ig. V I- 6 : The data below 28. 8 M eV a re from A lla s et a l, and the data above
this energy are those rep orted herein. The heavy and faint lines a re the resu lts o f c a l­
culations fo r oo=2. 8 and oo=l. 0 re sp e ctiv e ly , as in fig V I-4. No other param eter is
involved in the calculations. The absolute magnitude of the experim ental c r o s s section
shown here is probably ov er-e stim a te d by a factor o f approxim ately 1. 5, as noted in
the text.
EXCITATION
ENERGY
. MeV
F ig. V I-7 : Angular distribution coefficien ts for the y transition: A , A 3>
and A^. The data below 28. 8 M eV a re from A llas et a l, and those above 28. 8
M eV are ou rs. The calculations w ere p erform ed fo r 60=2. 8 (heavy lin e) and
co= l. 0 (fain t lin e) and involve no other param eter.
EXCITATION
ENERGY
, MeV
-247-
SNavaodom 10iJO
F ig. V I - 8 : A com pa rison o f the resu lts o f calculation and experim ent
is presented here for the ca se o f the y^ total c r o s s section . No data exist for
this transition in the energy region o f 20 to 27 MeV, because the corresp on d in g
peak in the spectrum disappears in the low energy background. A reson ance
corresp on d in g to the 19. 5 M eV peak has been, m easured by Feldm an et al and
has been found to con sist o f a stron g M l 3“ to 3“ d ecay, in a ccord a n ce with
our calculations. The param eter o> has no influence h ere, so that the ca lcu ­
lation involves no fre e param eter.
-248-
E XCITATIO N
ENERGY , MeV
Fig. V I-9 ( a ) : The e r r o r s quoted with these data do not include the
system atic e r r o r a risin g from the fact that the low energy background is not
exactly exponential, as it is assum ed in fitting the sp ectra; this type o f e r r o r
is n egligible in the ca se o f the y^ and y^ transitions.
0.4
0.2
-
0.2
-0.4
20
22
24
26
28
EXCITATION
30
32
34
ENERGY , MeV
36
38
Fig. V I-9 ( b ) : No data points a re shown h ere, sin ce the data w ere
with only the Pq and P 2 Legendre polynom ials; there is , h ow ever, a slight
w ard shift in the experim ental angular distributions which is consistent with
prediction s o f a sm all positive A-^ coefficien t in this energy region. A gain,
calculations involve no fre e param eter.
fitted
fo r ­
the
the
MICROBARNS
EXCITATION
EXCITATION
ENERGY
ENERGY
, MeV
, MeV
F ig . V I-10: The caption o f fig . V I-11 applies to the present figure as w ell.
-250-
r
i
i
i
i
i
i
i
i
i
0.4
B " ( p , y o ) C 12
ANGULAR
DISTRIBUTION
COEFFICIENT
A*
0.2
------ P U R E C O N F I G U R A T I O N S
------G I L L E T ' S W A V E F U N C T I O N S
o
to
<
-0.2
-
-0.4
. J. . _ i
20
22
1
24
. 1
26
i
i
28
30
i
i
i
i
32
34
36
38
E XCITA TIO N
E N E R G Y , Me V
EXCITATIO N
ENERGY
, MeV
Fig. VI-11: This is a com pa rison o f the resu lts o f calculations p e r fo r ­
m ed with G illet 1s wave functions and with pure j - j configurations. G ille t's o n e p article on e-h ole wave functions ( o f " approxim ation I " ) y ie ld the heavy line,
and when these states a re rep la ced with their dominant configuration (without
changing the en ergy ) the faint line is obtained. The two calculations have been
obtained for the ground state transition, with 40=1. 0; the sam e com pa rison for
40=2 . 8 would show even le ss d iscrep an cy between the two ca s e s.
-PARTIAL
MICROBARNS
-251-
EXCITATION
ENERGY
(MeV)
Fig. VI-12: P a rtial contributions from the variou s incom ing channels
to the total c r o s s section o f the ground state transition (b a sed on G illet's wave
functions and with to =2. 8 ). See also fig. V I-15.
2
3
II
15
10.0
s !/2
d3/2
1"
d5/2
1"
p 1/2
f 5/2
2+
2+
18 s 1/2 2~
19 d3/2 2 “
21 d3/2 3 "
22 d 5/2 3 "
29 f 5/2 4 +
34
36
1“
-2 5 2 -
-PARTIAL
MICROBARNS
B"(p,r,)c '2 1
b
20
22
24
26
28
EXCITATION
30
ENERGY
32
38
( M eV )
F ig. V I-13: P artial contributions to the total c r o s s section o f the transition to the 4. 44 M eV
fir s t excited state a risin g fro m the va riou s incom ing channels ( f o r G illet's wave functions and w=2. 8 ).
The num bers identifying the partial contributions a re the transition num bers used in the program .
See a lso fig. VI-16.
B"(p,r3 ) c 12
)
5
II
15
17
18
P3/2
d3/2
d3/2
d5/2
2+
2~
3“
3~
P3/2 3+
19 f5/2
2 0 f7/2
22 d5/2
23 f5/2
24 f7/2
3+
3+
4"
4+
4+
CO
z
01
< I
CD
-2 5 3 -
O
or
o
<
h
(E
<
0.
I
b
20
22
24
i
26
28
EXCITATION
30
ENERGY
32
34
36
38
(MeV)
F ig. V I-14: P artial contributions to the total c r o s s section o f the transition to the 9. 64 M eV
third excited state a risin g from the va riou s incom ing channels ( f o r G illet's wave functions and co=2 . 8 ).
The num bers identifying the partial contributions a re the transition num bers used in the program . See
also fig . V I-17.
100.0 I-
-2 5 3 -
24
26
28
EXCITATION
30
ENERGY
32
34
(MeV)
Fig. V I-14: P artial contributions to the total c r o s s section o f the transition to the 9. 64 M eV
third excited state a risin g from the va riou s incom ing channels ( f o r Gillet* s wave functions and to= 2. 8 ).
The num bers identifying the partial contributions a re the transition num bers used in the program . See
also fig. VI-17.
CTt o t
MICROBARNS
TO
CONTRIBUTIONS
SUCCES SI VE
-2 5 4 -
EXCITATION
ENERGY
(MeV)
Fig. VI-15: Same as fig VI-12, but the contributions were added successively; the
contributions of each entrance channel is represented by the corresponding area.
O y QT
TO
CONTRIBUTIONS
M ICROBARNS
SUCCESSIVE
-255-
EX C ITA T IO N
ENERGY
(M eV )
Fig. VI-16: Sam e as fig. VI-13, but the contributions were added successively; the
contribution of each entrance channel is represented by the corresponding area.
<rT0T
TO
CONTRIBUTIONS
M ICROBARNS
SUCCESSIVE
-2 5 6 -
E X CIT AT IO N
ENERGY ( M e V )
Fig. VI-17: S ame as fig. VI-14, but the contributions were added successively; the
contribution of each entrance channel is represented by the corresponding area.
-257-
however, meaningful error bars are shown with our data.
The absolute magnitude of the total cross-section for
our data points has been obtained temporarily through a
normalization to the work of Allas et al. pending the
results of careful measurement with a good quality target.
The measurements shown with open dots, between 11 and 14
MeV incident proton energy, were used to normalize our
results; these correspond to 90° measurements only and the
angular distribution measured by Allas et al. has been
assumed in obtaining the total cross-sections.
The measurements of Allas et al. include a great
number of data points, usually taken at intervals of 50
to 100 keV.
For the total cross section, all their data
points are shown, but for the angular distributions, we
have averaged in bins of 200 keV.
This has the disad­
vantage that it becomes more difficult to estimate their
error bars from the scatter of points, and that some very
narrow resonances in the angular distribution may be lost,
but it was not practical to reproduce all their data
points.
The reader is referred to their original article
for the finer details.
In fig. VI-4 to VI-9 there are in general two lines,
a heavy line and a fainter one, corresponding to two
different calculations.
The heavy line corresponds to a
value of the parameter co of equation (VI-27) equal to 2.8,
whereas the faint line has been calculated with c j = 1.0.
These two values of
CJ
represent extreme limits and a
-258-
reasonable value should fall between the two curves.
No data points are shown for the angular distribution
coefficient
in VI-9, because the
Y
angular distri♦
butions of this gamma ray were fitted with only two
terms:
a constant and a
7^
term.
The accuracy of the
measured differential cross-sections was not sufficient
to yield meaningful A l,
and / \ 3 coefficients.
The data,
however, are in general agreement with the prediction of
♦
a slight forward shift.
Fig. VI-10 and VI-11 show a comparison of the
predictions corresponding to Gillet's wave functions (heavy
line) and p u r e j l - f
of QJ -
configurations (faint line), for a value
1.0, in the ^
transition. This was done in order
to estimate the importance of the configuration mixing
predicted by Gillet, and to see if the use of his wave
functions substantially improves the results of calculations.
It should be emphasized, perhaps, that the results
represented by the faint lines are still based on Gillet's
calculations, up to a certain point, since they were
computed using the energies of Gillet's states; we merely
removed the configuration mixing in the wave functions.
It
is interesting to note that the only major effect of using
pure configurations is to destroy the resonance of 34 MeV,
which was populated through an impurity in the 34 MeV state.
This difference would not have appeared, however, if we had
compared Gillet*s wave functions with pure configurations
for a value of the parameter co
equal to 2.8; in that case,
-259-
there is no substantial difference in the two results, and
a
«
for all practical purposes Gillet's approximation 1 yields
negligible configuration mixing.
The shapes of the cross-sections and -angular
distributions calculated here are interesting from a purely
tecnnical point of view, in that they represent a good
example of how R
-Matrix theory in its full generality
can predict both direct and compound nucleus reactions;
the slowly varying background is essentially due to direct
reactions, although it is in principle more accurate than
a simple direct reaction calculation could predict, and
the 34 MeV resonance in
y
is a good example of a com­
pound nucleus resonance.
The low energy resonance of the y
cross-section
is also a good example of such a resonance, and it is
interesting to note that some data exists on a resonance
in this region; we have not had the opportunity to include
these data on the graph, however we shall discuss them
in the next chapter.
-260-
List of References for Chapter VI
(1)
V. Gillet, PhD Thesis, Universite de Paris, unpublished
(2)
M. Marangoni and A.M. Saruis, Nucl. Phys. A132, 649
(1969)
(3)
M.G. Mayer and J.H.D. Jensen, Nuclear Shell Structure,
Wiley, New York, 1955
(6)
J.M. Kennedy and W.T. Sharp, Atomic Energy of Canada
Limited, (Chalk River) Report CRT-580
(7)
H.J. Rose and D.M. Brink, Rev. Mod. Phys. 39, 306
(1967)
(8)
Albert Messiah, Mecanique Quantique, DUNOD, Paris,
1962. Quantum Mechanics, North Holland, Amsterdam
%
(9)
E.P. Wigner, Group Theory and its Applications to
the Quantum Mechanics of Atomic Spectra, Academic
Press, (1959)
(10)
V. Gillet and N. Vinh Mau, Nucl. Phys. 54, 321 (1964)
(11)
N. Vinh Mau and G.E. Brown, Nucl. Phys. 29, 89 (1962)
G.E. Brown and M. Bolsterli, Phys. Rev. Letters,
472 (1959)
(12)
A.M. Lane, Rev. Mod. Phys. 32^, 519
(13)
J. Fregeau, Phys. Rev. 104, 225 (1956)
-261-
(14)
R.G. Allas, S.S. Hanna, L. Meyer-Schutzmeister,
and R.E. Segel, Nucl. Phys. 58, 122 (1964)
-262-
CHAPTER Vn
DISCUSSION.
-263-
Before we can compare with some validity the
results of calculations with experiment, we necessarily
must discuss the very important problem of the normalization of the total cross-section, not only with regard
to our data above the giant resonance but also concerning
all the (particle,^) results published previously, and
in particular including the work of Allas et a l . ^ ^ in
the region of the C
12
giant resonance.
We have already mentioned, in chapters I and II,
that there is a possibility that a misunderstanding
of the response of large NaI(T^) crystals to high energy
gamma rays may have caused these earlier cross-sections
to be overestimated by as much as 50%, in some cases.
Peak shapes for high energy gamma rays can be obtained
quite accurately in the region of the peak, but it ±s
much more difficult to measure the low energy tail which
is usually seen associated with the peak; indeed it is
customary to assume this tail to be quite constant and
to extend all the way to zero pulse height.
When the
number of counts under the peak is determined, the area
under this constant background tail is typically integrated
with the peak to establish the relative Yield.
Fortunately, the uncertainty as to the response
of a large crystal to high energy gamma radiation has
been greatly reduced, by a series of recent measurements
at Livermore
(2)
, where a beam of monoenergetic photons
from positrons annihilated in flight was available and
-264-
has been used to carry out a series of precise peak shape
measurements.
The results, in all cases, show that the
low energy tail of the peaks consists of an exponential;
no constant background extending to lower energies has
been detected.
This is in contradiction with the above
mentioned assumptions; at the same time, however, the
reaction
, for example, which we have studied
for the purpose of examining peak shapes, shows very
clearly the traditional constant background. Our results
(fig. II-4) agree well with those of Allas et a l . ^ ^ on
this point.
To find the key to this apparent contradiction, we
must examine the experimental arrangement as a whole, and
not simply consider the crystal itself.
The major difference
between the typical (/^y) detection system and that tested
by Berman et al.
(2 )
'
consists in some 60 cm of paraffin which
we, and others, employ to shield the entrance face of the
crystal from fast neutrons.
In their measurements, the
group at Livermore were careful to have a tightly
collimated gamma ray beam on the axis of the crystal, but
with no absorber susceptible of degrading the beam, in
front of the crystal.
It must be remembered that because
paraffin is a low-Z material, gamma rays will Compton
scatter readily in it; 60 cm of paraffin represent almost
50% probability of interaction for a high energy gamma ray,
so that many of the lower energies gamma rays which are
detected in the typical (/»/) experimental arrangement have
-265-
already been Compton scattered.
Some of them were not
initially included in the solid angle defined by the
collimator and could be detected because they were
*
Compton scattered into it.
Fig. II-3 shows the various
possibilities; the details have already been discussed
in Chapter II.
In view of these circumstances, it seems reasonable
to adopt the following procedure:
first, we correct for
the total absorption due to the paraffin; this means that
all the gamma rays which had an interaction in it have
been taken into account and should not be counted as part
of the peak.
The peak is then fitted by a Gaussian with
a low energy exponential background and a constant tail.
The counts under the exponential background are
interpreted as genuine, in accordance with the Livermore
measurements, whereas the counts under the constant tail
are attributed to Compton scattering and are not included
in the integration under the peak.
The drawback with this
approach is that it is not possible to accurately decompose
the peak shapes into the two regions, but it is possible
to do it fairly uniquely, pending some eventual measure­
ment.
After analyzing all our data in this manner, it
was discovered that our results disagreed with those of
Allas et a l . ^ ^
, and with other previous measurements
12
in CA , by an approximate factor of 1.8.
While part
of this discrepancy could be attributed to the poor
-266-
accuracy of our knowledge of the target thickness, it
was unrealistic to ignore such a blatant discrepancy.
We attribute the disagreement with the previously reported
cross sections to the standard procedure which consists
in integrating under the low energy tail, all the way to
zero pulse heights.
Assuming our analysis of the situation
to be correct, we would expect essentially all the
(d ,y )
etc. excitation functions reported so far in the
literature to be too large, and the amount of the correction
needed to depend on the particular detector geometry used
for the experiment.
If this is so, and it should be checked
experimentaly, the situation is rather catastrophic, because
much of the physics is contained in the absolute magnitude
of the various total cross-sections which have been measured.
What is even worse, perhaps, is the fact that the correction
to be performed.is energy dependent, so that it will not
be sufficient to measure only one point on each of the
previously reported total cross-section measurements.
We should emphasize that in order to obtain a
satisfactory solution to this crucial peak shape problem,
measurements will have to be undertaken with monoenergetic
gamma ray beams for the geometry used in
experiments.
Because of the high flux of neutrons associated with the
nuclear reactions producing high energy gamma rays, it is
impossible to measure peak shapes satisfactorily with a
nuclear particle accelerator, since the paraffin moderator
is needed in this case.
A good example of the implications of these findings
-267-
is the necessity of revising our assessment of the importance
12
of isospin mixing in light nuclei, in particular in C .
In an article on C*2 and O3-6, Wu, Firk and Thompson^3 ^
had argued that the isospin mixing in C
12
is very large;
their arguments were based on a comparison of the
and { ^ ,7 % ,) cross-sections, through detailed balance.
This
result was rather mysterious*, because at comparable energies
16
the 0
case showed no appreciable isospin mixing, the
( y , j> ) and ( y , 77 ) cross-sections being equal within errors.
It is interesting to note that if our analysis of the peak
shapes is correct, the previously published B
X2
cross sections are too high by a factor of 1.5 or so, and
when they are corrected they become essentially equal to
the (^,77 ) cross-sections (through detailed balance), so
12
that isospin mixing would appear to be small in C
also.
For display purposes, and pending a more accurate
measurement which we' intend to perform with our new crystal,
we have normalized our results to those of Allas et al.^^;
in comparing the results of calculation with experiment,
however, it will be important to remember that the total
cross-sections are probably a factor of 1.5 lower than they
are shown to be.
In discussing the total cross-section, one is often
tempted to nelgect the effects of interferences between
the various resonances.
We are even aware that there have
been some attempts at fitting the total cross-sections with
*We wish to thank Dr. Firk for discussing these matters
with us.
-268-
Lorentz shaped curves.
This practice should be avoided,
we believe, in the case of the ground state decay in an
even-even nucleus, because essentially all ,the total crosssection originates in 1” states, and they all interfere
with each other in the total cross-section.
Not only
that, but the interferences are bound to be important.
The calculations of the differential cross sections
which are reported herein are based on an R-Matrix theory
approach, as described in details in chapters V and V I .
It should be emphasized, however, that this does not
imply that we have made the assumption of compound nucleus
formation; R-Matrix theory is perfectly general, and
encompasses both compound nuclear and direct processes,
and is expected to work even when both of these approxi­
mations are i n v a l i d ^ ^ .
For example, R-Matrix theory
is capable of reproducing a typical direct reaction cross
section with superposed compound nucleus resonances.
This
remark is important, because there is evidence that the
tail of the C
12
giant resonance is dominated by direct
reaction type processes, whereas the peak of the giant
resonance would correspond more closely to a typical
compound nuclear resonance; the R-Matrix theory used
herein should be expected to work well in both cases, and
in the intermediate region as well.
On the other hand,
we do not imply that the approach used herein is exact,
because of the truncation of the basis of stationary states
for the compound system.
-269-
The calculations take into account a total of 64
states (one-particle one-hole states of Gillet) and 60
channels, as described in chapter VI.
It is important to
*
stress that there are no free parameters involved, except
for the damping of the 34 MeV state, which has no consequences.
except in a narrow energy region centered around 34 MeV.
This includes the absolute magnitude of the calculated
cross section, which has not been normalized in any way,
contrary to a practice common in coupled channel calculation
which consists in introducing an absorbtion potential.
The
prediction of the absolute magnitude of the total cross
section is therefore unique, given Gillet*s wave functions.
Our calculations predicts that from 20 MeV up to
40 MeV excitation and above, the giant resonance state,
fp
which is mostly a
cross-section.
configuration, dominates the
Because the calculation reproduces well
the general shape of the total cross-section, as seen in
Fig. (vi -4), we are strongly tempted to believe that indeed
there is a single dominant state in the B
this would make of C
X2
reaction;
12 quite a unique case.
According to the uncertainty in the normalization of
the experimental cross section, as discussed above, we
believe that the
^
total cross section is probably lower
than what is shown for display purposes in fig.
a factor of approximately 1.5.
(vi-4)» by
This discrepancy on the
absolute magnitude could easily be attributed to the fact
//
that the B
ground state is only part of the time a hole
-270-
in
C
/J/z
Ip
, so that the one-particle one-hole states of
12 are not being fed as efficiently as the calculation
predicts.
(j^ y , )C^
This view is strongly confirmed by the
cross-section,, as we shall indicate below.
If this is so,
we should remember that away from the resonance the phase
associated with a state changes relatively slowly,whereas
it changes much faster when going through resonance;
the
peaks in the cross-section at 25.5 and 28 MeV should be
interpreted as interference shapes on the slowly varying
tail of the main component of giant resonance.
In particular,
the very sharp decrease in cross section at 24.5 MeV is
best explained as an effect of interference.
It is un­
fortunate that in the past, the necessity of taking into
account the interference which will clearly occur, even in
the total cross section, has not always been fully recognized.
As far as the main peak of the giant resonance is
concerned, we only need two states to satisfactorily explain
it; the 23 MeV main component, and another state at somewhat
lower energy (about 21.5 MeV) would be adequeate.
The sharp
decrease at 24.5 MeV and the peak at 25.5 MeV could be
explained by a single state at 25 MeV, and the apparent minor
peak at 28 MeV by another state in this region.
This last
peak is small and barely discernable with the presently
available data.
It is interesting to speculate on the probable nature
of the states at 21.5, 25 and 28 MeV, and examine whether
they are consistent with a one-particle one-hole picture.
-271-
First we should note that there are two 1",
J~-
1 one-particle
one-hole states involved in the calculations between 20 and
30 MeV; Gillet predicts one of them at 21.9 tMeV and the
other one at 24.2 MeV.
We have raised the energies of all
the 1“ states by a little over one MeV, in order to make
the comparison with experiment
easier.
This means that
in our calculations, there are states at both 23 and at
25.3 MeV and we could have hoped to reproduce the interference
structure around 25 MeV.
Unfortunately, with the wave
functions of Gillet, this does not occur.
25 MeV, where the
In fact, at
^/channel is estimated to contribute
80% to the total cross section, as seen in fig. (VI-12)
the 25.3 MeV state only accounts for 5% of the 80% in the
transition amplitude, whereas the main component of the
giant resonance, at 23 MeV, accounts for 90% of it (of the
80% in question).
More important, perhaps, is the fact
that the calculations predict that both states, in the
region of 24 to 26 MeV contribute almost in phase to the
transition amplitude in that channel.
This is simply
because the distance between the two resonances is less
than their width.
It is interesting to note that other
calculations^^ based on the one-particle one-hole picture
arrived at the same result, independently of Gillet*s wave
functions.
It would therefore seem that the 25 MeV state
represents a failure for the one-particle one-hole model.
Before we arrive at this conclusion, however, we have to
discuss the possibility that the resonance at 25 MeV is the
product of a
through its
~T~J~-
0 state, decaying to the ground state
1
admixture (or decaying to the
admixture of the C^2 ground state).
7
~=1
Such a possibility
must be ruled out for two reasons; the first' is that
isospin mixing in C
12
is very small, according to our
discussion above; in particular, the 25.5 MeV peak is
very well reproduced in the
) cross-section,'
The
second reason, perhaps more important, is that we expect
the one-particle one-hole
J ~ - 0 states to have particle
widths which are comparable to their
7 ~-1 counterpart, and
the 25 MeV state is much too narrow to be one of them.
We
are therefore forced to admit that the 25 MeV state is
dominated by configurations more complicated than oneparticle, one-hole.
These are not included within our
model calculations.
We shall see below, in the discussion of the (/»^/)
cross section that the evidence for such complex configura­
tions is excellent; in fact, we believe that the discrepancies
in the shape of the total cross section $nd in the angular
distributions for the (/>,^) reaction are so small, simply
because the ground state of C
12
is a relatively good
vacuum state, probably better than other calculations seem
to indicate.
that the C
12
We are aware that there is a good possibility
ground state is a poor vacuum state, as evidenced
for examply by the work of Cohen and Kurath
(12 ) ;
however, we
are not convinced that such calculations are necessarily
well founded, in particular considering the work on pseudonuclei
that a set of wave functions fit the data is not sufficient
-273-
evidence to believe that they are necessarily correct;
the work of Gillet referenced here is a good example of
such a situation.
Such purity of the ground state of C
12
would have the effect, that only the transitions from the
single-particle single-hole configurations would be
seen; this would imply that for the more complex configura­
tions to be observed, they would have to be mixed' in
approximately a 50%/50% ratio with a strong one-particle
one-hole state.
of the C
12
In other words, given a reasonable purity
ground state, the condition for the one-particle
one-hole model to work is simply that the one-particle onehole states be eigenstates of the Hamiltonian; it is not
necessary that they be the only eigenstates of the system.
It is probable that the 25 MeV state represents such a
mixture, in comparable proportions, of a one-particle onehole state with more complex configurations.
This
appears to be the only reasonable approach in explaining
the total width of the state and its very substantial
contribution to the ( p , Y 0 ) cross-section.
It is interesting to speculate on the nature of this
complex 25 MeV state, and we are helped in these attempts
by the B // energy spectrum.
a
We see that there is, in B ^ ,
excited state as low as 2.14 MeV I
We may add a
particle to the core, obtaining a two-particle two-hole 1~
configuration with an unperturbed energy comparable to that
of the giant resonance, which we may picture naively as a
I d . S^Z
particle coupled to the B 11 ground state.
These two
states could then mix to produce the observed giant resonance
-274-
at 25 MeV and another state at 25 MeV.
In fact, it is
much more probable that the situation is infinitely more
complicated, and that a large number of complex configurat
tions contribute small impurities to all the states, but
this discussion is mainly intended to show that in view
n
//
of the low-lying excited states of B and C
we would
expect that complex configurations in C
12
would start
being important at excitation around 20 MeV or lower.
The
two nucleon threshold in C ^
, defined as that
energy at which the emission of two free nucleons becomes
energetically possible, lies at approximately 28.5 MeV,
and R-Matrix theory ceases to be valid beyond this point.
We nevertheless expect that its results will remain
reasonable for several MeV above this energy, and we have
calculated cross-sections and angular distributions with it
at energies up to 40 MeV.
We would expect that past the
two-nucleon threshold, the experimentally observed crosssection for ( f a , f a ) would go down much faster than the
calculations predict, because the probability of twonucleon emission becomes very large and the emission of
gamma rays is proportionately reduced.
This is just what
is observed in practice; as shown in Figure (VI-4 ) the
experimental cross-section, from
23.5 MeV, starts de­
creasing much faster than the R-Matrix calculations predict.
This is in accordance with the well established phenomenon
threshold, the ( Y , Z w ) reaction becomes much more important
-275-
than the ( J f t W) reaction.
This competition has been observed
in heavy nuclei, where the Coulomb barrier inhibits the
emission of charged particles.
Fig* (/I-4 ) shows two theoretical predictions for the
total cross section.
As not^d above, the calculated cross
section has not been normalized; the absolute magnitude is
uniquely predicted.
The experimental cross section, however,
is believed to have been overestimated by a factor of 1.5,
and should probably lie much lower than it is shown.
The
apparent agreement between theory and experiment is there­
fore fortuitous.
The heavy line of Fig. ( VI-4) corresponds
to CO = 2.8, and the faint line to a value of 1.0 for the
same parameter.
This factor
CO
, it should be recalled,
represents the enhancement of the reduced widths for
configurations with a hole in the IS shell reflecting
that it is energetically possible to emit any of the / f t
shell nucleons, and it has been defined in chapter VI.
It
has been necessary to introduce a parameter for these re­
duced widths because the shell model cannot yield a
reasonable estimate for the reduced widths of the states
with a hole in the
that the
/
IS
-
!(l
Is
/ / V 'i
Ip
shell.
It is clear, for example,
configuration, which lies at approxi-
iJ/imately 30 MeV, can particle decay by emitting the I p
or
any of the
that it is bound.
nucleons, whereas the shell model predicts
Of course, when we introduce residual
interactions, the configuration will become mixed and will
be capable of decaying through its admixtures.
However,
Gillet had predicted very small mixing for this configuration
-276-
it was reported to be 95% pure, probably because all the
many-particle many-hole configurations and higher configura­
tions which could have mixed with it had been neglected.
In
*
any case, using Gillet's wave functions for this state and
the shell model prediction for the reduced widths (i.e.
considering the
/ f>
nucleons as bound) leads to a very
narrow resonance of 3 0 0 ^ / S peak cross-section for the 34
MeV state.
There are many ways of interpreting this 35 MeV
discrepancy; one of them is to consider that the system
clearly does not have a stationary state with a wave
function remotely like the Gillet prediction.
Another point
of view, which we have adopted for the purpose of discussion
is that there may be such a state, and the wave function
calculated by Gillet may indeed be a fair approximation
in the internal region of the compound nucleus; however,
when calculating the reduced widths, it is improper to use
the shell model estimate, because as soon as one of the
nucleons approaches the nuclear surface the remaining particles
rearrange themselves in a linear combination of excited
states of C /; or B ^ 1 , and the extra energy gained in this
rearrangement is used to push out the extra nucleon.
These
/5 \
ideas have already been discussed in a similar form by Lane' •
(p. 527) who suggests the introduction of an "energy-sharing"
region; on one side of this region, the wave functions would
be those of the compound system, whereas outside, we would
have to use the channel wave functions, which are essentially
-277-
direct products of the states of the residual nucleus
with the wave function of a free nucleon.
This energy-
sharing region would then be used to connect smoothly
the internal region with the channel region.* This idea,
; which is designed to compensate for the fact that in a
typical particle-hole calculation we do not include
a sufficient number of configurations to obtain areasonable accuracy for the wave functions near the
surface, is appealling in principle, provided we can
find a prescription for calculating what happens in
this energy sharing region.
Being unable to do this from
first principles, we simply have introduced a parameter,
the net effect of which is to change the total widths of
the states with a hole in the Is
shell, and therefore
to damp.the resonances associated with them.
The heavy
and faint lines of Fig. (vi-4 ) to (VI-9 ) represent two
values of the parameter; the faint line represent
an
unreasonably small value, and the heavy line an unreasonably
large value, so that we would expect the results to lie
in between the two curves.
The predicted resonance corresponding to this state
has never been observed.
liminary measurements
We had not seen it in our pre­
of the total cross-section; it has
not been seen in measurements of the total (J',/’) crosssection performed up to 50 MeV; we had hoped , however,
to detect a substantial change in the A2 coefficient of
the angular distribution which is predicted to be very
sensitive to the presence of this state.
The data of
-278-
fig. VI-5
clearly show that there is no evidence whatever
for a resonance in this region.
Since we would have to adopt an unreasonably large
value of the parameter cO
to obtain a vanishing contribution
from the 34 MeV resonance, we can safely conclude that there
is no such state in this energy region which has a wave
function of the type Gillet predicts.
The
fs ^
strength is most probably distributed among many complex
states, over an energy interval of some 20 MeV.
plex configurations, in turn,
These com­
are not reached efficiently
through the bombardment of B / f with protons, so that the
transition is simply never seen in our measurements.
This
picture is certainly more reasonable that the pure one-particle
one-hole state at 34 MeV; it is also consistent with the fact
that the
I5
hole strength is not observed.
calculation by Rowe et al.
(13)
An open shell
predicts this state at
higher energies.
The angular distributions for
^
) c ^ are repro­
duced with an amazing accuracy by the particle-hole model.
\
At first glance, it would seem that the A 2 coefficient is
predicted as much too large at low energy; it is measured
negative and predicted to be positive.
This, however, is
only an apparent contradiction, the source of which we shall
now discuss.
In order to give the particle-hole model an
opportunity to predict angular distributions correctly, we
should have replaced all the calculated energies of the C
12
states by their corresponding experimentally observed values,
-279-
in cases where such a one-to-one identification is possible.
There are only two clearly identified l“ one-particle onehole states in C
12
, above the proton threshold; these are
the giant resonance, at 23 MeV and another state at 17.2
MeV; the latter should clearly be identified with the pre­
dicted 17.7 MeV one-particle one-hole state of Gillet*s.
Instead of giving this state a 17.2 MeV energy and the giant
resonance an energy of 23 MeV, we have raised all the 1
states by an equal amount, (of approximately 1.1 MeV) in
order to line up giant resonance with its observed position,
thus hoping to make the the comparison of theory with
experiment easier.
The positive
maximum at 20 MeV in the
coefficient comes essentially
from the interference between the 17.2 (calculated as
18.9) MeV state with the giant resonance; the 17.2 MeV
state by itself would have a zero value of
why the maximum in
This is also
occurs at 20 MeV, whereas the
maximum contribution to the cross section is below 19 MeV.
By reducing the energy of the 17.2 MeV state to its
experimentally observed value in the calculations, we would
obtain excellent agreement between theory and experiment,
from 19 to 40 MeV.
In the region of 28 to 36 MeV, the
calculation does not quite match experiment, but this is
due to the effect of the 34 MeV state.
If we adopt the
point of view that this state is not reached in the
experiment, as discussed above, we obtain essentially the
heavy curve of fig. (VI-4) which agrees very well with
experiment.
-280-
For the A^ coefficient the agreement is excellent,
and at low energies
the same remark as above applies, with
regards to the position of the 17.2 MeV state.
The 20 MeV
,peak is again produced as an interference between the
displaced 17.2 MeV state and the underlying 14 2+ one-particle'
one-hole states (see Fig. 1-1).
For the A^ coefficient the
problem is not so severe, mainly because it is small at low
energies, and the fit is excellent over the complete
energy region.
The A 4 coefficient seems to show systematic
fluctuations below 28 MeV, but this probably reflects the
particular energy bins we have chosen in averaging the
data of Allas et al.^^; these fluctuations are not apparent
in the original data, where the larger number of points
permits an appraisal of the errors involved.
In any case,
we believe that the data are consistent with very small
values of A4 , as calculated.
The fact that the A4 coefficients
which were measured in the present experiment tend to be
negative may be an indication that the correction for the
target chamber anisotropy, though carefully measured, was
not accurate enough.
Part of the probable errors quoted
on these measurements account for this correction, and it
should be possible, with the new crystal and target chamber,
to measure the A 4 coefficient much more accurately.
In
particular, the possibility of structure in the A4 coefficient,
between 22 and 28 MeV, which is suggested by the data of
Allas et al. should be reinvestigated.
The results of the B
-281-
sense even more interesting than the ground state results,
because the evidence for complex configurations in the
compound system is extremely clear.
There are a minimum
of 8 to 10 states contributing to the cross-section.
The
immediate explanation is that since 3“ , 2 " as well as 1
states can contribute an El transition to the 2
+
•
excited
state, it is normal to expect a much more complex excitation
function.
However, this qualitatively correct reasoning
is in fact erroneous:
what one would expect from the one
particle one-hole picture is shown in the solid line of
Fig. (VI-6), with the various individual contriubtions
detailed in Fig. (VI-13 (the faint and heavy lines correspond
to the same values of the parameter aJ and from the above
discussion we will hereafter disregard the faint line).
At 22.4 MeV, 60% of the calculated cross-section is from
•
3/l
the d
3L
_
3
channel, and 24% from the d
remainder coming
2
from small contributions.
channel, the
It may be
interesting to note at this point that the present calcula­
tions take into account 246 transition pairs which
contribute a non-vanishing term to the A 2 coefficient,
each of the transition amplitudes having contributions
from many different states.
These two channels dominate
the total cross-section at all energies from 20 to 40 MeV,
and they remain in the approximate 2/1 proportion through­
out.
All the other states have negligible contributions!
This is an interesting result which would not have been
obvious from the usual treatment of the states as discrete
-282-
transitions; upon seeing a large number of peaks in the
( h X , ) total cross-section and a correspondingly large
number of 1” , 2~ and 3“ particle-hole states predicted
in the same energy region, the immediate ass'umption
would have been that "indeed, it agrees very welll"
when
in fact it does not agree at all; despite the fairly large
number of available transitions, the predictions for the
cross-section involve only one peak!
There is no hope
whatsoever of reproducing any of the observed fluctuations
on the basis of the model used herein.
The three most apparent differences between experi­
ment and theory in the { P y , ) cross-section are the
experimentally observed structure, the difference in the
magnitude of the cross-section, and the fact that the
experimentally observed giant resonance lies approximately
3 MeV above its predicted position.
We have already
discussed the structure to some extent, arguing that it
proves the presence of complex configurations; we shall
now consider the other two discrepancies.
In order to do_
so, we consider the following gedanken experiment:
a
( y p )
The
reaction on the first excited state of C
.
first excited state of
is predicted by Gillet to be
o
-3/ i
'/z
90 of |p
|p
character; this configuration corresponds
to
oAco
(no major shell jump) excitation, and it is
quite reasonable to assume that it constitutes the major
component of the lowest excited state.
The incident gamma
radiation, because it corresponds to a one-body operator,
can only lift one particle into a higher orbit; it can
-283-
lift the already excited
particles of the core.
/p
t f/l
particle, or any of the
When the /J b^1 particle is promoted,
*
'
12
we reach a one-particle one-hole state of C , and this
*
process is already covered by our particle-hole calculation
when one of the core particles is promoted, however, we
reach two-particle two-hole states, and this process is
not included in our particle-hole calculation.
The two
possibilities are represented below*
,3/2
I
-X-X -X -O X -X X X-
-X - X- X
X
•X -X
X
x-
We would expect, a priori, that the cross-section going
through two-particle two-hole states should be 7 times
as important as the cross-section resulting from the
promotion of the
/p ^2
particle and that it should lie
approximately 4 MeV above it.
The description which we
have just given is then consistent with the picture that
the y
giant resonance is really very much like the ground
state resonance built on the C3-2 first excited state.
Actually, we must revise the statement that the expected
-284-
enhancement is a factor of 7, because this should strictly
be true of a (^»/) experiment where all protons are detected,
whereas we are interested in the results of the
» f>0 )
t
gedanken experiment; it is clear that the one-particle onehole states will decay more rapidly to the ground state of
b
"
than the two-particle two hole states; by emitting the
single excited nucleon, the pure hole in
/p
is reached,
whereas the two-particle two-hole states will have to decay
to the ground state of
hole admixture.
through its one-particle two
In the (/»y ) experiment, we note that we
reach the one-particle one-hole states of C
pure hole component of the B t a r g e t ,
12
through the
and the two-particle
two-hole states through its one-particle two-hole admixture.
Taking into account the factor of 1.5 in the experimental
cross-section corresponding to the normalization correction
discussed above, we are left with a remaining discrepancy
of approximately a factor of two.
This is consistent with
a 10 to 15% admixture of [ O f * ) *
in the ground state of B 11
.
conf iguration
The discrepancy in the total -
cross-section for (/>^>) which is approximately a factor of
1.5, (after correction of the experimental normalization)
j
suggests that the
/
j j
/
component in the B
ground state
is approximately 65%; this would leave the possibility of
substantial admixtures of configurations other than the two
already mentioned.
As we would expect from the above, the angular distri­
bution predictions for
) do not fit very well; except
for the A^ coefficient, below 30 MeV, and the A 4 coefficient
-285-
which are both predicted and measured to be very small, the
theory and experiment simply do not agree.
behavior of the A. and A
•
The general
coefficients is not completely
t
fc
wrong, however, and this may reflect the fact that electric
transitions are fairly insensitive
to the spin, and
depend mostly on the orbital angular momentum of the pro­
moted
particle.
The fact that we calculate angular
distributions on the basis of the promotion of a /^^^particle
2/
when the dominant process involves the promotion of a // 2
particle does not change the angular distribution drastically.
It remains, however, that the agreement is nowhere as good
as in the case of the ground state transition.
In a recent calculation, Drechsel, Seaborn and
(7)
Greiner' ' considered the coupling of the giant resonance
in C
12
with the collective surface vibrational mode.
It
was found that other states carrying a substantial amount
of dipole strength appeared as a result of the coupling
of the modes of motion; in particular, the giant resonance
appeared to be split into 3 components, at 22 MeV, 23 MeV,
and 24.5 MeV respectively.
They are in agreement with our
remarks above, that the 25 MeV state must be a rather com­
plex configuration for the most part.
It is unfortunate
that the results of their calculation is presented as a
set of discrete transitions, preventing a meaningful
comparison, and it is unfortunate also that no prediction
is made for the
) cross-section, which is more
sensitive to the presence of states such as those described
-286-
by their calculation.
Nevertheless, we believe that their
calculation constitutes an important advance in giant
resonance theory.
t
Figs. ( v i - 8 ) to ( v i - 9 ) represent a comparison of
theory with experiment in the case of the third excited
state transition.
The second excited state is predicted
by Gillet to be a rather complex many-particle many-hole
state, and we would therefore expect the gamma transition
to this state to be rather small.
with experiment:
This is in agrument
we have seen no indication of a transition
to the 7.65 MeV state.
The transition to the 9.64 MeV 3“ third excited
state, on the contrary, is rather strong and relatively
easy to measure above 12 MeV of incident proton enery.
Below this, it disappears in the low energy background.
Even though we had
in the y
typically 1000 counts per spectrum
peak, the errors on the angular distribution
coefficients are rather large because the ^
very well resolved.
peak was not
In particular, the errors on the A^
and A 4 coefficients were so large, that the results were
not very meaningful.
We have fitted the y
distributions in two different ways:
included the PQ
angular
The first fits
to P4 Legendre polynomials, and the
second series involved only P Q and P^.
On the basis of
the observed chi-squares we present here only the results
of the latter type.
There was, however, some evidence of
a slight forward peaking, and this is consistent with the
-287-
calculations.
The comparison of the results of the calculation with
theory for the
'jfa
total cross section is very interesting*
taking into account the proposed reduction in the experi­
mental cross section by a factor of 1.5, as discussed above,
there exists an important discrepancy; the predictions are
too high by a factor of 5 !
Assuming the relative proportions
of the various particle-hole states predicted by Gillet to
be correct, we must conclude that this 9.65 MeV state is a
one-particle one-hole state only 20% of the time 1
There is
no other way to explain that the compound system does not
decay more strongly to the 3~ state.
Some indications that
this state is peculiar were already given by Gillet (see
p. 150 of his thesis^®^) in that he notes that the 3
state
is probably highly collective, because it is not fitted with
the same residual interactions as the others.
Nevertheless,
this case constitutes a good indication of the dangers of
fitting spectra with the use of an ad hoc residual inter­
actions; it is possible to obtain excellent fits and yet have completely false wave functions.
It is quite probable
that the, particle-hole composition of the state be predicted
correctly by Gillet, however, since the energy variation of
the total cross-section and the angular distribution are
predicted in reasonable agreement with experiment.
In
particular, a resonance which corresponds to the sharp
calculated peak at 20 MeV has already been measured at
Stanford; the measured peak cross-section is approximately
-288-
.10 microbarns, and the measured width is 0.5 MeV.
The
resonance has been identified as a strong Ml 3“ to 3“
decay.
This is in perfect agreement with the calculations,
which predict this transition as an Ml decay from a 3”
state.
The fact that the cross-section is predicted much
higher than it has been measured reflects many factors:
the complex wave function of the 9.64 MeV state accounts
for a factor of 5; the fact that the resonance has been
measured at 2.7 MeV, whereas the calculation predicts it
at 4 MeV also accounts for part of the discrepancy, through
Coulomb barrier effects.
The angular distribution at the
peak of the resonance is calculated
and a measurement
yields A
to have A2 - 0.25,
= 0.25 + .02 ;however, the
A^ coefficient is in complete disagreement; we predict
that it should decrease through the resonance, whereas
the angular distribution is found to be forward peaked
above the resonance and backward peaked below.
These are
two possibilities; the background El transitions may be
predicted wrongly, or the experimental curves reported
may have been mislabeled.
transition are
Fig. VI-10 to
The various contributions to the
detailed in Fig. (vi-l4 •
VI-11
show a comparison of the results
of a calculation based on Gillet*s wave functions, represented
by the heavy line, and a calculation with pure j-j configura­
tions.
In the latter case, the state energies were kept
according to Gillet*s predictions; the wave functions were
simply replaced by their dominant configuration.
It is
-289-
interesting to note that the changes are minimal, especially
in the total cross-section.
The comparison has been made
for a value of CL> = 1.0, and using pure configurations
t
makes the 34 MeV resonance disappear because the impurity
through which it was reached has been removed.
If we
had made the comparison at a different value of <^, large
enough to make the resonance disappear, the difference
would have been even much smaller.
This comparison was
intended to show the improvement in the predictions that
the mixing of configurations can give, and we hoped to
test the accuracy of Gillet's wave functions in the process.
The results are striking.
In all cases where there is a
substantial difference (and there are not many), using
pure configurations is an improvement over Gillet*s wave
functions.
energy
The 34 MeV state is a good example, the low
coefficient is another one.
Thus we may con­
clude that if Gillet*s wave functions work at all, it is
because they are not too different from pure j-j configura­
tions, and 7ft -Matrix theory with pure j-j configurationswork well in C
12
.
If Gillet obtains relatively good
energies for the states, it is because he started from
experimental energies and fitted for these energies the
parameters of his residualiihteraction; it is most probable,
however, that this residual interaction has no contact
with reality, and simply is a convenient way of parameterizing
a level fit.
It is doubtful if diagonalizing a carefully
optimized Hamiltonian to obtain one-particle one-hole wave
functions that do not work any better than pure configura-
-290-
tions is worthwhile; we have the distinct impression that
the place to stop is at the zero-range residual interaction
calculation of the 1” states, as performed initially by
*
Brown et al.
We believe, however, that this may not have
been obvious from the start, and that Gillet*s calculations
had the great merit of clearly indicating the limitations
of these techniques.
Before closing this chapter, we
shall discuss some of theoretical techniques used in a
typical particle-hole calculation.
In such a calculation
it is customary to assume a harmonic oscillator well for the
nuclear average potential.
As a result, all the states are
treated and predicted as bound states.
We do not deny that
this procedure may yield reasonable results for the states
which turn out to be truly bound, but there are only tv/o
truly bound states in C
12
.
This makes a direct comparison
between theory and experiment very difficult.
The
measurement consists of cross sections and angular distri­
butions, and the theory predicts a set of discrete transitions.
These discrete transitions are usually represented as
vertical lines, and the height of the line is made proportional
to the electromagnetic matrix element of the corresponding
transition.
This procedure is very misleading, for several
reasons;
the cross-sections due to various resonances
are not necessarily
proportional to the dipole matrix
elements; the resonances are in general overlapping and
interferences between transitions are important; also, it is
-291-
almost impossible to calculate (for a set of discrete
transitions) what the angular distributions should be.
Finally, it should be noted
that presenting the results
t
of calculations as a set of discrete transitions precludes
a prediction of the very important absolute magnitude of
the total cross-section »
The meaning of the word "state" , as it is currently
used in Nuclear Physics, is substantially different from
its formal definition in quantum mechanics.
When it is
noted that a "state" lies at 23 MeV, the usual implication
is that a resonance is observed at 23 MeV, nothing else.
What we mean by the word "state" depends on whether we are
describing a bound state or a resonance.
In both cases,
an eigenstate of the Hamiltonian is involved, but for the
bound states, the boundary conditions are unique, whereas
for the unbound states, they are arbitrary.
These bound
states have a precise physical meaning, whereas unbound
states are best thought of as an arbitrary basis for
expanding the scattering state of the system, i.e. a
mathematical tool.
In heavy nuclei, because of the high
Coulomb barrier, low lying unbound states can be success­
fully treated as bound states, in the sense that one can
conveniently forget their arbitrariness; this is not the
case in the region of C
12
, where a clear-cut distinction
between bound and unbound eigenstates is in order.
Among the infinity of possible boundary conditions
which can be used to define unbound eigenstates, one choice
-292-
has very special properties:
requiring that all the
logarithmic derivatives of the radial wave functions of
relative motion be equal to -1, on all the channel
♦
surfaces (of the open channels) leads to eigenstates of
the same energy as the corresponding resonances;
It is
therefore desireable to use this boundary condition when­
ever possible.
Otherwise, very substantial level shifts
can occur; a 5 MeV difference between the energy of the
eigenstate and the resonance energy" is typical for a
boundary condition parameter of +1 instead of -1.
The above remarks have very
profound
consequences
on the one-particle one-hole calculations using harmonic
oscillator wave functions; because of the peculiar
boundary conditions imposed on the wave functions by the
infinite harmonic oscillator well, we would expect the
resonances to lie very far from the calculated eigenstates.
To see this, we have assumed that the nuclear potential is
approximately of harmonic oscillator shape for r < Rc ,
and pure Coulomb (or zero) outside R . R was chosen for
c
c
12
potential continuity, ive. R c cr4 fermis in C . This
procedure has been used with some success by L a n e ^ ^
should yield a reasonable nuclear potential.
and
The infinite
harmonic oscillator eigenstates used by G i l l e t ^ ^ * ^ ^,
B r o w n ancj others have logarithmic derivatives at Rc
which vary between -4 and +1.
The level shifts resulting
from the use of such boundary conditions have been
computed, and they are of the order of + 6 MeV.
This
-293-
implies that if everything
was calculated consistently,
two eigenstates lying 10 MeV apart could produce resonances
at the same energy!
To summarize, let us say that assuming the energies
of eigenstates to correspond to observed resonance energies
is completely inconsistent with the use of the eigenstates
of the infinite harmonic oscillator.
Yet, Brown; Gillet
and others had great success in predicting the energies of
resonances.
How this was achieved is not a mystery:
they
simply took experimental resonance energies from neighboring
nuclei as the energies of their unperturbed configurations,
and then adjusted the parameters, of the residual interaction
so that the predicted eigenstates would line up with the
observed resonances.
Thus the whole scheme is completely
inconsistent and, needless to say, the residual interaction
used has little physical meaning.
In particular, we
should not expect the details of Gillet*s wave functions
necessarily to correspond to anything physical.
In fact,
it may be that the wave functions are nevertheless relevant,
because Gillet was careful to fit mostly low-lying or bound
state, for which the infinite harmonic oscillator
representation is reasonable.
We would also remark that the coupled-channel method
makes no use of the knowledge of the eigenstates of the
system; essentially, the Hamiltonian must be diagonalized
at each energy.
Thus the method combines in a single step
the work of Gillet and that of Boeker and Jonker, and
-294-
the full problem has to be solved for each energy.
the power of the ^ - M a t r i x theory is lost.
All
It is true
that the coupled-channel method is in principle more
4
accurate than is ^
-Matrix theory; unfortunately, in
practice, approximations of the same nature must be made.
The advantage of the 'JZ -Matrix theory approach is
that it is computationally simple; it essentially involves
the sum of the D -Matrix with a diagonal matrix, both of
.which are very easy to compute.
are inherent to the
The approximations which
-Matrix calculation are those used
in the one-particle one-hole and other shell-model
calculations; namely, the truncation of the basis in which
the state of the system is expanded.
-Matrix theory is
general, in the sense that it works as well for direct
reactions as for compound nucleus reactions; our calcula­
tions are a good example of ^
-Matrix theory used for what
is essentially a direct reaction.
The advantage of
^-Matrix
theory over a simple direct reaction approach is that
compound nucleus processes are not ruled out; a unique
calculational scheme works in the low energy region, the
high energy region, as well as in the intermediate region,
where the reactions are not compound nucleus, not direct,
but something in between.
The latter is the region where
the resonances start overlapping appreciably.
-295-
List of References for Chapter VII
(1)
R.G. Allas, S.S. Hanna, L. Meyer-Schutzmeister arid
R.E. Segel, Nucl. Phys. 58, 122 (1964)
(2)
B.L. Berman, Private Communication
R.L. Bramblett, S.C. Fultz, B.L. Berman, M.A-. Kelly,
J.T. Caldwell and D.C. Sutton, UCRL-70582 Preprint
(1967)
(3)
C.P. Wu, F.W.K. Firk, and T.W. Phillips, Phys. Rev.
Letters 20, 1182 (1968)
(4)
M. Marangoni and A.M. Saruis, Nucl. Phys. A132, 649
(1969)
(5)
A.M. Lane, Rev. Mod. Phys. 32, 519
(6)
R.C. Morrison, PhD Thesis, Yale University, unpublished
(7)
D. Drechsel, J.B. Seaborn and W. Greiner, Phys. Rev.
Letters 12, 488 (1966)
(8 ) V. Gillet, PhD Thesis, Unlversite' de Paris, (unpublished)
(9)
Feldman, Suffert and Hanna, Progress Report, Dept, of
Physics, Stanford University, August 30, 1968
( 10)
N. Vinh Mau and G.E. Brown, Nucl. Phys. 29, 89 (1962)
(11)
Claude Bloch, Nucl. Phys. 4, 503 (1957)
-296-
(12)
S. Cohen and D. Kurath, Nucl. Phys. 73, 1 (1965)
*
(13)
S.S.M. Wong and D.J. Rowe, Physics Letters 17,
488 (1966)
-297-
CHAPTER Vni
SUGGESTIONS FOR FUTURE EXPERIMENTS.
-298-
Suqqestions for Future Experiments.
The very important question of the absolute deter­
mination of the (particle, gamma) experimental cross
sections discussed herein clearly necessitates the
undertaking of two separate experimental programs:
the
determination of the response of large Nal(T^) crystals
(bare and with paraffin absorbers) will require investi­
gation which can only be performed with monoenergetic
gamma ray beams (i.e. from electron accelerators with
positron annihilation in flight) and all particle-gamma
results already published with necessarily have to be
reexamined for a possible (and in our opinion highly
probable) gross correction in the absolute magnitude of
the total cross section.
The second program involves
the measurement at several energies, for each nucleus,
of the particle-gamma cross sections, because the
correction will be energy dependent.
We are in contact
with the Livermore group regarding the first type of
measurements.
As we have demonstrated herein, the direct gamma
decay to excited states following particle capture has
great interest, and we believe that efforts should be
undertaken to consider the possibility of building a de­
tector system capable of resolving the decay to various
low-lying excited states whose spacing precludes their
separation with existing Nal crystals.
Such an apparatus
would also permit the study of the ground state decay of
-299-
those nuclei for which this transition cannot be resolved
at present.
Building such a system is certainly feasible
in principle; the question whether adequate counting rates
♦
could be obtained with it.
The answer may well hinge on
the availability of an on-line computer-based data
acquisition system, such as the WNSL system.
Having demonstrated herein the power of the radiative
capture studies in the examination of the microscopic
structure of C
12
,
it is clear that a vast amount of work
opens up in the entire remainder of the periodic table.
Extensions of the work initiated herein in 0
16
will clearly
be of great interest, as indeed will be the case in all
the regions of doubly closed shell nuclei.
This approach
however is not in any sense limited to these regions
although they will logically be the first to be included
in the ongoing program.
Following these however, the study
of both vibrational and rotational nuclei will permit much
further study of the coupling and microscopic structure
of nuclear collective phenomena than has hitherto been
possible.
-300-
CHAPTER IX
CONCLUSION.
-301-
The present work represents a detailed experimental
and theoretical study of the microscopic structure of C
II L
12
utilizing the B
C
reactions. The experimentally
12
difficult radiative capture experiment was performed in
order to benefit from the power of the electromagnetic
probe which enhances the role of the one-particle one-hole
aspect of the wave functions of the system, and thus
permits a meaningful test of the recently available oneparticle one-hole wave functions for C
12
.
The gamma transitions leading to the first four
12
states of C
were resolved; cross sections and angular
distributions were obtained for the transitions to the
0+ ground state, the 2+ first excited state and the 3
third excited state and an upper limit was set on the
transition to the 0 + state at 7.65 MeV, in the region of
14 to 22 MeV incident proton energy.
The technical
difficulty of extreme background was solved through the
design and construction of a new electronic counting
__
system which does not involve pile up rejection, with a
light pulser optically coupled to the Nal(T^) detector
and on-line control of the experiment by the laboratory
computer.
These techniques offer a wide range of
applicability to other types of experiments.
The spectra
obtained were fitted using a new algorithm which offers
extreme stability on small peaks, and the program which
uses this algorithm has been designed for on-line use.
-302-
The availability of the C
12
wave functions is not
sufficient to permit meaningful comparison of theory
with experiment; theoretical calculations of the cross
♦
sections corresponding to these wave functions are necessary.
In attempting to use Tft -Matrix theory for this purpose,
it was discovered, with some surprise, that no complete,
coherent theory existed; such calculations being extremely
sensitive to details (phases, constants etc.) it was found
necessary to start from basic quantum mechanics and to
develop a coherent scheme for calculating (/ , y )
cross
sections from the assumed knowledge of the stationary
states of the system.
This scheme was used to calculate the El, Ml and
E2 transitions to the ground state and excited states of
C
12
following proton capture, with the wave functbn of
Gillet's "approximation I".
The calculation involves
essentially no free parameter, and does not make the
restrictive assumption of compound nucleus formation or
of direction reaction mechanism.
excellent for the
^
The agreement is
transition, especially in the case
of angular distributions; small discrepancies
in the
total cross section are attributed to the admixture of
several 1“ many-particle many-hole states. These results
12
are consistent with a very well closed C
ground state,
which is in contradiction with recently published results
based on different models.
The agreement with the
^
transition is poor, and the enhancement of the experimental
-303-
cross section is consistent with the picture of a giant
resonance based on the first excited state ; these states
would involve mainly two-particle two-hole configurations.
Finally, the
y
experimental cross section is smaller
than the predicted values by a factor of five, and this
supports the view that the 9.64 3“ third excited state
of C
12
has only a small admixture of one-particle one-hole
configurations. The analysis supports the contention that
the radiative capture reaction involving the 2
+
C
12
state
at 4.43 MeV involves a giant resonance consisting of a
dipole excitation superposed on the 2+ state in direct
analogy to the usual giant resonance superposed on the
ground state.
The results of calculations include the relative
centributions of the various proton channels to the
total cross section, and a comparison of results obtained
with Gillet's wave functions and with pure j-j coupled
configurations.
The latter results are striking*
the
comparison shows that the mixing of configurations pre­
dicted by Gillet does not affect the calculated cross
section in a substantial way.
This implies that one-
particle one-hole calculations with ad hoc residual inter­
actions have probably already been pushed beyond their
range of validity (or at least beyond their range of
usefulness) and that if the wave functions work for the
transitions it is simply due to the fact that pure j-j
-304-
configurations describe this aspect of the C
fairly well.
12
situation
During the course of the experiment, it was
discovered that the method previously and generally used
♦
to extract cross section data from the experimental
spectra does not take into account the Compton scattering
o f ;gamma rays in the paraffin which is normally used to
shield the detector against fast neutrons; this leads to
errors ranging up to a factor of two (typically 1.5), and
suggests that essentially all the (particle, gamma) data
published previously over-estimate the total cross
sections by similar factors.
Measurements are in progress
in this laboratory, together with necessary complementary
measurements elsewhere using monochromatic gamma radiation
which hopefully will resolve this uncertainty in the total
cross section; these uncertainties have very important
consequences in any discussions involving the dipole
svim rule limits or the isospin purity of the states in­
volved in the radiative capture.
In the work reported herein we have developed a
selection of techniques, both experimental and theoretical,
I
which we hope will contribute in a significant fashion to
the developing potential of the higher energy radiative
capture studies as probes of nuclear structure.
We have
!
demonstrated their utility in the particular case of C
12 .
We would venture to predict that studies of this type
will play an increasingly important role in the develop­
ment of our understanding of the microscopic basis for
nuclear phenomena.
-305-
Appendix II-A:
i
The Fast Counting System.
In the early days of nuclear physics, electronic
pile-up usually was not a severe problem. The accelerators
I
operated below the Coulomb barrier, where typically only
a few channels are open.
The first reactions studiel
were clearly those with the highest yield, and to reduce
pile-up to an acceptable level it was sufficient to reduce
tiie beam accordingly.
In recent years, however, the
experimental situation has deteriorated rapidly; the
advent of more powerful, higher energy accelerations which
operate well above the Coulomb barrier, in a region where
thousands of channels are open, and the necessity of
studying
weaker and weaker reactions in order to answer
more detailed questions on the nuclear behavior, have
rendered electronic pile-up one of the major instrumental
problems encountered in the study of nuclear phenomena.
The reaction reported herein certainly constitutes
op
of the most dramatic example of such problems:
Very large Nal( n
a
) crystal is located a few feet from a
target which is bombarded with microamperes of 20 MeV
protons.
The detector is sensitive to neutrons emitted
!
from the target, as well as to cascading reaction gamma
rays, however we are -only interested in the very rare
events where a gamma ray is emitted directly from the
Q
capturing state (i.e. less than 1 in 10 of the photons
emitted from the target itself).
These events of interest
l
o
Ci
CO
10*
^ • •
•
10"
800
n B(Plr)l2c
..... %
< 10s
X
u
£T -GATE
500
•I
%
T RES HOLD
UJ
CL
%■•
%
o
r- \
PER
if)
z io2
3
O
(_>
400
COUNTS
Yo
VA .
300
y0 127.8 MeV)
i
9
Vr * a i
____ -
10
200
RUN* 7
Mev
•
•
•
z
P I L E - U P R E J E C T IO N
C IR C U IT -
600
CHANNEL
Ui
= td.O
•
» °
•
o
.*
E p = 15.0 MeV-
LP
•
•
■• ‘ 1
•
RUN *35
700
tiip .r .12
J L.
II ~
•
•
•
•
•
100
•
•
—
c
m •m
0
50
L...
40
ADC
50
60
CHANNEL
70
NUMBER
80
90
20
40
ADC
60
CHANNEL
80
100
NUMBER
120
ADC
CHANNEL
NUMBER
F ig. I I -A -1 : Spectrum a) was accum ulated in 20 m inutes using a conventional pile-u p r e je c tio n c ir cu it;
when the counting rate is in crea sed by a fa ctor o f fo u r , as in b), many counts a re r e je c te d but p ile-u p n everth e­
le s s com pletely d estroys the spectrum (cou nting tim e fo r b) was ten m inutes). Spectrum c)' has been obtained
in ten minutes with the fast counting system d escrib ed herein.
-307-
can be distinguished from the cascading gamma rays, because
hr
the direct decay, (
), results in higher, known, gamma
ray energies.
#
The first major breakthrough in fast counting
techniques was the introduction of the double-delay-line
amplifier, many years ago; its use is now familiar to
all nuclear experimentalists dealing with electronic
counting systems.
Then came a series of attempts at pile-
up rejection circuits, which are now available commercially,
or can be assembled from commercially available modules.
Two of the most popular types of pile-up rejection circuits
are respectively the inspector, and the circuit based on
a measurement of the delay between the leading edge and
the crossover of the electronic pulses from a double-delayline amplifier.
All these pile-up rejection circuits share a
very important drawback, however:
the correction for the
rejected pulses, which is always pulse height dependent,
is extremely difficult to perform accurately.
Furthermore,
they do not come close, in performance, to the system to be
described herein.
Fig. IIA-1 shows a comparison of results
obtained with a pile-up rejection circuit and with our new
counting system.
The counting time was 20 minutes in the
case of the pile-up rejection circuit, and only 10 minutes
for the fast counting system; yet, a factor of 8 more
counts could be accumulated in the latter case.
The pile-
up rejection circuit involved here was a complex assembly
of some 12 fast electronic modules designed to detect
GATE OPEN
TIME
PM
A NO DE
CRYST AL
LIGHT
OUTPUT
F ig. I I -A -2 : a) and b) rep resen t the lin ear signal b efore and after shaping re sp e ctiv e ly . F ig s. c) and d)
show how light pulses which overlap in tim e in the cry s ta l can still be analyzed c o r r e c t ly by the fa st counting
s y s te m , provided they a re separated by a tim e greater than the gate opening tim e.
-8 0 S -
CURRENT
TIME
-309-
pile-up from a measurement of the delay between leading
edge and cross-over.
This circuit was used in some very
early measurements; we succeeded in obtaining meaningful
#
12 with it, but
results in the giant resonance region of C
were not successful at higher energies, where the crosssection is lower and the background more intense.
(reduc­
ing the beam current does not help, because the counting
rate in the peaks then becomes lower than the cosmic ray
background; we did not, in these early measurements, have
an anticoincidence shield for cosmic rays).
We now discuss the principles which make it possible
to count at a rate forty times greater than is possible
with a double-delay-line amplifier, without necessitating
any pulse rejection.
The method basically consists in
treating the available information, i.e. the photomultiplier
anode current, more efficiently.
A high-energy gamma ray
pulse at the output of the photomultiplier, has the
following shape:
a fast rising (approximately 1 5 7?S )
leading edge, and an exponential decay with a time constant
of some 280 nanoseconds.
There are rapid fluctuations on
the pulses reflecting the statistical fluctuations in the
collection of primary electrons; for high energy gamma
rays however the fluctuations are relatively small in
proportion to the peak current, and we neglect them for
the moment.
Fig. II-A2-a.
The pulse^shapes therefore are shown in
Fifty nanoseconds or so after the start
of the pulse, we already know what the pulse height is,
and we can already estimate the energy of the gamma ray;
-310-
we do not need the whole pulse to make such an estimate.
Of course, if we base our estimate of the energy of the
gamma ray on the first 50 nanoseconds of the pulse instead
♦
of on the whole charge, we necessarily make a greater
statistical error, and the resolution of the detection
system is correspondingly impaired.
On the other hand,
we can afford to count faster, because pile-up with
another pulse occuring more than 50 nanosecond after the
start of the pulse of interest will not yield a false
estimate of the gamma ray energy.
entire operation depends
It follows that the
on a compromise'between
counting rate and resolution.
Optimum resolution demands
that we wait until the complete charge has been collected,
i.e. one microsecond or more; optimum counting rate demands
that we base our estimate of the gamma ray energy on the
first few nanoseconds of the pulse, in order to be in­
sensitive to pile-up from another gamma ray arriving a
short time later.
only
A decision on the best procedure can
be taken after the resolution and counting rate are
studied quantitatively as a function of the time taken to
estimate the gamma ray energy.
Before we proceed to make such an estimate, it is
necessary to discuss how, in practice, we have proceeded.
Since the decay of the pulses is very nearly exponential,
we can bring the pulse back to zero at any given time after
i
its initiation, by adding it to its reflection from a
suitably terminated line.
This is not clipping; a reflection
-311-
from a short-circuited line, as used in clipping, would
cause overshoot.
The properly shaped signal now is shown
in Fig. II-A-2b, in the case of a 2 5 w d e l a y cable.
By
opening a gate at the beginning of the pulse and closing
it after, and integrating the current between these two
times, we have an estimate of the gamma ray energy which
is based on the leading edge of the pulse.
The signal which determines the gate opening is a
logical signal which identifies the "interesting" pulses;
it can be a coincidence requirement, or in the case of the
experiment
reported herein, the output ofy a discriminator
with a high threshold (we are interested in the large
pulses only).
In this way, the gate only opens on the
few interesting pulses, so that at its output the overall
rate is considerably reduced and does not lead to
appreciable pile-up.
It is easy to see that if A k t
is
the gate opening time, pile-up will occur only between
pulses which are separated in time by less than £ \. t
For a A t
.
of 50 nanoseconds, we have therefore achieved
a counting rate improvement by a factor of approximately
40 over a double-delay-line amplifier, which requires a
two microsecond clearance.
It remains
to be seen what
the degradation in resolution would be.
In order to estimate the degradation of the
detector's resolution because of the pulse shaping intro­
duced above, it is necessary to know the average number
of primary photoelectrons produced in the
photomultiplier
per keV of energy deposited in the crystal:
the number
-312-
has been measured directly by Houdayer and Bell* and is
equal to 10 in the case of a good-quality detector.
We
also have to know the actual resolution of the detector
corresponding to a complete integration of the scintilla­
( A t ) due
tion pulses.. More precisely, the resolution
to the statistical fluctuations on the number of primary
electrons collected is equal to
rcM - M
where
A /C M )
is the average number of primary electrons
collected during the gate opening time
At
(this discussion
is valid for a given, fixed, gamma ray energy).
good crystal, at 20 MeV, we have
200000
f\/(A t )
^
For a
20,000 x 10 =
( A t ) ~ 0.53% . It is
and we expect
clear, therefore, that the measured resolution of a crystal
at 20 MeV, which is typically 5%, is not limited by the
statistical fluctuations on the number of primary electrons
collected.
,
We therefore introduce a new quantity,
which we refer to as the intrinsic resolution of the
crystal, and we postulate that the measured resolution of
the crystal at that energy,
7? (At) , is given by
This equation can be thought of as defining the
intrinsic resolution
, which is energy dependent but
hopefully independent of the gate opening time
*
A. Houdayer et al;
At ;
we have
Nucl. Instr. and Meth. 59, (1968)319
-313-
actually checked experimentally, in some selected cases of
interest to us, that this is the case.
duce the parameter
We shall now intro­
, a figure of merit for the detector,
" g 'f o o )
defined as
^
At any given energy,
X
"
is defined as the ratio of the
loss in resolution due to statistics to the intrinsic
Typical values for X
resolution at that energy.
range
from 0.5 for a small crystal at low energy, to 0.2 for a
In order to measure c ( for
good large detector at 20 MeV.
a crystal at a given energy, the procedure would consist
T ? (A t)
in measuring
t and
as a function of
fitting the obtained curve with the function
where
P 'U t)
is given by the equation above, and
/i/(A t) =
v e { i - j f At/z}
In the latter expression,
P
is the number of primary
photoelectrons produced per keV of energy deposited in
the crystal,
E
is the gamma ray energy in keV, and
/Vo. T ( T - & ) , approxi­
is the decay time constant of the
mately 280 nanoseconds.
to determine, therefore:
?
There are two unknown parameters
and
"R0 .
independent of the energy, whereas
7^ is presumed
7?0 is sensitive to it.
We have mentioned above that the parameter
P
has been
measured directly, but for technical reasons which we shall
P
not discuss here a measurement of
the measured
Tt(At)
based on a fit of
as a function of
At
meaningful and should probably be attempted.
it is possible to use values of
P
may be more
Nevertheless,
determined by the
-314-
RELATIVE RESOLUTION
oo
CM
m
o
CM
o
CO
in
o'
CM
to
Fig. I I -A -3 : Estim ating the gamma ray en ergies by using only the leading edge o f
the pulses in cre a se s the statistical uncertainty, and causes a degradation in resolu tion
which is given here as a function o f the gate opening tim e AL The param eter a defined
in the text is an energy dependent figu re o f m erit fo r the d etector, which is typically
equal to 0 .2 5 at 20 M eV.
-315-
direct measurement of Houdayer et al. and apply them to
a determination of the parameter
o(
done, with the results quoted above.
.
This has been
The measured
♦
resolution as a function of the gate opening time depends
on the parameter c(
only, and we have, explicitly:
where the relative resolution 7?% measures the deterioration
in the resolution corresponding to the gate opening time
A t
•
We have plotted
and for
0^
o( ® 0.25" .
= 0. Z £
in fig. II-A-3 for
o( - /
in a typical high-energy situation,
and we see that we can use a gate opening
time of 50 nanoseconds with only a slight degradation of
resolution; indeed, we could use a gate opening of 100
nanoseconds, and the corresponding loss in resolution
would be too small to be measurable.
It is interesting to note that the system described
herein permits resolution in time of scintillations which
overlap in the crystal.
Fig. II-A-2c shows three light
pulses which occured almost simultaneously, and we may
assume, for example, that the second pulse is the one of
interest.
The circuit is still capable of extracting
correctly the energy of this gamma ray, provided it is
separated by at least
At
from the two background pulses,
as shown in fig. II-A-2d.
s
There remain two technical details which we should
FAST DC
AMPLIFIER
(x 4 )
5 0 & PASSIVF
FANOUT
DELAY
>-
TERMINATION
INTEGRATING
LINEAR GATE
(LG 102)
GATE
OPEN A
SIGNAL'
v
/N
OUTPUT TO ADC
------- > ------- O
DELAY
(BUSY-VETO)
DELAY
>-3 1 6 -
FAST
DISCRIMINATORTRIGGER
(T 101)
A
FAST DC
AMPLIFIER
(x4 )
FAST
COUNTING SYSTEM
F ig. I I -A -4 : B lock diagram showing the fa st am p lifier o f appendix I I -C , the shaping netw ork,
linear gate and the h igh -level d iscrim in ator which opens it to integrate the la rge pulses o f interest.
the
-317-
discuss before introducing the practical circuits which
will perforin the above-mentioned operations.
We have
completely disregarded, thus far, the effect of the
fluctuations on the electronic pulses.
These fluctuations
are an intrinsic characteristic of the detection circuit,
of course; they are not due to noise, "ringing" or other
electronic problems, but simply originate in the statistical
fluctuations in the number of primary electrons collected.
The main effect of these fluctuations is that the shaping
circuit cannot bring pulses back exactly to zero; if a
background pulse occurred before the pulse of interest,
there will be pile-up with the fluctuations of the back­
ground pulse.
) experiment, however, the
In the
background pulses are usually small, and their fluctuations
are another order of magnitude smaller, so that this effect
can be neglected.
The other problem concerns the rise­
time of the detector, which we have completely ignored in
the discussion above.
In practice* however, the risetime
for large detectors is never very good, because the light
output of very large crystals rises rather slowly; the
detector risetime therefore is likely to become the practical
limit on the gate opening time
to have
At
t
, because it is impractical
smaller than approximately four times the
risetime of the detector.
Fig. II-A-4 constitutes the block diagram of the fast
counting system used for the acquisition of the data
presented herein; the preliminary data taken with the
5" x 6" crystal were obtained with an even simpler circuit,
-318-
consisting of a discriminator and a gate only.
operation of the circuit is very simple:
The
The photo­
multiplier, which is 50-ohm back terminated feeds the
input of a specially designed highly stable DC-coupled
amplifier (see appendix II-C); this high output
impedance amplifier has its output shaped by the delay
line with the special termination; the pulse is- subse­
quently split equally into a logic signal and a linear
signal.
The logic signal, after amplification, is used
to trigger a high-level discriminator, and this discrimina­
tor opens the linear gate on the large pulses of interest,
which are integrated inside the gate, and give rise to a
slow output with a height proportional to the charge fed
in at the gate input.
It is the discriminator delay
cable which determines the gate opening time, while the
linear gate is prevented from opening a second time on
the same pulse through a "busy" signal fed to its "veto"
input.
The system is .very simple in principle; however, in
order to attain the highest counting rates we have used
DC coupled electronics throughout.
This procedure com­
pletely eliminates baseline shift, but it also introduces
two new problems, namely low frequency noise pickup and
zero temperature shift and time drift; these two problems
practically do not exist in AC coupled electronics, of
course.
The noise was reduced through extreme care in
eliminating pick up, and the special amplifiers described
in appendix II-C were designed to alleviate the zero
stability problem.
The net result was that in the final
-319-
circuit used during the actual data acquisition, both the
noise and the zero shifts were completely negligible.
-320-
Appendix II-B:
The Photomultiplier Control Unit.
The severe background problems encountered in the
course of this experiment necessitated the on-line com­
puter control of the photomultiplier (PM) gain.
The
stabilization was effected through readjustments of the
PM high voltage, in accordance with a computed correction
based on the position of the centroid of the light pulser
peak.
The correction appears at the output of the
computer interface as a low-level DC signal which must be
amplified; the present appendix describes this special
circuit which has been designed and built as part of the
experiment reported herein.
Because of the DC-coupled electronics, it was
desirable to use the PM at a high output current level,
in order to minimize noise and shift problems in the
electronics, and the average anode current had to be
monitored during the run.
Further more, a protection
system which automatically drops the PM high voltage in
case of a sudden increase in radiation was needed, be­
cause such an increase in background radiation is likely
to occur while the beam is optimized, and it could result
in a catastrophic PM failure.
Figs. II—Bl to II-B3 represent respectively the PM
high-voltage control, the PM current measuring device
and the protection circuit with adjustable threshold; the
three circuits were build in a single enclosure, but are
logically distinct, except for several interconnections.
PH O TO M ULTIPLIER CONTROL U N IT :
S E R V O HIGH V O L T A G E S U P P L Y
180
F ig .I I -B -1 : This c ir cu it p rovid es a va ria ble negative high voltage o f 0-100 v o lts , which is c o n tro lle d by the
0-10 volt output o f the com puter in terfa ce, to p erm it on -lin e adjustm ents o f the photom ultiplier voltage. The output
o f this c ir c u it is added to a fixed high voltage supply, and it is dropped autom atically when the protection c ir cu it cuts in.
PHOTOMULTIPLIER CONTROL UNIT:
C U R R E N T M E A S U R IN G S Y S T E M
+12
F ig. I I -B -2 : The photom ultiplier average anode cu rren t, which m ust be m onitored
during the run to avoid ov e rloa d , is m easu red with the p resen t c ir cu it; six s ca le s a re p r o ­
vided and the output feeds the p rotection c ir c u it o f fig . II -B -3 .
P H O T O M U L T IP L IE R
P R O TE C TIO N
+ 12
CONTROL U N IT :
C IR C U IT
-H.V.
15 K
220
-3 2 3 -
-H.V.
2 0 0 1 2 ,1 W
Q O U T
^'STOP'
-12
F ig. I I -B -3 : This p rotection c ir c u it autom atically drops the high voltage on the photo­
m ultiplier when the average anode cu rren t exceed s a p re se t le v e l; a stop signal is a lso sent to
the com puter.
-324-
Appendix II-C:
Fast DC-Coupled Amplifier.
In the design of the fast counting system, DCcoupled electronics were used throughout, in order to
♦
eliminate baseline shift at the highest counting rates;
it was found that the temperature stability of the linear
gate used to integrate the pulses is marginal, and further­
more, it was desirable to obtain the signal from a high
impedance source in the control room, for shaping purposes.
No amplifier meeting the various requirements of frequency
response, output impedance and zero stability was
commercially available, and therefore such an amplifier
was necessarily designed and fabricated as part of this
experiment.
The specifications are as follows:
the amplifier
is DC coupled, with a risetime of 3 nanoseconds; its
gain is 4.2 into 50 ohm.
The design minimizes output
variations with a change in supply voltage; the output
and both inputs are temperature compensated, and the
temperature coefficient is easily rendered smaller than50
V/°C referred to input by suitably choosing the input
transistors.
The output drift with time has been measured
to be approximately 45pL V/day.
The amplifier shows
excellent linearity in the range t- 120 mV of input signal;
and its output impedance is 800 ohm.
The design is actually
that of a differential amplifier, and the inverting input
v has been used successfully to cancel some ground loop
noise, by grounding it near the source of the signal to be
amplified.
-325-
FAST
STABLE AMPLIFIER
(D C C O U P L E D )
vcc
NON-INVERT
56.2
INVERTING
OUTPUT
500
BALANCEI
N9I4
q 6
F ig. I I -C -1 : This D C differential am p lifier was designed and built to
m eet the sp ecification s o f r is e tim e , stability and high output im pedance requ ired
by the fast counting system . Ultim ate tem perature stability is achieved by p roper
ch o ice o f the input tra n sistors.
-326-
Fig. II—Cl is the circuit diagram of the amplifier
the unmarked components must be chosen to optimize
performance:
for example,
X,
and
are adjusted to
#
obtain a 50 ohm input impedance.
Vcc is
\/
cc
_
and
is -I- 12 VDC
— 12 VDC , both regulated to 0.02%/°C .
All resistors marked with 3 significant digits are 1%
metal film % watt resistors; their value is given in
ohm, unless otherwise noted; similarly, all capacitors
are given in picofarad.
through
The transistors marked
are active, but
through
for temperature compensation purposes.
^
<?,
are used
-327-
Appendix IV-A;
The Data Analysis Program.
In the chapter on data analysis , the numerical
methods of peak fitting were discussed at some length;
the object of this appendix is to briefly describe the
actual program which was used to analyse the data presented
herein.
The program is self-checking, in the sense that one
of the phases is used to simulate a gamma ray spectrum;
this spectrum is composed of a known number of peaks of
known intensities and positions, with a low energy expoential
background and a cosmic ray background.
The various
parameters such as peak heights and positions are entered
as data.
One then proceeds to fit this known spectrum
with the program, and the answers obtained by the fit are
compared with their known values.
In order to allow the use of complex peak shapes
and still keep a reasonable execution time, we have chosen
to store the peak shapes instead of computing them each
time they are needed; more precisely, they are computed
once for each major iteration, i.e. six or seven times in
the course of a fit.
In order to save storage space, the
peak shapes, as well as most of the other arrays are stored
as integers of length two (i.e. two bytes long, where one
byte is 8 bits).
This allows a maximum of approximately
i
30,000 counts in a channel, and is quite adequate for high
energy gamma ray spectra.
Keeping the number of counts per
channel in integer storage has some drawbacks, however:
-328-
it is exceedingly easy to accumulate round off errors.
For
example when the cosmic ray background is subtracted, we
obtain a number of counts per channel in the corrected
spectrum which is clearly not necessarily an integer.
It
is not sufficient to insure that the number of counts
be
rounded off; even rounding off can cause severe'
accumulation of errors.
Instead, one must keep track of
the fractions dropped in rounding off the neighboring
channels when deciding on which way any given channel is
to be rounded.
The same problems
and their solution is similar.
arise during the fit,
It is not clear in retro­
spect whether the space saved by using storage in the
"integer of length two" mode was worth the extra difficulty
of careful handling of round off errors.
The spectra are usually read from a magnetic tape
produced by the data acquisition program.
these spectra, several parameters are read:
Together
with
the run
identification number, the beam energy and the angle of the
detector are among these.
for the fit.
Other parameters must be entered
In order to proceed as rapidly as possible,
the light pen is used whenever it is practical to do so;
otherwise, the parameters are entered via the typewriter
keyboard.
The entry of parameters on the typewriter is
made both easier and safer through use of a subroutine
which eliminates the need of a specific format.
The para­
meters are entered as unsigned integers, between commas.
-329-
DATA
H : ERRORM
C A L C U L A T E S AND P R IN T S
E R R O R M ATRIX
ANALYSIS
G:INPET
'
H: R T
IN PU T DATA FROM T A PE
K IN E M A T IC S CALCULATIONS
E N T E R RUN PA R A M S
ON T Y P E W R IT E R ; P E A K
SH A PE PA R A M ET ER S
DATA A N D P A R A M S
INPUT A N D O U T P U T
H: M I N R U N
AND
f it t in g p a r a m .
W ITH C U R S O R :
BO U N D A R IES ,
STA RTING V A L U E S ,E T C .
en ter
FITTING
H :L P C
F iR S T O RD ER CORRECTION
OF P IL E U P USING LIGHT
P U L S E R SPEC T R U M
H: I N R U N
E N T E R FITTIN G P A R A ­
M E T E R S W ITH LIGHT
PE N :BO U N D A R IES,
STA RTING V A L U E S . ETC.
H: I N T G
N O N -LIN EA R L E A S T SQ UA­
R E S P E A K FITTING ROUTINE
H: D R O P
H: S H I F T L
L IS T S C H AN N ELS FOR
W HICH R E S ID U A L S A RE
L A R G E R THAN A LIM IT
H-.CRC
TRA N SLA TIO N OF L A B
SPEC TRU M TO C O M P E N S A ­
T E FOR Z E R O O F F S E T
G: G P A R A
COSM IC RAY
SU BTR A C T IO N
H: F I N A L
INPUT OF PA R A M S
A B S O R B E R S .S O L ID
A N G L E,T A R G ET THICK
PR IN T A N S W E R S
CRO SS-SEC TIO N S AND
P E A K PO S IT IO N S
H: A P P
MODIFY C U R R E N T E S T IM A T E
OF P E A K PO SIT IO N S
Fig. IV-A-l
G : I<£UNIT
H : SIM
A S S IG N S IN PUT AND
OUTPUT FORTRAN UNITS
DURING EXEC U T IO N
PREPA RES
DATA
ANALYSIS
A FA KE RUN
FOR PROGRAM T ES T IN G
H: S H O W
D IS P L A Y O R IG IN A L
LA B SPEC TRU M
H
CAN CEL
JO B
A: D D U M P
\
DUMP U SIN G D IS P L A Y
AND LIGH T P E N
H: S H O W
I
D IS P L A Y C O RREC TED
S P E C T R U M AND F IT
G: T P H D L
PO SITIO N DATA TAPE
AT TH E BEG IN N IN G OF
A G IV E N R U N
AUXILIARY
A N D DISPLAY K E Y B O A R D
F ig. IV -A -2
FUNCTIONS
-330-
Blanks are ignored; the subroutine checks that no
illegal character is present and that the proper number
of parameters has been entered.
When two successive
#
commas are entered (or commas separated by blanks only)
the corresponding parameter is not changed.
The format
of the data on the tape will not be discussed, since
any potential user would probably modify it to suit his
own needs.
An interlock system is provided to prevent
the operator from calling a subroutine before all the
prerequisite phases have been executed.
This is
accomplished by having a variable called "LEVEL" in common
between all phases, and changing its value according to
the present status of the fit.
The value of LEVEL starts
at zero and increases up to 90, which corresponds to a
completed fit.
Each phase requires that the present values
of LEVEL be within some limits, otherwise it is not executed;
upon returning, it modifies LEVEL according to the new
status of the fit.
The interlock still leaves considerable
choice to the operator, while forbidding any move whichwould result in cancellation of the job due to an illegal
operation.
A list of phases with their purpose and parameter
options is given on p.332 to p.334.
The numbering of
the parameter keys corresponds to figs. IV-Al and IV-A2.
The required "LEVEL" for execution is also given when
t
applicable, and the modification to "LEVEL" upon successful
completion of the phase is given in the next column.
-331-
The simplest fitting sequence which could yield
some results is the following: INPET, GPARA, RT, SHOW, INRUN,
L P C , CRC, INTG, SHOWl, FINAL, ERRORM.
The next fit could
start at RT, assuming no change in the parameters entered
in INPET and GPARA.
The only phases which require substantial
time is INTG, the fitting routine.
-332-
LEVEL
INPUT OF PARAMETERS
IN
GPARA:
(
) Thickness o f a b s o r b e r s , solid angle, target thickness
10
OUT
20
♦
IN P E T :
( 0 ) Peak shape param eters (o n ly )
60
( ) Z and A o f target, number o f peaks, excitation en erg ies,
peak shape p aram eters.
INRUN:
Enter param eters fo r the fit ( Light p en ): Fitting
boundaries, starting v a lu e s, etc.
40
50
( 1 ) update m ode
( 2 ) light pen mode
APP:
M odify curren t values o f peak positions and non-linear
70
param . o f exp. background: Enter new values in 10F10
form a t, only for those which req u ire changing.
A blank
field w ill resu lt in the corresp on d in g param eter staying
unchanged.
DISPLAY SUBROUTINES
SHOW:
( ) D isplay origin al laboratory spectrum .
(7 )
30
a lso adjust sca le using light pen
( 6 ) also locate channel number and contents with light pen
(4 )
SHOW1:
also change sca le using typew riter
( ) display c o r r e c te d lab. spectrum
(7) also adjust sca le using light pen
( 6 ) also locate channel number and contents with light pen
(5 )
also display the fit to the spectrum for com parison
40
40
-333LEVEL
INPUT AND OUTPUT OF DATA
RT:
Input o f data from tape (sp e ctra and run
IN
p a ra m eters)
display title, re la tiv is tic kinem atics.
DROP:
(x )
*
20
OUT
30
OX*
90
L ists channels fo r which the weighted squares o f
the resid u a ls is la rg er than x , w here x is the inte­
ger rep resen ted by the param eter keys. A minus
sign means the exp. spectrum is low er than the fit.
FINAL:
P rints answ ers: c r o s s s e c tio n s ,e tc .
70
80
ERRORM:
Com putes and prints e r r o r m atrix at cu rren t position.
80
90
50
59
Top half o f fir s t colum n is absolute e r r o r s on peak
p ositions and exp. bgn. n on-linear param .
Bottom
half is relative e r r o r s on peak heights and exp. bgn.
height (std . d evia tion s).
Other colum ns are the
corresp on d in g correla tion s.
CALCULATIONS AND FITTING
Light pulser (p ile -u p ) c o r re c tio n
LPC:
( ) p erform it.
( 0 ) avoid it.
CRC:
C osm ic ray background subtraction
59
60
INTG:
Fitting routine; 12 iterations is the m axim um , and
60
70
when it is reach ed , con vergen ce did not o ccu r.
( 0 ) linear fit only, and returns fo r approval o f the
starting value.
( ) com plete fit over all p aram eters.
T o com pensate for z e r o offset o f e le ctro n ics .
SHIFTL:
(x )
Shift experim ental spectrum x channels to the right
( 0 key up) o r to the left ( 0 key dow n).
Im portant: use im m ediately after R T .
-334MICELLANEOUS SERVICES
IOUNIT:
A ssign F ortran input and output units during execution.
(0 ) Initialization, then norm al I/O assignm ents.
( ) Norm al I/O assignm ents.
SIM:
P rep are a fake run fo r program checking.
TPHDL:
To position the data tape at the beginning of any given run.
EXPLANATION OF SYMBOLS:
The underlined nam es like GPARA r e fe r to phases
not subroutines
(although there is a lso a subroutine by that name which is included in the p hase).
A phase is an absolutely located program ready fo r loading and execution in one
p iece.
Loading and execution are accom plish ed by d ep ressin g a keyboard key
and button com bination which has been assigned to that phase by suitable con trol
statem ents.
In parenthesis ( ) appears the param eter key of fig. IV -A -1 which
m ust be down fo r the phase to be executed with the given option d escribed .
som e c a s e s , entering a binary number
In
( x ) on the param eter keys is involved.
When a blank is shown between parenthesis, it means that all param eter keys
m ust be up; when the parenthesis a re om itted, the param eter keys a re not used.
The "L E V E L IN " colum n gives the condition the param eter LEVEL
m ust satisfy fo r the phase to be executed and the "L E V E L O U T" colum n shows
the condition LEVEL w ill satisfy upon su ccesfu l com pletion o f the phase.
For
exam ple, after execution of phase SHOW1, if LEVEL is le s s than 40, it w ill be
set to 40; oth erw ise, it is left unchanged.
-3 3 5 IN PU T/OU TPU T STANDARD UNIT ASSIGNMENTS.
c
a)
a>
o
55
o
o
^
°
<u
6
5
Cj
.s
d
o
^
-o
55 co
L-H <D
° 3
<D d
S u
d
£ >
S)'
CG
M
Use of the input/output unit in the program .
<
*d
+->
1 IOUNIT
LOG
TYPE
PRINT
IGN
2
2
6
11
Enter code nam es.
O n-line partial listing o f I/O assignm ents (one b lo ck only)
O ff-lin e com plete listin g o f I /O assign, upon returning.
W rites blanks to clea r buffer.
2
READ
M
N
11
2
W rites blanks to clea r buffer.
Enter lis t of values fo r v a ria b le s, between com m as.
3
PfflD
PRINT
TYPE
6
2
O ff-lin e phase identification.
O n -lin e phase identification.
LOG
IER
2
2
W rites channel number and contents upon requ est (light pen)
E r r o r return from PLOT.
TAPE
TYPE
PRINT
LOG
4
2
6
2
R ead sp ectra to analyse and p relim in ary inform ation.
Output run num ber.
Output run number and re fe re n ce inform ation.
E r r o r return from DESSIN or TE XT.
LOG
TYPE
PRINT
2
2
6
W rites Q -value o f rea ction , and e r r o r return.
M essa g es requesting entry o f s p e c ific p aram eters.
W rites value o f param eters o ff-lin e , for re fe re n ce .
LOG
L
N
M
2
6
2
E r r o r return from DESSIN, INRUN.
Com m ent "u p d a te" (INRUN). W rite no. of chann. shifted
E r r o r return (LPC ).
(SHIFTL)
W rites how much background subtracted, and e r r o r (CRC);
p ile-u p c o r re c tio n param eters (LPC)
XFORM
LOG
PRINT
CHK
2
6
11
E r r o r return.
N orm al return; list p aram eters.
Check output; w ill print peak shapes.
9 INTG
(DROP)
LOG
PRINT
TYPE
LOG
11
2
6
Was used to print gradient on -lin e for debugging and checking.
W rite check on perform an ce o f fitting routine.
W rite number o f iterations and ch i-squ ared.
W rite list o f channels with large resid u als (DROP).
11
R ead input param eters to make fake spectra.
Dump param s and prepared sp ectra on tape.
P rin t ch i-sq u a red after random ization o f spectra.
P rin t p repared spectra for checking.
4
SHOW
SHOW1
5 RT
6
IN PET
7 INRUN
(SHIFTL)
(LPC)
(CRC)
8
10 SIM
6
6
N
M
INT
IGN
-33611 FINAL
LOG
2
(GPARA) PRINT 6
(ERRORM)
READ
5
E r r o r return.
W rites o ff-lin e answ ers; c r o s s s ection s, etc. (FINAL)
W rites e r r o r m atrix (ERRORM). W rite param s (GPARA).
R ead param eters: target thickness, solid angle etc
(GPARA).
12 A P P
O n-line w rite.
O n-line read new starting values.
W rites blanks to clea r buffer.
J
L
IG
2
2
11
-337List of Programs
Table IV-A3 shows how each phase is composed in
terms of "modules"; in other words a list of subroutines
♦
is given for each phase.
Table IV-A4 is a list of main
program, functions and subroutines in alphabetical order;
they are listed by function name or subroutine name,
except for the main program which is called "WAIT".
Every­
thing is included, except the subroutine "EIGEN" which
performs the diagonalization of a symmetric matrix; its
description is available in IBM's "Scientific Subroutine
Package".
The display functions are also missing of
course; they are part of the system at NSL.
A brief
description is given below for those who have not had the
benefit of using them.
The display functions are a set
of integer functions which are used to create and manipulate
display buffers in storage while the display is on.
ASSIGN and UNASGN assign and unassign display numbers
respectively; AWAKE and RELEAS create and destroy display
buffers; ACTIVE and ERASE put on the screen or wipe out
all the points associated with a given display number;
finally, PUTPT is used to put points in the buffers, that
is to say define a set of coordinates X Y and Z (intensity)
tfhich will correspond to the points displayed.
The X and Y
coordinates range from 0 to 1023 and Z is between 1 and 7.
The light pen is controlled by another function called
TRACK.
All these functions return an integer, which is
usually the display number when the operation was
ROOT PHASE:
MAIN
BLOCK
WAIT,
PLOT,
LFIT,
IOUNIT
INPET
CHECK
SIM
RT
INTG
INRUN
SHOW
SHOWl
TPHDL
GPARA
FINAL
APP
SHIFTL
DROP
LPC
ERRORM
CRC
DATA
PHID, READ, CPEAK, DESSIN
DELAY, TEXT, RSHIFT, LSHIFT,
XFORM, ElGEN, EPEAK,
IOUNIT
IN PET
CHECK
SIM
RT
INTG
INRUN
SHOW
SHOWl
TPHDL
GPARA
FINAL , CROSSC
APP
SHIFTL
DROP , LFITP
LPC , SEEK,
ERRORM
CRC
TABLE IV-A3
-339-
BLOCK DATA
CHECK
CPEAK
CRC
CROSSC
DELAY
DESSIN
DROP
ElGEN (IBM'S SSP)
EPEAK
ERRORM
FINAL
GPARA
INPET
INRUN
INTG
IOUNIT
LFIT
LFITP
LPC
LSHIFT
PHID
PLOT
READ
RSHIFT
RT
SEEK
SHIFTL
SHOW
SHOW1
SIM
TEXT*
TPHDL
TRACK
WAIT
XFORM
TABLE IV-A-4
Subroutine "BITSCN" called by TEXT has been obtained
from J. Birnbaum in the form of a compiled module.
The subroutine text should be replaced by the new
system's routine which makes use of the character
generator hardware.
-340-
successful, and a negative integer when it could not be
completed.
/
-3 4 1 -
c
0001
C
C
C
TO MODIFY THE STARTING VALUES OF THE PEAK POSITIONS AND EXPONENTIAL
BACKGROUND NON-LINEAR PARAMETER INDIVIDUALLY USING THE TYPEWRITER.
SUBROUTINE APP( ARG)
C
C
C
C
C
C
C
C
C
C
THE EXP BGN NON-LIN PARAM COMES FI RST (SCALED SO THAT ITS I NI T I A L
VALUE AS GIVEN IN INRUN CORRESPONDS TO
0 . 5 ) THEN THE POSI TI ON UF
EACH PEAK.
ENTER ONLY THOSE WHICH YOU WISH CHANGED, THE OTHERS WILL
REMAIN THE SAME.
THE SUBROUTINE IS USED BEFORE CALLING THE FI TTING ROUTINE TO MAKE
SURE THE STARTING VALUES ARE GOOD WHEN. A VERY SMALL OR HAROLY
DISTINGUISHABLE PEAK MAKES CONVERGENCE VERY LOCAL AND DEPENDENT ON
THE STARTING VALUES.
0002
COMMON/KE/K
COMMON/ START/ PPI1 0 )
COMMON/LEVEL/LEVEL
COMMON/I0C/I0C(44),J,L»IG
REAL P B I 1 0 )
INTEGER*2 ARG
0003
0004
0005
0006
0007
C
0008
0009
I F I L E V E L . L T . 7 0 ) RE7URN
KS=K+1
W R I T E t J , 1 I P P ( K S ) , ( PP11 I , I = 1, K>
1 F0RMAT(F10.5,9F10.3)
WRITE!IG,2)
2 FORMAT(120XI
READ(L » 3 ) P B( K S ) , ( P B ( I ) , I = 1 , K I
3 FORMAT( I OFI O. O)
0 010
0011
0012
0013
0014
0015
C
0016
0017
0018
0019
o oo oo o
4
0001
0 0 4 1 = 1 , KS
I F ( P B ( 1 ) . GT. 0 . )
RETURN
END
P P ( 11 = P B ( 1 1
ALL COMMON BLOCKS MUST BE MENTIONED IN THE BLOCK DATA SUBPROGRAM
OTHERWISE THE SYSTEM' S PROGRAM ' SETUP' WHICH SUPERVISES DISPLAY
AND KEYBOARD OPERATIONS WILL BE RELOCATED.
BLOCK DATA
C
0002
0003
0004
0005
0006
0007
0008
0009
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
002 3
0024
002 5
0026
0027
0028
0029
0030
0031
0032
0033
0034
0035
0036
0037
0038
0039
COMMON/ LP/ LPI 5 1 2 )
COMMON/LAB/LAB( 5 1 3 )
C0MM0N/ B/ L8. RB
COMMON/ PS/ RES, RTB, RFET, TAI L
COMMON/ RDATA/ Q, RKE, AT, ZT
COMMON/ KE/ K, EEXCI1 0 )
COMMON/SNAMES/SNAMES, IMAX
R E A L* 8 S NA ME S ( 1 2 > / ' I 0 UN1 T
','REAO
'/
COMMON/LISE/EP,CHARGE
COMMON/ CORR/ANGLE,DT,ICC
COMMON/PE/P E( 2 0 1
©
REAL P E / 2 0 * 0 . /
COMMON/RAT/RAT
COMMON/RELAT/BETA, GAMA, ANGCM
C0MM0N/ BGN/ KX1. KX2
COMMON/ C/ LC, RC
COMMON/ START/ STARTXI1 0 )
COMMON/LEVEL/LEVEL
INTEGER L EVEL / O/
COMMON/ I ON/ I DN1, I DN2, I ON3
INTEGER I D N 1 / 0 / , I D N 2 / 0 / , I D N 3 / 0 /
COMMON/ CLAB/CLABI3 0 1 2 )
COMMON/ INT/ INTI 10)
COMHON/FS/FS
REAL F S / 5 0 0 . /
COMMON/CPLOT/ION
INTEGER I D N / O /
COMMON/ SPECT/SPECT( 5 1 2 )
COMMON/ST A T / S T A T ( 5 0 0 1
COMMON/LDN/LON
INTEGER LDN/ O/
COMMON/ ANS/V( 1 0 )
COMMON/ TIK/ TKLEO, TKIRO, TKPAR
COMMON/ SCEF/OFCRSC( 1 0 )
COMMON/PARA/SOLANG,PRTCHG,TGATNB
COMMON/NC/NC( 1 0 )
COMMON/IOC/ IOC
INTEGER I 0 C ( 4 6 ) / 5 , 2 , 6 t l i t
11,5,-1,-1,
1
2
0040
,4,2,6,2,
,5,4,6,11,
END
2,11,6,-1,
2 , 6 , 5 , —1 ,
6,2 ,-1,-1,
2, 6, 2, 6,
2 , 2 , 1 1 , —1 /
2,2,-1,-1
2,6,11,-1,
11,6,2,6
OOo o o
-3 4 2 -
• SIM* IS TOO LARGE TO FIT IN THE KEYBOARD AREA AND FOR THIS REASON
IT I S LOADED INDIRECTLY (OVER THE ROOT PHASE) THROUGH ' CHE CK ' .
0001
OOO
SUBROUTINE CHECK!ARG)
KEYBOARD SUBROUTINE TO CALL SIH
0002
INTEGER*2ARG
CALL LOAOI' S I M ' )
CALL SIM
RETURN
END
ooooooooo
O
0003
0004
0005
0006
« * » » * » * * BE CAREFUL. THIS SUBROUTINE HAS BEEN FIXED SO THAT
THE PEAK SHAPES IT PRODUCES VARY SMOOTHLY WITH INPUT ARGUMENTS
IF ANY CHANGES ARE MADE IT IS ESSENTIAL THAT THE SMOOTHNESS BE
CAREFULLY CHECKED BEFORE USI NG, OR CONVERGENCE COULD b £ IMPAIRED.
PRINCIPLES OF PEAK SHAPES HAVE BEEN COVEREO IN MY THESIS
C. BRASSARD, NSL, YALE ( 1 9 7 0 ) SECTION ON DETECTORS.
0001
OOOOOOOOOOOOOOOOOO
SUBROUTINE C P E A K I P E , P P , I S T , P E A K , L , I N T I
COMPUTES PEAK SHAPES
LAMBDA
RES
RTB
RFET
INT
P P , PE
1ST,L
PEAK
C
CP
TAIL
WI . NI
W
WM
WP
0002
PARAM IN Y=A*EXPI - ( X - X O I * * 2 / L A M B 0 A I
RESOLUTION FOR CS137
RATIO TOP/BGN TAIL
RATIO FIRST ESCAPE/TOP
INTEGRAL (SUM) UNDER PEAK BEFORE ADDING TAIL
PEAK POSI TION, CH#, ENERGY, KEV
PEAK SHAPE OF LENGHT L , TOP N0M1N AT P E A K I P P - I S T )
PEAK SHAPE RETURNED,1*2
NOMINAL PEAK HEIGHT
RAW PEAK HEIGHT AT JUNCTION OF EXPON AND GAUSSIAN
ENERGY AT WHICH JUNCTION OCCURS AT 1 / E OF TOP, KEV
JUNCTION POSI TI ON, R*4 AND 1* 4
R * 4 DISTANCE BETWEEN PEAK AND JUNCTION
R * 4 SLOPE OF LOGARITHM OF EXPONENTIAL
R * 4 PEAK POSI TI ON, RELOCATEO BY 1ST
COMMON/PS/RE S, RTB ? RFET, TAI L
INTEGER*2 PEAKI 2)
REAL LAMBOA
OOO
0003
0004
COMPUTE FULL ENERGY PEAK
LAMBDA=RES/ 2. *PP
LAMBDA=LAMBDA*954/PE*LAMBDA
OO
0005
0006
WP=P P—I ST
C=20000.
W=SORT(LAMBOAI/ PE*TAIL
CP=C*EXP(-TAIL/PE*TAIL/PE)
WI = P P-W—I ST
WM=2*W/LAMBDA
NI=WI
0007
0008
0009
0010
0011
0012
O
0013
IF(N I.L E.0)NI=0
I F I N I . E Q . O ) GO TO 11
0014
0015
0 0 7 1 = 1 , N1
7 P E AK( I ) = CP* EXP( WM* ( I - W l ) 1 + 0 . 5
11 N I = N I +1
0016
0017
0018
0019
DO B I=N1,L
0020
0021
TEM=( I - WP)
TEM=-TEM/LAMBDA*TEM
PEAK!I>=C*EXP(TEM)+0. 5
8 CONTINUE
0022
oon
0023
0024
0025
0026
0027
0028
0029
0030
0031
AOD FIRST ESCAPE PEAK CONTRIBUTION
1 FI=511.*PP/PE
IF = FI
CF=FI —1F
RF=1.-CF
J=L-IF-1
DO 2 I = 1 . J
IFIF=IF+I
2
PEAKI I> = PE AK( 1 I + R F E T * ( R F * P E A K ( I F 1F) + CF*PEAK( IF I F + l I 1 + 0 . 5
-3 4 3 C
C
0032
0033
0034
SUM UNDER PEAK BEFORE ADDING TAI L
INT=0
00 5 1= 1, L
5 INT=INT+PEAK<I )
C
C
C
0035
0036
0037
0038
0039
0040
ADO LOW ENERGY TAI L
TEM=2 0 0 0 0 . / RTB
J=WP
CF=WP-J ■
RF=1.-CF
TEN=RF*PEAK{ J) +CF *PEAK<J + l )
I F ( T E N . L T . 1 0 0 0 0 ) GO TO 3
C
0041
0042
0043
0044
0045
0046
0047
DO 4 1 = 1 , J
X K = 1 . - P E A K ( I I/ TEN
XK=XK*XK
PEAK( I ) =PEAK( I ) + T E M* X K + 0 . 5
4 I F ( PEAK( I ) . L T . O ) GO TO 3
I F ( P E A K ( J + l ) . L E . O ) GO TO 3
RETURN
C
0048
0049
0050
3 INT=-1
RETURN
END
C
C
C
0001
COSMIC RAY BACKGROUND SUBTRACTION.
SUBROUTINE CRC( ARG)
C
C
C
C
0002
THE STANDARD COSMIC RAY SPECTRUM FOR YOUR CRYSTAL MUST BE STORED
IN COSMS IN BINS OF 2 MEV.
COMMON/START/PP
COMMON/PE/PE
COMMON/ST A T / $ T A T ( 5 0 0 ) . S T A T C ( 5 0 0 )
COMMON/ CLAB/CLABI1 0 2 4 ) , S T AT ( 1 0 2 4 ) , STATCI 1 0 2 4 I . T A B L E ( 5 0 0 )
COMMON/LEVEL/LEVEL
COMMON/ I0C/ I0C(26)»N, M
COMMON/ C/ LC.RC
C0MM0N/ B/ LB. R8
INTEGER»2C0SMS ( 3 1 ) / 3 8 0 , 3 9 1 , 4 0 1 , 4 1 1 , 4 2 2 , 4 3 2 , 4 4 3 , 4 5 3 , 4 6 4 , 4 7 4 , 4 8 4 ,
1 495,505,516,526,536,547,557,568,578,588,599,609,620,630,641,651,
2 661,672,682,693/
INTEGER RB.RC
INTEGER*2 ST AT , STATC, A R G, S T AT , STATC, CLAB
0003
0004
0005
0006
0007
0008
0009
0010
0011
0012
C
I F I L E V E L . L T . 5 9 ) RETURN
I F. ( \ . EVEL. LT. 60 ) LEVEL = 60
CALL P HI DCCRC
•)
0013
0014
0015
C
C
C
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
COSMIC RAY SUBTRACTION
35 ECAL=PE/PP/1 0 0 0.
LIMT = 6 0 • /ECAL
L 0HL=1./ECAL+1.
IF(UIMT.GT.1024)LIMT=1024
I F I R C . L E . L I M T ) G 0 TO 41
WR1 TEI N. 4 2)
4 2 FORMAT!' R C . G T . 6 0 MEV' )
LEVEL=40
RETURN
41 ELC=LC*ECAL
ERC=RC*ECAL
C
0027
0028
0029
0030
0031
0032
0033
0034
0035
0036
0037
0038
0039
0040
0041
0042
0043
0044
Jl=ELC/2.+l.
J2=ERC/2.+l.
T E M = ( 2 * J 1 - E L C ) * C O S M S ( J 1 > / 2 . + < E R C - 2 * ( J 2 - 1 ) ) *COSMS<J 2 ) / 2 .
I F ( J 1 . E Q . J 2 ) T E M = T E M - C 0 S M S ( J 1)
4 3 J1=J1+1
1 F I J 2 . L E . J 1 ) GO TO 44
TEM=TEM+C0SMS(J1)
GO TO 43
4 4 NRCS=TEM+0. 5
NRCL=0
0 0 4 5 I =LC, RC
4 5 NRCL=NRCL+STAT( I )
NRCL=NRCL-S TAT ( R C )
C
«
FRAC = 0 .
SMUL=ECAL/ 2. *NRCL/ NRCS
0 0 4 6 I = L0WL t LIMT
E=I*ECAL
JE =lE +l.l/2.
-3 4 4 0045
0046
0047
0048
0049
0050
0051
0052
0053
0054
EJ=(E+1.I/2.-JE
TEM=SMUL*( EJ*CUSMS( JE+1) + ( 1 . - E J ) *COSMS ( JE ) I+FRAC
J =TEM+0. 5
FRAC=TEM-J
CLAB( I ) = ST A T ( I ) - J
46 S T A T C I I ) = S T A T C ( I > + J
TEM=SQRT(l./NRC L+l./lNRCS+100))
AVER=1.*NRCL/(RC-LCI
WRI TE( M, 47) TEM, AVER
47 FORMAK T 4 5 , ■REL ERR ON BGN SUBT . = • , F 5 . 3 . • AVERAGE SUBTRACTIONS
1
t F6 •2 t 1 PER C H . ' )
0055
0056
0057
0058
0059
DO 38 I = L B . R8
J=I-LB+1
$TATC(J)=STATC(I)
$TA T ( J ) = C L A B ( D + STATCI I )
3 8 I F U T A T I J ) . L E . O ) $ T A T ( J I =1
0060
0061
o o o o o o o
RETURN
END
0001
*#<.»*»****
WARNING
**=*«**#=**
INCLUDES A CORRECTION FOR TARGET CHAMBER ANISOTROPV IN TERMS OF AN
ADDITIONAL THICKNESS OF IRON AND LEAD. REMOVE BEFORE USING WITH
DATA TAKEN WITH A DIFFERENT TARGET CHAMBER.
o o o o o o o o o o o o o o o o o o o n o o o o o n o o o
SUBROUTINE CROSSC
0002
0003
0004
0005
0006
0007
0008
0009
0010
0011
0012
o
0013
o o
0014
SUBROUTINE TO CALCULATE THE DIFFERENTIAL CROSS SECTION STARTING
WITH THE ROW NUMBER OF COUNTS CORRECTED FOR OEAO TIME ANO
ELECTRONIC PILE UP
AMULED IS THE ABSORBTION COEFFICIENT IN LEAD
AMUIRO IS THE ABSORBTION COEFFICIENT IN IRON
AMUCAR IS THE ABSORBTION COEFFICIENT IN CARBON
AMUHYD IS THE ABSORBTION COEFFICIENT IN HYDROGEN
AMUPAR IS THE ABSORBTION COEFFICIENT IN PARAFFIN
PARAFFIN ABSORBTION COEFFICIENT IS CALCULATED FROM CARBON ANO
HYDROGEN COEFFICIENT ASSUMING THE PARAFFIN USED IS C10H22
PRTCHG IS THE CHARGE OF INCIDENT PARTICULE GIVEN INMUCOULOMB
PTCN8 R IS THE NUMBER OF INCIDENT PARTICULE FOR A GIVEN CHARGE
CHARGE
IS BEAM CHARGE ACCUMULATED, MICROCOULOMBS
ANGL
IS LAB ANGLE OF DETECTOR
K
IS NUMBER OF PEAKS IN THE FIT
NC
IS NUMBER OF COUNTS IN EACH PEAK, CORRECTED FOR DEAD TIME
ANO PI LEUP.
SOLANG IS SOLID ANGLE OF COUNTER
TGATNB
I S NUMBER OF TARGET ATOMS PER SQUARE CM OF TARGET
TKLED, TKI RO, TKPAR
THICKNESS OF LEAD, IRON (STEEL) ANO PARAFFIN
ABSORBERS
G/CM2
EGR
ARE ENERGIES OF GAMMA RAY LINES
DFCRSC
DIFFERENTIAL CROSS-SECTION COMPUTED BY CROSSC.MICROBARNS.
ANY CORRECTIONS OUE TO TARGET CHABBER ANISOTROPY ARE ADDED
TO THE TKLED, T K I R ^ A N O TKPAR RE SPEC TI BELY
COMMON/ I OC/ I OC( 4 0 I , LOG, T YPER
COMMON / L I S E / E P, C HAR GE
COMMON /CORR/ ANGL , D T , I C C
COMMON / K E / K
COMMON / N C / N C ( 1 0 1
COMMON / PARA/ SOLANG, PRTCHG, TGATNB
COMMON / T I K/ T KL E D , T KI R O, T KP A R
COMMON / S C E F / U F C R S C I 1 0 )
COMMON/ PE/PE( 1 0 )
REAL*4 E GR ( 1 0 ) , TOTYLO( 1 0 ) , TRYLO( 1 0 )
1NTEGER*2 EP. CHARGE, ANGL . O T . I C C
I NT EGER S K, TYPER, NC
ESTIMATION OF THE NUMBER OF INCIDENT PARTICLE
PTCNBR=CHARGE/PRTCHG
ENERGIES OF THE VARIOUS GAMMA RAY LINES
1 1= 1 , K
1 EGR( I ) = P E ( I ) / 1 0 0 0 .
IN THE SPECTRUM
00
0015
0016
C
0017
0018
0019
0020
0021
0022
0023
0024
002 5
0026
002 1= 1 , K
I F I E G R l I ) . L T . 1 0 . I GO TO 100
I F I E G R l I I . G T . 5 0 . ) GO TO 100
I F I E G R l I ) . G T . 1 5 . ) GO TO 102
AMULED= < 0 . 0 0 1 3 4 * E G R ( I ) ) + 0 . 0 3 5 3 0
AMUIRO=(0 . 0 0 0 2 0 = EGR( I ) 1 + 0 . 0 2 7 6 0
AMUCAR= ( - 0 . 0 0 0 5 2 * E G R ( I ) 1 + 0 . 0 2 4 7 0
AMUHYD=(-0.00142*EGR(I)1 + 0 . 0 4 6 6 0
AMUPAR=( ( 1 2 0 . / 1 4 2 . ) * AMUCAR) + ( ( 2 2 . / 1 4 2 . )*AMUHYO)
GO TO 110
0027
0028
0029
0030
0031
0032
0033
0034
0035
0036
0037
0038
0039
0040
0041
0042
0043
0044
0045
0046
0047
0048
0049
0050
0051
0052
0053
0054
0055
0056
0057
0058
102
I FI EGR! I I . G T . 2 0 . 1 G O T O 103
AHULE0 = 1 0 . 0 0 1 1 6 * E G R ( I ) 1 + 0 . 0 3 8 0 0
AMUI R 0 = ( 0 . 0 0 0 2 V > E GR < I I 1+ 0 . 0 2 7 0 0
A MU CA R =( - 0. 00 0 52 *E GR ( I) 1 + 0 . 0 2 0 8 0
A MU HY D = ! - 0 . 0 0 0 7 8 * E G R I I 1 1 + 0 . 0 3 7 0 0
AMUPAR=(( 1 2 0 . / 1 4 2 . l * A M U C A R ) + ( ( 2 2 . / 1 4 2 . 1 *AMUHYD1
GO TO 110
103 I F ( E G R ( I 1 . G T . 3 0 . 1 GO TO 104
A MUL E D = ! 0 . 0 0 0 8 6 * E G R 1 1 1 1 + 0 . 0 4 4 0 0
A MUIR0=( 0.00024* EGR(I 1 1 + 0 . 0 2 7 0 0
AMUCAR= < - 0 . 0 0 0 10=>EGR( I 1 1 + 0 . 0 1 7 6 0
AMUHYD=(-0.00040*EGR(I)1+0.02940
AMUPAR=( ( 1 2 0 . / 1 4 2 . 1 * A M U C A R ) + ( ( 2 2 . / 1 4 2 . ) * AMUHYD 1
GO TO 110
104 I F ( E G R ( I 1 . G T . 4 0 . 1 GO TO 109
A MU L E D= ( 0 . 0 0 0 6 3 * E GR ( I 1 1 + 0 . 0 5 0 9 0
AMUIRO=(0.00021*EGR(I)1+0.02700
A M UC A R = ( - 0 . 0 0 0 0 4 * E G R ! I 1 1+ 0 . 0 1 5 8 0
AMUHYD=( - 0 . 0 0 0 2 1*EGR<I 1 1 + 0 . 0 2 3 7 0
A M U P A R = ! ( 1 2 0 . / 1 4 2 . 1 * A M U C A R ) + ( ( 2 2 . / 1 4 2 . 1 *AMUHYD1
GO TO 110
1 0 9 AMULED=( 0 . 0 0 0 4 6 * E G R ( I 1 1 + 0 . 0 5 7 7 0
AMU1R0=(0.00017+EGR!I)1+0.02950
AMUCAR=Q. 0 1 4 2
AMUH YD = ( - 0 . 0 0 0 2 1=*EGR ( I 1 1 + 0 . 0 2 3 7 0
AMUPAR=(( 1 2 0 . / 1 4 2 . ) * A M U C A R ) + ( < 2 2 . / 1 4 2 . 1 *AMUHYD1
110 TOTABS=-AMULED*TKLFD-AMUIRO*TKIRO-AMUPAR$TKPAR
I F ( A N G L . E 0 . 3 0 0 ) TOTABS=T0TABS- 1. I *AMULEO- 2. *AMUIRO
I F ( A N G L . E Q . 4 5 0 . 0 R . A N G L . E Q . 1 3 5 0 ) T0TABS = T 0 T ABS - 0 . 4 * AMUI R0
I F I A N G L . E Q . 1 4 5 0 ) TOTABS=TOTABS- 0. 9*AMULED- 2. 2*AMUI R0
TOTYLD( I 1 =NC ( I 1 / E X P1 T0 TA B S )
T R Y L O t I 1 = T 0 T Y L 0 ( 1 1/SOLANG
0059
0060
00 6 1
0062
0063
0064
0065
0066
0067
0068
0069
O F C R S C t I ) = T R Y L 0 ( I 1 / ( TGATNB*PTCNBR1
I F ( l . E O . l ) WRI TE( TYPER, 11 11T0TABS
11 1 FORMAT! ' + • , T 2 0 , ' T O T A B S = ' , F 7 . 4 )
GO TO 2
1 0 0 WRI TE( TYPER, 10 11
WRITE( L O G , 1011
OF C R S C ! 1 1 = 0 .
101 FORMAT!• * ENERGY NOT IN 1 0 - 5 0 MEV INTERVAL,
2 CONTINUE
RETURN
END
CROSSC F A I L S '
C
0001
SUBROUTINE OELAY(N)
TO PROVIDE DELAY FOR LIGHT PEN
C
C
0002
0003
0004
0005
0006
0007
0008
IF(N)1,2,3
1 K=-N
DO 4 1 = 1 , K
DQ 4 J = 1 , 5 0 0 0
4 L =l./2
CALL A C TI V E I 01
RETURN
C
0009
0010
0011
0012
0013
3 00 5 1 = 1 , N
DO 5 J = 1 » 5 0 0 0
5 L =l./2
2 RETURN
END
C
0001
SUBROUTINE DESS I N( I ON , N , * , * 1
C
C
C
C
DIRTY WORK OF DI SPLAYI NG.
RETURN 1 IS NORMAL, RETURN 2
0002
INTEGER ASSI GN, ERASE, RELEAS, AWAKE, ACT1VE.DCLEAR
C
0003
0004
0005
IF(IDN)7,11,12
11 I F ( N ) 7 , 7 , 1
12 1 F I N 1 3 , 2 , 4
C
0006
0007
1
IDN=ASSIGN(0,0,0)
1F ( ION 1 6 , 6 , 5
C
0008
0009
2 IER=ERASE( I ON)
I F ( I ER- I ON 1 6 , 7 , 6
C
0010
0011
IS ERROR.
,
3 IER = ACT I V E ( I ON 1
IFIIER-I0N)6,7,6
-3 4 0 C
0012
4
0013
0014
0015
I ER=ERASE( ION)
IFHER.NE.ION.ANO.IER.NE.-3)
IER=DCLEAR( ION)
IF t 1 ER - I ON1 6 , 1 0 , 6
GO TO 6
C
0016
0017
0018
0019
5
10
IER=AWAKE( I DN, N)
IF(IER-IDN)6,10,6
I ER=ACTI VEl I ON)
1F ( I E R - I D N > 6 , 7 , 6
C
0020
0021
0022
6 RETURN 2
7 RETURN 1
END
C
C
C
C
C
C
0001
SUBROUTINE DESIGNED TO FIND CHANNELS WHICH ARE VERY FAR FROM THE
F I T , TO LOCATE POSSIBLE HARDWARE ERRORS. THE CHANNELS FOR WHICH THE
SQUARE OF THE WEIGHTED RESIOUALS IS LARGER THAN A CERTAIN LI MI T
FIXED BY ' ARG1 ARE PRINTED.
SUBROUTINE OROP(ARG)
C
0002
COMMON / L A B / LA B I 1 0 2 4 )
COMMON/CLAB/CLABI 1024 I , T ( 5 0 0 0 )
COMMON/ B/ LB,RB
COMMON/ IOC/IOC I 35 I , LOG
COMMON/KE/K
COMMON/ STAT/ STAT(5O0)
EQUI VALENCE( W( 1) , JK)
I NT EGERS L A B , C L A B , A R G * T , W( 2 ) , J V * 4 I 8 I , R B * 4 » S T A T
REAL R F I T I 5 0 0 )
0003
0004
0005
0006
0007
0008
0009
0010
C
0011
0012
LT=50 0
KS=K+1
L-R6-LB+1
1 1=41
W( 2) =ARG
JK=JK/256
0013
0014
0015
0016
C
0017
CALL P HI OI 1 DROP
')
C
CALL LFI TPt C L A B ( L B ) , STAT I 1 1 , KS , L T , T ( 1 1 I , L , R F I T ( 1 > I
0018
C
WRITE ( L O G , 5 )JK
5 FORMAT!• + • , 1 1 0 X , • LEVEL= • , 1 3 )
0019
0020
C
0021
0022
J=0
00 1 1= 1,5 10
I F I A B S I R F I T I I I I . L T . J K ) GO TO 2
J =J+1
R F 1 T ( J ) = RF1TI I )
JV(J ) = I+LB-1
2 IF ( J . L T . 8 . A N D . I . L T . L I G 0 TO 1
I F ( J . E Q . O ) RETURN
WRITEI LOG, 4 I ( J V ( I I ) , R F I T ( 11 I , I I = 1 , J I
4 F0RMATI8II7.F8.2))
I F ( I . G E . L ) RETURN
9
J =0
1 CONTINUE
END
o o o
0023
0024
002 5
0026
0027
0028
0029
0030
0031
0032
0033
0034
0001
SUBROUTINE TO COMPUTE EXPONENTIAL BACKGROUND.
SUBROUTINE
EPEAK( TT, PEAK I
C
0002
INTEGER*2 P EAK( 1 ) , L * 4 / 4 6 0 /
C
0003
0004
0005
0006
0007
0008
T EM=20000.
DO 1 1=1,L
PEAK! I ) = T E M + 0 . 5
1 TEM=TEM*TT
RETURN
END
-3 4 7 -
0001
S U B R O U T I N E E R R OR MI A RG )
S U B R O U T I N E TO C A LC U L A T E E RR OR M A T R I X AT M I N IM UM
0002
0003
0004
0005
0006
0007
0008
0009
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
002 3
0024
002 5
0026
002 7
COMMON/LAB/LAB(512)?XL
COMMON/SPECT/M,EM
C O M M O N / S T A R T / P P (10)
C O M M O N / C L A B / C L A B I 1 0 2 4 ) , T (5000)
COMMON/KE/K
C0MM0N/STAT/STAT(500)
C O M M O N / 8 / L B,RB
C O M M O N / A N S / V I 10)
COMMON/LEVEL/LEVEL
C O M M O N / I O C / I O C ( 41),PR INT
I N T EG E R* 2 C L A B ,T ,S T A T ,ARG
I NT EGER RC,PRI(.T,X,RB
REAL M ( 2 1 0 ) ,E M ( 2 1 0 ) , R ( 4 0 0 ) , F ( 1 0 ) , F P ( 1 0 ) , A A ( 10,10),
1
A B ( 1 0 , 1 0 ) , B B ( 1 0 , 1 0 ) , S 0 2 / 1 .4142136/
E Q U I V A L E N C E < R ( 1 ) , A A( 1 , 1 ) ) , ( R ( 1 0 1 ) , A B (1,1 ) ),(R ( 2 0 1 ) , B B (1 ,1))
IF ( LE VE L . L T . 8 0 > RE TURN
L EV E L = 9 0
HR I T E (P R I N T ,2)
C AL L P H I 0 ( 'ERRORM
')
H R I T E (P R 1 N T ,1)
! FORMAT!'21)
1 FORMAT!' + • , T 2 0 , ' E R R O R MATRIX
1 , ' RELATI VE ERRORS ON HEIGHTS,
ERRORS ON POSITIONS
CORRELATI ONS' / )
KS=K+1
KSP=2*KS
L T= 5 0 0
I REL=40
L=RB- LB+1
ZEROING
0028
0029
DO 3 1 = 1 , 4 0 0
3 R (I)=0.
CALCULATION OF F,
0030
0031
0032
0033
0034
0035
0036
0037
0038
FP,
FPP.
00 4 X=1,L
H=STAT( X )
I AD=IREL+X
00 5 1 = 1 , K
FI 1 ) =T ( I AD)
F P ( I ) = ( TI I A D - 1 ) - T ( I A O + 1 ) 1 / 2 .
5 1AD=IAD+LT
F ( KS) =T 11 AO )
F P ( K S ) = { X —l l / X L 4 F ( K S ) / P P ( K S )
COMPUTE MATRIX AB
0039
0040
0041
0042
DO 6 MU=1 , KS
DO 6 NU=1, KS
TEM=FIMU) / H*FP( NU>
6 AB(NU, MU)=ABINU, MU)+TEM
COMPUTE MATRICES AA AND BB
0043
0044
0045
0046
0047
DO 7 NU=1, KS
DO 7 MU=NU,KS
AA( NU, MU) = AA( NU, MU) + F( MU) / H* F ( NU)
7 BB(NU,MU)=BB(NU,MU)+FP(MU)/W*FP(NU)
4 CONTINUE
COPY MATRICES AB, AA,ANO BB INTO M
0048
0049
0050
0051
0052
0053
DO 8 J R = 1 , KS
DO 9 J C = 1 , K S
9 MlJR+IJC+KS)*!JC+KS-1)/2)=AB(JR.JC)
00 8 J C = J R , KS
M l J R + J C M J C - 1 ) / 2 )=BB( J R . J C )
8 MI JR+KS+<JC+KS>M J C + K S - 1 ) / 2 ) = AA( J R , J O
DIAGONALIZE MATRIX M
0054
CALL E I G E N ( M , R , K S P , 0 )
FIND EM,
0055
0056
0057
0058
0059
0060
0061
INVERSE OF M
0 0 1 0 NU=1, KSP
0 0 10 MU=NU,KSP
I=NU+( MU- l)«MU/ 2
EMI I ) = 0 .
DO 10 J = 1 , KSP
JJ=IJ-1)*KSP
10 EMI I ) = E MI I ) + R( NU + J J ) * R ( MU + J J ) / M I J * ( J + l ) / 2 )
IN CHANNELS,
'
o o o
-3 4 8 TAKE THE SQUARE ROOT OF DIAGONAL ELEMENTS OF EM
0 0 11 I = 1 , KS P
J=I*(I+1I/2
I F(EM( J ) 1 1 2 , 1 2 , 1 3
12 E M I J ) = - S Q R T ( - E M I J I I
GO TO 11
13 EMIJI = S OR T ( E MI J ) I
11 CONTINUE
o o o
0062
0063
0064
0065
0066
0067
0068
0069
0070
00 71
0072
0073
0074
FIND CORRELATIONS
o o o
KSP1 = KSP-1
00 15 NU= 1 t KS Pl
J=NU+1
DO 15 MU= J , K S P
I=NU+MU*( MU—1 1 / 2
15 EMI 1 ) = E M ( I ) / E M ( N U * ( N U + l ) / 2 >/ E M ( MU*( M U + 1 I / 2 I
0075
0076
0077
0078
0079
SCALE DIAGONAL ELEMENTS OF EM
ooo
0 0 1 6 I = 1 ,KS
J=I*(I+1I/2
EMU ) = E M ( J ) / V ( 1 I / S 0 2
J = ( I + K S ) * ( I + K S + 1 1/2
16 E M ( J ) = E M ( J ) / V ( I I / S 0 2
0080
00 8 1
0082
0083
0084
WRITE ANSWERS
o o o
DO 1 7 NU=1, KSP
17 WRI TE( PRI NT, 1 8 ) ( EM(NU+MU*( MU-1 ) / 2 ) , MU=NU, KSP)
18 FORMAT!10 ■ , 1 2 F 1 0 . 4 / ( T 1 5 , 1 0 F 1 0 . 4 ) I
RETURN
END
0001
FINAL CORRECTIONS,
OOOOOOOOOOOOOOOOOOO
SUBROUTINE
0002
CALL TO CROSSC TO COMPUTE CROSS-SECTIONS
FI NAL( ARG)
PRINTS THE ANSWERS.
AC08
IS THE JACOBIAN OF t he LAB TO CM TRANSFORMATION ( R E L A T I V I S T I C )
BETA
I S THE RELATIVE VELOCITY OF THESE FRAMES/ Vt L OF LIGHT
ANGCM
IS THE CENTER OF MASS ANGLE OF THE DETECTOR.
CRNBR
IS THE FRACTION OF THE LIGHT PULSER COUNTS WHICH ACCUMULATE
IN LOW CHANNELS DUE TO DEAD TIME OF THE ADC.
USED FOR
DEAD TIME CORRECTION.
ANSWERS PRINTED FOR EACH PEAK INCLUDE
( IN THAT ORDER) EXCITATION
ENERGY OF FINAL STATE, CENTER OF MASS PEAK ENERGY, LAB PEAK ENERGY,
FITTED PEAK POSI TI ON, CHECK ON LINEARITY (SHOULD ALL BE ONE)
LINEAR PARAMETER Or F I T * 1 0 0 0 0 0 0 , NUMBER OF COUNTS UNDER FITTED PEAK,
DIFFERENTIAL CROSS/ SECr I ON IN MICR08ARNS/ STERAOI AN.
THE SUBROUTINE CROSSC I S CALLEO TO COMPUTE THE CROSS- SECTI ON.
IT
PRINTS THE NATURAL LOGARITHM OF THE ATTENUATION DUE TO ABSORBERS
IN FRONT OF THE CRYSTAL (TOTABS)
COMMON/ LP/ LPI 1 0 2 4 )
COMMON/ CORR/ANGLE,DT,ICC
COMMON/ NC/ NCI1 0 )
COMMON/ INT/I NT( 1 0 I
COMMON/ ANS/ VI1 0 )
COMMON/ SCEF/ DFCRSC(10)
COMMON/ KE/ K, EEXC( 10)
COMMON/LEVEL/LEVEL
COMHON/ PE/ PEI1 0 ) , PECM( 1 0 )
COMMON/RELAT/BETA,ACOB,ANGCM
COMMON/ START/ PPI 10)
COMMON/ I OC/ I OC( 4 0 ) , LOG,PRINT
REAL CHECK! 1 0 1 , 8 ( 1 0 )
INTEGER*2 ARG, L P . ANGLE, 0T, ICC
INTEGER PRINT
LOGICAL FLAG
0003
0004
0005
0006
000 7
0008
0009
0010
0011
0012
O
0013
0014
0015
0016
0017
FLAG=. TRUE.
I F I L E V E L . L T . 7 0 ) RETURN
LEVEL=80
CALL PH10 ( ' FINAL
')
0018
0019
o OO
0020
0021
0022
0023
COMPUTE CORRECTION DUE TO
ISUM=0
00 1 1 = 1, 1 02 4
OEAO TIME
-3 4 9 0024
0025
0026
0027
1 ISUM = ISUM+LP<I I
IF C A 8 S ( 1 . * I C C / I S U M - 1 . ) - . 0 0 2 ) 2 , 2 , 3
3 FLAG=. FALSE.
2 ISUM=0
00 5 1 = 1 , 1 0
5 I SUM= ISUM+LP( 1 )
CRNBR=1 . * I SUM/ I CC
002 8
ooo
0029
0030
0031
0032
FIND NUMBER OF COUNTS IN EACH PEAK
o
DO 8 1 = 1 , K
8 NCI I ) = V < I > * I N T ( I ) * A C O B / ( 1 . - C R N B R 1 + 0 . 5
0033
0034
0035
PRINT ANSWERS
q
ooo
CALL CROSSC
DO 9 1 = 1 , K
9 OF C R S C I I ) = D F C R S C ( 1 1 * 1 . E + 3 0
0036
0037
0038
TEM=PEI 1 1 / P P 1 1 )
00 6 1=1,K
o
6 CHECK! I ) = P P ( I I / P E I I ) * T E M
0039
0040
W R I T E ! P R I N T , 1 5 ) ACOB,BETA,ANGCM,CRNBR
1 5 FORMAT!' 4 ' , T 6 0 , •J A C O B I A N = * , F 6 . 3 , • BETA=1 , F 6 . 4 , '
ANGCM=1 , F 5 •1,
1
• CRNBR=1, F 6 . 4 )
WRI TE( PRI NT, 1 3 )
13 FORMAT( '0 EEXC' , 5 X , ' PECM' , 8 X , * P E 9 X , ' P P ' , 5 X C H E C K • , 9 X ,
1
• V , 8 X, M M T ' , 6 X, ' C T S ' , 7 X , 'OFCRSC ■ )
00 4 1 = 1 , K
4 $( 11 =1 0 0 0 0 0 0 .*V<I)
W R I T E ! P R I N T , 1 0 ) I E E X C ! 1 ) , PECM( I ) , PE I I I , P P ( I ) . C H E C K ! I ) , t ( I ) , I N T ! I ) ,
1
N C ( I ) , O F C R S C ! I ) , 1 = 1 , K)
10 FORMAT! ' 0 1 , 6 G 1 0 . 5 , 17 , I 9 , G 1 7 . 4 I
I F ( FLAG I RETURN
0041
0042
0043
0044
0045
WRI TE( PR I NT, 1 1 )
WRIT E( L O G , 1 1 )
11 FORMAT! 1 * * * * I C C . N E . S U M ( L P ) ' )
WRITE(L0G,12)
ICC. ISUM
FORMAT! • 1CC = * , Z 8 , •
1 , ' I SUM= • , 2 8 )
RETURN
END
O
0048
0049
0050
0051
0052
0053
0054
m
p
0046
0047
0001
o o o o oo ooo o o o oo o O
SUBROUTINE GPARA(ARG)
0002
o o
0010
0011
0012
oo
0020
THE THICKNESS OF THE LEAD ABSORBER
THE THICKNESS OF THE IRON ABSORBER
THE THICKNESS OF THE PARAFFIN ABSORBER
ALL IN INCHES.
AVONB IS THE AVOGAORO NUMBER
TGTIK IS THE TARGET THICKNESS IN MG/CM SO
AMTN IS THE ATOMIC MASS OF THE TARGET NUCLEUS
PRTCHG IS THE PROTON CHARGE ( PROJECTI LE)
IN MICROCOULOMBS
S I S THE RADIUS OF THE COLLIMATOR IN FRONT THE CRYSTALAND
D I S THE DISTANCE BETWEEN TARGET AND CRYSTAL IN THE SAME
UNITS
IF(LEVEL.LT.IO)
IFILEVEL.LT.20)
RETURN
LEVEL=20
CALL PHIDI 'GPARA
' )
READ!REAOER, 1 I TKLEO, TKIRO, TKPAR
1 FORMAT( 3 F 1 0 . 0 I
READ! REAOER, 1I TGTIK
READ( REAOER, 1 1 0 , S
0013
0014
0015
0016
0017
0018
0019
TKLED I S
TKIRO IS
TKPAR IS
COMMON / TI K/ T KL ED , T KI R O, T KP A R
COMMON / PARA/SOLANG,PRTCHG,TGATNB
COMMON/RDATA/Q,RKE,AMTN
COMMON/LEVEL/LEVEL
COMMON/ I OC/ I OC( 4 1 1.TYPER, REAOER
REAL R A T I O / 2 . 5 4 / , D L E D / 1 1 . 3 4 / , D I R O / 7 . 8 6 / , D P A R / . 9 / ,
INTEGER*4 READER, TYPER, AMTN, ARG*2
0003
0004
0005
0006
0007
0008
0009
PROGRAM
TO CALCULATE THE SOLID ANGLE,THE NUMBER OF ATOMS IN
THE TARGET
THTS SUBROUTINE MUST BE CALLED BEFORE THE SUBROUTINE
CROSSC
ESTIMATION OF THE SOLID ANGLE
P I = 3 . 1 4 159
C= ( 0 * 0 I + ( S * S )
C = SQR T I C )
COS = D/C
S OL A NG= ( 2 * P 1 1* ( l . - C O S )
AVONB/ 6. 0 2 5 E + 2 3 /
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WRI T E( T YP ER, 2 > TGTIK, SOLANG, TKLED, TKIRO. TKPAR
2 FORMAT! ■ + • , 2 0 X , •T GT I K= •, F 8 . 4 , '
SOLANG=• , F 6 . 4 , 1 0 X , 1 T K L E 0 = ' , F 5 . 2
1
•
TKIRO=',F5.2 ,'
TKPAR= 1 , F5 .2 )
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-3 5 1 C
C
0042
0043
0044
0045
0046
0047
0048
0049
0050
0051
0052
PEAK SHAPE PARAMETERS.
16
I F( LEVEL.GT.60)LEVEL=60
WRI T E I T Y P E . 1 2 )
12 FORMAT!' * ENTER RES, RTB, RFET , T AIL ' )
CALL R E A D ! I A , 4 , 6 1 3 , 6 3 )
13 R E S = I A ( 1 ) / 1 0 0 0 .
RTB=IA(2)/1000.
RFET=IA(3)/1000.
TAIL=IA(4)
WRITE(PRI NT, 1 4 ) RES,RTB,RFET,TAIL
14 FORMAT(• R E S = ' , F 6 . 3 , <
R TB=',F6.3,'
RFET=• , F6 . 3 , •
RETURN
TAIL=',F6.0>
C
0053
0054
3 WR I T E ( L OG, 1 5 )
15 FORMAT I ’ 6 ERROR DETECTED IN " R E A D "
1
/*
SUBROUTINE RETURNS' )
RETURN
END
n o o n
0055
0056
KEYBOARD SUBROUTINE TO INPUT INFORMATION FOR F I T OF ONE RUN
USES LIGHT PEN
0001
SUBROUTINE
C
C
C
C
C
C
C
C
C
C
C
C
C
INRUN(ARG)
INFORMATION INVOLVED INCLUDES FI TTI NG BOUNDARIES, STARTING VALUES
FOR THE POSI TION OF THE HIGHEST ENERGY PEAK AND TWO POINTS ON THE
EXPONENTIAL BACKGROUND (ALSO AS A STARTING VALUE) AS WELL AS LIMITS
FOR THE COSMIC RAY BACKGROUND EVALUATION.
CAN BE USED IN TWO MODES ACCORDING TO PARAMETER KEYS
UPDATE AND LIGHT PEN.
ON LIGHT PEN MODE THE VALUES ARE ENTEREO FROM SCRATCH.
UPDATE MODE I S FASTER, THE PROGRAM ASSUMES THESTARTING VALUES ARE
RELATED TO THE RESULTS OF LAST FIT
TIC MARKS ARE OISPLAYEO FOR I DENTIFICATION OF THE CHANNELS WHICH
HAVE BEEN CHOSEN.
0002
COMMON/ B/ LB,RB
COMMON/BGN/KX1 , KX2
COMMON/ C/ LC,RC
COMMON/ ION/ 1DN1, IDN2
COMMON/LEVEL/LEVEL
COMMON/RAT/RAT
COMMON/ START/PP( 1 0 )
COMMON/ LAB/LAS( 1 0 2 4 1 , XL
COMMON/KE/K
COMMON/PE/PEI 1 0 )
COMMON/ I OC/ I OC( 2 4 I , LOG,NN
INTEGER J N / Z 0 0 0 0 4 0 0 0 / . K N / Z 0 0 0 0 2 0 0 0 / , L A B * 2
INTEGER R B . R C . T E S T ,
TRACK, ARG*2, IM( 7 ) , PUTPT, AND
1 , K M ( 2 1 ) / ' E N T E R LB
ENTER RB
ENTER LC
ENTER RC
2RTXENTER KX1
ENTER KX2
•/
LOGICAL UPDATE,CTRACK
0003
0004
0005
0006
0007
0008
0009
0010
0011
0012
0013
0014
0015
C
I F(LEVEL.LT.40I
LEVEL=50
0016
0017
RETURN
®
C
KS=K+1
L=ARG/ 2 5 6
IF(L 1 1 , 1 , 3
3 UPOATE=. FALSE.
CTRACK=. FALSE.
TEST=ARG
I F ( A N D ( J N . T E S T I . N E . 0 IUPDATE= . TRUE.
I F I R A T . L E . 0 . 5 )UPDATE=. FALSE.
I F ( AND( KN, TE S T ) . N E . 0 I CTRACK=. TRUE.
IF(RAT.LE.0.5)CTRACK=.TRUE.
I F ( . NOT. ( UPDATE. OR. CTRACK) ( RETURN
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
C
CALL PHI0 ( • INRUN' I
0029
C
0030
0031
0032
0033
0034
21
1F( UPDATE) CTRACK=. FALSE.
C0EF=1.
I F ( UP DA T E ) C 0 E F = 1 . / R A T
I F ( UP DATE) WRI T E( NN, 2 1 )
FORMAT! •+ ' , T 1 0 0 , ' U P D A T E ' I
C
0035
0036
0037
0038
0039
0040
00 4 1
OR NUCLEUS NOT IN TABLE*
IM(l)=LB*COEF+0.5
1 MI 2 ) = R B * C OE F + 0 . 5
IM ( 3) =LC*C0EF + 0 . 5
I H(4)=RC*COEF+0. 5
IM(5)=PP(l)*C0EF+0.5
I M( 6 ) = K X l * C 0 E F + 0 . 5
1M( 7 ) = KX2 * C 0 E F + 0 . 5
ENTER STA
-3 5 2 0042
0043
0044
0045
0046
0047
0048
0049
0050
0051
0052
0053
0054
0055
0056
0057
0058
0059
0060
4
6
7
8
20
9
CALL D E S S I N I I D N 2 , 1 2 0 , 6 4 , 6 5 )
1=0
1 = 1+1
CALL DESSI NtI ON 1 , 2 0 0 , 6 8 , 6 5 >
CALL T E X T ( I D N 1 , 4 0 0 , 6 0 0 , 2 , 2 , ' H O ' , 1 2 , < M ( 3 = 1 - 2 1 , 6 5 )
I F ( .NOT. CTRACKIGO TO 20
IDN=TRACK( I X , I Y , I Z )
CALL DELAYC- 3)
I F < I D N . L E . O ) GO TO 7
I F( ION.NE.I ONI ) I M (I )= 1X
IY=I* 0 ,4 9
I F ( I . G E .6 )IY=-1
IY=(7+3*IY)»4+1
00 9 IX=1, IY,4
I E R = P U T P T ( I 0 N 2 , I M ( I ) , 8 0 0 - I X , 1)
I F ( I E R . N E . I 0 N 2 ) GO TO 5
IER = PUTPT( I DN2, I M< I ) , I X , 1)
I F C I E R . N E . I D N 2 ) GO TO 5
I F I I . L T . 7 ) GO TC 6
C
00 61
CALL D E S S I N I I O N l , 0 , 6 1 7 , 6 5 )
C
0062
0063
17 DO 18 1 = 1 , 7
18 IFI IM( I ) . L E . O )
GO TO 11
C
0064
0065
0066
0067
10 I F ( I M ( 2 ) - I M < 1 ) - 1 0 ) 1 1 , 1 2 , 1 2
12 I F ( I M ( 4 ) - I M ( 3 ) —10 ) 1 1 , 1 3 , 1 3
13 I F ( I M ( 5 ) . L T . I M ( 1 ) . 0 R . I M ( 5 ) . G T . I M ( 2 ) ) GO TO 11
IF < I M ( 7 ) - I M ( 6 ) - 1 0 ) 1 1 , 1 4 , 1 4
C
0068
0069
0070
00 71
0072
0073
5 WRI T E( L OG, 1 5 )
15 FORMAT! • = ERR IN DISPLAY S R . RETURN' )
RETURN
11 WR1TE( L O G , 1 6 )
16 FORMAT!' * INVALID ENTRY. RETURNED' )
RETURN
C
0074
0075
0076
0077
0078
0079
0080
14 LB=IM(1)
RB = I M( 2 )
LC=IM(3)
RC= I M( 4 )
PP!1)=IM(5J
KX1=I M( 6 )
KX 2 = I M( 7 )
C
00 81
0082
0083
0084
0085
0 0 23 1 = 1 , X
23 P P ( I ) = P E ( I ) / P E ( 1 ) = P P ( 1 )
PP( KS)= 0 .5
X L = ( K X 2 - K X 1 ) / A L OG( LABIKX1 ) * 1 . / L A B ( KX2) ) * 0 . 5
RETURN
C
0086
0087
0088
OOo
■
0001
1 CALL DESS I N ( ION2 , L t 6 2 , 6 5 )
2 RETURN
END
THIS
IS THE FI TTI NG ROUTINE
SUBROUTINE
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
I TSELF.
INTG(ARG)
THE PEAK SHAPES ARE PREPARED BY CALLING
XFORH.
THE ALGORITHM IS AS FOLLOWS.
THE CHISQ
I S CONSIDERED
AS A FUNCTION
OF THE NON-LINEAR PARAMETERS ONLY, THE LINEAR PARAMETERS ALWAYS
BEING ASSUMED OPTIMUM.
THIS FUNCTION OF MANY VARIABLES SHOWS A
MINIMUM WE WISH TO FI ND.
THE FIRST DERIVATIVES ARE COMPUTED AT
VARIOUS POINTS ANO THE SECOND DERIVATIVE IS OBTAINED BY FINITE
DIFFERENCES.
T he m a t r i x of s e c o n d DERIVATIVES I S DIAGONALIZED
AND THE DISPLACEMENT NEEDED IN EACH OF THE EIGcNDIRECTIONS IS
COMPUTED.
IF THE DISPLACEMENT NEEDED IS UNREASONABLY LARGE, I T IS
ARBITRARILY LI MI TED. THIS IS THE SECRET OF THE REMARKABLE STABI LI TY
OF THE METHOD ON SMALL PEAKS.
WHEN THE SUBROUTINE IS CALLED WITH A NEG ARG, THE F I T IS DONE OVER
LINEAR PARAMETERS ONLY, THE VALUE OF CHISQ I S PRINTED AND THE FI T
CAN BE DISPLAYED ON THE SCREEN.
IF THE
OPERATOR IS HAP=Y WITH THIS
STARTING VALUE, HE CAN THEN PROCEED TO FI T FOR NON-LINEAR PARAMS.
THIS I S DONE BY CALLING INTG AGAIN WITH A POSITIVE ARG.
THE FIT IS LIMITED
TO 12 ITERATIONS AND THE NUMBER OF ITERATIONS IS
PRINTED ON-LINE WITH THE CHI SQ.
IF THE NUMBER OF ITERATIONS IS SMALLER THAN 12 THE OPERATOR CAN
SAFELY ASSUME THAT IT Dll) CONVERGE
THE SUBROUTINE IS PREVENTED FROM CONVERGING AS LONG AS THE
CORRECTIONS IN EACH STEP ARE LIMITED
SO THAT EVEN IF THE CHISQ
SETTLES TO A MINIMUM IT DOES NOT MEAN THAT CONVERGENCE OCCUREO.
THE LI ST OF SUCCESSIVE CHISQ ARE PRINTED OFF- LINE W(TH THE PEAK
POSITIONS FOUND.
-3 5 3 -
0002
COMMON/LAB/LABI 1 0 2 4 ) , XL
C0MM0N/ SPECT/ W( 1 0 2 4 )
COMMON/ CLAB/ CLABI1 0 2 4 ) , T ( 5 0 0 0 )
COMMON/ B/ LB»RB
COMMON/KE/K
C 0 MM0 N/ S T AR T / P P ( 10)
COMMON/ ANS/V( 1 0 )
C0MM0N/STAT/STAT(500),STATC(500)
COMMON/LEVEL/LEVEL
COMMON/ I OC/ I OC( 3 2 ) . L OG, PR I NT,TYPE
INTEGER*2 LAB, CLAB, ARG, T , W, S TAT , STATC
INTEGER R B , T Y P E , P R I N T , X , I E R 1 / ' L F I T 1/ , STARTX, RB1
LOGICAL FLAG
REAL GRAD( 1 0 ) , GRAS( 1 0 ) , G M ( 1 0 , 1 0 ) , DEV( 1 0 ) , CHI SOI 1 1 )
REAL A ( 5 5 ) , R ( 1 0 0 ) , TR1 / 0 . 1 / , T R 2 / 3 . 0 /
REAL* 8 L A B E L I 4 ) / '
P P ' , ' DEV• , •
GRAD','
CHISQ'/
EQUIVALENCE ( R ( 1 ) , GM( 1 , 1 ) )
0003
0004
0005
0006
0007
0008
0009
0010
0011
0012
0013
0014
0015
0016
0017
0018
C
0019
I F I L E V E L . L T . 6 0 ) RETURN
I F I ARG. GE. OI CALL P H I DI ' I N T G
0020
•)
C
0021
0022
ITER =1
STEB=-.01
L T= 50 0
KS=K+1
L=R8 - L B + 1
11 =4 1
J2=I1-1+K*LT
LB1=LB—1
RBI = RB+1
0023
0024
002 5
0026
0027
0028
0029
C
0030
0031
0032
33 CALL XFORM(ARGI
I F ( L E V E L . L T . 7 0 ) LEVEL=70
IF(ITER.EQ.l.AND.ARG.GE.O)
GO TO 23
C
0033
32 CALL L F I T ( C L A B I L B ) , S T A T ( 1 ) , W ( L B ) , K S , L T , T ( 1 1 ) , L , A X , V , C H I S O ( I TER) ,
1
GRAD)
GRAD(KS >= - G R A D ( K S l / P P I K S I / X L
IFIAX-.O Ol)2,2,5
2 IF(ARG> 7 6 , 1 , 1
o o o
0034
0035
0036
00 37
0038
0039
0040
0041
0042
0043
CHECK STATUS
o o o
1 IF(( ITER+41/8-1) 4 , 4 1 , 2 3
4 1 I F ( F L A G ) GO TO 4
I F ( A B S ( C H I S O ( I T E R ) - C H I S O ( I T E R - 1 ) ) . G T . . 0 0 2 ) GO TO 4 0
I F ( A B S ( C H I S O ( i T E R - l ) - C H I S Q ( I T E R - 2 ) ) . L T . . 0 0 4 ) GO TO 23
4 0 DO 4 2 1 = 1 , K
4 2 I F ( A B S ( D E V ( I ) I . G E . 0 . 5 ) GO TO 4
I F ( A B S ( D E V ( K S ) ) . L T . 0 . 0 1 ) GO TO 22
0044
0045
CALCULATE MATRIX GM OF SECOND DERIVATIVES
o
4 DO 6 1 = 1 , K
CALL L S H I F T ( 1 . , T ( L T * ( I - 1 I + 1 I )
1 5 CALL L F I T ( C L A B ( L B ) t S T A T ( l ) ,
G R A S ( KS ) = —G R A S ( K S ) / P P ( K S ) / X L
IFIAX-.OOl I 8, 8,5
- 1 , K S . L T , T ( 1 1 1, L , AX, V, OSQ. GRASI
o
0046
0047
0048
8 00 7 J=1,KS
7 GM( I , J I = G R A 0 ( J I - G R A S ( J )
0051
6 CALL R S HI F T ( 1 • » T ( L T * ( 1 —1 1 + 1 ) I
0052
o o o
0049
0050
CALL EPEAK1<P P ( KS I + S T E B I * * < 1 . / X L I , T ( J2 + 1 ) )
16 CALL L F I T ( C L A B ( L B ) , S T A T ( 1 ) ,
- 1 , K S . L T , T ( 1 1 ) , L . AX, V , OS Q. GR A S )
GRAS <KS)= —G R A S ( K S ) / ( P P ( K S ) + S T E B I / X L
IFIAX-.OOl)11,11,5
o
0053
0054
0055
11 0 0 10 J= 1 , KS
10 G M ( K S , J ) = ( G R A S ( J I —G R A O ( J ) ) / S T E B
o o o
0056
0057
SYMMETRIZE GM,
PUT
IN A,
0 1AGONAL IZ E
0 0 6 0 1 = 1 , KS
0 0 60 J = I , KS
60 A( 1 + J * ( J - l > / 2 ) = ( G M ( 1 , J I + G M I J , I ) 1 / 2
on
0058
0059
0060
CALL E l G E N ( A , R , K S , 0 )
o o o
0061
0062
0063
0064
0065
0066
CHECK FOR LOW EIGENVALUES,
i
FLAG=. FALSE.
22 DO 13 1 = 1 , KS
TR=TR2
GRAS( I ) = 0 .
TEM=0.
COMPUTE OEV.
-3 5 4 0067
0068
0069
0070
0071
0072
0073
0074
0075
00 6 6 J = 1 , K S
TEN=R(J+(I-1)*KS>
GR A SI I >= GRAS ( I ) + T E N » G R A 0 <JI
TEN=TEN*TEN
I F ( T E M- T E N ) 1 9 , 6 6 , 6 6
19 LL=J
TEM= TEN
6 6 CONTINUE
I F( LL. EQ. KS )TR= TRl
C
0076
0077
0078
0079
0080
0081
0082
N=I * ( I + 1 ) / 2
IF(A (N )-1.£-20)3,14,14
14 G R A S ( I ) = G R A S ( I ) / A ( N )
I F ( ABS( GRAS( I ) ) - T R ) 1 3 , 1 3 , 3
3 G R A S I I ) = G R A S < I ) / A B S I G R A S I I ) ) *TR
FLAG=. TRUE.
13 CONTINUE
C
0083
0084
0085
0086
0 0 17 I = 1 , KS
DEVI I 1=0.
00 17 J = 1 , K S
17 DEVI I ) = O E V t I I + R I 1+ 1 J - 1 ) * K S ) * G R A S I J )
C
0087
0088
0089
2 1 DO 38 1 = 1 , KS
38 P P I I ) = P P ( I l - O E V I I )
IFIPPIKS) . L T . . 0 2 ) PP (K S)= .02
C
ITER=ITER+1
1F I CHI S QI I TE R- 11 - 1 . 5 1 3 1 , 3 1 , 3 3
31 0 0 3 4 1 = 1 , L
STATtI)=H(I+LB-1)+STATC(I)
34 I F ! S T A T { I ) . L E . O I S T A T I 11=1
0090
00 91
0092
0093
0094
C
GO TO 33
0095
C
C
C
0100
0101
ERROR RETURNS
5 H R I T E ( P R I N T , 2 7 ) IER1
WRI TE( LOG, 2 7 ) I ERl
27 FORMAT! 1 * = 4 = MATRIX INV FAI L
IF(ARG>76,76,78
78 I F ! I T E R - 1 ) 2 3 , 7 5 , 2 3
0096
0097
0098
0099
C
C
C
C
',A4)
NORMAL OUTPUT
2 3 W R I T E I P R I N T , 7 7 ) L A B E L ( 1 ) , IPPI I ) , I = 1 , K S )
IF I I T E R - 1 1 7 5 , 3 2 , 7 5
0102
C
75 WRITE I P R I N T , 7 7 ) LABEL I 2 ) , I DEV I I ) , I = 1 , K S )
W R 1 T E I P R I N T , 7 7 ) L A B E L ( 3 ) , I GRAD I I ) , I = 1 , K S )
W R I T E ! P R I N T , 7 7 ) LABEL( 4 ) , ( C H I S Q I I ) , I = 1 , I TER)
7 7 FORMAT! A8 I 1 X . 8 G 1 4 . 5 ) )
0103
0104
0105
0106
C
7 6 WRI T E( T YP E, 4 4 ) I T E R , C H I S QI I T E R )
WRITE(L0G,77)LABEL(3),(GRAD(1),I=I,KS)
4 4 FORMA T( I 4 . F 1 0 * 5)
0107
0108
0109
C
0110
0 0 46 1 = 1 , LB1
4 6 W11) =0
0 0 47 I = R B 1 , 1 0 2 4
47 HI I ) =0
RETURN
END
0111
0112
0113
0114
0115
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
0001
9
THIS PHASE PERMITS REASSIGNING FORTRAN INPUT AND OUTPUT UNITS DURING
EXECUTION.
WHENEVER AN I / O OPERATION IS TO BE PERFORMED, THE UNIT
ON WHICH IT MUST BE DONE
IS TAKEN FROM A COMMON ARRAY
NAMED ' I O C '
THE PURPOSE OF IOUNIT IS
TO SELECTIVELY ALTER THIS ARRAY.
THE FI RST
TIME IT IS USED, IT MUST
BE I N I T I AL I ZED BY CALLING IT WITH A NEG.
ARG, AND IT WILL READ A LI ST OF CONTROL NAMES. EACH NAME WILL BE
ASSOCIATED WITH A BLOCK OF FOUR ASSIGNMENTS.
THESE CONTROL NAMES
CAN BE CONVENIENTLY TAKEN TO BE THE NAMES OF THE PHASES WHICH HAVE
THEIR I / O UNITS CONTROLLED BY A GIVEN SECTION
OF I OC. ( I OC IS
DIVIDED INTO 12 SECTIONS OF FOUR WORDS EACH) .
A LI ST OF FOUR BLOCKS
WITH THEIR ASSOCIATED PHASES AND NORMAL ASSIGNMENTS IS GIVEN TN THE
THESI S.
I NI TI AL ASSIGNMENTS ARE
CONTROLLED
BY THE BLOCK DATA
I NI T I AL I ZAT I ON OF IOC.
TO BYPASS I OUNI T, IFDESIRED, THE PROPER
I NI TI ALI ZATI ON MUST BE USED IN THE BLOCK DATA SUBPROGRAM AND
IOUNIT
NEEDS NEVER BE CALLED.
SUBROUTINE
IOUNIT(ARG)
C
0002
0003
0004
0005
0006
REAL*B NAME, END/ ' *ENO
• / , SNAMES112 )
INI EGER T Y P E , P R I N T , I O C ( 4 8 ) , ARG*2
COMMON/ IOC/IOC
COMMON/SNAMES/SNAMES,IMAX
EQUIVALENCE I IOC I 1 ) , L O G ) , I IOC 12 I , TYPE) , 11OCI 3 ) , PR I NT I , I IOC 14 I , I GN)
0034
0035
0036
0037
0038
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IFI FLAG) GO
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-J Cl
-3 5 7 -
0012
1F! LEVEL. I . T . 5 0 )
LEVEL=59
CALL PHI D( 1LPC
0013
0014
RETURN
' I
C
0015
o o o
IF (ARG)32,32,33
P1LEUP CORRECTION.
ESTIMATION OF TAI L PARAMETERS.
33 L I M I T = 2 . * S E E K ( L P , 1 0 , 1 0 2 4 , . 7 5 l - S E E K <L P , 1 0 , 1 0 2 4 , . 2 5 )
JT = 2 . 5 + 2 . 8 0 * S E E K ( L P , 1 0 , L I M I T , . 7 5 1 - 1 . 8 0 * S E E K ! L P , 1 0 , L I M I T , . 25 I
0018
0019
NTAIL=0
DO 1 0 1 = J T , 1 0 2 4
10 NTAIL=NTA1L+LP<11
I F ( NT A I L . LE. 5 ) GO TO 3
0020
o
0016
0017
n
0021
0022
E X P C = S E E K ! L P , J T , 1 0 2 4 * . 6 3 2 1 —JT
JP=SEEK(LP,1 0 , 1 0 2 4 , 0 . 5 1 + 0 . 5
I F ( E X P C . L T . 1 5 . I GO TO 3
KE3=4.*EXPC+1.
I F I K E 3 . G T . 5 0 0 ) KE3=500
o
0023
0024
0025
0026
0027
0028
0029
0030
o
C= NTA I L / E XP C * E XP < ( J T - J P I / E X P C )
ACCU=0.
00 13 I =J T , 1 0 2 4
13 A C C U= A C C U+ C * E X P ! ! J P - I I / E X P C )
0031
0032
0033
0034
o
J J S= 0
DO 14 1 = 1 0 , 1 0 2 4
14 J J S = J J S + L P (1 I
C=C/JJS
0035
0036
ooo
DO 31 1 = 1 , KE3
31 T A B L E ! I ) = C * E X P ( - I / EXPCI
0037
0038
0039
0040
0041
0042
EVALUATION OF SPECTRUM OF PI LE UP
ISUM =0
DO 20 1 = 1 0 , RC
2 0 1SUM=1SUH+LAB( 1 )
0 0 21 1 = 1 , RC
STATC( I >=0
2 1 STAT! 1 I =LAB ( I I
C
I TER= 1
27 LSUM=0
0 0 22 1 = 1 0 , RC
22 LSUM=LSUM+STATC( 1 )
C0 EF=1. *LSUM/ I SUM
0043
0044
0045
0046
0047
C
FRAC=0.
00 2 3 1 = 1 0 , RC
CORR=—STATCI I ) + C OE F * S T A T ( I l + F R A C
J=CORR+0•5
FRAC=CORR-J
23 STAT( I ) = L A B ( I I + J
0048
0049
0050
0051
0052
0053
C
0054
IF!ITER-2)2,2,28
C
0055
0056
0057
0058
0059
0060
0061
0062
0063
0064
0065
0066
0067
0068
0069
0070
2 FRAC=0.
1=8
30
1= 1+ 3
I F ! I . G T . R C ) GO TO 4 0
JR=I —KE3
I F ( J R . L T . I O ) JR=10
CORR =0•
11=1-1
DO 2 4 J = J R , I I
2 4 CORR=CORR + S T A T ( J ) *T ABLE (1 —J 1
CORR=CORR+FRAC
S TATC( 1 - 1 ) = C 0 R R + 0 . 5
S TATC! I I=CORR + 0 . 5
STATC( 1 + 1 ) = C 0 R R + 0 . 5
FRAC= CORR- S T ATC( I )
GO TO 30
C
40
0071
ITER=ITER+1
GO TO 27
0072
C
0073
0074
0075
0076
0077
0078
0079
2 8 TEM=COEF+COEF*COEF
FRAC =0 .
00 4 1=10,1024
C0RR=TEM*LA8 ( I I+FRAC,
J =C0RR+0.5
FRAC =CORR—J
4 STATCI I ) = 2 . * S T A T C ( I I + J
-
0080
0081
0082
358 -
PERC=1. *LSUM/ I SUM
WR I T E ( M , 9 ) J T , P E R C , E X P C , N T A I L
9 FORMAT!' + ' , T 7 0 , ' J T = ' , 1 4 , •
PERC=' , F 7 . 5 , '
1
• NTAIL=' , 1 4 )
RETURN
0083
3 WRITE( N , 2 9 )
29 FORMAT!' UNUSUAL LP SPECTRUM' )
WR I T E ( M, 1 ) J T , N T A I L , E X PC
1 FORMAT!• + • , T 7 0 , ' F A I L I U R E
J T = ',I4 ,'
LEVEL=50
RETURN
0090
0 0 91
0092
0093
0094
0095
0096
32 0 0 3 4 1 = 1 , 1 0 2 4
STATC( I ) = 0
34 S T A T I I ) = L A B ( I )
WRI T E ( M, 3 9 )
39 FORMAT!• + • , T 8 0 , ' N O
RETURN
END
n o o n
0084
0085
0086
0087
0088
0089
0001
EXPC='.F6.2,
N T A IL =',I4,'
EXPC=' , G 1 0 . 4 )
PILEUP CORRECTI ON' )
SUBROUTINE TO SHIFT PEAK SHAPES TO THE LEFT .
o
SUBROUTINE L S HI F T ( DEV, A )
0002
I NT EGERS A ( 1 )
IOEV=DEV
N= 5 0 0 - 1 DEV
FR=DEV-IOEV
FRC=1 . O- FR
o
0003
0004
0005
0006
0007
1 DO 3 1 = 1 , N
1 1 = 1 + 1 OEV
3 A ! I ) = FR*A<11+ 1 >+FRC*A(11 1 + 0 . 5
RETURN
n
0010
0011
o
0008
0009
IF(FR)5,2,1
0012
2 DO 4 1 = 1 , N
4 A! I ) = A( I + IOEV)
5 RETURN
END
o
0013
0014
0015
0001
o o o
SUBROUTINE PHIOIPHN)
0002
SUBROUTINE
FOR PHASE
IDENTIFICATION
REAL* 8 PHN
INTEGER PRINT, TYPE
COMMON/ I OC/ I OC! 8 ) . P R I N T , T Y P E , I OP( 3 8 )
WRITE( P RI NT, 1 ) PHN
WRITE( TYPE, 11PHN
1 FORMAT!' * ' , A 8 )
RETURN
®
END
o
0003
0004
0005
0006
0007
0008
0009
0001
o ooonooooo
SUBROUTINE PLOT<N , F S , / A / , / B / , * , * )
0002
IMPLI CI T INTEGER( A - Z )
COMMON/CPLOT/ IDN
REAL C. TEM. FS
INTEGER*2 A ( 1 ) , B ( 1 ) , K U T * 4 / 4 /
non
0003
0004
0005
non
0010
ERASE PREVIOUS DI SPLAY,
START NEW DISPLAY
CALL O E S S I N f I O N , 2 * N + 2 0 , 6 2 , 6 1 )
0006
0007
0008
0009
DISPLAY SUBROUTINE (LOGARITHMIC) FOR TWO SUPERPOSED SPECTRA A,
N
NUMBER OF CHANNELS TO DISPLAY IN EACH SPECTRUM
FS
# OF COUNTS CORRESPONDING TO I Y = 7 9 9
C
CONSTANT IN FUNDAMENTAL EQUATION IY=C*LOGI A ( I X ) )
RETURN I IS NORMAL RETURN, RETURN 2 IS ERROR RETURN.
A AND B ARE FIRST AND SECOND SPECTRA TO BE DISPLAYEO
A IS INTENSITY 3 AND B I S INTENSITY 1 .
ALL CHANNELS HAVING LESS THAN KUT COUNTS ARE OMITTED
PUT POINTS
IN BUFFER
2 00 8 I X = l , l i
IY = 80 * ( I X - 1 )
IER=PUTPT( I D N . O , I Y , 3 )
8 I F ! 1 E R . N E . I O N ) RETURN 2
B
-3 5 9 -
00 LI
0012
0013
0014
0015
0016
0017
0018
0019
4
0020
0021
0022
0023
0024
0025
0026
0027
o o o o o
5
7
1
I F I F S . L E . K U T + 1 0 . ) FS=KU7 + 10.
C= 7 9 9 . / A L 0 G ( F S / K U T >
DO 5 IX = 1» N
I F ( A ( 1 X ) . L T . K U T I G 0 TO 4
IY=C=ALOGIl.*A(IX)/KUT)+0.5
I F ( I Y . G E . 8 0 0 ) GO TO 4
IER=PUTPT( I D N , I X , I Y , 3 )
I F C I E R . N E . I U N ) RETURN 2
I F I B ( I X J . L T . K U T ) GO TO 5
I Y=C=ALOG<l.*B(IX)/KUT1+0.5
I F ( I Y . G E . 8 0 0 ) GO TO 5
I ER=PUTPT( I O N , I X , I Y , 1 I
I F { I E R . N E . I D N ) RETURN 2
CONTINUE
RETURN 1
RETURN 2
END
THE PURPOSE OF THIS SUBROUTINE IS TO HAKE THE INPUT OF DATA EASIER
ANO TO
PREVENT BLOWING UP DUE TO ILLEGAL CHARACTERS IN AN I OR F
FORMAT FIELD
0001
o o o n o o o o o o o o o o o o
SUBROUTINE READI INF, I ED, * , * )
ROOT PHASE SUBROUTINE TO READ FROM TYPEWRITER. READS IN >A'
FORMAT AND CONVERTS TO 1 * 4 . THE NUMBERS MUST BE UNSIGNED INTEGERS
SEPARATED BY COMMAS.
INF
DIMENSIONED 4 0 , CONTAINS 1=4 INFORMATION READ IN
IF ALL CHARACTERS BETWEEN COMMA I ANO 1+1 ARE BLANKS
THE VALUE OF I N F I I + l ) IS LEFT UNCHANGED.
I
CHARACTER BEING PROCESSED
S
RANK OF DIGIT PROCESSED IN CURRENT WORD
L
WORD PROCESSED
J-l
VALUE OF DIGIT PROCESSED
IED
NUM8 ER OF INTEGERS TO BE READ
IF THE NUMBER OF INTEGERS REAO I N=I ED, RETURN TO FIRST STM NO
IF AN ILLEGAL CHARACTER IS PRESENT OR IF IEO I S DIFFERENT FROM
THE NUMBER OF CHARACTERS READ IN, RETURN I S HADE TO SECOND STM. NO
0002
I MPLI CI T I NT EGER*4 I A- Z I
INTEGER I N F I 4 0 I
INTEGER*2 TYPE{ 8 0 I , COMMA/•, ' / . S P A C E / 1
'/,
1 NUM(IO) / ' 0 ' , ' 1 ' , ' 2 ' , ' 3 ' , ' 4 ' , ' 5 ' , ' 6 ' , ' 7
COMMON/ IOC/IOC 1 4 ) , M, N, I OP I 4 2 )
0003
0004
o o
000 5
0006
0007
0008
0009
10 0
0010
0011
0012
101
0013
0014
0015
0016
0017
0018
0019
11
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
0030
00 31
12
o o o
10
S=0
L=1
WRI TEI M. l OO)
FORMAT I T 8 1 )
R E A DI N , 1 0 1 1 TYPE
FORMAT( 8 0 A 1 )
DO 10 1 = 1 , 8 0
I F I T Y P E U l . E Q . S P A C E ) GO TO 10
I F ( T Y P E ! I ) .NE.COMMA) GO TO 11
L=L+ 1
I F I L . G T . I E D ) RETURN 2
S=0
GO TO 10
DO 12 J = l , 1 0
I F I T Y P E I I I . N E . N U M I J ) I GO TO 12
S=S + 1
I F (S .E Q .l ) INF(L)=J-1
I F ( S . E Q . l ) GO TO 10
INF(L)=10*INF(L)+J-1
GO TO 10
CONTINUE
RETURN 2
CONTINUE
I F ( L . E Q . I E D ) RETURN 1
RETURN 2
ENO
SUBROUTINE TO SHIFT
PEAK SHAPES TO THE RI GHT.
SUBROUTINE R S H I F T ( DEV, A)
0001
C
0002
INTEGER*2 At 1)
IDEV=DEV
N = 5 0 0 - I DEV
FR=DEV-1DEV
FRC=1.-FR
0003
0004
0005
0006
C
IF(FR)5,2,1
0007
C
■*
','8
','9
•/
-3G 00008
0009
0010
0011
0012
1 DO 3 1 = 1 , N
J=501-I
I1=J-IDEV
3 A ! J ) = F R * A( I 1 - 1 ) + F R C * A ( 1 1 1 + 0 . 5
RETURN
C
0013
0014
0015
0016
0017
2 DO 4 1 = 1 , N
J=50l- I
4 A ( J ) = A ( J - 1 DEV)
5 RETURN
END
C
C
C
SUBROUTINE TO READ TAPE AND PREPARE TITLE FOR DISPLAY
00 01'
SUBROUTINE RT(ARG)
C
C
C
C
C
C
THE DATA TAPE IS READ FIRST ( I T IS ASSUMED CORRECTLY POSITIONED
FI RST BY TPHDL) .
THE RELATI VI STI C KINEMATICS COMPUTE THE CENTER
OF HASS ANGLE AND THE CORRECTION OF SOLID ANGLE,
AND THE TITLE
I S DISPLAYEO ( RUN NUMBER, ANGLE, ENERGY, AND TARGET NUCLEUS 10)
0002
0003
0004
0005
0006
0007
0008
0009
0010
0011
0012
0013
0014
0015
0016
0017
0018
COMMON/LAB/LAB
COMMON/LP/LP
COMMON/ LISE/EP,CHARGE
COMMON/ CORR/ANGLE.OT,ICC
COMMO N/ P E / P E ( 1 0 ) , P E C M( 1 0 )
C0MM0N/ KE/ K, EEXC( 10>
COMMON/ RDATA/ Q, RKE, AT, ZT
COHMON/LEVEL/LEVEL
COMMON/RELAT/BETA, ACOB,TETA
COMMON/RAT/RAT
COMMON/LON/LDN
COMMON/ I OC/ I OC( 1 6 ) , TAPE, TYPE, PR I NT.LOG
INTEGER RUN, TAPE, TYPE, P RI NT , AT, Z T , BTOOEC
I NT EGER S E P, CHARGE , CODE, ANGLE,OT , ICC , BEAM, TANT, LPE . PMHV, LPFC ,ARG
I NT EGERS FRSTCH.LASTCH, LAB ( 1 0 2 4 I ,LP< 1 0 2 4 )
REAL M 1 , M 2 , M 3 , M A S S = > 8 , E Q U I V * 8 , D P 6 / 1 0 0 0 0 0 0 . /
L0GI CAL*1 T I T L E 18 0 I / ' RAD I AT IVE CAPTURE OF PROTONS
TARGET ' ,
1
' A=
Z=
KEV
OEGREES' /
EQUIVALENCE I T I T L E ( 7 3 ) , CODE) , ( T I T L E ( 7 5 I , RUN)
0019
C
0020
0021
0022
ARCCOS(ALF)=CONV*ARCOS(ALF)
AC0S ( AL F I = COS ( AL F / CONV)
CONV= 5 7 . 2 9 5 8
C
0023
0024
IFILEVEL.NE.20.AND.LEVEL.NE.90)
LEVEL=30
RETURN
C
0025
0026
0027
WRI TE( PRI NT, 1 0 )
10 FORMAT( • 1 ' )
CALL P HI D C R T
C
C
C
READ DATA FROMDATA ACQUISI TI ON
0028
0029
0030
0031
0032
0033
1
2
3
0034
0035
0036
•)
4
PROGRAM
READt T A P E . l 1 C 0 D E , RUN, EP. ANGLE. CHARGE, OT.BEAM, TANT, I C C , LPE, LPFC,
1 FRSTCH.LASTCH,PMHV
WRI T EI T YP E. 2 ) CODE,RUN
9
W R I T E ( P R I N T , 3 ( CODE, RUN, BEAM, TANT, LPE, PMHV, ANGLE, OT, ICC
F O R MA T ! A 2 , A 4 , 2 0 A 2 )
FORMAT(• * ' , A 2 , A 4 >
FORMAT!• + • , 8 X , A 2 , A 4 , •
BEAM=',I5,'
TAN T=',I5,‘
LPE='.I5,
1
• PMHV=1, I 5 , 1
ANGLE=' , 1 5 , 1 D T = , , I 5 , '
ICC=',I5)
REA0(TAPE, 4)LP
READ! TAPE, 4I LAB
F ORMAT! 1 80A2)
C
0037
0038
I F I A N G L E . L T . 170 IANGLE = ANGLE*10
I F ( ANGLE . G T . 1 7 0 0 ) ANGLE=ANGLE/10
C
C
C
C
0039
0040
0041 '
0042
0043
0044
0045
0046
0047
0048
0049
0050
0051
0052
0053
RELATI VI STI C KINEMATICS
EQV= EQUIV( 0 I
M1=(MASS( 1,1) +DP6)*EQV
M2 = ( MA S S ( Z T , A T ) + A T * D P 6 )*EQV
M3=(MASS(ZT+1,AT+1)+(AT+1)*0P6)*EQV
PHI = 10*E P
E=M1+PHI
PPR=SQRT! 2 * M 1 # P H I + P H I * P H I )
BETA=PPR/ ! M2+EI
GAMA=1,/SQRT!1.-BETA*BETA)
ET=GAMA*! M2+E-BETA*PPR)
DO 11 I = 1 ,K
11 PECM! 1 ) = ( E T * E T - ( M 3 + E E X C ( I I ) * ! M 3 + EEXC! 1 ) ) 1 / 2 . / E T
TETAL=ANGLE/ 1 0 .
UPDN=ACOS( T ETALI
UDN=GAMA»( -BETA+UPDNI
-3 6 1 0054
0055
0056
0057
0058
0059
0060
0061
0062
0063
SDN=UDN/ SQRT( 1 . +UDN*UDN-UPDN*UPDNI
TETA=ARCC0S(SDN)
OLDPE = PE( 1 )
TEM=GAMA*( 1 . +BETA*SDN)
DO 12 I = 1, K
12 PEI I l = T £ M* P E C M( I )
SPDN=GAMA* IBETA+SDNI
SPSQ=1. -SDN*SDN+SPDN*SPDN
ACOB=GAMA*I 1 . +BETA*SDN) / S PS Q/ S ORT I S P S Q )
RAT = 0 L D P E / P E ( 1 )
C
C
C
0064
0065
0066
0067
0068
DISPLAY TITLE
KTNEP=10*EP
I=BT0DEC(AT,4,TITLE(40),LN)
I=BT0DEC(ZT,4,TITIE(47),LN)
I = B T O D E C ( K T N E P f 6 , T I T L E I5 1 ) tLNI
I=BTODEC IA N G L E / 1 0 14,T IT LE ( 61 ), LN )
C
0069
CALL D E S S I N ( L D N , 9 0 0 , £ 6 , t 5 )
C
0070
0071
0072
0073
0074
0075
0076
0077
0078
0079
0080
6 IF IARG1 9 , 9 , 8
9 CALL T E X T ( L D N , 0 4 5 , 1 0 0 0 , 1 , 2 , ' HO 1 , 2 8 , T I T L E , 65 I
B CALL T E X T ( L D N , 1 2 0 , 9 6 0 , 1 , 2 , ' H O ' , 2 0 , T I T L E ( 3 1 1 , £ 5 I
CALL T E XT ( L DN, 0 9 0 , 9 2 0 , 1 , 2 , ' H O ' , 2 3 , T I T L E ( 5 0 ) , 6 5 )
CALL T EXT ( L DN, 1 4 0 , 8 6 0 , 2 , 4 , • HO• , 6 , T I T L E ( 7 3 ) , £ 5 I
RETURN
5 WR1 TE( LOG, 7 I
WRI TE( PRI NT, 7 )
7 FORMA T( ' * ERROR RETURN FROM OESSIN OR T E X T ' )
RETURN
END
C
C
C
C
C
C
C
0001
FUNCTION TO ANALYSE THE POSI TI ON OF A PEAK IN SPECTRUM A FROM
CHANNEL K TO CHANNEL L.
SEEK WILL BE PUT EQUAL TO THE POSI TION CORRESPONDING TO
HAVING A FRACTION F OF THE COUNTS BELOW SEEK.
FUNCTION S E E K ( A , K , L , F >
C
0002
INTEGER*2 A ( 1)
ISUM=0
DO I I = K , L
1 1 SUM= ISUM+A ( I I
U=F*ISUM
ISUM=0
DO 2 I = K , L
ISUM=ISUM+A( I I
1F(ISUM-U)2,3,3
2 CONTINUE
SEEK=L
RETURN
3 SEEK=I+1.-(ISUM-U) / A ( I I
RETURN
END
0003
0004
0005
0006
0007
0008
0009
0010
0011
0012
0013
0014
0015
0016
C
C
C
C
C
C
C
C
C
0001
THE ANALYSIS PROGRAM ASSUMES THAT THE SPECTRUM IS LINEAR AND THAT
THE ZERO CORRESPONDS TO CHANNEL ZERO.
IF THIS
I S NOT TRUE, THE
SPECTRUM MUST BE SHIFTED TO THE LEFT OR TO THE RIGHT USING SHI FTL.
TO FIND OUT IF THE ZERO IS PROPERLY PLACED, THE ANALYSIS CAN BE
PERFORMED IN A PRELIMINARY WAY AND THE RESULT ' CHECK' OBTAINED.
THE AMOUNT OF SHIFT IS SET WITH THE PARAMETER KEYS.
SUBROUTINE S HI F T L ( ARGI
C
COMMON/LAB/LABI 1 0 2 4 )
COMMON/LEVEL/LEVEL
COMMON/ I OC/ I OC( 2 5 > , L
I NT EGERS ARG,LAB
0002
0003
0004
0005
C
IFILEVEL.LT.30)
LEVEL=35
0006
0007
C
0008
0009
0010
0011
0012
0013
0014
0015
0016
N=ARG/ 256
IF(N)1,2,3
1 M=1024+N
DO 4 1=1, M
4 LABI 11=LAB( I - N )
M= M+1
DO 7 I=M, 1 0 2 4
7 LAB( I 1=0
GO TO 2
RETURN
-
362 -
C
0017
0018
0019
3 M=1024- N
DO 5 1 = 1 , M
5 LAB(1 0 2 5 - 1 )=LABIM-I+1)
0020
DO 8 1=1,N
0021
8 LA8 < 11=0
C
0022
0023
0024
0025
0026
2 CALL P H I D ( ' S H I F T L
•)
WRI TE( L , 6 ) N
6 FORMAT!• + ' , 1 2 5 , ' CHANNELS’ )
RETURN
ENO
C
0001
SUBROUTINE SHOW(ARG)
C
C
C
C
C
C
C
C
C
C
C
C
KEYBOARD SUBROUTINE TO DISPLAY LAB
FS
# OF COUNTS CORRESPONDING TO I Y = 7 9 9 , FULL SCALE.
I FS
TEMPORARY INTEGER STORAGE TO ENTER FS
I D N , I X , I Y , I Z ARGUMENTS FOR TRACK
I N , J N , L N , T E S T , I T ARE USED TO TEST THE KEYBOARD PARAMS KEYS.
PARAMETER KEYS OPTIONS
1
CHANGE SCALE WITH LIGHT PEN
2
OBTAIN CH # ANO CONTENTS ON
8
(FOURTH KEY FROM RIGHT) SET
0002
0003
0004
0005
0006
0007
0008
0009
LOG
SCALE ON ' READ'
UNIT
I MPLI CI T I NTEGER( A- Z)
COMMON/LAB/LAB
COMMON/FS/FS
COMMON/LEVEL/LEVEL
COMMON/ IOC/ I OC( 1 2 ) , LOG, I E R , I OP ( 3 4 )
I NT EGERS ARG, LAB( 1 0 2 4 )
INTEGER I N / Z 0 0 0 0 0 1 0 0 / , J N / Z 0 0 0 0 0 2 0 0 / . L N / Z 0 0 0 0 0 8 0 0 / . N 0 C H / 6 0 0 /
REAL FS
C
0010
0011
0012
0013
I F I L E V E L . L T . 3 0 ) RETURN
I F I L E V E L . L T . 4 0 ) LEVEL=40
TEST=ARG
CALL PHI 0( ' SHOW
')
C
0014
CALL PLOT( N O C H , F S , L A B , L A B , 6 1 , 6 2 )
C
C
C
0015
0016
0017
0018
0019
0020
0021
0022
LIGHT PEN ROUTINE TO CHANGE SCALE.
1
I T = A ND( I N, TE S T )
I F ( I T . E Q . O ) GO TO 3
ION = T R A C K ( I X , I Y , I Z )
CALL DELAY( - 3 )
FS=LAB(IX)
FS=FS*FS
I F ( I X . E Q . O ) FS=I Y
I F t F S . L T . 1 0 . ) FS=10.
-
C
0023
CALL PLOT( NOCH, FS, LAB, L A B , 6 3 , 6 2 )
C
C
C
0024
0025
0026
0027
0028
0029
0030
003L
LIGHT PEN ROUTINE TO OBTAIN CHANNEL # AND CONTENTS.
3 IT = AN0( J N. T E S T )
I F I I T . E Q . O ) GO TO 5
I 0N=TRACK( I X , I Y , I Z )
CALL DELAY( - 3 )
I F ( I X . E Q . O ) RETURN
K=LAB( I X)
WRITE(L0G,7) IX, K
7 FORMAT!' * LAB I • , 1 4 , • ) = • , 1 6 )
C
C
C
0032
0033
0034
0035
0036
OBTAIN SCALE FROM TYPEWRITER.
5 IT = ANO(LN»T EST)
IFI I T . E Q . O ) GO TO 4
10 CALL R E A D ! I F S , 1 , 6 1 1 , C 1 0 )
11 FS=I FS
I F I F S . L T . 1 0 . ) FS=10.
C
0037
0038
0039
0040
0041
CALL P LO T ( NO C H , F S , L A B , L A B , 6 4 , 6 2 )
2 WRITE!IER,8 )
8 FORMAT!' * ERROR OETECTEO IN " P L O T * " )
4 RETURN
END
-
0001
C
C
363 -
KEYBOARD SUBROUTINE TO DISPLAY CLAB,
ANO SPECT.
SUBROUTINE SHOWl(ARG)
C
C
C
C
FS
# OF COUNTS CORRESPONDING TO I Y = 7 9 9
I N , J N , K N . T E S T . I T ARE USED TO TEST THE KEYBOARD PARAHS KEYS
I D N , I X . I Y , I Z ARE ARGUMENTS FOR TRACK
C
C
C
C
C
0002
c
c
c
c
c
c
DEFI NI TI ON OF PARAMETERS SIMILAR TO SHOW
CLAB I S THE LABORATORY SPECTRUM CORRECTED FOR COSMIC RAYS ANO PI LEUP.
SPECT I S THE FIT TO CLAB.
PARAMETER KEYS OPTIONS
1 CHANGE SCALE WITH LIGHT PEN
2
OBTAIN CH # ANO CONTENTS I SPECTRUM IS I DENTI FIED)
3
WHEN THIRD KEY FROM RIGHT I N, BOTH CLAB ANO SPECT DISPLAYED
OTHERWISE, ONLY CLAB
I MPLI CI T 1NTEGER( A - Z )
COMMON/ CLAB/CLABt1 0 2 4 ) , T ! 5 0 0 0 )
COMMON/ FS/FS
COMMON/ SPECT/ SPECT( 1 0 2 4 )
COMMON/LEVEL/LEVEL
COMMON/ I OC/ I OC( 1 2 ) , LOG, I E R , I 0 P I 3 4 )
REAL FS
I NT EGER S ARG, CL AB, S P ECT , T
INTEGER I N / Z 0 0 0 0 0 1 0 0 / , J N / Z 0 0 0 0 0 2 0 0 / , K N / Z 0 0 0 0 0 4 0 0 / . N 0 C H / 6 0 0 /
0003
0004
0005
0006
0007
0008
0009
0010
0011
0012
I F I L E V E L . L T . 4 0 ) RETURN
2 TEST =ARG
CALL PH1 0 ( 1SH0W1
•)
0013
0014
0015
IT=AND( KN, T EST)
I F ( I T . E O . O ) GO TO 8
0016
0017
CALL PLOT( NOCH, FS, C L A B , S P E C T , 6 4 , 6 9 )
8 CALL PLOTI NOCH, FS, CLAB, CL AB , 6 4 , £ 9 )
LIGHT PEN ROUTINE TO CHANGE SCALE.
0018
0019
4 IT = ANO< I N , T E S T )
I F ( I T . E O . O ) GO TO 3
IDN=TRACK( I X , I Y , I Z )
CALL DELAY( - 3 I
F S = C L A B ( I X)
FS=FS»FS
I F I I X . E O . O ) FS =I Y
I F f F S . L T . 1 0 . ) FS=10.
0020
0021
0022
0023
0024
002 5
CALL P LOTI NOC H, FS, C L A B , S PE C T , £ 3 , 6 9 )
0026
LIGHT PEN ROUTINE TO OBTAIN CHANNEL # AND CONTENTS.
3 I T = AND( JN, TEST)
I F ( I T . E O . O ) GO TO 5
ION=TRACK(IX,IY,IZ)
CALL DE L A Y1 - 3 )
I F ( I X . E Q . O ) RETURN
I F C I Z . N E . 3 ) GO TO 10
K=CLAB( I X)
WRIT E ( L O G , 7 ) I X, K
7 FORMAT!' * C L A B ! • , 1 4 , • ) = • , 1 5 )
10 I F ! I Z . N E . l ) RETURN
K= SPECT( I X)
WRI T E t L OG, 12 I I X, K
12 FORMAT!' * CH # = ' , 1 4 , 1 7 , ' COUNTS' )
5 RETURN
9 WRI TE( I E R , 1 )
I FORMAT! • * ERROR DETECTED IN " P L O T " ' )
END
0027
0028
0029
0030
0031
0032
0033
0034
0035
0036
0037
0038
0039
0040
00 4 1
0042
0043
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
SUBROUTINE TO PREPARE A FAKE SPECTRUM
TO TEST THE FI TTI NG ROUTINE
WILL SIMULATE THE DATA ACQUI SITI ON PROGRAM BY WRITING ON A DATA SET
ALL THE DATA WHICH ARE USUALLY WRITTEN BY THE DATA ACQUISITION
PROGRAM.READS PRELIMINARY INFORMATION ANO TRANSCRIBES I T ,
READS THE THE POSI TI ON, ENERGY AND HEIGHT OF EACH PEAK, CALLS CPEAK
TO MAKE THE PEAKS, READS TWO POINTS ON THE EXPONENTIAL BACKGROUND
AND THE INTENSITY OF THE COSMIC RAY BACKGROUND AND AODS THESE TU THE
PEAKS.
A STARTING VALUE ANO A MULTIPLIER FOR THE RANDOM NUMBER
GENERATOR ARE READ (ANY TWO LARGE INTEGERS WILL DO ) AND THE SPECTRA'
ARE RANDOMIZED TO LOOK LIKE REAL SPECTRA. THE FAKE GAMMA RAY
SPECTRUM THUS CREATED I S WRITTEN OUT.
A FAKE, NON- REALI STI C LIGHT PULSER SPECTRUM IS CREATED ANO WRITTEN
ITS PURPOSE I S TO CHECK THE PI LE- UP CORRECTION SUBROUTINE LPC.
SOME PARAMETERS AND THE SPECTRA ARE PRINTED (UNLESS SUPPRESSED)
THE PERFORMANCE OF THE DATA ANALYSIS ROUTINE AS A WHOLE CAN BE
CHECKED BY COMPARING ITS ANSWERS WITH THE KNOWN PEAK HEIGHTS AND
0001
c
c
c
c
f
c
r
b
C
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
0002
0003
0004
0005
POSITIONS WHICH SHOULD AGREE WITHIN ERRORS HALF THE TIME OR SO.
FOR
ANY BIAS WILL REST BE DETECTED BY KEEPING THE PEAKS SMALL .
THE PURPOSE, THE RANDOMIZATION CAN ALSO BE DISABLED BY PUTTING
KSTART TO ZERO.
c
c
SUBROUT INE SIM
RUN
RUN NUMBER, EBOI C, 1=4
K
NUMBER OF PEAKS IN SPECTRUM,1=4
PEAK P O S I T I O N , C H # , R # 4
PP
PE
PEAK ENERGY, KEV, R=4
PH
PEAK HE I GHT, I * J
KX1.KY1 FIRST POINT ON EXPONENTIAL BACKGROUND,CH#,# C T S , I * 4
KX2. KY2 SECOND PT ON EXP BGN. CH# , # C T S , I * 4 , K X i n K X 2
CONSTANT OF COSMIC RAY BGN, R * 4 .
OR
NON-LINEAR EXP BGN PARAMETER R*4
D
JX
CH # WHERE THE COMPUTATION OF THE BGN STARTS
STARTING VALUE FOR THE RANDOM NUMBER GENERATOR, I *4
KSTART
MULTIPLIER FOR RANDOM I
I GEN, 1=4
KRNG
WEIGHTED SUM OF SQUARES OF THE RESIDUALS BETWEEN CH # LB
. CHISQ
ANO RB,FOR WHICH A l l ) KCR+5, / NCH, R#4
NCH
NUMBER OF CHANNELS OVER WHICH THE SUM IS PERFORMED
DUMMY ARGUMENT TO FI LL OUTPUT L I S T , 1*2
$
CODE
1*2 TWO CHARACTERS TO IDENTIFY THE SERIES OF RUT|S
1=2 PROTON LAB ENERGY, IN UNITS OF 10 KfcV
EP
1=2 DETECTOR ANGLE TENTHS OF DEGREE
ANGLE
CHARGE
1=2 BEAM CHARGE,MICROCOULOMBS
DT
1*2 #BCI REJECTED/ #BCI TOTAL*IOOOO
BEAM
1=2 NOMINAL BEAM CURRENT,NANOAMPERES
TANT
1=2 LIVE TIME, SECONDS
ICC
1=2 NUMBER OF LIGHT PULSER EVENTS
LPE
1*2 LIGHT PULSER DIAL SETTI NG, FULL SCALE=10 TURNS=IOOOO
LPE
1*2 LIGHT PULSER DIAL SETTING 10 TURNS=FULL S C A L E , = 1 0 0 0 0
PMHV
1=2 PM HIGH VOLTAGE, TOTAL. VOLTS
TEM
R* 4 TEMPORARY STORAGE
I,J
INDICES
Al 1 0 2 4 ) 1*2 SPECTRUM
B ( 1 0 2 4 ) 1*2 RANDOMIZED SPECTRUM
L P I 1 0 2 4 ) LIGHT PULSER SPECTRUM
N
1=4 FTN REF # , SOS I P T , READ PARAMS
1* 4 FTN REF # , TAPE
OUTPUT PARAMS, SPECTRA
INT
FTN REF # , SOS LOG PRINT NCH, 0 , CHISQ
IGN
1=4 FTN REF # , I G N
PRINT SPECTRAC
M
TO SUPPRESS RANDOMIZATION,
READ KSTART = 0
C0MM0N/ B/ L8, R8 •
COMMON/ I OC/ I OC( 3 6 ) , N , M , L N T , I G N , I 0 P < 8 )
INTEGER RB,RUN,RC
INTEGER*2 ARG,CODE, E P , ANGLE. CHARGE, DT, BEAM,TANT, I C C , LPE, PMHV,
1
$ / 0 / , A ( 1 0 2 4 ) , B ( 1024) , L P ( 1024 )
REAL F LOAT ! 1 0 2 4 1 / 1 0 2 4 = 0 . /
INTEGER»2C0SMS ( 3 1 ) / 3 8 0 , 3 9 1 , 4 0 1 , 4 1 1 , 4 2 2 , 4 3 2 , 4 4 3 , 4 5 3 , 4 6 4 , 4 7 4 , 4 8 4 ,
1 495,505,516,526,536,547,557,568,578,588,599,609,620,630,641,651,
2 661,672,682,693/
EQUIVALENCE I F L O A T ! L ) , A l l ) ) , ( F L O A T ! 5 1 3 ) , L P ( 1 > )
0006
0007
0008
C
0009
CALL P HI OI ' SIM
C
C
C
0010
0011
00L2
•)
INPUT ANO OUTPUT PRELIMINARY
INFORMATION AND NUMBER OF PEAKS
READI N, I ) CODE, RUN, K, EP, ANGLE, CHARGE, DT, BEAM, TANT, I CC, LPE, PMHV
1 F0RM AT(A2,A 4,14/(15))
REWIND M
WRITE<M, 2 >CODE, RUN, EP, ANGLE. CHARGE, DT, B E A M, T A NT , I C C , L P E , $ , $ , $ , PMHV
2 F0RMAT(A2,A4,20A2)
0013
0014
C
C
C
0015
0016
0017
0018
0019
REAO EACH PEAK POS I TI ON,
ENERGY,
HEIGHT.
ADO TO SPECTRUM A
DO 3 1 = 1 , K
REAU(N,4) PP,PE,PH
4 F0RMATI3F10.0)
CALL C P E A K ( P E , P P , 0 , B , 1 0 2 4 , I N T )
DO 3 J = 1 , 1 0 2 4
3 FLOAT( J ) = F L O A T ( J ) + B ( J ) * P H / 2 0 0 0 0 .
0020
C
C
C
READ K X 1 . K Y 1 , K X 2 , K Y 2 AND COMPUTE EXP BGN
0021
0022
READI N, 5 ) K X 1 . K Y 1 , K X 2 , K Y 2 , C R
5 F0RMAT(4I5,FI0.0)
0023
0024
0025
0026
0027
D=AL0G(KY1*1./KY2)/(KX2-KX1)
I F ( O . L E . O ) GO TO 6
JX=KX1-AL0G(3 0 0 0 0 . / < Y 1 1 / 0 + 1 .
I F I J X . L T . l ) JX=1
0 0 7 1=J X , I 0 2 4
7 FLOAT!I)=FLOATlI)
+ KY1 * EXP ( - 0 * ( I - K X 1 ) )
002 8
C
REAO KCR ANO ADO COSMIC
-3 6 5 ECAL=PE/PP/1000.
LC=l./ECAl+l.
RC=60./ECAL
1F(RC.GT.1024)RC=1024
0 0 18 I = LC, RC
E= I *ECAL
JE =(E +l.)/2.
EJ=(E+1.I/2.-JE
TEM=CR* (E J* C0S MS (J E+ l) +( 1. - E J I * C 0 S M S ( J E ) )
18 FLOAT( 1 ) = FLOAT( I I + TEM
0029
0030
0031
0032
0033
0034
0035
0036
0037
0038
C
C
C
REAO RANDOMIZATION PARAMETERS AND RANOOMIZE SPECTRUM
6 R E A D( N , 8 > KSTART,KRNG
8 FORMAT( 2 1 1 0 )
KRNG=8*KRNG-3
CHI S Q= 0.
NCH=RB-LB+1
0039
0040
00 4 1
0042
0043
C
0 0 9 1 = JX , 1 0 2 4
TEM=0.
0 0 10 J = l , 1 0
KSTART=KSTART*KRNG
10 TEM=TEM+KSTART/ 2. 147E+9
TEM=TEM*SQRT(0. 3*FL0AT( I) I
B(I)=FLOAT(II+TEM+0.5
T E M = B ( I ) - FLOAT( I )
I F ( I . G E . L B . A N D . I . L E . R B ) CHISQ=CHISQ+TEM*TEM/ FLOAT( I )
A(1) =FL 0AT (I 1+0.5
9 1F ( B ( I I . L T . O ) B ( I ) =0
CHISQ=CHISQ/ NCH
0044
0045
0046
0047
0048
0049
0050
0051
0052
0053
0054
0055
C
C
C
0056
0057
0058
0059
0060
00 61
0062
0063
C
MAKE LIGHT PULSER SPECTRUM
DO 12 1 = 1 , 1 0 2 4
12 L P ( 1 1 =0
LP(509l=ICC/4-58
L P ( 5 1 1 ) = L P ( 509 I
L P ( 5 1 0 1= I C C - L P ( 5 0 9 I - L P ( 5 1 1 ) - 5
L P (5151=63
L P ( 5 3 5 I = 34
L P ( 5 5 5 I = 16
C
C
OUMP SPECTRUM ON TAPE
W R I T E ( M , 1 1 ) LP
WRI TE( M , 1 1 1 B
11 FORMAT( 180A2I
REWIND M
0064
0065
0066
0067
C
C
C
PRINTED OUTPUT
13 WRI TE( LNT, 1 6 ) NCH. CHI SQ. D
16 F 0 R M A T ( T 5 0 , ' N C H = ' , 1 5 , '
C H I S 0 = •, F 1 4 . 5 . T 8 5 , • D = » , E 1 3 . 7 1
I F ( I G N . E O . l l ) RETURN
WRI T E ! I GN, 1 7 I A
WRI TE( I G N , 1 7 ) B
17 FORMAT!• 1 ' , 1 5 , 1 9 1 6 / ( 2 0 1 6 ) 1
RETURN
END
®
0068
0069
0070
0071
0072
0073
0074
0075
C
C
C
C
C
OLD VERSION OF CHARACTER GENERATOR SUBROUT I l € FOR DI SPLAY.
MAKE USE OF THE CHARACTER GENERATOR HARDWARE.
SHOULD BE REPLACED EVENTUALLY, TO SAVE ON STORAGE.
0001
CCCC
CCCC
00 0 2
0003
0004
0005
0006
0007
0008
0009
0 010
0011
0012
0013
0014
0015
0016
C
DOES NOT
SUBROUTINE TEXT( I ON, XO, YO, Z , M , D I R , L S , T I R T , # )
NON STANDARD VERSION OF TEXT. REQUIRES EXTERNAL BUFFER TREATMENT
INTEGER XR. YR
EQUIVALENCE ( C H . C t l l )
ROUTINE TO PLACE A STRING OF ALPHAMERIC CHARACTERS ON THE CRT
INTEGER*2 X , Y
EQUIVALENCE ( NEROt 1 ) , HERO I
LOG I C A L * 1 NERDI4I
INTEGER*4 HERD
LOG I C A L * 1 T I R T ( L S ) , C ( 2 >
L0GI CAL*1 T E R T ( 5 0 I , P 0 I N T S ( 3 0 )
INTEGER FOOL I T ( 2 0 I
EQUIVALENCE( TERT( 1 ) , FOOL I T ( 1 1 1
INTEGER X O, Y O, Z
I NT EGERS C H / O / , D I R , V E E / ' V E ' /
INTEGER
ASSIGN, AWAKE, PUTPT, PUTOAT.GETDAT
INTEGER ACTIVE
INTEGER*4 PUNC( 5 2 1 / Z 2 0 0 0 0 0 0 0 , 2 * 0 , Z 0 0 0 8 E 2 0 0 , 1 7 * 0 , ZOOOOEOOO, 1 0 * 0 ,
* Z 2 0 8 C 0 0 0 0 , 1 7 * 0 , Z00004210,Z038OEO0O /
,
*ALPHA<57> / Z y C 7 F 1 8 F C , Z F 4 6 3 E 8 F 8 , Z 7 4 6 1 0 8 B 8 , Z F 4 6 3 1 8 F 8 , Z F C 2 1 F 8 7 C ,
* Z 8 4 2 1 F 8 7 C , Z 7 4 6 7 0 8 B 8 , Z8C63F 8 C 4 , Z 7 1 0 8 4 2 3 8 , 7 * 0 , Z 6 4 8 4 2 1 3 C , Z 8 C A 9 C 9 4 4 ,
*ZF4210840,Z8C6B50C4,Z8C655CC4,ZFC6318FC,Z843F18FC,Z6CA818B8,
-30G-
*Z8CBF18FC , 8 * 0 , ZF84 3 F 8 7 C , Z 2 10 8 4 2 7 C , Z F C 6 3 1 8 C 4 , Z 2 2 9 4 A 8 C 4 , Z8EEB58]C
* Z 8 A 8 8 4 5 4 4 , Z 2 1 0 8 4 5 4 4 , l r AO8 4 1 7C , 6 * 0 ,
*Z746318B8,Z21084210,ZFC20FOFC,ZF842FOFC,
*Z103EA308,Z7443E87C,ZFC7F087C,Z210428FC,Z7462E868,ZF843F8FC
/
C
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
0017
0018
55
C
0019
1212
0020
0021
0022
002 3
0024
0025
0026
0027
002 8
0029
0030
00 31
0032
0033
0034
0035
0036
0037
0038
0039
0040
0041
0042
0043
0044
0045
0046
0047
0048
0049
0050
0051
0052
0053
0054
0055
0056
0057
0058
0059
0060
0 0 61
0062
0063
0064
0065
0066
0067
0068
DEFI NITI ON OF PARAMETERS
XO, YO= LOCATION IN SCOPE COORDINATES OF LOWER LEFT CORNEROF
FI RST CHARACTER IN THE STRING
Z = SCOPE INTENSI TY, 1 TO 7
M= SI ZE OF CHARACTERS, IN MULTIPLES OF 6X5 RASTER UNITS
THAT I S , M= 4 RESULTS IN A CHARACTER SIZE 24 X20
0 I R = ' H 0 ' OR ' V E ' FOR HORIZONTAL OR VERTICAL STRING, RESP.
LS=LENGTH OF CHARACTER STRING SPECI FIED IN ' TE X T '
■TEXT' = ALPHAMERIC TEXT TO BE OISPLAYEO
ENTRY 1
ALL PARAMETERS GIVEN.
CREATE DAT
DO 55 1 = 1 , LS
TERTI I ) = T I R T ( I I
COMMON ROUTINE
CONTINUE
ID1R=0
NSL0G=2
I N I T I A L I Z E SPACING COUNTER
I SP=M*7
IBLK=M*7
X = XO
Y=YO
I BCT=0
CREATE DISPLAY BUFFER
LOAD DAT
DETERMINE IF HORIZONTAL OR VERTICAL
4 0 I F ( D I R . E Q . V E E ) IDIR = 1
TEXT ROUTINE
C
OETERMINE CHARACTER INVOLVED ANO OBTAIN ITS BI T PATTERN
C
DO 60 KKK=1, LS
4444
C ( 2 1= TERT(KKK)
I F I C H . E 0 . 6 4 ) GO TO 6 5
I F I C H . G E . 1 2 9 . A N O . C H . L E . 1 6 9 ) CH=CH+64
I F I C H . G E . 1 9 3 ) GO TO 70
SPECIAL CHARACTER
IB IT S=PUNC( CH—7 4 1
GO T 0 80
ALPHAMERIC CHARACTER
C
IBITS=ALPHA(CH-192)
70
GO TO 80
CHARACTER I S BLANK— SPACE LOCATION CTR ANO RETN FOR NEXT CHAR
C
I F ( I D I R . M E . 0 ) GO T 0 6 7
65
X=X+IBLK
GO TO 6 0
Y=Y+I BL
67
GO T 0 6 0
BI T PATTERN OBTAINEO— SETUP DISPLAY POINTS
C
CALL B I T S C N l I B I T S , P O I N T S )
80
YR = Y
DO 1 1 0 1 = 1 , 6
L= 5*I
XR=X
0 0 95 J = 1 , 5
IF(.N0T.P0INTSIL+J-5))
GO TO 9 4
I F ( I D I R . N E . O ) GO TO 102
96
I F ( X R . G T • 1 0 2 3 ) GO TO 9 4
I F I Y R . G T . 1 0 2 3 ) GO TO 94
I F ( X R . L T . O ) GO TO 94
I F ( Y R . L T . O ) GO TO 94
I R N= PUT P T( I DN, XR , Y R , Z >
9995
IF ( I R N . N E . I D N ) RETURN 1
102
94
95
97
110
60
GO TO 9 4
I RN=PUTPT( I DN, Y R , XR , Z >
XR=XR+M
CONTINUE
1 F ( I O I R . N E . O ) GO TO 9 7
YR=YR+M
REMOVED TEMPORARILY SO THAT LETTERS HAVE GOOO SHAPE
YR=YR+2*M
GO TO 11 0
YR=YR—M
CONTINUE
X=X+I SP
CONTINUE
RETURN
END
-3 0 7 C
-
JOOL
_
_
SUBROUTINE TPODL(ARG)
C
C
C
•
0 0 02'
0003
"
PROGRAM TO LOCATE S P t C I E I C
'
-
-
- -
RIJNS ON THE DATA
TAPE
I
”
" INTEGER*2' AKG, CODE, CIDEP "
*
INTEGER RUN , P.Ui-lP , R I NP , T A P E / 0 4 / , DTOB I N , L O G / 0 2 /
CALL~ PH I o f ' TPHUL
'T ~
'
C
0 0 0 5____ __________
RE AO I LOG, 1 ) COOE , RUN
0006
"
‘ T' F0R' . - I AT( A2" , A4)
'
0007
CALL DTTJB IO ( RUN D, 4 , RUN, LNI
C
0003
~'i RE'a'd(TA‘ p e 7 6 , E NU = 3 ) CI UEP, ' kI NP
0009
6 F O R M A T ( A 2 . A4 , / / / / / / / / / / / / )
0 0 1 0 ____ ___________
I F ( CUDE-CI0EPJ2, 4 , 2
001*1
■ 4" IF ( ( RUNP-P.INP ) *( RUN- RI NP ) ) 2 , 5 , 2
C 0012
5 00 7 1 = 1 , 1 3
0013"
7 b a c k s PACE " T ape '
*
0014
WRI fE(LUG, 8>C10EP, RI NP, RINP
00 15
3 FORIiATI1 * ' , A 2 , A 4 , 3 X , Z 8 >
0016
......................... RETURN' '*
'
0017
3 REWIND TAPE
0018
RETURN
00i 9
~
END -
"
•
0004
'
'
‘
'
‘
!
'---------------------
'
'
'
•
'
'
C
MAIN PROGRAM ' WAI T '
C
C
ALL THE PHASE S' WHICH CONST IT U T E T H IS OATA ANALYSTS PROGRAM ARE
C
KEYBOARD PHASES WHICH ARE LUADcO ANO EXECUTED WHEN
THE CUKRESPON IMG
______________ ,_ C ____ KEY I S^DEPRESSEp^EXCEPT _FOR_THE_ROOT PHASE,.
.r_H.ISjM«INI .PROGRAM_______
C
IS CA. LED ■ IAIT 1 , IT IS Pa r t d F THE RUOT PHASE and
SIMPLY p r o v i j s
C
A WAIT LOOP FUR THE CPU TO EXECUTE BETWEEN TWO KEYBOARD PHASES
C
0*001
* '
" WR I T E ( 2 , 2 ) * '
0002
2 FOR RAT( 1 * COMPUTER WAITING FOR KEYBOARD PHASES' )
0003
______ _______ 1 GO TO 1 ___________________ ______________ _______________________________________ ___________
0004'
Ei'10
0001
SUBROUTINE XFORM(ARG)
KEYBOARD SUBROUTINE TO CREATE PEAK SHAPES, EXPONENTIAL BACKGROUND
AKlD SU3TRACT COSMIC RAY BACKGROUND FROM LAB (TO PRODUCE CLA b )_____
LAB _ _
K
EEXC
___________ ___________
1=2 GAMMA RAY SPECTRUM
_
*1=4 NUMBER UF PEAKS IN SPECTRUM
R= 4 R EXC I TAT IUM cNERGIE S , KEV, CURRE SPONO ING TO THE PE
LOW-LYING S T AT E S . F I R S T _______ ___________________ __________________
_
R * 4 6 - V a LUc"* FUR I HE REACTION, KEV
RKE
R=4 RA rIU t c m / t l a b
L lWRB _
1=4 LEFT AFID RIGHT FI TTI NG BOUNDARIES IN CLAB______________
' PP'
*R=4' PEAK POSI TI ONS, START f.NG VALUES, CH~j"
NON-LIN
PARAM UF EXP BGN
PP( KS)
SCALING FOR THE NOi-l-LIN PARAM UF EXP BGN
XL
_
ST ART X
" 1 = 4 EUUI Va LENT" *TU P P ( 1 ) ( USE J TO* FLOAT)*' "
1=2 ENERGY OF INCIDENT PROTON IN LAB, UNITS OF 10 KEV
EP
CLAB
_
1=2 LAB CORRECTED FOR COSMIC RAYS _
_
____
' 1 * 2 STORAGE FOR" PEAK SHAPES CALCUL AT ED ' BY "XFORii
‘ PEAKS "
1=4 SUM UNDERPEAKS AS RETURNED BYCPEAK
INT
KX1 , KX2
1=4 CH ?)_0' : 2 Pi'S IN LAB ON EXP^ BGN .
_
I * 4 t_c FT Alio*'RIGHT Cl ISM1 C RAY* BDUTlOAK 1 c S
LC , lie
R*4 IRESULUflON FDR CS137
RES
|R= 4 RATIO FD TOP TU BACKGROUND TAIL
RTB _
R=4 Ra T I O' QF "FIRST ESCAPE TO TOP
RFE T
LCIG
SO SLOG, ERROR RETURN
C _ PRINT SDSOPT, NUR.tAL RETURN, PARAMETERS LI ST___
CHK
"
1=4 IGN, CHECK UUTPUl WHEN C r l K . N J . l l
PE
R=4 PEAK E JcRGI c S , IN U10ER UF DECREASING EN. IN KEV
_ I DEL
I -•4 ItLll Cn Ti DM FOR Pl.AK SHAPES
_
“ ‘ PEAKS! IP.2L + 11 CORRESPONDS TO CL Ad ( LB )
1ST
1 = 4 PEAK SHAPE COVERAGE STARTS AT CLA lMI S T + 1)
ARE 1*4 INDICES AMO DUMMY .ARGUMENTS.
1,J,J1,J2,K1
KS
THROUGHOUT THE PRIJG1AM IT IS ASSUMED THAT ADC CHANNEL 0 CDRRESPU US
TO ZEIU POLsL HEIGHT. (AND 11IAT THE _J\UC IS LINEAR)
IF MOT USE SHIFTL
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-369-
Appendix V-A
Properties of States Under Time Reversal.
The importance of the "time reversal" operator in
*
ensuring a set of convenient phases cannot be over­
emphasized.
Its exact definition as well as some of its
remarkable properties will be found in an appendix of
(17)
Wigner's book'
' on Group Theory.
An excellent account
of the time reversal operator properties may also be
found in Messiah's chapter XV.
Very briefly, the time reversal operator 9
is an
antilinear, antiunitary operator whose exact form depends
on the representation being used:
If
K
is the complex
conjugation operator associated with our representation ,
defined as leaving the basis vectors invariant, i.e.
AC /A s < r )>
-
A ( d e f i n i t i o n )
°
(V — A — 1)
/(c/*s<r>=
(antilinearity)
C*/Aso>
O
Then the time reversal operator for a particle with spin '/z
has the form,
- i<Tg K a
in Messiah's phase convention.
(V-A-2)
Here
Cfj
is the second
Pauli spin operator
Qj =
(?
o )
~ l ° s =
The antiunitarity character of
follows:
(/
o )
9 can
(V-A-3)
be proved easily as
If f t ) and /U > are two kets, we wish to show
that the length of their scalar product is conserved under
-370-
time reversal, i.e.
I < 1 Im > I =
f[e /t> ] f B lu > l
(v-A-4)
but we have
[$ lt> ]r $lu> = [~i<Z
if
5
l u>]
iK jP lK ja > ]
=
=
J L
(V-A-5)
<//«>*
By definition, an antilinear operator which conserves
the length of the scalar products is antiunitary.
Now to prove the property of the time reversal
operator which is of importance in chapter V.
phases of the basis vectors
If the
and
are
chosen according to our standard definition
9
/
^
»
( ' 0 * ^ ’/ ) / - * > >
-
„
6
I X j* ' *
=
(-f)r
(V-A-6)
/*'/-»>
their scalar product must be real.
as follows:
This can be proved
From antiunitarity we have (V-A-5)
[ 0 / A f m > ] f ( B l X j w > )
=
< ) j r n l X f 7 r > y
On the other hand, following our phase convention
[ $ / A i 7 » ' > ] f( 0 I X j r » > )
*
= ( - ' / ” < ) / - * /
(
0 2
/£ /- m >
(V-a-7)
-371-
but if we assume a standard angular momentum representation,
the scalar products are independent of 7 n , and
[ & /A y m A ? ^ (o lA 'y r r i? ) =
C A j j m J A 'j rn >
so that finally,
(V-A-8)
< C A jrn ! A'yrn A - C A i'yr? / A
All the other scalar products between the vectors
of basis A and A
other than those corresponding to the
same irreducible representation y
vanish.
, and the same line V r i ,
This proves the stated property that all scalar
products will be real.
Our phase convention is not the
only one having the property,but it is the one most
commonly used, and the one which Rose and Brink^^adopted.
It has the advantage of being invariant under angular
momentum coupling with real coefficients.
i
-372-
Appendix V-B:
Normalization of the Spherical Waves to
Unit Flux.
That the unit flux spherical waves (V-15) indeed
carry unit flux can be proved by integrating on a sphere
of radius
large enough so that the asymptotic form is
valid, the normal component of the flux:
the flux operator:
, .. __.
f *
T it*
where
We introduce
( ~ t
V
J f
^7
implies "real part of" and
the velocity operator.
is
The gradient operator, in spherical
coordinates, is
f
t
\ dn,
-L 1
,
i a * ?>&
-3
)
/i sc*tO b(p j
- i(k/L fSfa
Applying this last operator to -- ^
we
obtain the three spherical components
— k , (k.A.) * & ^ ^
k (k x f t
£^
i(la fS ,) J
}
Y jW
k ( h ) ' 2 e i(k /t4 S ,)x k e S p
Y jW
component along / L
along &
direction
along V
auction;
At large distances, where we choose to integrate, the £
and
<f>
components vanish and the flux is normal to the
surface of the sphere.
Thus
i(k/?■*&)
FLOX
=
..
^
S
LV*xn)^
Tf t t J j vr 1 X*(X
l)~ k7 T
i
fal*) A * il
[ - u a x U ^ f o * ) ] e'M
n u x *
-
-373-
Repeating the same procedure on the outgoing waves yields
Flux = +1.
-374-
Appendix V-C
Derivation of the Blatt and Biehenharn
Formula for Gamma Rays.
From (V-41), introducing the transition matrix
(V-31) we have:
(V-C—1)
The Wigner-Eckart theorem is now used to introduce reduced
matrix elements.
These may be defined in many different
ways, and to be consistent with Rose and Brink we use
their convention (3.25)
The Wigner-Eckart theorem is the quantitative statement
of the fact that the matrix elements <22T,M , / TLf1
(for the same
by rotation.
, /,
and *JZ ) are related to each other
If we know any of them (non-vanishing we
can easily compute all the others.
Introducing the transition amplitude
Thus
,
(V-C— 3)
-375-
where
is the "transition" index which includes the
following explicit quantum numbers
{Z T 'ir ; t s j l ; X L
/=
A/
#
sa -
/ ACUz
r f r t fc - r ' d n )
^ r
JL M w -tf
*
o
t o t;
a 4 *™
"
•
!
( v ‘
c '
4 >
From (V-26) we obtain the explicit value of the incoming
wave amplitudes, and
c f< r
/
V'
w
/ V
X+L+s#-
* 5 “ i F f P - A l l ? CO
4 + f+ s
(-0
I-»
(-0
*
LMw
/7~ - — a
■ ty J ’
f ) L
V * /
\ f
[o < r - < r ) { ( r x y u . J J t
/
Expanding the squared sums,
T
c**.
CUL
Zk T P
T
A
J
U 'M M 'w w '
r - t
tf/ t7'
&
( X L J )(M i )
t *
i f ' (o')
K
M 'jA ) \° ( r - < r J
(s f J ) r )
[o 'U y u .J J tJ
W #
^
f/J
(-/)
J.v
t f 'T
x
( X L 'T
\ f M '-p 'T O & -<r)[o'i>-y& ')
9
The convenient rotation matrices coupling formula may be
introduced,
<
(Messiah C-ll)
I
P
*
■
j
-376-
Combining these two,
f\ ^ ' *
M '-f
L> K
A
J)
JO, *■(-•) /C//A7
A (rf)
A7^
The sum over
<y
)(L L ‘ K
) tf7 f ) K
K
H fJ
(V-c%8)
is performed first, remembering that
takes only the values +1 and -1 ,
Z f
kvh/'
<z / / V # 1
ft -
L ' K ) - f - t ) " /+ w ' F
C->f (f-f-A f) ~ I/'/ 0/
A/= O
With
^
(-//O/
(v_c_g)
being the only contributing term.
After a few trivial manipulations to regroup the terms,
dcr
f
a s i' W
p F
V
/
1
T
£ . , l
I
A
s
tct? tiff/*■
/*■
’
( X L *
V A L 'J' )
x
(fM -y a jifM ^ I
M M 'H
(V-C-10)
( ) ■
The summation over t
and
t'
runs over all transitions
which can contribute to the cross-section.
Performing the sum o v e r / , M
rr
S
& )
fMM*
and M ' first,
Z '( X L J )(X L 'J ')(Jit-'K \ .
[sh -p M
tl'-p W -r t'-H j
(V-C-ll)
tO
lu 'y u H /lji'lX J
/
This result is obtained with the use of a formula first
-377-
published by Biedenharn, Blatt and Rose
(ig\
which can
be found in a much more convenient form, involving 3-j
and
JL
6-j
coefficients, in Messiah appendix C-7; namely,
*
\7?l-y)27»J J \77z -)^?r>y ) K
C ~ //
, : ■ ■ s r ■ ■ ■1
f\r»t
t ' m* 2*yn4 J
This formula and a similar one for
(V-C-12)
9-j
coefficients
will be used repeatedly in this chapter.
The expression for the cross-section becomes
ctJ2
/
2 iz
y
t t 'K
/j'cta-) f J s / )
/
I
Ifcrcr'K }
J \ l' £ X J
( r s f x U
y
x +m '+ w
£ _ , £ )
V O -H /*/*-'
U
\ ( f
Next comes the sum over / / . p i ' and i )
i le s s o rs ? •
v
X t /A '*
X
.
(V—C—13;
IC r.) £>
:
r ^
&
s ix
m
(V-C-14)
Finally the sum over O ' :
<y<rx <r3
(V-C-15)
M -s+ K
- o )
The remaining sum over H
(JU K )
J*£ 'K \
\oonJ
iws / * /
contains only one non-vanishing
-378-
term, namely //= O
.
Collecting all terms,
tt K
fft.Ccos£>)
and since
where &
is the angle between
the direction of incident beam and the emitted
_ray»
t t 'K
The formula is put in a more convenient form as (V-42) in
section B of chapter V, and a summary of the notation is
also given there.
-379-
Appendix V-D:
Particles and Holes
It will be convenient, in chapter V and VI, to
use the word "configuration" to designate a direct product
of single-particle states.
The state of the system will
then be expressed as a linear combination of these
"configurations"
.
This interpretation is somewhat
different from what can be found in the literature
(26)
,
where the word configuration usually applies to a set
of single-particle states with all quantum numbers,
except the magnetic one, the same.
Initially we assume a set of numbered single­
particle states.
{
j /2>^ 132 ....
/«>... }
(V-D-l)
The set of A/-particle configurations:
/ /k t )
9
<*> ! k X
/£> 0
(cv& w . M l k{ OM*. + t *
spans the space of /V-particle states.
(V-D-2)
If we assume
the particles to be identical fermions, we require the
totally anti-symmetric linear combinations of the above
configurations, i.e. antisymmetric under exchange of
any two particles.
For example,
i
I > =
IO ,lk\
k*£
(V-D-3)
-380-
is antisymmetric in particles 1 and 2 because if
7^
is
the operator which exchanges particle 1 and 2,
5 /> =
!k \ //>, -
/k > , =
The notation
- (ne>, t-t \ -
U>, !k>t ) * - I >
(V-D-4)
not standard but extremely
convenient; it represents the single-particle state con­
sisting of particle n
in single particle state Jk .
Also, for simplicity, the direct product notation
&
has
being replaced by simple ^Juxtaposition.
Thus
fk >W> = /*> « H>
(V-D-5)
The totally antisymmetric states are formed by writing a
Slater determinant.
and k
For three particles in s.p. states
, for example, the following antisymmetric
state can be constructed
(V-D-6)
That the state is antisymmetric under exchange of any two
particles is obvious since interchanging two particles
corresponds to interchanging two rows of the determinant.
If the three single-particle states are different, the
configuration above is. normalized; otherwise, it vanishes.
Any other totally antisymmetric configuration made from
states / i)> ,
, I
with particles 1, 2 and 3 will not
be linearly independent of the one above.
This suggests
-381-
a much more convenient notation j
The left-hand side is the new notation; the list of
particles involved,
will usually be implicit.
The right-hand side is simply the definition of a
determinant, with the proper normalization.
is over all permutations cf> of I
( P =o
9
is the parity of
'■
,f
The sum
, and k.
, and 7^
for even, 1 for odd
permutations).
A general definition can be given along these lines:
(V-D-8)
From this definition we have in particular:
(V-D-9)
where the + sign holds when (j> is an even permutation and
the minus sign when
Cf
is an odd permutation.
Before introducing creation and annihilation operators,
particle-hole states must be defined.
Frequently the
situation where some of the single-particle states are
almost always filled and the others almost always
empty is encountered. • Let
F
be the set of single particle
states which are almost always filled.
We assume a finite
-382-
number
A /
of these, which hereafter will be referred
to as being "below the Fermi surface".
Since the
division in two sets is arbitrary and merely for convenience,
it is important in each particular case that
F
be defined
clearly by giving a list of single particle states; in
practice, the Fermi level is always defined between two
subshells.
F ~ axFttouvvf M t o f
/\f
S. p s&6c£teA> = { I k p . „ . l k
y ]
(V-D-10)
In defining hole states, the difficulty comes
essentially from the phases.
We wish to define these
states in such a way that the holes will behave, as much
as possible, like particles.
that the states
two holes.
For example, we require
be antisymmetric under the exchange of
For this purpose, it is essential to*
introduce a standard numbering of the single-particle
states in
F
, say 1 to
.
We then define a particle-
hole state, with the semi-colon notation:
— ho/es —7 ' - ^ - p a n i t c / e s c t A / x u . &
_
=
f / , 2 t ... C k , £
... n / s s / n f ) . . .
(v-D-ii)
tfF - / 3 A/f , < * , £ ,
... >
If the holes are not in their standard order, the particle
hole ket is defined
as +
standard order, depending
the particle-hole ket
on the parity ofthepermutation
which restores the standard order.
Thus if
F
of
is given by
-383-
F
=
{
/»>,
(V-D-12)
ts>, /&>}
n > ,
we have, for. example
I Z , 4 ; C > =
N , z , /; 9 > *
I/,3 ,S ,C .,? >
-//,z ,a ;4 >
9 )
=
(v-d-13)
The particle and hole creation and annihilation operators
is the number of particles in F
may now be defined ( A /
0%
The operations (Z
and
b
~
O
x /
i x l
)
o c c u ^ o id >
increase the number of particles
by one, and it is assumed the particle created is the last.
by one, and it is assumed that the last particle has been
removed.
This remark is crucial, because without it the
phase of the operators is not well defined, nor is the
result of the operation.
It must be remembered that
although the particles are physically indistinguishable
we do distinguish between them in the formalism.
The
physical indistinguishability of the particles is taken
-384-
care of in the formalism by the Pauli principle, not by
loosing track of the identity of the particles involved
in the states.
It is for this reason that<we must treat
a notation such as
l i , f ,.
..., * j >
with extreme care.
This state is only well defined if
the particles involved (as well as in what orders they
are involved) are specified in the text.
/ L>
The notation
••• > k , £ \ 2,
is much superior, but we only use when there is a risk
of confusion.
6
f k
Thus the creation operator bt- above is
;*.... p j
¥
I i ,;.... k
k ; ° < .../> ^
^
(V-D-15)
We shall now illustrate the operation of the particle
and hole creation and annihilation operators, assuming
that £
is given by (V-D-12).
<?//;>
=
We have,
/ ;*>
as/3;7>=o
a,/2;K-r>’ -a7lz,z?>- -/z;g>
2,
is not defined
6 ? /
g 7>
-
//,z ;g ,r >
(V
_D
_16)
b3 l z , 2 , s ; > - - 4 / 2 , 2 , s ; > -- - / 2 , S i >
From the definitions above we get the anticommutation
relations:
C a „ tft - «
C * ,A %
fc/Al - *
H
4 ^
' o
- *
( V _ D .
1 7 )
-385-
Any particle operator anticommutes with any hole operator.
They would obviously commute except for the added phase
, .A/ '
C ~/J
in the definition of the particle * creation and
A
annihilation operators. This { • / )
factor has been in­
-
cluded, despite the increased calculational complications,
so that our definitions might agree in phase with the most
popular usage.
Anticommutation relations involving particles only
follow from the definition:
(V-D-18)
Similarly for hole operators?
’ °
‘ °
iK
(V-D-19)
a
,. t p , ■ 4
7
'
V
The operators defined so far are all independent.
One
set operates only above and the other set only below the
Fermi surface; all are required, to permit excitation of all these
states.
It is interesting to derive the relationship
between these operators and the particle creation and
annihilation operators C * and
C
which would be defined
in the absence of a Fermi Surface (i.e. if we choose to
keep track of all particles and not introduce the concept
of holes) i.e.
;
-386-
c f / / i ..
u)
/A y U -
s
=
=
/ yU . . . . j £ >
o
A X o r? < rt o c c u ^ e M .
(v-d-20 )
As an illustration, let us assume again that F " mis given
by (V-D-12), and we have
c3f /1,9,3 > =
C/
13,1,9,37 = -/3,/,9,3> * O
ll,9,Z> => /3J, 9,Z>
l l, 9 ,Z > - - < Z z l z , 9 l > -
-/< ? />
(v -D -2 1 )
We have
A/
a « H ,f,
k ;
(■ /)
/i,/,
; /,
/V-7
/
= £/.) A - k 'x n d u x a z -» ^
(T, / ♦ “ N 'jm d ic to )->^
a&
/
s e F ,
./•>
^
(V-D-22)
aX f s £ £
jy tc e p Z i }j \ ■■■>&
The two states being operated on all the same, and the
results are the same, hence it may be concluded that the
operators are identical
&o( r
C cJso
-
C 't )
(V-D-23)
This is the usual definition of the operators <2^ . This
/y '
also justifies inclusion of the phase ( / )
in our
definition of
given above; it would otherwise have
appeared here instead.
-387-
Between the c 's and the P 's
as obvious.
the connection is not
In the following, we assume that all the
states are given in standard order, and that s.p. states
*
above the Fermi surface are all empty.
The latter do not
interfere with operations below the Fermi surface so
that the proof is still generally valid.
(V-D-24)
aM
s e f & d y i.J k ...
where ??/ and
p?(-
a l l s s f / u l -■ /& , -
are the number of holes and particles
respectively below the s.p. state
It follows that
(V-D-25)
(V-D-26)
In the same way, we obtain
(V-D-27)
This extremely important factor of C - /) 1' 1 , which comes
naturally
in this derivation from the requirement that
states be antisymmetrized in holes below the Fermi sur(25)
face is qualified by Brown'
' as "coming from the
iriur&y depths of G r o u p ,Theory".
\
The proof that
C
/
3 '* /
•= £ / )
should be illustrated
-388-
by an example.
4 !za ,
=*>
Taking
F
7> = - /z, s; <iy
4
(V-D-28)
c J / 1 , 4 , < 1 ,9 >
l 3t
=
b f IG 1 7 >
*
■»
^
o , & , 9 > - - t l 33 , a , 6 , * >
6^ ’ }
On the other hand, for
=>
as in (V-D-12), we have
=
; 7>
! ,> 2>3 ' a ‘ s , 7 '> "
q, I l , Z , 3 , U , S ^ y
= '
O
f
7>
Repeating for all possible states yields
(V-D-29)
^
etc.
The behavior under rotations of the particle-hole
states defined above remains to be investigated.
The
point of view adopted here is that of group theory.
We
assume that the single-particle states from which the
antisymmetrized particle-hole states of the system are
constructed transform under rotations according to welldefined irreducible representations; these are the
rotation matrices
j -
(Messiah uses
rfn
M /P
instead of
as defined in Messiah^^
j-
Mn
To denote the vacuum, the notation / 0 ) > is used.
The state which corresponds to all the s.p. states below
the Fermi surface filled, and all the others empty is
called the particle-hole vacuum and we use the notation
-389-
f c 'y
f o r its
(V-D-30)
two sets of single particle states which transform as
described above,
•/” > ' -
tft* 71^
where
? > /” > -
fs
i s
n
y
Jt m ,> Q
rotated
i
™
state,
(V_D' 31)
the operator
, and the J D ^ C % ) are Messiah’s
of finite rotation
rotation matrices.
X
777
Then the ( 2 J M ) states
-
f a
<f
7 ^
t o y
ol
transform according to the irriducible representation
J O 7.
In the case where the two sets above are the same,
IS My vanishes
for odd values of
for even values of
different the
S
.
IS My
S
and is not normalized
if the two sets above are
never vanish and are always
normalized,even if f a - f a
•
If the two f a s are the same we
must use a slightly different notation which distinguishes
between the two sets above - for example an additional
index.
Putting
& *>
f t ~ f t i~ fat
*
( i, 1/ m)
-
T
c j . mi /o p
(I, ir i ) f t
ft
/O)
(V-D-33)
-390-
Interchanging dummy indices
and rrtz
, and the order
of the operators,
/cw> -
frfX e frr (* ,& )
/o>
cv-D-34)
Which implies that //^vanishes for odd \T , if ' < X ° f2 %
In this case, the
(O ' A O
for even
J
are not normalized;
indeed,
< j-M lJ tf/
< c ^c) Mi c^
= J7 2T ( £ ; ip Z t X - n ) < o lC r f C w
; cP Z d o ^ ' < 0 ^
^
m
c ? c * lo >
'
= (7+C-/f)
<V D 35)
(V-D-36)
It is because the normalization is lost when single­
particle states, which transform among each other by
rotations, are coupled together, that coefficients of
fractional parentage are introduced.
For one-particle one-
hole states there is no need to introduce them explicitly,
and although we shall consider them later, only very
simple cases will arise.
The proof that the /TOtransform among each other
according to the irreducible representation £ )
simply based on the assumption that
form according to I ) * ' and
is
and ]
trans­
respectively; before
demonstrating this a more general result is derived.
-391-
To investigate the behavior under rotations of states
involving many particles (like hole states), it is very
convenient to establish the transformation properties
t
of the creation and annihilation operators.
>
cl
-»
?1?
c irn
t
.
*
( v -
D
-
3 7 )
This is equivalent to establishment of the commutation re­
lations for the creation and annihilation operators with
the rotation operator
.
This is interesting because
when many particles are involved, the creation
operators
become extremely complicated, yet intuitively it might
70^ ,
be expected that it should transform according to
as it obviously does when operating on the vacuum:
2 c'" 2 " / o >
* J
*
-
% N m >
V
= T
777'
! / > • '>
"
(V-D-38)
It remains to prove a very powerful theorem,
namely, that the above result holds even if the operators
do not act on the vacuum.
?
CL
0
?"
-
In general,
Z &
■#)>
j K 7 c t .
777™
J *7
This may be demonstrated as follows:
(V-°-39)
let / J f f ) b e an f ij- t
particle state (completely antisymmetrized) of good
angular momentum v7" and projection M
.
(This is not a
-392-
restrictive assumption , because any state may be expanded
into a linear combination of such /J ~ M >
states.
(V-D-40)
Using the definition of
and expanding the determinant
yields
M>
-j.
.
(V-D-41)
1
Summing over
t i ' and reintroducing
C^
/...i-tj i h
we have
? 5i n * »
which holds for the complete set o f f / X H A j s o r all CT and M
;
thus,
?
Cf w
?
^
Cjn ) '
J T
(v-d-43)
The transformation properties of the destructive operators
are simply obtained by remembering that they are the adjoints of the creation operators.
?
?
But the (linear) rotation operators R
r
- <?<£ r f -
x x > L w
J
(V-D-44)
are unitary, and
$m '
(V-D-45)
-393-
The set of &&■ / states
ATM > .
T
(V-D-46)
& ct!o >
W ,M Z
v
0* 1
j'
trrnsform as ! D • this is now easily obtained:
/i
.
% u m > -
T
c i f ^
r
(i,
c L f f l Z c L , ? 7 % > °>
0
7 rtjrt2
*
(V-D-47)
And using the reduction formula for rotation matrices
(Messiah appendix C-12)
T £ ] ? t f > - T £ ? ,(£ IJ M ' )
A'
(v-D-48)
Most of these proofs for two-particle systems may
be obtained in a much simpler way.
For example, the
transformation properties under rotation of the operators
C jn
and
c jt f
are not really needed.
The case of
coupling of two particles from the same subshell for example
can be dealt with much more simply because
AM>
=
T (t, 4 d f) / f ”0, p ,
(V-D-49)
-394-
is already antisymmetric in particles 1 and 2, and normalized
for even J"
(vanishes for odd J" ), and creation operators
4
do not have.to be introduced.
We have, however, exchanged
some of this simplicity, which exists in the two-particle
case, for a more general, much more powerful approach,
which will be needed in dealing with the holes.
When considering the particle-hole scheme, we have
CL ,C L ^ , />
the operators
and
6 ^ instead of C and C * .
Everything noted above concerning the operators C and C 7
is true for <L and o f
, provided that due care is
exercised, in :defining the Fermi Surface, not to divide
a subshell.
If a state is included in
F
, the subspace
into which it transforms by rotation (subshell) is
assumed to be included in
t>S
F
completely.
To define the
, a standard order must be defined for states of
F.
We assume they are ordered in increasing values of the
magnetic quantum number 70
///>
t
Because
(V-D-50)
always even for nucleons, it does not
matter how these sub sets are ordered among themselves and
we have, according to our previous definition,:
t
v
-
ft)
C j+ n t + f
V
l i t A C W -A )
(V-D-51)
-395-
To maintain compatibility with the standard literature
7/i
is changed to
in the labeling of
b-
.
This is
not a fundamental change hence
/
<
-
/"»
, f
7"
c .
< ' ’- D - 5 2 >
It must be emphasized that
4,creates a hole in s.p.
v
* not l y '™ } ! Similarly
state
b
.
(V-D-53)
V 7”
The transformation properties of the
P s under rotations
are easily obtained by their definitions and the symmetry
properties of the rotation matrices.
/
_
■
.
c - f t
70
=
( - o '” ' ” ’&
?!*.?• I
&- m<- mr' >
(f
( r n F
Q L < ’>
(V-D-54)
Similarly,
(V-D-55)
' 7E? Y
2 " -
t
£ )m* m , ( * " ) kj „7” '
-396-
Therefore it follows that the hole creation and destruction
operators transform like the particle creation and
destruction operators:
this result is obtained simply
by using a standard order of increasing values of 771 •
This works a great simplification in all subsequent
formalism.
In particular, one-particle one-hole states
of good angular momentum j j r i )may be obtained simply as
.
lz r h > -
27O )
(1 1,J‘) a f Af
(v-D-56)
27777)'
where we have observed our standard coupling order:
particle first, hole second.
-397-
Appendix V-E
Many-Particle Matrix Elements in Terms of
The Single-Particle Matrix Elements.
Part 1:
Decay to the Ground State.
We must evaluate
!p /C >
T
=
M7n>
(£ ? m ‘7 2 )\fT
9 %
In principle, the sum o v e r ^ and
When < y /
single particle states.
))
however,
~
*
V
(V-E-l)
) 1 0
involves all the
is applied to ( C ) ,
must operate below the Fermi surface and / x
above, otherwise the expression vanishes.
% { lu lS Pl » cjc„
f
<plS'lv> £ % } l c >
Thus,
-
(V-E-2)
/* • *
X
(< a t s f o
& & * /» > s y - n
4
and the sums over y£t and
P
p
j 10
now run over all single-particle
states above and below the Fermi Surface respectively.
We have
.
~ . f <c l L 4 „ 4 „ . L j C>
*
/ V
“ (~ i)
*
*
.
(V-E-3)
y -M O
And similarly for the neutron part, so that
-398-
< p 'J M
ov
± 1 * 1 0
=
T £ / / *
7»m '
T
^
. C -'F
c rA 4 q *
S f'y rt !
defining
I f ' ' *> ’>
S
y
S
*
_
a v d
iT
=
—
;V—E—4)
”
2
c?A „ <T”
— r—
-
(v-ii6)
we obtain formula (V-115)
Part 2:
Decay to Excited States.
We here must evaluate
&
*
*
* K
*
(Z r /iis ^ ify
$ ! cy + <j»/5’7 r > c j c „ J
* .>
x
>
>«-=>
All terms of the sum to be evaluated involve a product
of 6 operators of the form :
l>a c^c a *&f
There are basically two possibilities:
The *
term is, or
is not, the adjoint of the C term, i.e. ( c rC = / ) or ( c fC ° - c c i)
-399-
Collecting all terms of the first type involves
6 a a f6* X
<y*IS*ly>
and simply gives the expectation value of
closed shell multiplied by the scalar p
r
for the
o
d
u
c
t
^
We are only interested in the case for which these two
states are different, such that the scalar product
vanishes.
For this reason the terms with
dropped.
The term in <£C { b & C * C <Z*l> I
C *C - /
are
will give
no contribution unless each proton creation operator is
cancelled by a proton destruction operator , and similarly
for neutrons.
in acting on
This is because the operator
1C )
b&c^c
must re-create ( C l if the matrix element
is not to vanish, and therefore the number of protons
(or neutrons) should not be changed by its operation.
Since they appear already paired, the only non-vanishing
possibility is that in which all 6 operators belong to
protons or all 6 operators belong to neutrons.
Basically
this happens because the electromagnetic interactions,
unlike the weak interaction, does not transform protons
into neutrons or vice-versa.
It is still true, however,
that the electromagnetic interactions are not isospin
conserving, since the interaction of protons and neutrons
with the field are not identical.
alone gives
The proton contribution
-400-
2 Z ft C O
J jJ t
(^ r r i p t l j ( 7^x rr)p M t )
*
S < c l L , a. c ' c , a * b[ jc > C/PSpi*>
7> ™>
»
& * , j x mi
_, ,.
(V-E-6)
The remark above concerning protons and neutrons applies
equally to particles and to holes.
Each hole created
must be destroyed and each particle created must be
destroyed if
S C l b C L C ^ C CZ>*b */
For this reason, cfa
is not to vanish.
must both be particle operators
or both hole operators.
aj>
S
c»
/-#*
j'- m '
h. A s ^ / k J rO
ft,™ !
1"
J ”
j »> -
.
I , ,
.
! 0 < t r » l S tfa -o o 'fa
/
*
(V-E-7)
The first sum runs over states above the Fermi surface,
and will yield the "particle jump" where the hole acts
merely as spectator, and the second sum runs over the
states below the Fermi surface and will yield the "hole
jumps".
f a
^
/
More precisely,
< C lf t n ! */>'
‘f t j n f a m ; 1 0
»
(V-E-8)
-401-
The contribution from the neutron jump is computed
in the same way, and the final answer is given in V-D-2,
as (V-117).
-402Appendix V-F:
Elimination of the Hole
Part 1:
Decay to the Ground State
The purpose of this appendix is to compute the
reduced matrix elements of a tensor operator JE>L in terms
of the single particle ope. ators
We used the definition of reduced matrix elements (V-C-2)
introduced in appendix V-C on the Blatt and Biehenharn
formula.
The sum runs over three indices, not two,
which explains the
J"
in the denominator.
From (V-116), we have
<#’<r»*ts£/c> ■M?/4 ( 0 T
0 0 *
Also, from (V-C-2),
Combining the three results,
This expression may be reduced to a 6-jcoefficients, but
more elementary mathods may also be used.
Since
X ^ A>
After introducing this, it becomes possible to use the
unitarity property of the Clebsch Gordan coefficients, and
JL (V-F—2)
-403-
Finally,
Expressed in terms of a 6-j coefficient (to compare with
(V-119), which involves decay to excited states)
' ^ / / y no - ^ ^
Part 2:
(V_F"3)
Decay to Excited States.
We first calculate the particle jump contribution:
from the definition of the reduced matrix elements, (V-C-2),
( /£
,) < ? *? ” -
'
/
(v-p-4)
and from (V-117)
j-k+M+f-h'+S
j j > i 'x r * > p - 1 ( - 0
L" *
W
l f
*
j7
*
(V-F-5)
the definition of reduced matrix elements enters again :
- £ ' tLrf
(£&)<!•&/•>
Therefore
< < A jt H iS tJ } ' t , x * >
'
P
(r * ■ -* ) ( t
"
2 7 fa )
f .H M '/A /c 'y y
W
*
k
r Jn ) ( t ' r X
f) ( t f°o ) ( y v ( ) < / ? V >
<—
8>
-404-
By introducing a sixth 3_j
coefficient,
/L L O )
(V-F-7)
- T
j - i f
w
< p s - //p
(V -F -8)
^ / * y “ ¥ X ~r * * - ’? + L ~M '
f-A + u X
(-’)
= ( - //
but
for all non-vanishing contributions.
the
/" In
9-j
We then recognize
coefficient defined as:
1,3
f '3 ]
(fa
f a Its )
(fa
fa fa )
y
A
\% rti27
)/3J [7Jlt 7>tl7J1}/ \n v 7>n r>33/
acc v
-
y*
(
fafa J( f,2 f a
l»/r *2 , W
fa )(fa
(V-F-9)
fa fa )
[ */« *11 * 3 i ' ' * '3 *13 f a t
This extremely useful formula will be used repeatedly
through the end of this chapter; it is not given
explicitly in Messiah, but can be found in Rothenberg’s
, (18)
tables
Finally,we obtain the particle jump contribution
of formula (V-119).
This contribution may also be
written in terms of 6-j
coefficients (Messiah C-9)
" " " n
It is also possible to derive this formula directly in
terms of 6-j
coefficients, but going through a
9-j
-405-
coefficient instead makes it substantially easier to
keep track of signs.
It is instructive to compare this last formula
with the one obtained for ground state decay.
Apart
from the phase, the reduced matrix elements differ by a
factor of &
y
.
The transition rate for a decay to
the ground state, everything else being identical, would
z(zpO
therefore be
faster than for decay to an
excited state!
The physical reason for this effect becomes clear
if one looks at the inverse reaction.
This then just
says that absorbtion of gamma rays by a subshell filled
with
Z (Z y jt)
particles will be
Z ( Z f l'- H )
times
more probable than if one one particle was present in that
subshell.
The approach of Lane
wayss
(13)
differs from ours in many
His treatment makes use of isospin formalism but
not of the concept of holes.
It is instructive to compare
his formulas #81 and #82 with our results.
Whether or
not the decay is to the ground state is taken into account
in his formulas by the factor
n < A / } * , > <A7)«.>
where 71
is the number of particles in the subshell, i.e.
for Lane since he uses isospin formalism.
X X 1}
> and
are equal to
are coefficients of fractional parentage and
I
'/( f it
f°r excited configurations, and 1
when the particle fills the hole (ground state).
Thus we
-406-
get back the factor of
l/£ ^
on the amplitude or
in the transition rate.
It is interesting to
note, therefore, that we have been calculating coefficients
of fractional parentage implicity.
For
O
, this facto." of
Z ( Z f * J / ) is equal to 8.
Its presence accounts for the ground state decay being
much stronger than the decay to any of the excited states
when averaged over energy.
jump will now be derived.
<yA m
t is i y i
The contribution from the hole
From (V-117),
*
= " z o f f jx
( i k J )(F
= 5
( ',)V ’l< 6 - ’i ,i S t i h - i >
x, W
? ° )
The reduced matrix elements for S
A*
< k 'i 'i s t ik i>
can be expressed in
terms of the reduced matrix elements for
the same way as for the particle-jump part.
Eckart theorem twice).
in exactly (Apply Wigner-
After introducing an extra
coefficient with a zero for recoupling,
.
//////•/ //V/ - 7
,
V*
For all non-vanishing contributions to the sum,
XiM-tL+H'-l-lt-l'-fp/t
L-fjfk'-XFinally,
9 s f /J
contribution of (V-119).
we obtain the hole jump
’
-407-
In terms of
6_j
coefficients,
<rjk7m%ij>/L'xt>/i *
7
I
7
(xJ j J jp s*n >
< v
.
F-13)
-408-
Appendix V-G:
Electric Single-Particle Matrix Elements.
The operator
_
Z
<«
f ,
TL T f = &Td// [ZLCZI^)
. r .i,iy )
in
(V-G-l )
can be considered as consisting of the product of three
factors:
A statistical factor which is the same for
neutrons and protons, a constant
^
which depends on the
nature of the nucleon, and the operator itself,
# *1 *
.
A
The first task involves the evaluation of
where ( - i f A and
and
are single particle states,
S ' are the orbital angular momentum quantum
S 1
numbers; S and
equal to
the spins of the nucleon are both
and have been omitted;
y
("and y / ) are
the result of coupling X ,S (and C ) s ') .
From the definition of reduced matrix elements,
(V-C-2), we have
< 4 / H
i V ; N
p
=
I
decoupling X s
■
< V - G '
2 >
and < s ' yields
7YI rn'O'O'
(V-G-3)
’>
-409-
But
« < s ( r l < Y m l i LJ i}£li W w O / s'o “>
where /s<r>is the spin state and / - / m b carries the space part
(orbital and radial).
The operator does not act on spin,
t
< S (r .(S ,c r''> *
so that
,
but it is preferable to intro­
3-j
coefficient to replace c$\ , &
J
ss'
The state //r»> has a wave function of the form *
duce a
where
0,
and
^^G cp)
are the spherical position coordinates
of the particle and
CG^CSi)
is the radial wave function.
■jS.
The phase I
is necessary to ensure the previously
assumed time reversal properties which guarantee real
scalar products,
(V-A-6),
4 -m
0 /Y w >
= (-*)
//-»?>
(V-G-4)
It should be pointed out that although we have suppressed
all quantum numbers other than £
and ^
there are many states with the same £
for simplicity,
and
y
and the
other quantum numbers needed to label them (principal
quantum number 71 for example) would appear labeling
(for example,
)•
We proceed to evaluate
< 4 m ln U ‘ /
K P in / Z i*'
YL tf
/ Z ' m 'b
:
//V> (V-G-5)
-
i u
£ U [Y m
Yw
LM
Y (s l) ^
e'rr,'
f V A ) aj* - >
J *
4
The integral over three spherical harmonics is well known
(Messiah appendix C-3);.
fY (J l)Y (/I) Y
'
(SL)dfl -- (0 o o /(nrrtnt)
T^ p
(V_G_ 6)
-410-
After
collecting all terms,
P l s T n o r l i L/LL ) / 1 -I's'W es'1)> =
LM
.
(V-G-7)
T 7T T M e ,L\('J-e,L Ys s '°] („ „
1
M
"
^ .o o )
This matrix element is, of course, real;
L +4 -4
unless
l LA L
is even.
\
^
( 'c ,o o )
vanishes
H
The reduced matrix element
H P / ) becomes
J iy u '+ Z /u + S - O ' j l + t - - e ' + m
,
< t/ln
.Li l YLHN cj ‘> - I
0
.M -4
j2
I
H
i
n/vir
£0
x
t f &>&"/>*> s r t '
)
[ u u . ^ c U
X
Koco/ J 1 r
(V-G-8 )
i'LS) /d s j \ f J 's 'i' \ ( d £ L \ ( s s
\ Jtl M -/X / \ 7 r )O '-/l/\ m ,0'-jU'
We introduce one more
3-j coefficient for recoupling,
like in (V-F-7); with this extra factor,
'//h l l V //J' v\
lH N
? >
-&*■£-4'
=
/JI'L\ Cfi,
\F * r
/, ,,
(a o
o ) J aj ur A
<*-
x
1
„
2 7 (rU
/V'z / ) f 1 5 t \ f 4 ' s
J L m (E y u ,) (
4*tu,
+7n+S-o'-M
y
\f s s'° ) f L 1 ° )
) \'7 n w ' M ‘) (.f r ,- o r ‘k / V M - M ' k J
The sum runs over all 9 magnetic quantum numbers.
For all non-vanishing contributions to the sum,
i * +A 'srrt*s -O'- M
C 'l)
'
(V~G_9)
',hS
(" F
-411-
and using formula (V-F-9), for a
g_j
coefficient gives
rT T T J
1 i L*.LY
L t I'P =
i LtUe
*
(V-G-10)
d
o
t
)
d
t
o
]
Introducing the constants immediately yields the expression
for the reduced matrix element, (V-127).
-412-
Appendix V-H:
Single-Particle Magnetic Transitions.
Part 1:
Preliminaries.
The magnetic operators are somewhat more difficult
to evaluate than are the electric ones.
~J~
butions (V-123)
an<j
(space)
The two contri-
T~^ ( s p / p )
are of the
same order of magnitude (although usually
(spin)
dbminates) and unlike the electric case, it is not
advisable to neglect one of them.
In this appendix we
derive some preliminary results needed to evaluate the
matrix elements of these operators, and proceed to
calculate the space part.
The spin part is the object
of appendix V - I .
a) Definition of standard components from cartesian components
(Messiah Appendix C-14)
(V-H-l)
where
the cartesian components of
the axis of quantization.
form like
b)
A
} \(
and \ ( are
V and ^ has been chosen as
These standard components trans-
(Spherical tensor of order 1)
The gradient formula:
(V-H-2)
It is easiest to prove it in cartesian coordinates:
-413-
raf * a 3 / d?
x i
i 2>X3 dyx dp J J
-
Lax*
J
Z 2
^ax
and likewise for the y
c)
and ^
components.
Scalar product of two operators of order 1:
A -s
-
^ ( r O * A „ S'
A
'S *
( C/SftAbL X t 'S =
(V-H-3)
A XSX A A yS y 7 J ljS j )
This property can be obtained from Messiah appendix C-15,
or proved directly by substituting the expression for the
standard components, as defined above.
d)
A # , A *" ^
Evaluation of the operator
XZ^ is a standard component of A . and
A
, where
is its length.
From the definition of the spherical harmonics
( /g * ) ^
(Messiah and of appendix B-10) we have
Z
=
7
c
^ = /-/
.
For
(£ « -%
L -/ 3 z - L
Of the three possible values
we retain only
, r H
L
, the
^
YfH
* tllus
) Y
x n
and Y * /2 + t
3-j coefficient
vanishes; this can easily be seen from the parity of the
spherical harmonic
fp .
to
'(.h
^l h
iS
*
^
which is
( 0
.
The parity of
and it must therefore be orthogonal
A more subtle reason cancels the contribution
-414-
from the */= L f i /
Vp2A. * " * '
term:
¥
„= O
L+> » l
(Messiah B-10)
In all the following derivations, it will be understood
that
^ L -l
,
*
Keeping only the non-vanishing terra,
- (f)*
The Laplacian
o f H (ft» $ ) f a
? r
P7
<v-H-5>
.
may be expressed in terms of the
orbital angular momentum operator &
; in invariant fashion
we may write
s.*
[/2 =
(V_H_6)
(vz)U- F) - Z*
which in spherical coordinates takes the form (Messiah,
appendix B-ll, footnote)
U 2
V
~
1
7i
— 2 A. —
F V " /
A2
Z ( Z I * 0 A 1" Y
-
And finally,
Ya/ a a ' Y i "
( ? )
2
Z t? Z &
f *
( f t f f -n l ) A * f a
e) Matrix elements of the spin operator^
The spin operator &
a tensor of order
IV - H - 7 )
/ S 'c r'ft .
is a tensor of order 1, and
~
Zi,
.
The procedure here consists
in evaluating one of the matrix elements, and in using
the Wigner-Eckart theorem to find the others.
immediately yields
This method
-415-
< C s o -/£ J s 'c r'A = ( - / )
< S '& l t ? l s '& >
s - r / S ' i s ) (s' i s
u r W v \ A o -'a )
&
-
...
* k t $ > !s
(tt o - t )
Therefore,
„ £-&• r^-i
X s c r /^ /s 'o " > * ( -/)
/s ' / s )
\ < y'jj-o )
(V-H-8)
f )
Expansion
We
A
of
J
have
Proceeding
in
=
the
k
same w a y
< 4 k 1 ^ 1 4 m A = £i)
We
immediately have
the
3-j
appendix
]T
in
terms
as
described
//
k>.
above,
k+i M 1 4 \ f J t d )
L ^ k ) (/o-iJ <4iU J 4 »
is
more
of
course,
difficult.
/g , / )
C-5),
the
M k X U ! <, U m A
X 4 /1 4 o ! 4 fA - /
coefficient
of
a
( / O-/' * C~0
From
but
(Messiah
[J C 4 h ) ] ^ 2
-—
And the desired expression follows immediately:
=
Part
2:
Space
The
operator
L
k
7” ° ( w ;
the
-
4
y /M n )
(W i t - k l
(V_H _9 )
Contribution.
space
has
0 )
contribution
form
(V-123)
to
the
magnetic
transition
-416-
This operator may be considered as consisting of three
parts; a constant independent of the nature of the
particle, a constant ^
which depends on whether the
particle is a proton or a neutron and the operator itself,
XKAS'XJ-I.
The first step consists in evaluating the matrix
element
(V-H-10)
y-E s-rricrl
) - J U s 'T n 'y ^ ^ K s c r ! s y > y / r n l( z [ A L ] /I A l - p T r , ' )
where
L -f
(V-H-2).
Noting that 17 s i 1 ) ( H
,
and we have used the gradient formula
vanishes identically,
(Messiah B-10) and using the formula for scalar products,
(V-H-3), the following expression is easily obtained for
the above matrix element:
J
f(t sX)
I
SC
V
The operator involved in this matrix element has
been calculated at (V-H-7), and
% (i f )
* (v-h-11)
**C
At this point we can use (V-H-9), which leaves us with the
matrix element of
/
; this is evaluated in the same
way as for the electric operator:
-417-
F A s -m p I
V ( j i u Y ] ^ M ) ■4
/ ■ t's 'w o '1'} =
1 ? \J3 l *
v y /f t
( H
e ,n
,)
a
m
x
’z
v
h
I )
m
'
i )
*
We Already have the required sum over three magnetic
quantum numbers, and it is advisable to evaluate it
immediately in terms of a 6-j
5 V
2 (~ 0
l
coefficient,
£ )(£ ■ /* •)
( L S I )/ L
(-y r j k W / O M - y t l) \.7 T ) ~ t 2 - k J
v rfk
£
lv - H -1 3 )
/ J
L
using a formula in Messiah, appendix C-7.
Finally,
F t s m c r j V (s iLi* Y „ ) • £ I i s ' - m ' c ' } -
Lrt
( S
iZ
t iZ
X
X
flim
^
Y *
£•/)
x
s € ~ i'
:
L.
From the definition of reduced matrix elements and the
expansion of
IZ j^ x n
in terms of the
terms of the
f'lsm o'P
and /
,
yU yt'M /
/m tri o'o'
(J s p
U s n A V x
(V —H—1 5 )
? ? „ ) - * ! 2 's n ^ >
-418-
where we have assumed
Z + -£ -■ £
ducing an extra 3-j
coefficient, we obtain a sum over
to be always odd.
Intro­
all 9 magnetic quantum numbers, which may Tae reduced to
a 9-j
coefficient:
<ijUWi*YL)-Hep m*.i
U
TTFrip ( i Z i H W } & )
x
*(V'H'16)
For all non-vanishing contirubtions to the sum,
Z ' - t i ' + S - O ' + r v j l-t M
c-/f
=
Jt-s+L
or
The final result is given in (V-128), after introduction
of the various constants.
-419-
Appendix Vi:
Single-Particle Magnetic Transitions, Spin
Contribution.
The spin contribution to the magnetic transition
operator is given by (V-123),
& Z 7 )H
-
rn c r /
To evaluate
W U Z L J O /^ S s
i L '(fan)
k M '
^
I - f ’s ' m ' o ' ' }
we proceed
substantially in the same way as in appendix V-H:
After
using the gradient formula and the expression for the
scalar product, and eliminating the various vanishing
contribution we get
Slsm or l(\7/zl i L~* )faM) •&! t's'm 'r'ft) = Z *
7 ?
(m
i
*
)
(V-i-l)
where
<< - / - ~ /
as usual.
< ( S C , / S i / / s '0 " f t >
The expression
(V-H-8); the matrix element of
was calculated as
2 -^
is evaluated
in the same way as for the electric case.
Therefore
pm
-O '
4'jC ys'fS ) f / L * )f S /'J
(V-i-2)
) fu u toS'Z
The reduced matrix elements of this operator between states
/■ /j , y
and
/ P fa 'P
are obtained from the same
formula used for the space part,
-420-
rm 7 » '0 ‘0-
(P C C /) < &
^
( T y u
v ,n
w x fr)
I f f
s u -s '„ ',> >
(v-i-3 )
Putting together the last two formulae gives us a sum
over six
3-j
coefficients.
And since for all non­
vanishing terms in the sum,
M + s -O '+ p y c * .' _ ^
& # '- s
it immediately follows from the expression of a 9-j
coefficient in terms of
3-j
d i l l v (X i‘" ) f ) - s t t y >
tc
J.
t \
coefficients, that
-J ? 1
-
j 'M A c 1
.
*
(V-i-4)
With the proper constant, we obtain (V-129).
In this case
it is not possible to avoid the use of a 9-j coefficient
without seriously complicating the derivation.
For Ml
transitions however,
coefficient
again reduced to a
== /,“/ = O and the
6-j
coefficient.
9-j
Thus for practical
computations involving felectric multipoles of any order
and Ml multipoles, the
avoided.
9-j
coefficients may be totally
Nevertheless, their higher symmetry makes them
suitable for manipulations of formulae.
-421-
Appendix VI-A
Conversion from Isospin Formalism to
Proton-Neutron Formalism
Protons and neutrons being distinguishable particles,
it is never necessary to antisymmetrize the wave function
for an exchange of a proto.. and a neutron.
Thus the
state of the nucleus can be written as a linear- combina­
tion of direct products of antisymmetrized states of the
protons times antisymmetrized states of the neutrons.
precisely the state of a system of ^
protons and
More
72-
neutrons will be a linear combination of states of the
form
states respectively, and the hat on the creation operators
distinguishes the neutron creation operators from the
proton creation operators.
Similarly we use hats on the
neutron annihilation operators and on the neutron hole
creation and annihilation operators to distinguish them
from their proton counterparts.
It is well known that it is also possible to
think of protons and neutrons as being two states of
the nucleon, and use the isospin quantum number to label
these two states.
The proton and neutron form a doublet
and therefore have isospin 1/2.
We use the convention
that isospin "up" (projection si 1/2) means a neutron,
-422-
and isospin "down" (projection = -1/2) will denote a
proton.
Antisymmetrization must be carried out for all
nucleon pairs, and the state of the nucleus will be
linear combinations of states of the form
h
c !r
*/
where
A
vacuum,
AA
/o >
c / r
is the number of nucleons, /Ob is the nucleon
/■
C
creates a nucleon with isospin projection
in single particle state / .
Although we introduce
no distinction between the proton operators
the nucleon operators
C^
c £
and
, no confusion should arise
because they cannot be used in the same context.
The proton-neutron scheme and the isospin formalism
are completely equivalent in all respects, therefore
it should prove possible to translate from one to the
other.
This translation is needed here, because Gillet
gives particle-hole kets in isospin formalism, whereas
all our derivations were based on the proton-neutron
scheme which we chose for compatibility with the article
of Rose and Brink^^, and for simplicity.
Except for the simple case of two particles, we
shall not attempt to prove here that the two languages
are equivalent; instead, a one-to-one correspondence
between states in the proton-neutron scheme and states
in the isospin formalism will be assumed to exist, and
its exact nature will be derived.
-423-
The relationship between the proton-neutron scheme
and the isospin formalism is rather subtle, and it is
best to start with an example.
particle states f k )
and
We assume Jtwo single­
/ft)
4 states can be obtained.
to be filled; obviously,
In the proton-neutron shceme,
these are
4
(tk > h
)
vS (/*>„
- !*>„ ! 4 > J
+
< V I- A ' 2 )
/* > „ , n > p
7s ( !k>h U \ , - M>„ H>h )
The first and second states are respectively states of
two protons and of two neutrons.
is bbe single­
particle state consisting of proton #1 in
so on.
/M )
, and
Both are antisymmetric upon exchange of the two
particles.
For the last two states we have arbitrarily
taken the symmetric and antisymmetric linear combinations,
of the basic states
tty ,
t i> „
/k > „ ,
which represent respectively a proton in / M )
neutron in
J ft)
, and vice-versa.
with a
These four states
can also be written in isospin formalism:
& c
^
-/k \ //>;
( l b , 4 lt>2//>,)
//p (2
Ioo>n
T} - 1 /
.*(vi_a_3)
-424-
where
//7f”^/2
with isospin
is the isospin state of nucleons 1 and 2
/
and projection
the isospin state with isospin
^
T
_/'/* fA
/ r p /2 - X/)* T ( t,
7*= O
, and
:
/O O ^
is
in general
T )
(VI-A-4)
The two-neutron and two-proton states are easily trans­
lated into the
respectively:
is needed.
7W
T. = /
and
7"W , 7 1 * -/
states
* J
&
in this case only a change in notations
For example,
becomes
,
etc... so that
A
/& -*))
i v J _ A. 5)
* vk
and similarly,
n » a
On the other hand, translating the two
( v i _ a 6)
7j - o
,
one-proton one-neutron states involves some difficulties
which originate in the different methods of labeling the
particles.
In the proton-neutron scheme, the protons and
the neutrons are labeled separately.
Thus we have proton
1, proton 2, proton 3, .... and neutron 1, neutron 2 ...;
-425-
in the isospin formalism, however, all nucleons are
labeled together:
difference is of
nucleon 1, nucleon 2...
This
fundamental importance and represents
the different requirements of antisymmetrization.
The
case of two particles we are presently dealing with
involves a proton (proton 1) and a neutron (neutron 1)
in the proton-neutron scheme, or equivalently nucleon
1 and nucleon 2 in isospin formalismm:
The translation
of the one-proton one-neutron states proceeds as follows:
we could write
(VI-A-7)
if we choose to call the proton "nucleon 1" and the neutron
"nucleon 2".
On the other hand, we could choose to call
the proton "nucleon 2" and the neutron "nucleon 1" , in
which case
That these two formulae are not equivalent is quite
obvious since the states on the right hand side are
orthogonal.
However, it is easy to see that the latter
can be obtained from each other simply by exchanging
nucleons 1 and 2.
Remembering that the state in isospin
formalism must be antisymmetric under exchange of the
two nucleons, we see that the proper linear combination
-426-
of the two states above must be used.
i*> m u > J -
i(ik \r h 7 ,\
Therefore,
n i'in y ,
±
(VI-A-9)
7/k>JU\ tfy i-'i),)
We consider the phase of the isospin state to be defined
by the requirement that
lk > k s t k \ /& -O ,
and there remains a sign ambiguity which depends on how the
proton and neutron are labeled when they are treated as
indistinguishable nucleons; we arbitrarily take as positive
the term which corresponds to labeling the neutron as
nucleon 1 and the proton as nucleon 2.
The expression
factors to give
fe(lk\
pt
111
t lk\ H>. ) = i
"I
u\ + lk\ Ii>,)
'
*
(VI-A-10)"
(l'A fi) \‘U-‘h \ I I'll V jfrO , )
or equivalently,
A
(%
/
/
>
„
,
+
»
>
m
)
=
m
(
*
m
w
)
I
o
o
\
-
(VI—A—11)
k
( If y
- ! k \ , u>f , ) - - k
O
h
- !k \ U >, )h ° > n
This completes the set of formulae needed to translate
from one scheme to the other in the simple case of two
particles.
briefly.
The more general case will only be treated
-427-
The states
< - •
<
* » ' *
5a".'"
protons in
< - ■
•"'
* , > > "
V *
/ k ,> .. /4 > and
(vi— a — 12 )
are both states of
/0;>
neutrons in
UR . . . . /&)
and
therefore must be linearly dependent; that is to say
equal within a phase.
That setting this phase is not a
trivial problem can be easily demonstrated as follows:
the two-particle state in the notation of second
quantization is written as
ck / o y
®
ce t o y
(VI-A-13)
/o>
c k-'/z
L . c£.*fi
Clearly we could have either
c f / c / • c /lo y
1
4W »
c * /o f
.
c/_6 c; t /o>
(VI-A-14)
c;h
/o>
but certainly not both, because these two equations aremutually exclusive.
_t
commute, and
C . ,,
* A
This is clear since
anticommutes
f
with c
f'/z
and
c j
.
Thus
c * lo t ® c f /o f = c f /o f s c //o /
whereas
t
CX
. t
J
°
y
'
-
/
Ct H C L
,^\
(VI—A—15)
,6>
Instead of tackling directly the problem of translating
an antisymmetric state in proton-neutron scheme into an
antisymmetric state in the isospin formalism, we will show
-428-
how a simple direct product of single-particle states
can be translated, and give the recipe for antisymmetrization in both proton-neutron scheme and isospin formalism.
#
After having done the translation through the intermediate
of a simple direct product state, the general translation
formula will become obvious.
The direct product of single-particle states with p
protons and 71
neutrons,
(VI-A-16)
cp(t>)
where
Cf>
and
Lp
q ,0 )
^ y > K ip ( n )
are permutations on {/■■■
respectively and
and//---71./
means the single particle
state of proton # <p (p in
» as defined above, can
be expressed in isospin formalism provided we arbitrarily
choose a new name for each particle.
We therefore agree
to arbitrarily call neturon #1 "nucleon #1", neutron #2
will become "nucleon #2", and so on.
The protons will
be labeled, in a similar way, from "nucleon#( 71+ I ) "
to "nucleon # (71+f> )" .
With this in mind, the state
above is rewritten as
-■
<
* w
assuming specific phases for the isospxn states
and
f /i li} .
% *>*<»-■ ■ ■
I'U'/i)
-429-
By antisymmetrizing these direct product states
in a consistent manner,
(by this we mean that anti-
symmetrization be a linear operation) we will obtain
the same antisymmetric state in proton-neutron scheme
and isospin formalism.
A to
Using
denote the anti-
symmetrization projection operator, we have for-the
proton-neutron scheme
A ^ 9 0 )
...
“^
\
(VI-A-18)
(fi(p ^ ®( A
< f ( 0 ----
{ p (fij)
since proton and neutron parts are antisymmetrized
separately.
From (V-D-8) we obtain the antisymmetrization
operator*, and
A
~L
0 0 ) ---■^
y?a)
^
"*
(VI-A-19)
{< P 'l
/ ( f '^ 0 )
c p 'tp Q )
where the sum is carried over all / / permutations of
{ A, X —
(T ffy )
/
J , and
=0
T P fc p ) is the parity of permutation
for even Cf> , 1 for odd
runs over all permutations,
).
Cf .
Because c f '
also runs over all
permutations and the summation index can be changed.
This yields
^
^ 7^
, ) ------
(VI-A-20)
'
'- ''W o j
/A>'/ y e p )
*
The antisymmetrization operator has this form only
when acting on direct products of single-particle
states.
-430-
The antisymmetrization of the neutron part proceeds
similarly to yield
A /I, \tpoj
'jrC
" "Tift')'
^_ t
- /)
(VI-A-21)
7 n f~ fp j ^
.... /? » > „ (£ '(*> )
ivy
The state written in isospin formalism can be antisymmetrized
with respect of all the nucleons; if CO
on
H, —
is the permutation
A/// such that
C O (i)
IC O
»
(VI-A-22)
c o /m ^ ) = Y p ffJ + n
/-/
the state becomes simply
^
^
*
/Jet> ,
,/% - '/,> ,
.
Cofa+O
6u (rt+i)
...
/ A > , z)
r c o (v + p )
( V I _ A . 2 3 )
co(r,+-t>)
and can be antisymmetrized in the same way as the neutron
or proton states. ■
Antisymmetrization yields, in a more compact nota­
tion,
,
. ........
y
^
,
&
.v - f w j
u * * i w
.........
where the sum runs over all permutations c o ' of
*
< v i_ a _ 2 4 )
-431-
Once the rule for relabeling the protons and neutrons
as nucleons has been chosen, the translation of a simple
direct product state from proton-neutron scheme to
isospin formalism is unique.
Furthermore, antisymmetriza-
tion in both cases is also uniquely defined.
We therefore
are allowed to consider the antisymmetrized states
,
¥
{ If y c p o )
(VI-A-25)
....
and
A { it. % a v ,
/4 *
0
(VI-A-26)
as being the same state, expressed in proton-neutron
scheme and isospin formalism respectively.
since £ - j )
»
£ ./)
eft.... eft loft
4
In particular,
we have
0
eft.... Sft / o f =
*!>
(VI-A-27)
c f .... cfa c [ .... c !
toy
4 ,'L
6? '/i
fy -h
Applying the commutation rules on each side yields this
general rule:
A state in proton-neutron scheme can be
transcribed directly, term by term, into isospin
formalism, provided all the neutron creation operators
be on the left of all proton creation operators, and the
order among proton creation operators and neutron creation
operators respectively be the same in both cases.
For
-432-
example, we have
t d
•
c / , o /
-
(VI-A-28)
^
however
q fc /c j /o f » c / i o f a
$4 cj.,/t cA
cA ,
C}c p I o ( • cf/of a cJA cJ.A 4+
/0 >
4+
(VI-A-29)
/o>
In the first case, the neutron operator has been left on
the right-hand side of the proton operators, and in the
second case, the order among proton operators has not
been preserved in translating.
The notation used so far in the proton-neutron
scheme can be simplified by simply writing
with a slight risk of confusion.
It should be remembered
that the proton operators commute with the neutron
operators:
for example,
E(
*r
I/
r<
i s» ,
r t lo 'i
+ 4/ C
- /j. r5 f' q'/°>
(VI-A-31)
The translation from proton-neutron scheme to
isospin formalism offers more difficulties when particlehole states are involved.
Since we are interested in
♦
one-particle one-hole states, we have to investigate
the effects of introducing a Fermi surface.
The method basically consists in writing the
state as a set of creation operators acting on the
vacuum; this effectively removes the Fermi surface, and
the translation proceeds as described above.
The Fermi
surface is then simply re-introduced in the new formalism.
One problem arises when introducing the Fermi surface
in isospin formalism; the standard order has to be
defined.
In order to keep intact the angular momentum
coupling properties of the hole states, it is advisable
to still order the single-particle states in increasing
values of angular momentum projection vn.
I
(VI-A-32)
There are
+(
magnetic substates, and 2.^ + 1 is always
even; therefore, it does not matter how these subshells
of 2 y -f- /
states are ordered among themselves.
particle-hole operators will thus be defined
The new
(c k Y t O ff):
-434-
Notice that the isospin projection - Z
has been used to
label the hole operators instead of Z .
The change in
sign is for the same reason as in the case' of »?; it is
just a convention.
I
jm z
¥C
f-n
-f”c -0
'/ f t
C f
It is important to remember that
,
in
9e n e r a l .
f
Let us translate the particle-hole state '
+
CL-
/ f
£>.
, !c >
n
®
/X
(VI-A-34)
JCA
/* ”
into isospin formalism; we define the closed shells
/C A
and
(C )t
by
f>
/■
fp
proton closed shell /C }-= C. .
C.
!o )
f f
f t
(in order of
neutron closed shell / C A V
lo T - increasing m
t r " ' rr
(VI-A-35)
A more general clCsed shell with more-than one subshell
could be used with the same results; we keep only one
subshell ^
for simplicity.
Similarly, the nucleon
closed shell
!c /
0 /c>”
/c >
*
(VI-A-36)
.
C .r, „ .... C f .
f t '/,
f j'h
C f .........C [ .
j'f - t ,
W
/o >
' 11
in order of increasing rri , all neutron operators
on the left of all proton operators.
-435-
Then,
f
, t
. f
a-m
.
n
/c > • /c>
(VI-A-37)
'
r
cr r
~
<? «.
&
^
• ' c> \
This can be translated into isospin formalism, and we
obtain
&!■
c
i*
f .
W
&
, t c f * > /C ? =
c f
c f
c f
*
C f
/?'* f ™ - '''
(VI-A-38)
cp (!m......................................
c f .. . /o>
'+ O ~ /t ..
All operators on the right-hand side of the equal sign
now have three indices, and parenthesis were used to
avoid confusion.
introducing the operator
C
- '- Y i
fm
\\
'-y m '-'/t
and using the anticommutation property of the operators
y
i e
l <
i S
6^m
, /cZ/O”
' C~ °
cp
«
Therefore,
a f.
, b f
=
.,A t y y * ""
,,
Jc7
J0>
(VI-A-39)
-436-
ap f 6y $m . IcF&ic')
= /
a]rfn -'h
/y m ' f,o
(VI-A-40)
ic>
A similar result is derived for a neutron particle-hole
pair, in exactly the same way.
&
t *
/c > v
0/
c / *
t
a t
b f-
(VI- A- 41)
IC >
Certainly, this result was obvious from the start, except
for the sign.
We do not know of a simple way of proving
that the sign in the above formulae is always ~h ; in
fact, it needs not necessarily be always •/ , if some of
the conventions used in the proof are dropped.
sign conventions come in at various places:
Arbitrary
the
relabeling of protons and neutrons as nucleons, the
ordering of states below the Fermi surface (i.e. definition
of the hole operators in both the proton-neutron scheme
and isospin formalism) and the definition of
/ O
/O,
/C }
and
all bring in sign conventions; furthermore, the
fact that each subshell y
consists of an even number of
single-particle states is used extensively in the proof.
-437-
Appendix VI-B
Standard Configurations Used in the Theory
and the Calculations.
1) The Single-Particle States.
The standard single-particle states of chapters S
are defined by
In ijm y
where
$
-
Z
& /)
f srns l
/S T t y 'y ls the spin
is the nucleon spin, and
state of the nucleon.
(vi-b-i)
/5w,) is further defined by
its transformation properties under time reversal,
^
@ /S W S > - C O
with 0
N
(VI—B—2)
IS 'M s >
as the time reversal operator.
Chapter XV and Wigner^8 ^ chapter 26)
(Messiah^8 ^
Both
{ / / and
ft/SYf?s )J are defined to form a standard representation of
angular momentum, so that they transform among each other
under rotations with the use of the rotation matrices,
/p im p
--
(vi-b-3)
m4 '
and similarly for the
/S m s y
.
"P is the operator
A
for rotation R ; it transforms the state
J 7 l/rr? 4 y into
the rotated state
is the rotation
matrix for rotation
appendix C.
*
”
R
m tn
, as defined in Messiah's
f
(Messiah writes the symbol 7ft
instead of
m 'm
-438-
our
,
)•
In terms of Messiah's spherical har­
)Z
, the wave function ( A M P tr J ^ y
tm
Z w P iW g b
is simply given by
monics
ket
Z A /v P m ^ b
-
for the
(VI-B-4)
a n £ (J i)
^
in spherical coordinates, where the coordinates of >2
(/2f J 7 ) and
J2 =
are
stands for the azimuthal angle 0
and the polar angle ( f
.
The radial wave function
is the solution of the radial equation for the spherically
symmetric modified Gaussian potential of (VI-5).
The sign
conventions are those indicated by fig. VI-2 and VI-3;
the normalization is in the interval
0 < A e < /Re
only.
Applying the time reversal operator yiedls
P -rrij
£ /vP m .b
-
O )
/v P -m b
(VI—B—5)
& I t* Pj-rn) 9 (r1)
For simplicity,
abbreviated
and
MPj-nb
I'A P m ^ b
/■P y m b
an<^
will be
respectively, when
there is no risk of confusion.
2)
The Standard Configurations
We call configurations the states of the system
obtained by coupling to good total angular momentum the
single-particle states
/7 ) P j7 n b
.
The stationary states
of the nucleus will be linear combinations of these config­
urations of the same angular momentum and parity, assuming
-439-
the Hamiltonian to be invariant under rotations and
reflections.
The word "configuration" as used herd corresponds
to the term "particle-hole kets" used by Gillet, and
dirfers slightly from its usual meaning in the literature.
We call "standard particle-hole configurations"'the states
defined below;
it is in terms of these that the theory
is derived in chapter V; Appendix II-C establishes the
relationship between Gillet*s states for C
12
and our
standard configurations.
The standard one-particle one-hole configurations
are defined as
/ V O ’/it) *■ 2
0
where
i
V//
' / //>' *7*) 7=p
?
<r-x
(-')
(m m '-m ! J
mm'
Cc. •
and
o ,.
,
respectively create a proton
lT )4 y m )
and a proton hole in states
defined above.
\
and
The exact definition of operators
t l f ^ i" //
a '
a f &
will be found in appendix V-D.
The + signs
refer to the symmetry of the state; the + sign corresponds
to the symmetric (
T"~ O )
state and the -sign identifies
the antisymmetric ( T"- / ) state.*
The particle-hole vacuum
P
)
I c y
«•
/ c
f
®
/C )
is defined simply as
/ c y ’
where
/C
is the proton's particle-hole vacuum and
*
Correct as stated here.
-441-
Appendix VI-C:
; The Use of Gillet's
The States of
Wave Functions.
Gillet defines his configurations, or particle-hole
kets, at p. 95 of his thesis
ar L.icle^^^.
(1)
or as equation (5) of his
Rewritten in a more convenient notation, they
are
I n ' ( € 's ') f '
,
i-m
;
T ic e s )j :< ?
m
t
o
]
*
'h 't
£
C 'O
m rm 'T1
Z'-ZJTO )
x
(VT-G-1)
/ ( t s ') j " m ‘ T / & ) / » > >
! '/ t c > * / * o
u „ ,e , u „ e
The primes refer to the hole and the unprimed quantum
numbers belong to the particle.
This definition is given with a warning that this ket
transforms like a bra in space rotations, but no other
explanations are given.
We have some unpleasant comments about this
definition:
the first such comment is that the
themselves are not well defined.
It has been possible
to infer indirectly from Gillet's thesis that they are
of the form
(VI-C—2)
where the wave functions associated with
are the
spherical harmonics of Messiah (or Condon and Shortly).
The use of these spherical harmonics without the factor
l
A
, and the fact that the time reversal operator seems
-442-
not to have been mentioned in his work, indicate that
Gillet was not using the properties of the states under
time reversal to ensure real scalar products.
This brings
♦
us to the definition of
/ S w O" } ; we assume that these
states form a standard representation of the angular
momentum operators, but unfortunately, we also need to
know how they transform under time reversal.
This, we
can only guess.
Another problem which is also related to phases
concerns the meaning of the asterisk in (VI-C-1).
quite clear that Gillet meant by this
complex conjugation.
It is
the operation of
Unfortunately, complex conjugation
is not well defined for kets unless the standard representa­
tion with which it is associated is explicitly identified.
In other words, the operation of complex conjugation on
kets is not unique.
(Messiah XV-5).
Since Gillet does
not introduce another basis, we could assume that the
complex conjugation operation denoted by
one corresponding to the basis
/(P 5 )jJ w A
is the
.
Unfortunate
ly, this leads to a contradiction, because the complex
conjugation operator associated with a representation is
defined as the antilinear operator which leaves the hasis
vectors of this representation invariant; this means the
operation
V
would do nothing to
/' ( 4 s ) y m
, and the
particle-hole ket would not transform as a bra as Gillet
claims it does.
-443-
Gillet's references on the phase conventions are:
"The particle-hole ket is defined with the choice of
phase of Bell
Physics 1 2
27
" and reference #27 is "J.S'. Bell Nuclear
(1958) 117".
This is an excellent article,
but we have been unable to find any reference to a
phase convention in it.
Another peculiarity of the particle-hole ket
definition of Gillet's is its transformation properties
under rotations.
Most writers define a hole ket by the
method outlined in the theory chapter ( c k 'g ) .
yields kets which transform as kets.
This method
Instead of doing
this, Gillet produces hole kets which transform like
bras, then redefines each particle ket by changing the
sign of the 7/L quantum number and introducing a phase
(-/)a
, forcing it to transform like a bra.
Having
done this, he is then capable of coupling them together
to make a particle-hole ket which transforms like a bra.
The whole procedure, as we hope to have made
clear, is apt to generate confusion.
However, by per­
forming a transformation on this particle-hole ket definition
we have been able to clarify the situation somewhat.
The definition of particle-hole ket above is identical
with
(-/)
( - ,)
! v e t 's ') f ; n (4 s )j ; j r - n T o f
' L ( f i m ' m t j n ) ('A Z i ? z i r o ) [ ( / / m i r j '- v A
r n m 'Z f
,
/{ j
! % ?> U t f f
(t)1
=
/ 'A - t 'f J
This simply comes from the symmetry properties of the
Clebsch-Gordan coefficients (Messiah’s definition) and
changing signs of the dummy indices.
) 7)(fts)j j J'MT’O J
Since the
' transform
like bras, according to Gillet, we would expect the
f
/
f
'
/
7
)
j ' j r)
t
o
transform like the
hermitian conjugates of bras, i.e. like kets; indeed
we see that
which is an ordinary ket, fulfills
the requirement. Therefore, we can safely assume.that
f t -7%'
jf
the ( - / ) *
' 7 P ')
transform like kets also.
Assuming as before that the single particle states are
given by
/ft/m ) where
(vi-c-4)
111#)/ftrne) Jsms)
are the spherical harmonics,
where we again have used the symmetry properties of the
Clebsch-Gordan coefficients and assumed the complex
conjugate of a direct product to be the direct product
of the complex conjugate of the two factors.
of the dummy indices 7Y)^ and
Guessing that by the &
7T?S
The
signs
were changed.
operation on
illet just
means to take the complex conjugate of the wave function
Y j> ' **»
€ -m e
•
and since
y
*
^
-445-
The next point is crucial:
/P th)
what was meant by
/ /
\*
I S -70s /
?
(hopefully); but what about
The most important clue is that to be
consistent, the ( - / )
kets.
we have been able to determine
S /S '-r tfs X
must behave like
This means that
I s - ct(s')!s‘w>
(V I-C -7 )
s'
where
o(
is a phase factor which depends only on
Fortunately (and otherwise we would have problems)
can take only one value, 1/2.
matter what
oC
s'
S/
Therefore it does not
is, and we can set it equal to 1.
This
brings
S '-m ,
#
[ $ ir)s b
~
& )
IS -ffls b
(VI-C-8)
We have therefore proved (painfully) that the only way
to be consistent is to define the action of the operation
on spin states, as in (VI-C-8).
phase of the
define
*
fS 'rV j b
» if
if the phases of the
This would set the
was defined, dr it would
I S l 7ns y
were known. ~
But whatever it is, we would not call it a complex
conjugation operator, so that the notation
*
without
any comments is in bad taste.
Unfortunately, we have
(2)
seen it used by other authors from Saclay
.
The purpose of the present derivation is to establish
the relationship between our well defined configuration and
Gillet*s states; these states are given in terms of the
particle-hole kets we have just studied, by the formula
-446-
jXSMTO] =
*
j
f
. fa
■„ W r > , ;
C - ' f t f S / v a ' s ’J / ; » ( & ) / ; J M T o J
v e -v tj?
The
? •
/ •
( V I . C _ 9 ,
/A S M T O )
are Gillet's wave functions; they are
\s 3 " T
given in terms of a table of real coefficients
The above equation constitutes the definition of these
■ f '/S '
coefficients. The phase £ ■ /) *
has been added arbitrarily
by Gillet to improve symmetry of the equations under
particle-hole exchange (Nuclear Physics 54(1964) p.332, top
of page).
Since
diagonal in
( - /)
y J -T
A )■„/£'*)/>,'
A j 71 f <7 v
(J
and
7^ we
is independent of
M
and
can also write
I X J - H T o )
o r
=
._ v .
c r - r t iT ip s '
ft fa, Yi-tf*'(■/)
(VI-C-10)
ln'(Mfi;n(tslfi;J-MTo)
which is a relation between kets which transform as kets,
instead of kets which transform as bras, under rotations.
The next problem encountered in dealing with the wave
functions of Gillet comes from their properties under
exchange of two nucleons.
It is clear that they are
not antisymmetric under exchange of the nucleon above the
Fermi surface with any one of
face.
the nucleons below the sur­
Using these wave functions as such leads to
, neglecting the exchange integrals between the excited
nucleon and the core.
-447-
Before using the states we must somehow antisymmetrize.
Thus
[ft)'r
O f * / ' / , - t ' f ] /+/»>/?, O
(vi-c-ll)
will be replaced by
aj j m t * ' f/) '* r
r ) 10
lvi-c- 12 )
where the particle and. hole creation operators now refer
to our single-particle states as defined in appendix 2-B.
.-t?
.J >
The appearance of I
and
2
essentially comes from
I
the fact that we use
as orbital wave functions
and Gillet uses simply
The disappearance of (■ > )*
.
-ro '
is linked with the sign in
b jj'j'r n 'Z '
~
From (VI-C-3) we have
i f n
ynrrji
m
x
m rr w
r t f k </mrlrpm
. r
Introducing numerical values for the
CJ-M+7
(-1)
Iri'C fs 'J j ; TiC&y
- 4
O
737 YT}'
)
7
( U
;
.
J -M 7 0 ]
- n j 1
3-j
ivi-c-isj
coefficients
=
a
(VI-C-16)
{a*.
+
0 ? a *
b f
] ^ >
-448-
Using appendix VI-A to translate into proton-neutron
scheme,
z r-M -t r
(-!)
h 'C f s O /'; n ( / s ) j j J - r f T O ) -
(H i) k
( ■ / " ,
d /m
1
/ 70'
'
d /m i
(VI-C-17)
Because we should not be allowed to talk in terms of
isospin when using proton neutron scheme, it is advisable
to write this equation in a slightly different form:
ct- m
€ -/)
+ P
+r ,
h ' ( S s ) f ' ; r x t s ij
t c / 3
j
J
J - M T O J
'
(£ • & -% )&
(VI-C-18)
here the upper sign applies for the
and the lower sign applies to the
7= O
7”=/
configuration
configuration.
Except for a sign, and the order of coupling /
and
we recognize our standard configurations of appendix VI-B.
Changing the order of coupling brings in a sign
and
&/)
l
i r ' e c -,)f t r J
h ' C M y } vets)/; J-rr TO) *
I
( V
I _
C _
1
9
This finally permits us to write Gillet's states in
terms of our configurations and his tables of amplitudes,
namely (VI-C-10),
)
-449-
(-/)
„
M 7-H
C T -///7 -
to}
\/TT
X
;
=
Sts' TH
X
,
c -0 *
c -o
*
Sit
'
*
i
/j f jm
A, vfynfy
>
X!
( v i - c - 20)
It is always possible to change the overall phases
of the states without affecting the differential crosssection, since these phases are arbitrary.
with our states
lA J T ft}
defined as
pf
IA?ni>=(-i)
where
Therefore,
with + sign for T * 0
i/AJ-MTo}
state
T,
and - sign for 7 = /
+ +
state
is the parity of the state ( 1P = 0 for even, 'T f
= 1 for odd states) we have
»>*■>■ 7 >T{j t S ’ i * " '
X "
,
n fy
m
™
>
(VI-C-21)
n T f
This formula in effect converts Gillet's wave
function into our notations.
A brief summary of the
origin of the various terms may be useful. ( ~ 0 ^
is
/ tJ-S'
of Gillet with the change
in coupling order of the particle and the hole;
from Gillet's use of
angular momentum; I
instead of
, where
77^
I ^
I
arises
for orbital
is the parity of the state
-450-
and configuration has been added for convenience ( 1
always real).
Finally,
/
4
f
~
)
>
are the configura­
tions defined in appendix VI-B, and the has nothing
#
tt
tt
to do with parity;
r
just means the symmetric state
r
(or
ff
= 0) and
T =1).
—
0
stands for the antisymmetric state
With these conventions, the formula above
defines the state
/ \Z T M i
of the system in terms of
Gillet's tabulated particle-hole amplitudes.
One last remark in closing this appendix:
Gillet
seems to have omitted the correction for the recoil
of the other nucleons, in the electromagnetic transition
matrix elements.
for
7= 0 to
7”=
For example, he predicts large
0 transitions;
El
widths
it is a well known selection
rule that the recoil correction (sometimes called "effective
charge") makes such transitions vanish in self-conjugate
(i.e. / E - E
) nuclei.
-451-
A P_P_ENDJX
yi_-_D_
The R -M a trix P rog ra m for Calculations of D ifferential
C r o s s Sections.
-452C
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REDUCED WIDTHS PROGRAM
PROGRAM TO COMPUTE RADIAL WAVE FUNCTIONS AND RADIAL
FOR A REALI STI C CARBON-12 NOCLEAR POTENTIAL.
(NOW USES FREGEAU' S MODIFIED GAUSSIAN POTENTIAL)
READS THE WELL DEPTH VZERO (MEV) AND SIZE PARAMETER AZER0 (FERMI)
AND COMPUTES THE POTENTIAL FOR NEUTRONS AND PROTONS.
PRINTS ANU
PLOTS BOTH WITH THEIR DIFFERENCE (THE COULOMB POTENTIAL)
READS ENERGY E (MEV) ON UNIT 2 . THE DIFFERENTIAL EOUATIUN IS
SOLVED FOR L = 0 , l , 2 AND 3 .
THE WAVE FUNCTIONS ARE NORMALIZED.
IF E IS GREATER THAN ZERO, THE LOGARITHMIC DERIVATIVE OF THE Y ' S
AT RC ARE PRINTED.
IF E IS LESS THAN ZERO, THE VALUE OF THE Y ' S
AT R = 7 . 0 FERMIS ARE PRINTED.
RC IS DEFINED IN THE PROGRAM TO BE 4 . 5 FERMIS BUT CAN EASILY BE
CHANGED.
THE Y ' S Ar.i DEFINED AS
R TIMES THE RAO IAL WAVE FUNCTIONS
WHEN A DI GIT J I S ENTERED IN THE FIRST COLUMN ON UNIT 2 , INSTEAD OF
THE ENERGY, AND THE REST OF THE FIELD IS BLANK OR ZERO, THE Y ' S
CORRESPONDING TO J ARE ENTERED IN A SPECIAL TABLE S FOR LATER PRI NT.
THE CORRESPONDANCE IS AS FOLLOWS.
J=1
L=0
(FOR I S )
J =2
L= 1
(FOR I P)
J=3
L=2
(FOR I D)
J =4
L= 3
(FOR I F)
J=5
L=0
(FOR 2S>
J =6
L=2
(FOR 2 P )
WHEN 1 0 0 0 . IS ENTERED INSTEAD OF THE ENERGY, THE INPUT IS TRANSFEREO
TO UNIT 5 WHICH NOW READS THE ENERGY E .
THE Y ' S CALCULATED AT
THOSE ENERGIES ARE PRINTED AND PLOTTEO.
WHEN 1 0 0 0 . IS ENTEREO
INSTEAD
OF THE ENERGY, ON UNIT 5 , THE Y ' S WHICH HAVE BEEN COPIED
IN TABLE S ARE PRINTED AND PLOTTED.
ALL
THE RELEVANT MATRIX ELEMENTS (FOR El AND E2 TRANSITIONS)
ARE
COMPUTED AND PRINTED, WITH A DIMENSIONING CONSTANT OF 1 . 6 1 F .
CONTROL
IS RETURNED TO UNIT 2 .
IF IT I S DESIRED TO RETURN
CONTROL
TO UNIT 2 AT ANY TIME A / * SHOULD BE READ ON UNIT 5 .
TO TERMINATE EXECUTION OF THE PROGRAM, ENTER A LETTER ON UNIT 2
AND THE SUPERVISOR WILL CANCEL THE JOB.
THE
CALCULAT IOMS ARE PERFORMED FOR PROTONS ONLY OR FOR NEUTRONS
ONLY.
TWO CARDS HAVE TO BE CHANGED TO SWITCH FROM ONE TO THE
OTHER (SEE ASTERISKS BELOW)
PRINTING I S DONE ON UNIT 6 AND ALL PLOTTING ON UNIT 3 .
0001
0002
0003
0004
0005
0006
o o o
.
0007
0008
0009
REAL V P ( 7 0 0 1 , VNI 7 0 0 ) , COUL( 7 0 0 ) , I BUFFI 2 0 0 0 )
REAL TP I 7 0 0 I
REAL SI 7 0 0 , 6 I , ENSI 6 )
REAL Y ( 7 0 0 , 4 I , O L ( 4 )
REAL* 8 W ( 7 0 0 , 4 ) , W V , W P , W D , W V 1 , W P 1 , W D I , R R , R P , P 0 1 / 1 . 0 - 2 /
LOGICAL FIT
COMPUTE PROTON ANO NEUTRON POTENTIALS
10 REAO( 5 , 1 IVZERO.AZERO
WRITE( 6 , 2 ) VZERO,AZERO
1 FORMAT( 2 F 1 0 . 1 1
2 FORMA T( ■ 1 V Z E R 0 = ' , F 8 . 3 , •
0010
AZERO=• , F8 . 4 / / I
C
0011
0012
ASQ=AZERO*AZ£RO
00 3 1 = 1 , 7 0 0
R = I /1 00.
3 VNII) = - VZERO*!l.+R*R/0.75/ASQ)*EXP(-R*R/ASQ>
V P ( l ) = 0.
00 4 1 = 2 , 7 0 0
R = I /1 00.
4 V P (I)= V P (1 - 1 )+0.01*VN(I)*R*R
VN0RM=1 5. 84 /VPI 70 0)
00 5 1 = 1 , 7 0 0
5 V P ( 11=VP( 11*VN0RM
0013
0014
0015
0016
0017
0018
0019
0020
0021
C
COUL( 700 I = 1 5 . 8 4 / 7 . 0 0
DO 6 1 = 1 , 6 9 9
R = 0 .0 1 * (700-1 I
6 COULI700-I)=COUL(700-I+1)+0.01*VP(700-Il/R/R
0022
002 3
0024
0025
C
DO 7 1 = 1 , 7 0 0
7 VP( I ) =VN( I I +COULI I I
0026
0027
C
0028
0029
0030
0031
00 3 2
0033
INTEGRALS
CALL P L O T S ! I B U F F , 2 0 0 0 , 3 )
CALL SHOW( VN, 1 0 . , 3 I
CALL SHOWICOUL,1 0 . , 3 I
CALL S H 0 W ( V P , 1 0 . , - 3 )
WRI TE( 6 , 8 ) 1 I , V N I I ) , C OUL ( I I , V P ( I I , I = 1 0 , 7 0 0 , 1 0 )
8 FORMAT! 11 0 , 3 F 2 0 . 5 I
-453-
0034
0035
0036
0037
0038
0039
0040
0041
0042
0043
0044
0045
23
F 1T = . T R UE .
READ( 2 , 2 4 ) E
2 4 FORMAT( F 1 0 . 5 >
I F I E . E Q . 1 0 0 0 . ) GO TO 13
I F I E . L T . 1 0 0 0 0 . ) GO TO 25
J=E/10000.
L L- J
lF(LC.GT.4)LL=LL-4
DO 3 1 1 = 1 , 7 0 0
31 S ( I , J > = Y ( I , L L )
ENS ( J ) = EOLO
GO TO 23
0046
0047
004 8
0049
0050
0051
0052
0053
0054
0055
13 F I T = . F A L S E .
READ(5,11,END=23) E
I F I E . N E . I O O O . ) GD TO 32
DO 33 J = l , 5
33 CALL S H O W I S I 1 , J ) , 0 . 2 , 3 )
CALL S H O W ( S ( l , 6 ) , 0 . 2 , - 3 )
WRIT E ( 6 » 3 4 ) ENS
3 4 FORMAT( • 1 * , 9 X , 6 E 1 8 . 6 / / 1
WRITE(6,35> ( I , ( S < I , J > , J = 1 , 6 ) , I = 1 0 , 7 0 0 , 1 0 )
3 5 FORMAT) 1 1 0 , 6 E 1 8 . 6)
COMPUTE MATRIX ELEMENTS (WITH A L P HA= 0 . 6 2
0056
0057
0058
0059
0060
0061
0062
0063
0064
0065
0066
0067
0068
0069
0070
0071
0072
0073
0074
0075
0076
0077
0078
0079
40
42
43
44
41
DO 41 L = 0 , 2
WRITEt 6 , 4 0 I
FORMAT! * 4 ' I
DO 41 J 1 = 1 , 6
DO 41 J 2 = J 1 , 6
L1 = J 1
I F ( L l . G T . 4 ) L l =H - 4
L2 = J 2
IF(L2.G T.4IL2=L2-4
Ml= I ABS I L 1—L 2 )
I F ( L . E Q . 1 . A N D . M 1 . N E . 1 ) GO TO 41
I F I L . E Q . 2 . A N D . M I . N E . O . A N D . M 1 . N E . 2 ) GO TO 41
SUM=0 .
DO 4 2 1 = 1 , N
TP< I ) = S < I , J 1 ) * S ( I , J2 I * ( 0 . 0 0 6 2 * 1 l * * L
DO 4 3 1 = 3 , N , 2
SUM=SUM+(TP( I - 1 I + T P I 1 + 1 ) + 4 . * T P ( I ) I / 3 0 0 .
WRIT E( 6 , 4 4 ) L , J 1 , J 2 , SUM
FORMA T ( 3 I 8 , F 1 2 . 6 / I
CONTINUE
GO TO 23
32 WRI TE( 6 , 1 2 ) E
11 FORMAT( F 1 0 . 1 1
12 FORMAT! • 1ENERGY E = ' , F 1 0 . 3 / / I
SOLVE DIFFERENTIAL EQUATION
0080
0081
0082
0083
0084
0085
0086
0087
0088
0089
0090
0091
0092
-\
0093
0094
0095
0096
0097
0098
0099
0 100
0101
0102
0103
0104
0105
0106
)
2 5 0 0 14 LL = 1 1 4
L =LL- 1
EOLO=E
WV=1.DO
WP=0.DO
DO 15 1 = 1 , 7 0 0
RR=( 1 - 1 ) *P01
R P = I * P01
I F ( I . E Q . l ) R R = P01
W D = - 2 . 0 * L L * W P / R R + 0 . 0 4 4 * ( V P ( I I - E I *WV
C**** WD=-2.0*LL*WP/RR+0.044#(VN(Il-E)*WV
WP1=WP+P01*WD
WV1=WV+WP*P01+WD*P01*PO1 / 2 . 0 DO
WD1=-2.0*LL*WP1/RP+0.044*(VP(I)-E)*WVl
C**** WD1=-2.0*LL*WP1/RP+0.044*(VN(I)-E)*WVl
WD=( WD+ WDl >/ 2 . 0 0
WPl=HP+WD*P01
W V 1 = W V + ( WP + WP 1 ) * P 0 1 / 2 . D 0 + WD * P 0 1 * P 0 1 / 2 . D 0
W(I,LL)=KV1
WV=WV1
WP= WP1
15 CONTINUE
14 CONTINUE
0 0 16 1 = 1 , 7 0 0
RR = I * P 0 1
RP = RR
DO 16 L L = 1 , 4
W( I , L L ) = W ( I , L L ) * R P
,
16 RP=RP*RR
o o
0107
0108
0109
NORMAL I ZE
0123
0124
0125
RC=4.5
N= 100 . * R C + 0 . 5
N2=N/ 2
N=2*N2
DO 2 0 L L = 1 , 4
SUH=0 .
21 I = 2 , N2
M=2*I-1
21 S U M = S U M + ( W ( M - 1 , L L ) * W ( M - 1 , L L ) + W ( M + 1 , L L ) * W ( M + 1 , L L ) + 4 . * W ( M , L L ) *
1
W(M,LL)1 / 3 0 0 .
SUM=SQRT( SUM)
0 0 22 1 = 1 , 7 0 0
22 Y ( 1 , LL I =W( I , L L ) /SUM
2 0 CONTINUE
I F I F I T ) GO TO 26
PAUSE
CALL SHOW( YI 1 , 1 ) , 0 . 5 , 3 )
CALL SHOW! Y U , 2 ) , 0 . 5 , 3 )
CALL SHOW I Y { 1 , 3 ) , 0 . 5 , 3 )
CALL SHOWIY( 1 , 4 ) , 0 . 5 , —3)
0126
0127
0128
0129
0130
0131
0132
0133
0134
0135
0136
0137
0138
WRI TE( 6 , 1 7 ) 1 I , ( Y ( I , L L I , LL = 1 , 4 ) , I = 1 0 , 7 0 0 , 1 0 1
17 FORMAT! 1 1 0 , 4 0 2 0 . 7 )
2 6 0 0 27 LL = 1 , 4
IF(E .L T.0.)DL (L L )=Y(700,LL >
IF(E.G E.O .)OL(LL)=(Y(N,L L) -Y ( N - 1 ,L L ) ) * 1 0 0 ./Y ( N ,L L )
2 7 CONTINUE
WRI TE! 6 , 2 8 I E , ( O L ( L L ) , L L = 1 , 4 )
2 8 FORMAT!* E = ' , F 1 0 . 3 , '
LOG OER= • , 4E20 . 4 >
I F I F I T ) WRITE ( 2 , 2 9 ) ( D U L L I , L L = 1 , 4 )
2 9 FORMA T( 4 E 1 1 . 3 I
I F ( FIT I GO TO 23
GO TO 13
ENO
0110
0111
0112
0113
0114
0115
00
0116
0117
0118
0119
0120
0121
0 122
C
C
C
C
C
C
C
C
C
0001
0002
0003
0004
0005
0006
0007
0008
0009
0010
0011
0012
0013
0014
0015
0016
0017
PLOTTING SUBROUTINE
A IS THE ARRAY TO BE PLOTTED
S I S A Y - AXI S SCALE FACTOR.
IF S IS NOT EQUAL TO 1 0 . ,
THE EXCURSION
OF THE Y AXI S WILL BE LIMITED TO - 5 TO +5
IPEN DETERMINES IF THE NEXT LINE WILL GO ON THE SAME GRAPH
OR NOT
PLOT I S THE L I B RA R Y ' S CALCOMP PLOTTING ROUTINE.
SUBROUTINE SHOW I A , S , IPEN )
REAL A l l )
CALL PLOT( 0 . 0 , A ( 1 1 / S , 3 I
00 1 1=1,700
Y=A( l l / S
I F I S . E Q . 1 0 . I GO TO 3
1 F I Y . L T . 5 . ) GO TO 2
CALL PLOT( I / 1 0 0 . , 5 . , 3 I
GO TO 1
2 I F I Y . G T . - 5 . I GO TO 3
CALL P L O T ! 1 / 1 0 0 . , - 5 . , 3 )
GO TO 1
3 CALL P L O T ! 1 / 1 0 0 . « Y . 2 >
1 CONTINUE
I F ( I P E N. L T . OI C A L L PLOT( 9 . 0 , 0 . 0 , 9 9 9 I
RETURN
ENO
-455C
C
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0001-
PHASE ' QN'
FIRST
MAIN PROGRAM
(OUT OF FOUR)
READS PARAMETERS,LIST OF CONFIGURATIONS, STATES AND CHANNELS, WAVE
FUNCTIONS, REDUCED WIDTHS, BOUNDARY VALUE PARAMS, RADIAL INTEGRALS.
THESE ARE REAO UN UNITS 3 ANO 5 , ANO A CHECK OUTPUT IS PROVIDED ON
UNITS 4 ANO 6 .
COMPUTES REDUCED WIDTHS MATRIX AND PRINTS IT ON UNIT 4 .
ALL CONFIGURATIONS, STATES ANO CHANNELS ARE NUMBERED IN ORDER OF
APPEARANCE IN THE INPUT DATA. ALL THE T=0 STATES SHOULO APPEAR BEFOR
THE T=1 STATES OF THE SAME SPIN AND P A RI T Y . THERE MUST BE THE SAME N
NUMBER OF T=0 ANO ■=1 STATES OF SAME SPIN ANO PARI TY.
COMMON BLOCKS.
COMMON BLOCKS OF ALL 4 MAIN PROGRAMS ARE IDENTICAL
PERMIT SEPARATE LINKAGE EDI TI NG.
THE MEANING OF THE VARIABLES I S AS FOLLOWS.
RC
B
MBKS
WIDTHS
HF IN
JFI N
PARF
OL( I )
DC( I )
WFP
IN OROER TO
/ PARAM/
CHANNEL RADIUS
MULTIPLIER FOR BOUNOARY VALUE PARAMETERS BPAR.
NUMBER OF BLOCKS (A BLOCK CORRESPONDS TO A S P I N- PARI TY
COMBINATION)
MULTIPLIER FOR THE REDUCEO WIOTHS WTS.
STATE NUMBER OF FINAL STATE. ( 0 FOR GROUND STATE)
SPIN OF FINAL STATE
PARITY OF FINAL STATE
NUMBER OF STATES IN I TH BLOCK (ALSO NMBR OF CONFI G. )
NUMBER OF CHANNELS IN I TH BLOCK
DAMPING PARAMETER FOR STATES WITH A HOLE IN I S 1 / 2
/QNS/
QN REFERS TO THE TABLE OF QUANTUM NUMBERS FOR STATES ANO CONFIG.
QC REFERS TO THE TABLE OF QUANTUM NUMBERS FOR CHANNELS.
QT REFERS TO THE TABLE OF TRANSI TI ONS.
PARL( L)
PARITY OF STATE L AND CONFIG L .
$JL(L)
SPIN OF STATE L ANO CONFIG L .
*T(L>
I SOSPIN OF STATE L AND CONFIG L .
H( L )
H ( L ) = 1 MEANS CONFIG L HAS A HOLE IN IS 1 / 2
AND H ( L ) = 2
MEANS THE HOLE IS IN THE IP 3 / 2 LEVEL.
NIL)
PRINCIPAL QUANTUM NMBR OF CONFIG L .
LBU)
ORBITAL ANG. MOM. OF CONFIG L .
JB(L)
TOTAL ANG. MOM. OF PARTICLE OF CONFIG L .
EL( L )
ENERGY OF STATE L.
' PARC( C)
PARITY OF CHANNEL C
tJC(C)
SPIN OF CHANNEL C.
TAU( C)
TAU=0. 5 OR 1 . 0
IS NEUTRON CHANNEL WITH RESIOUAL NUCLEUS IN
IP 3 / 2
ORI S 1 / 2
HOLE STATE RESPECTIVELY.
T A U = - . 5 OR - I .
IS PROTON CHANNEL WITH RESIDUALNUCLEUS IN
I P 3 / 2 OR I S 1 / 2
HOLE STATE RESPECTIVELY.
LC(C)
ORBITAL ANG. MOM. OF NUCLEON IN CHANNEL C
JC(C)
TOTAL ANG. MOM. OF NUCLEON IN CHANNEL C .
QT( T , 1 )
CHANNEL INVOLVED IN TRANSITION T
QT(T,2)
=1 WHEN E l , 2 WHEN E2 AND 3 WHEN Ml TRANSITIONS.
STS
GAM
RADINT
ME
.
MCC(K)
TT( I )
WT( I )
WTS
BPAR
/S T S /
WAVE FUNCTIONS (STORED BY BLOCKS OF A GIVEN SPIN AND P A R . )
REDUCEO WIDTHS MATRIX (STORED BY BLOCKS OF A GIVEN SPI N- PAR)
RADIAL INTEGRALS.
ELECTROMAGNETIC TRANSITION REDUCEO MATRIX ELS.
/WT/
NUMBER OF COEFFICIENTS IN WT TABLE WHICH CORRESPOND TO K.
PAIR OF TRANSI TI ONS.
(STORED COMPACTLY)
ANGULAR DISTRIBUTION COEFFICIENT W CORRESPONDING TO
A PAIR OF TRANSITIONS T T ( I )
/ B W/
REDUCEO WIDTHS.
BOUNDARY VALUE PARAMETERS.
I MPLI CI T REAL! I , J , K , L , N )
C
0002
0003
0004
0005
0006
0007
A
COMMON / P A RAM/ RC, B, MBKS, WIDTHS, M F I N , J F I N , PARF, DL( 1 2 ) , 0 C ( 1 2 ) , WFP
COMMON / O N S / Q N 1 7 0 , 8 ) , Q C ( 7 0 , 5 ) , O T ( 5 0 , 2 )
COMMON / S T S / S TS( 3 0 0 ) , GAM( 5 5 0 ) , RAO I NT( 6 , 6 I , ME( 7 0 , 3 )
COMMON / WT/ MCC( 5 ) , T T ( 5 0 0 0 ) , WT( 5 0 0 0 )
C0MM0 N/ 8 W/ WT S < 6 ) , B P A R ( 6 )
INTEGER*2 TT
-456-
0008
0009
LOG 1CAL TAUL
INTEGER D L » D C » P L , P C , P L L , P C C , PLC, RELOC
INTEGER C , C $ , L A , B E T A
REAL PARLI 1 > , $ J L ( 1 ) , t T ( 1 ) , H( I ) , N( 1 ) , L B ( I I , J B I L ) . EL( I >
REAL PARC( 1 ) , i J C ( 1 >, TAU<1 ) , L C < 1 ) , J C ( I )
0010
0011
0012
C
EOUI VALENCE
( QN( 1 , 1 ) , P A R L I 1 ) ) , (ONI 1 . 2 ) . $ J L ( 1 ) ) , ( ON( 1 , 3 ) , S T ( 1 ) )
1.
(Q N (1,4),H (1)),(Q N (I,5),N (1)),(0N (1.6),L B (1)).
1
(QN (1,7),JB (1)),(0N (1,8),E L(1>>.
1
(QC( 1 , 1 ) , P A R C ! I ) ) , ( Q C ( 1 , 2 ) , £ J C ( 1 ) ) , ( Q C ( I , 3 ) , T A U ( 1 ) ) ,
1
< QC ( 1 , 4 ) , L C < 1 ) ) , ( Q C ( 1 , 5 ) , J C < 1 ) )
0013
C
0014
0015
0016
0017
R E A D( 5 , 3 ) MB KS , MF I N, R C , B , WI DTHS , WF P
3 FORMAT! 1 5 , I 4 . 4 F 1 0 . 3 )
WRI TE( 6 , 4 ) MBKS. RC. B. WI DTHS. MFI N, WFP
4 FORMAT( ' 1MBKS=' , 1 5 , 1 R C = ' , F 5 . 2 , >
B = ',F 5 .2 ,'
1
'
MFIN= ' , 1 4 , 1 W F P = ' , F 5 . 3 )
READ( 3 , 1 ) ( OL( MK) , MK= 1 , NBKS)
WRI TE( 6 , 1 ) ( OL( MK) , MK=1 , MBKS)
READ ( 3 , 1 ) (OC( MK) , MK=1, UBKS)
WRI TE( 6 , 1 ) <0C( MK) , MK=1, MBKS)
1 FORMAT( 1 2 1 5 )
0018
0019
0 020
0021
0022
WIDTHS=' , F 5 . 2 ,
C
0023
o o o
I F ( RELOC( MB KS ) . LT . O) PA US E ' ER R RELOC*
0024
0025
READ AND WRITE QUANTUM NUMBERS
WRIT E( 4 , 5 )
5 FORMATI / / 30X, ' QUANTUM NUMBERS OF STATES AND CONFIGURAT IONS' / / 17X,
1
'PA R ',8X ,' £ J ',U X ,'$ T ',9 X ,'H ',9 X ,'N ',8 X ,'L B ',8 X ,'J B ',9 X ,'E '/)
M1 = P L ( MB KS , 0 L ( M3 K S ) )
I F ( M l . G T . 7 0 ) PAUSE ' QN TABLE 0 " F '
DO 6 M2 = l , Ml
READ( 3 , 7 ) ( Q N ( M 2 , M 3 ) , M 3 = 1 , 8 )
6 WRITE(4,8)M2,(QN(M2,M3),M3=1,8)
7 FORMAT( 8 F 1 0 . 3 )
8 FORMA T ( I 1 0 , 7 F 1 0 . 1 , F 1 0 . 3 )
o
0026
0027
0028
0029
0030
0031
0032
0033
0034
0035
0036
0037
0038
0039
o
I F I M F I N . E O . O ) GO TO 23
JFIN=$JL(HFIN)
PARF= PARL( MFI N)
GO TO 24
23 JFIN=0.
PARF =L•
2 4 CONTINUE
0040
0041
WRI TE( 4 , 1 2 )
12 FORMAT( ' 1 ' , 3 OX,' CHANNEL QUANTUM NUMBERS• / / 1 7 X , ' PAR' , 8 X , ' $ J ■, 7 X ,
1
' T A U ' , 8 X , ' L C ' , 8X , ■J C • / I
K1=PC( MBKS, DCI MBKS) )
I F ( M l . G T . 7 0 ) P A U S E ' Q C TABLE 0 ' ' F *
00 9 M2=l,MI
READ! 3 , 7 ) ( QC( M2 , M3 ) , M 3 = 1 , 5 )
9 WRITE( 4 , 8 ) M2 , I QC( M 2 , M 3 ) , M3=1 , 5 )
o o o
0042
0043
0044
0045
0046
READ AND WRITE WAVE FUNCTIONS
WRITE(4,10)
10 FORMAT!' 1 • , 3 0 X , ' WA V E FUNCTIONS’ / / )
DO I I MK=1,MBKS
MD=DL( MK) / 2
DO I I M5=1 , 2
0 0 11 M l = 1 ,MD
M6 = ( M 5 - 1 ) * M D
M2=PLL( MK, 1+M6, M1+M6)
M3 = M2+M0-1
I F ( M 3 . G T . 3 0 0 I P A U S E ’ STS TABLE 0 " F '
REAO( 3 , 7 ) ( S T S I M4 ) , M4 = M2 , M3 )
11 WRITE! 4 , 7 ) (.STSIM4 ) , M4=M2, M3)
non
0047
0048
0049
0050
0 0 51
0052
0053
0054
0055
0056
0057
0058
READ AND WRITE THE RADIAL
INTEGRALS
0059
0060
0061
0062
0063
0064
R E A O ( 5 , 2 5 ) WT S , B P A R
2 5 FORMA T( 6 F 10 . 2 )
WRITE! 6 . 2 6 1 WT S
WRITE(6,27) BPAR
26 FORMAT! ' 6 WIl>THS=' , 6 F I 0 . 3 )
2 7 FORMA T ( ' 6B0UN0ARY VALUE PARAMS=•, 6 F 1 0 . 3 1
0065
0066
0067
0068
0069
R E A D! 5 , 2 0 ) RAOINT
FORMAT(6 F 1 0 . 3 )
0 0 21 M l = 2 , 6
0 0 2 1 M2 = 1 ,M 1
2 1 RADI N T ( M1 , M2 ) = R A 0 I N T ( M2 , M1 )
20
-457-
' 0070
0 0 71
22
WRITE(4,22) RA( )INT
FORMA T ( ' 4 ' , 1 8 X , ' RAO IAL
INTEGRALS• / / ( / 6 F 1 0 . 4 ) )
C
C
C
0072
0073
0074
0075
0076
0077
0078
0079
0080
0081
0082
0083
0084
0085
0086
0087
0088
0089
0090
0091
0092
0093
0094
0095
0096
CALCULATE REDUCED WIDTHS MATRIX
0 0 13 MK= 1 » MBKS
ML = OL(MK)
MC=DC( MKI / 2
0 0 1 4 M l = 1 .ML
LA=PL( MK, M1I
DO 1 4 M2=1,MC
C = PC(MK,M2I
TAUL=. TRUE.
I F I A B S I T A U I C ) I . G T . 0 . 7 ) TAUL=. FALSE.
M4=-LC(C)
I F ( TAULIM4=M4+1
I F(M4/2*2.NE.M4)M4=M4+1
M4=M4/2+JC(C)+0.5
GG=0 .
0 0 15 M 3 = l , ML
BET A = P L ( MK, M3)
I F ( $ T ( B E T A ) . N c . i T ( L A > ) G 0 TO 15
1 F ( L B ( B E T A ) . N E . L C I C ) I GO TO 15
I F ( J B ( BETA I . NE. J C ( C ) I GO TO 15
SIGN = 1 .
M8 = N ( B E T A I + l .
IF(M8/2*2.NE.M8)SIGN=-SIGN
M9=4.*N(BETA)-3.+LB(BETA>
I F ( H ( B E T A ) , E 0 . 2 . . A N O . T A U L ) G G = GG+ S TS( PLL( MK, M 3 , Ml I I *SI GN* WTS( M9)
I F I H I B E T A I . E Q . I . . A ND. . NOT. TAULI GG=GG+WFP*STS <PLL ( MK,M3 »M1 ) )*SI GN
1 *WTS(M9)
15 CONTINUE
IF(M4/2*2.NE.M4)GG=-GG
GG=GG* WI DT HS / 1 . 4 1 4 2
M7=PLC( MK, M1 , M2+MCI
I F ( M 7 . G T . 5 5 0 ) P A U S E ' G A M TABLE 0 " F *
GAM( PLC( MK, M l , M 2 ) >=GG
IF(STILA).EO.l.)GG=-GG
GAM(M7) =GG
14 CONTINUE
13 CONTINUE
0097
0098
0099
0 100
0101
0102
0103
0104
0105
0106
C
0107
0108
0109
WRI TE( 4 , 1 6 )
16 FORMAT( ' 1 ' , 3 0 X , •REDUCED WIDTHS M A T R I X ' )
DO 2 MK= 1 , MBKS
WRITE(4,17)
1 7 F ORMAT ! ' O' I
MC=DC(MK)
ML=DL(MK)
DO 2 C=1, MC
M 1 = P L C ( MK, 1 , C )
H2=PLC( MK, ML, CI
M3 = PC ( MK, C I
2 WRITE( 4, 18IM3, ( GAM( LAI, LA=M1, M2>
18 FORMAT! 1 5 , 3 X , 1 4 F 8 . 31
0110
0111
0 112
0113
0114
0115
0116
0117
0118
0119
C
0120
0121
CALL L I N K ! ' ME• I
END
C
SUBROUTINE SBF( R HO , F J , F N , F J P , FNP)
0001
C
C
C
C
/
COMPUTE SPHERICAL BESSEL FUNCTIONS
F J , FN SPHERICAL BESSEL FUNCTIONS AND FJP . F NP
REAL F J ( 4 ) , F N ( 4 ) , F J P ( 4 ) , F N P ( 4 >
0002
C
0003
0004
0005
0006
0007
0008
0009
41
0010
0011
0012
42
0013
0014
0015
0016
F J( I)=SI N(RHO) /RHO
FN ( 1 1 =COS ( RHO ) /RHO
F J ( 2 > = F J ( 1 I / R H O - F N I 1)
FN(2> = F N ( 1 1 / R HO+ F J I I I
DO 4 1 L = 1 , 2
F J ( L + 2 ) = - F J ( L >+ <2*L«-l ) / RHO*FJ ( L + l )
F N (L + 2 ) = - F N ( L ) + ( 2 *L+l )/ RH0n- FN( L + l )
0 0 42 1 = 2 , 4
FJP( I ) = F J ( I - 1 ) - I / R H 0 * F J ( I )
FNP( I ) = F N ( 1 - 1 ) - I / R H O * F N ( I I
F J P ( 1 ) = - F J (2 I
FNP (1 ) =-FN ( 2 )
RETURN
END
THEIR DERIVATIVES
0001
O OOOOOOOOOOOOOOOOOO
-458-
0002
O
0003
0004
0005
0006
0007
0010
0011
0012
O
0008
0009
0014
P O
0013
0015
0016
PHASE
'ME*
SECOND MAIN PROGRAM
(OUT OF FOUR)
COMPUTES REDUCED MATRIX ELEMENTS FOR TRANSITIONS TO THE GROUND
STATE OR TO EXCITED STATES.
CALLS SUBROUTINE ' ELM' TO CALCULATE
SINGLE- PARTI CLE MATRIX ELEMENTS.
THE ENERGY OEPENDENT FACTORS HAVE BEEN REMOVEO ANO PUT IN ' C S '
THE
THE MATRIX ELEMENTS BETWEEN PURE CONFIG. ARE COMPUTED FI RST ANO
THE MATRIX ELEMENTS BETWEEN TWO STATES ARE OBTAINED FROM THEM
WITH THE USE OF THE WAVE FUNCTIONS OF THOSE STATES.
ME(L,M>
IS THE MATRIX ELEMENT FOR THE El TRANSITIONS ( M= 1) OR
THE E2 TRANSITIONS (M=2) OR THE Ml TRANSITIONS ( M=3) FROM STATE
L TO STATE MF1N.
^
I MPLI CI T R E A L ! I , J , K , L , N )
COMMON / P A R AM/ RC. B, MBKS , WI DT HS, MF I N. J F
, P A R F , 0 L ( 1 2 ) , DC( 1 2 ) ,WFP
COMMON / Q N S / Q N ( 7 0 , 8 ) , Q C ( 7 0 , 5 ) , Q T ( 5 0 , 2 )
COMMON / S T S / S T S ( 3 0 0 ) ,GAM( 5 5 0 ) , RAO I NT( 6 , 6 ) , ME( 7 0 , 3 )
COMMON / W T / M C C ( 5 ) , T T ( 5 0 0 0 ) , WT( 5 0 0 0 )
C 0 MM0 N/ B W/ WT S ( 6 ) , BP A R ( 6 )
INTEGER*2 TT
INTEGER O L , D C , P L , P C , P L L , P C C , P L C , R E L O C
INTEGER C , C $ , LA, BETA
REAL PAR ( 1 ) , $ J L ( 1 ) , * T ( 1 ) , H ( 1 ) , N ( 1 ) , L B ( 1 ) , J B ( 1 > , E L < 1)
REAL WORK( 7 0 ) , ME
REAL F I N S ( 1 0 )
EQUIVALENCE
( Q N ( 1 , I ) , P A R ( I ) ) , ( QN( 1 , 2 ) , t J L ( 1 ) ) , ( ON( 1 , 3 ) , * T < 1 ) )
1,
( O N I 1 , 4 ) , H ( 1 ) ) , ( Q N ( 1 , 5 ) , N < 1 ) ) , ( Q N ( 1 , 6 ) , L 8 <1 ) ) ,
1
(Q N (1,7),JB (1)),(Q N (1,8),E L (1M
F(X)=SQRT(2.*X+1.)
ooon
CALL RELOC( MBKS)
MN=PL(MBKS, DL(MBKS) )
0017
0018
0019
GROUND STATE DECAY
I F I M F I N . N E . O ) GO TO 3 0
0 0 3 1 BETA = 1 , MN
DO 31 MODE=1 , 3
M=$T( BETA)
JL=MODE
IF(M0DE.EQ.3)$L=l.
TEM=0.
I F ( $ L . N E . t J L ( B E T A ) ) GO TO 32
TEM=ELM(MODE » M, N( B E T A ) , L B ( BETA) , J B ( BETA) , 1 . , H( BETA) - 1 . , H( BETA) - . 5 ,
1 RAOINT)
TEM=TEM*(-1.4 142*F(JB(BETA))/F (tL))
32 ME( BE TA,MOOE)=TEH
31 CONTINUE
GO TO 29
0020
0021
0022
0023
0024
002 5
ooo
0026
0027
0028
0029
DECAY TO EXCITED STATES
0030
0031
0032
0033
0034
0035
0036
0037
0038
0039
0040
0041
0042
0043
3 0 0 0 33 MK=1 , MBKS
ML=DLIMK)
0 0 33 L A = 1 , ML
I F ( M F I N . E Q . P L ( M K , L A ) ) GO TO 34
3 3 CONTINUE
PAUSE* ERR " M E " '
3 4 ML=ML/2
MF C = P L ( MK , 1 ) - 1
IF (LA.GT.ML)MFC=MFC+ML
M1 = P L L ( M K , 1 , L A ) - 1
DO 35 M2 = 1 , ML
35 F I N S ( M 2 ) = S T S ( M 2 + M 1 )
WRI TE( 6 , 3 6 ) M F I N , ( F I N S ( M 2 ) , M2=1, ML)
3 6 FORMAT( ' 4 M F I N = « , 1 4 , 8 X , ' W A V E FUNCTION OF FINAL STATE 1 S ' / 1 0 F 1 2 . 4 )
0044
0045
0046
0047
0048
0049
0050
0 0 4 5 M2=1, ML
M6=MFC+M2
M4=(H(M6)-1.-LB(M6))
1F(M4/2*2.NE.M4)M4=M4+1
M4= M4 /2+ JB( M6) +0. 5
4 5 I F ( M 4 / 2 * 2 . N E . M 4 ) F I N S ( M 2 ) = - F I N S (M2)
WRITE(6,36)MFIN,(FINS(M2)»M2=1,ML)
0051
0052
005 3
0 0 3 7 BETA=1,MN
DO 37 M00E= 1 , 3
TEM=0.
moo
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0 2 0 - 4 - 4-4 —
m m 71 w o
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H 0 3 H II II II 3 ** II 2 T
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w m N H m O fg <- 71 —• o:
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—
2 -1 2 m r o' v r — 03 —
or a
3
C > C 3 S (-fy — x m cd
— 3 CCP
m - m + — — ft C&_ —i m
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71 3 <- rv m 3 > -4
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W -> >
— O 71
2 —
O — o 71• H
O- 2 0 - 2 > w • w — 2 O -m -♦2
m (z m w m — 12* ♦ CA m w
— « <- ♦ + o m z 2 — « c.
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0054
0055
0056
0 OOO0 O O000 O O OO0 0 0 0 0 0 0
►
— — h- h- H H* >- H- — — — 0 0
>—
— f— H* — O O O O 0 0 O O O O O *0
h- ►- ►- »—h- ►
CD•*4 O' VI * UJ IV K O O 00 —JO' VI * cp rv *- 0 'O a»
o o o o o o o o o o o o
•W T 1
THIRD MAIN PROGRAM
(OUT OF FOUR)
COMPUTES ALLOWED TRANSITIONS (ALREADY SUMMED OVER STATES OF
THE COMPOUND SYSTEM) AND THE ANGULAR DISTRIBUTION COEFFICIENTS
ASSOCIATED WITH THEM FOR K = 0 , 1 , 2 , 3 , ANO 4
THESE COEFFICIENTS ARE STOREO IN ARRAY W, ANO THEY ARE USED IN THE
BLATT-BIEDENHARN FORMULA.
I MPLI CI T R E A L ! I , J , K , L , N )
o
0001
0002
PH ASE
COMMON /PARAM/RC,8,MBKS,WIDTHS,MFIN.JF ,PARF,DL(12),DC(12I,WFP
COMMON /QNS/QN(70t8)rQC(70,5),QT(50,2)
0003
0004
0005
0006
o
COMMON / S T S / S T S I 3 0 0 ) , G A M ( 5 5 0 ) , RAO I NT( 6 , 6 ) , ME( 7 0 , 3 )
COMMON / W T / M C C ( 5 ) , T T ( 5 0 0 0 ) »W ( 5 0 0 0 )
C0MMON/8W/WTS( 6 ) , 3 P A R ( 6 )
0007
INTEGER Q T , T , T $ , C H ( 1 > .MODE( 1 )
INTEGER C , C $ , L A , B E T A
INTEGER D L , D C , P L , P C , P L L , P C C , P L C , R E L O C
I NT EGERS TT
REAL PARC( 1 ) , A J C ( 1 ) , TAU( 1 ) , L C ( 1 ) , J C ( 1)
o
0008
0009
0010
0011
0012
EQUIVALENCE ( Q T ( 1 , 1 1 , CH( 1 1 ) , ( OT( 1 , 2 ) .MODE( 1 I I ,
(Q C (1,1),PA R C (1)),(Q C (1,2),SJC (1)),(Q C (1,3),T A U (1)),
(Q C (1,4),LC (1)),(Q C (1,5),JC(1))
o
1
1
0013
0014
o o o
F I { X ) = S QRT ( 2 . * X + 1 . )
CALL RELOC(MBKS)
0015
0016
0017
0018
0019
T=0
MN= PC( MBKS,OC( MBKSI I
DO I C = l , MN
IF ( TAU(C I . N E . - 0 . 5 ) GO TO 1
IF ( T . G E . 4 8 ) PAUSE •OT TABLE 0 " F '
CALL T R I A L ! 1 . , J F , t J C ( C I , £ 2 )
o
0020
0021
0022
OBTAIN A SET OF CONTRACTED TRANSITIONS
I F ( PARF. EQ. PARC ( C ) I GO TO 2
T = T+1
QT(T,11=C
QT(T,2>=1
2 CONTINUE
o
0023
0024
0025
0026
0027
0028
0029
0030
0031
o
CALL T R I A L ! 2 . , J F , $ J C ( C ) , 8 3 )
I F ( P A R F . N E . P A R C ( C l ) GO TO 3
T = T+1
Q T ( T , 1 ) =C
QT( T , 2 ) = 2
3 CONTINUE
CALL T R I A L ! I . , J F , $ J C ( C ) , C 4 )
I F ( P A R F . N E . P A R C ( C ) ) GO TO 4
T = T+1
QT{ T , 1 ) =C
QT ( T, 2 I =3
4 CONTINUE
o
0032
0033
0034
0035
0036
0037
1 CONTINUE
o
0038
MT=T
WRITE(4,5)(T,QT(T,1),QT(T,2),T=1,MT)
5 FORMAT! ' 4 ' , 8 X, ' T * ^ X . ' C 1 , 6 X , 'MOOE • , / / ( 3 1 1 0 ) )
o
0039
0040
0041
MCTT =0
DO 7 MK=1, 5
K=MK—1
MCT=0
DO 6 7 = 1 , MT
00 6 T$=l»T
non
0042
0043
0044
0045
0046
0047
OBTAIN QUANTUM NUMBERS
PI=0 .
$L = 1 .
I F ( M O D E ( T ) . E Q . 2 ) $ L =2 .
IF(M 0DE(T).EQ.3 )PI=1.
0048
0049
0050
0051
C
0052
0053
0054
PI$=0.
*L*=1.
IF(MOOE(TS).EQ.2)SLt=2.
-461-
0055
IF(M0DE(TS).EQ.3)PIS=1.
C
M1 = SL+PI +SLS+PI S+K
I F ( M 1 / 2 * 2 . N E . M 1 I GO TO 6
0058
0059
0060
006L
0062
0063
L=LC ( CHIT ) I
LS = L C ( C H ! T S ) I
J=JC( CH(T1)
J $ =J C ( C H I T S ) )
$J=SJC(CH(TI)
$JS=SJC(CH(TS))
non
0056
0057
0064
0065
0066
0067
CHECK IF W VANISHES BECAUSE OF ONE OF THE TRIANGULAR CONDITIONS
TRIAL<$L,$LS,K,C6 )
T RI AL) L« L S , K , £ 6 )
TRIAL(SJ,SJS,K,C6 )
TRIAL!J,JS,K,6 6 )
o
CALL
CALL
CALL
CALL
0068
0069
0070
0071
0072
MCTT=MCTT+1
I F ( M C T T . G T . 5 0 0 0 ) P A U S E ' W TABLE 0 " F *
MCT=MCT + 1
T T ! MC T T ) = 2 5 6 * T + T S
W( MCTTI = F I ( L > * F I ( L S ) * F I ( J ) * F I ( J S ) * < 2 . * S J + 1 . ) * ( 2 . * S J S + 1 . I *
1
< 2 . * K + 1 . ) * C 3 J ( $ L , S L S , K , - 1 . , 1 . , 0 . >*C3J( L , LS,K , 0 . , 0 . , 0 . I*
1
C6 J ( $ J , S J S , K , S L S , S L , J F ) * C 6 J ( J S , J , K , S J , S J S , 1 . 5 I *
1
C6J(L,LS,K,JS,J,0.5>
I F!T.NE.TS) W! MCTT)=2. * W(MCTT)
M1 = J + J $ + S J S - S J + J F - 1 .
IF(M1/2*2.NE.M1)W(MCTT)=-W(MCTT)
6 CONTINUE
7 HCC(MK) = MCT
WRITE( 6 , 8 )MCC
8 FORMAT! >1MCC = • , 5 1 1 0 / )
WRITE( 4 , 1 2 )
12 F O R M A T ( 4 X , ' T ' , 3 X , ' T S • , 1 4 X , • W• / I
DO 9 M l = l , MCTT
M2= TT( M11 / 2 5 6
M3=TT(M1) -M2*256
9 WRI TE(4 ,10) M2 ,M3 ,W! M1 )
10 F 0 R M A T ( 2 I 5 , F 1 5 . 7 )
n
0078
0079
0080
0081
0082
0083
0084
0085
0086
o
0073
0074
0075
0076
0077
0087
0088
.
.
o
CALL L I N K ! ' C S » )
ENO
000L
noooooon
SUBROUTINE CPRI NT( MBKS, A , N , TEXT, DC I
0002
0003
SUBROUTINE TO PRINT REAL OR IMAGINARY PART OF AN ARRAY STORED
IN MBKS BLOCKS UNDER CONTROL OF RELOC
N=0 IMAGINARY PART
N=1 REAL PART
TEXT MUST BE 24 CHARACTERS LONG
DC CONTAINS DIMENSIONS OF BLOCKS
INTEGER DC 11 ) , C , P C , P C C
REAL* 8 A I 1 ) ?T EXT I 3 I
0004
0005
WRITE(6,24) TEXT
24 F O R M A T ! ' l ' , 3 0 X , 3 A 8 I
0006
0007
0008
0009
DO 2 5 MK=1»MBKS
MC = DC (MK)
DO 2 5 J = 1 1MC
C = PC! MK, J I
M1=2*PCC(MK,1,J)-N
M 2 = 2 * P C C ( MK, MC , J ) - N
2 5 WRITE! 6 , 2 6 1C, ( A! M3) , M 3 = M 1 , M 2 , 2 )
2 6 FORMAT! 1 4 , 4X, ( 8 0 1 4 . 7 1 I
RETURN
ENO
0010
0011
0012
0013
0014
0015
-462C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
0001
PHASE ' CS*
FOURTH AND LAST MAIN PROGRAM.
THE THREE PREVIOUS MAIN PROGRAMS WERE EXECUTED ONLY ONCE, BUT ' C S '
IS EXECUTED ONCE FOR EACH ENERGY.
A SET OF OPTIONS IS READ FI RST ON UNIT 2 (OPTIONS ARE CALLED FLAG)
THERE ARE 10 OPTIONS POSSIBLE BUT NOT ALL ARE ACTI VE.
THEY ARE
ENTERED IN FORMAT 1 0 L I .
NORMAL RUNNING USES
FFFFFTFFFF
OPTIONS
FLAG(l)
WRITE ARRAYS 0 ANO Y
FLAG( 2 )
WRITE ARRAY
RO
F LAG! 3 )
WRITE ARRAY
RI
FLAGI4I
WRITE ARRAY
WK
F LAG! 6 1 WRITE CONTRIBUTIONS OF EACH PAIR OF TRANSITIONS TO THE
TOTAL CROSS-SECTION AND ANGULAR DISTRIBUTION COEFFICI ENTS,
OPTIONS
1 THROUGH 4 ARE FOR DEBUGGING PURPOSES ANO MAY BE REMOVED
RD IS R-MATRIX PLUS DIAGONAL MATRIX TO BE INVERTED, AND D.CONTAINS
THE ELEMENTS WHICH WERE AOOEO TO THE DIAGONAL OF THE R-MATRIX
RI CONTAINS THE INVERSE OF RD AND WK THE PRODUCT OF MATRICES
RI, RD(COMP. CONJG> AND Y
( ESSENTI ALLY WHAT I S CALLED TRANSFER
MATRIX IN THE THEORY)
THE ENERGIES AT WHICH THE CALCULATIONS ARE TO BE PERFORMED ARE.READ
ON UNIT
5 , IN TERMS OF FI RST ENERGY, ENERGY
STEP AND LAST ENERGY.
IF THE ENERGY STEP AND LAST ENERGY ARE OMITTED (OR ZERO)
A LIST
OF CONTRIBUTIONS OF ALL LEVELS TO THE TRANSITION AMPLITUDES AND A
LI ST OF CONTRIBUTIONS OF THE TRANSITION PAIRS
(TO THE TOTAL
CROSS-SECTION AND ANGULAR DI STRIBUTI ON COEFFICIENTS) ARE PRINTED
ON UNIT 6 .
THE ENERGY DEPENDENT FACTORS OF THE ELECTROMAGNETIC MATRIX ELEMENTS
WHICH WERE OMITTED IN ' ME ' ARE COMPUTEO.
COULOMB PHASE SHI FTS, COULOMB FUNCTIONS AND SPHERICAL BESSEL
FUNCTIONS ARE COMPUTED (SUBROUTINES C0UL2 ANO SBF)
THE ELEMENTS OF THE DIAGONAL MATRIX TO BE ADOEO TO THE R-MATRIX
ARE COMPUTEO FROM THESE, THEN THE R- MATRI X IS
COMPUTEO ANO THE
SUM I S PERFORMED.
THE RD MATRIX I S INVERTED
(CALL TO C 1 6 ) AND THE
TRANSFER MATRIX COMPUTED.
THE CONTRACTED TRANSITION AMPLITUDES ARE CALCULATEO, STORING
CONTRIBUTIONS OF ALL THE LEVELS INVOLVED.
FINALLY, THE TOTAL CROSS-SECTION AND ANGULAR DISTRIBUTION COEFF.
ARE COMPUTED AND PRINTED ON UNIT 6 , THE TRANSITION PAIRS ARE PUT
IN ORDER OF DECREASING IMPORTANCE AND THE LARGETST CONTRIBUTIONS
ARE PRINTED ON 6
( I F F LAG( 6 I I
I MPLI CI T R E A L l I . J . K . L . N )
C
0002
0003
0004
0005
0006
COMMON / PARAM/ RC, B , MBKS , WI DTHS . MFI N, JF , PARF, OL( 1 2 I , 0 C ( 1 2 1 , WFP
COMMON / O N S / O N ( 7 0 , 8 ) , 0 C ( 7 0 , 5 I , QT( 5 0 , 2 )
COMMON / S T S / S T S I 3 0 0 ) , G A M ( 5 5 0 ) , RAO I NT( 6 , 6 1 , ME ( 7 0 , 3 )
COMMON / WT/ MCC( 5 ) , T T ( 5 0 0 0 ) , W ( 5 0 0 0 )
C0MM0N/8W/WTS ( 6 ) , BPAR ( 6 )
C
\
)
0007
0008
0009
0010
0011
0012
0013
0014
0015
0016
■ 0017
0018
0019
0020
0021
0022
INTEGER OL, DC, P L , P C , P L L , PLC. PCC
INTEGER C . C t . L A
REAL EL( 1 1,ME
REAL PARC( 1 1 , S JC ( 1 ) , T A U ( 1 ) , L C ( 1 ) , J C ( 1 )
INTEGER O T , T , T $ , C H ( 1 ) , M D ( 1 )
I NTEGERS TT
LOGI CAL* 1 F L A G ! 10)
COMP LEX*16 R D 1 5 0 0 ) , R I ( 5 0 0 ) , W K ( 5 0 0 ) , C T E M
REAL F ( 4 > , G ( 4 ) , F P ( 4 > , G P < 4 ) , D R < 1 ) , R Y ( 1 )
COMPLEXES S I G ( 4 ) , CTM, 0 ( 7 0 ) , Y ( 7 0 ) , S ( 8 0 )
REAL C S C I 5 ) , A ( 5 ) , A B ( 5 ) , S T ( 1 0 0 )
INTEGER*2 TU( 1 0 0 ) , T S 2 , B F ( 7 0 )
L0G1CAL*1 BLANK/ 1 ' / . A S T / ' a ' / . C C
INTEGER MY( 7 ) , M Z ( 7 ) , MPH(1 4 1
REAL FAC( 3 ) , A R ( 1 4 ) , S L R ( 2 8 )
COMP LEX* 8 S L I 1 4 )
C
002 3
0024
EQUI VALENCE! D( 1 I , DR( 1 ) I , ( Y ( 1 I , R Y ( 1 1 I
EQUIVALENCE _ ( Q T ( 1 , 1 ) , CH( 1 ) ) , ( QT( 1 , 2 I , MO( 1 ) ) ,
1
( Q C < 1 , 1 ) , P A R C< 1 ) ) , ( Q C ( 1 , 2 ) , $ J C < 1 ) ) , ( O C ( 1 , 3 ) , T A U ( 1 ) ) ,
1
(OC( 1 , 4 ) , L C ( 1 ) ) , (Q C <1 , 5 ) , J C ( 1 ) ) , ( Q N ( 1 , 8 ) , E L ( 1 ) I
EQUI VALENCE! SL( 1 ) , S L R ( 1 ) )
0025
C
0026
0027
0028
CALL RELOC1MBKSI
READ! 2 , 9 1 EL AG
9 FORMAT! 10L1)
non
-403-
0029
0030
0031
0032
0033
0034
0035
0036
0037
0038
CHECK ON BLOCK DIMENSIONS,
BLOCK FINDER BF
non
0 0 7 0 MK=1,MBKS
I F ( 0 L ( M K ) . L E . 1 4 ) GO TO 70
PAUSE' NMBR OF STATES IN A BLOCK EXCEEDS
7 0 CONTINUE
M1=0
DO 71 MK= 1 , MBKS
MC=DC( MK)
DO 71 C=1, MC
M1=M1+1
7 1 BF ( Ml >=MK
0039
0040
14*
ENERGY STEPS
o
3 REA0(5,1)ESTART,ESTEP,EFIN
1 FORMA T I 3 F 1 0 . 0 )
0041
0042
0043
0044
non
E=ESTART-ESTEP
2 E=E+ESTEP
I F I E . G T . E F I N . A N D . E F I N . N E . O . ) GO TO 3
I F I E F I N . E Q . O . I EFIN =—1•
I FI MFI N.EQ.O) WN=E/197.3
I F ( M F I N . G T . O ) HN = ( E - E L I MF IN I I / 1 9 7 . 3
IF(WN.LT.0.)WN=0.
TEM=SQRT ( WN/2 • )
FAC ( 1 I=WN*1 . 61*TEM
FAC( 2 ) =2. 6000#WN*WN*TEM . .
FAC(3)=WN*T.EM
. .
....
o
0045
0046
0047
0048
0049
0050
0051
COMPUTE SOME ENERGY DEPENDANT FACTORS
0052
0053
0054
non
EP=12.*(E-15.961/11.
ETA=0.789/SQRTIEP>
RH0=.2098*RC*SQRT(E-15.96)
COMPUTE COULOMB PHASE SHIFTS SIG
SIGI11=( l . , 0 . )
0 0 7 MML=1, 3
CTM=CMPLX( 1. *HML, ETA)
7 SI G( MML+1) =SI G( MML) *CTM/ CABS( CTM)
non
0055
0056
0057
0058
0059
COMPUTE VECTORS 0 AND Y
o
CALL C0 UL2 I ETA, RHO. F, G , F P , GP)
COULOMB FUNCTIONS
MC=PC(MBKS, DC(MBKS))
DO 11 C=1, MC
ML L = L C ( C ) + 1 .
I F ( T A U ( C ) . G T . 0 . ) GO TO 11
U=B*BPAR(MLL)
1 F ( ML L . EQ . 1 ) U= B * B P A R ( 5 >
I F ( MLL. NE. 2 ) GO TO 9 8
B1=B*BPAR( ML L )
B2= B * B PA R ( 6 )
U =(B 1+B 2)/2.+(B 2-Bl)/2.*SIN ((E-32.1/7.6)
I F ( $ J C ( C ) * P A R C ( C ) . E Q . - 1 . ) U=BPAR( MLLI*B
9 8 D ( C ) = 1 . / ( U—RHO*CMPLX(GP( MLL) , F P ( MLL) ) / CMPLX( G ( MLL) , F ( ML L ) ) )
Y(C)=CMPLX(F(MLL),G(MLL))*SIG(MLL)/SQRT(RHO)/CONJG(D(C))
11 CONTINUE
o
0060
0061
0062
0063
0064
0065
0066
0067
0068
0069
0070
0071
0072
0073
RHO=. 2 0 9 8 * R C * S O R T ( A B S I E - 1 8 . 7 2 ) I
I F I E . L T . 1 8 . 7 2 ) RH0 = 0 .
0076
CALL S B F ( R H O , F , G , F P , G P )
n o
0074
0075
ooo
0077
0078
0079
0080
0081
0082
0083
0084
0085
0086
0087
0088
0089
SPHERICAL BESSEL FUNCTIONS
0 0 12 C=1,MC
MML=LC( CI+1•
I F ( T A U ( C ) . L T * 0 . ) GO TO 12
U=B*BPAR(MML)
1 F ( MM L . E 0 . 1 ) U = B * B P A R ( 5 )
I F ( M M L . E 0 . 2 . A N 0 . 4 J C ( C ) * P A R C ( C I . N E . - l . IU = B*BPAR( 6 I
0 ( C ) = 1 . / ( U - 1 . - R H 0 * C M P L X ( G P ( M M L ) , F P ( M M L ) I / C M P L X ( G ( MML) , F ( MML I ) )
Y ( C ) = SORT(RHO)*CMPLX(F(MML) ,G(MML) )/C0NJG(0(C) I
12 CONTINUE
COMPUTE RD MATRIX
0 0 1 6 MK=1,MBKS
ML=OL(MK>
MC=DCIMKI
0 0 17 C=1, MC
-464-
0090
0091
0092
0093
0094
0095
0096
0097
DO 17 C $ = 1 , C
CTEM=( 0 . D O , 0 . 0 0 )
DO 1 8 L A= 1 ,ML
18 CTEM=CTEM+GAM(PLCIMK, LA, C) ) * G A M ( P L C ( M K , L A , C t ) ) / ( E L ( P L <MK, LA> ) - E )
I F ( C . E O . C t ) CTEM=CTEM+D( PC( MK, C) )
RD( PCC( MK, C, C$>) =CTEM
17 R D ( P C C ( M K , C * , C ) >=CTEM
16 CONTINUE
C
0098
0099
0100
0101
0102
22
0103
0104
0105
0106
0107
21
23
20
I F ( . N O T . F L A G ! 1 ) ) GO TO 20
WRI TEI 6. 22)
FORMAT! 1 1 1 , 8X , ' C ' , 1 6 X, 1 RE D ' , 1 6 X , ' I M D ' , 1 6 X , ' R E
MC=PC(MBKS, DC(MBKS) I
DO 21 C = 1 , MC
M1=2*C
M2=M1-1
WRI TE! 6 , 2 3 1 C, D R M 2 ) , D R ( M 1 ) , R Y ( M 2 ) , R Y ( M 1 >
FORMAT!1 1 0 , 4 E 2 0 . 51
CONTINUE
Y ',16X,»IM
Y '/l
C
0108
0109
0110
0111
I F ( F L A G ( 2 ) ) C A L L C P R I N T ( MB KS , R D, 1 » ' R E A L PART OF RO MATRIX
I F ( FLAG! 2 ) ) CALL CPRI NT( M B K S , R D , 0 , ' IMAG PART OF RD MATRIX
C
C
C
',DCI
'.DC)
INVERSION OF MATRIX RO
DO 30 MK= 1,MBKS
MI=PCC(MK,1, I>
T 0 L = 1 . E—8
CALL C I 6 ( R 0 ( M 1 I , W K ( M 1 I , R I ( M l ) , 0 C ( M K > , T 0 L , C 3 1 >
GO TO 30
3 1 WRITE( 2 , 3 2 ) TOL
32 FORMAT!' T 0 L = ' , E 1 0 . 2 )
3 0 CONTINUE
0112
0113
0114
0115
0116
0117
C
0118
0119
0120
I F ( FLAG( 3 ) ) CALL CPRI NT( MB KS , R I , I , • REAL PART OF MATRIX R I ' . O C )
I F ( FLAG( 3 ) )CALL CPRI NT( MBKS, R I , 0 , • I MAG PART OF MATRIX R I ' . D C )
C
C
C
COMPLEX MATRIX MULTIPLY
DO 3 4 MK= 1 , MBKS
MC = DC (MK)
0 0 3 4 C=1, MC
0 0 34 C i = l , M C
CTEM=( 0 . 0 0 , 0 . 0 0 I
DO 35 M1=1,MC
35
CTEM= CT EM+ RI ( P CC( MK, C, M1 ) ) * DC0 NJ G( RD( PCC<M K , H 1 , C S I ) )
I F ( C . E Q. C t > C T E M= C T E M- < 1 . 0 0 , 0 . 0 0 )
34 WK( P CC( MK, C, CS) I =CT EM* Y ( P C( M K , C * ) )
0121
0122
0123
0124
0125
0126
0127
0128
C
0129
0130
I F ( FLAG( 4 ) )CALL CPRI NT( MBKS, WK, 1 , ' REAL PART OF MATRIX WK' . DC)
I F ( FLAG! 4 I ) CALL CPRI NT( MBKS,WK, 0 , • IMAG PART OF MATRIX WK' . OC)
C
C
C
C
0131
0132
0133
0134
0135
0136
0137
0138
0139
0140
0141
0142
0143
0144
0145
0146
0147
0148
0149
0150
0151
0152
0153
0154
0155
0156
0157
0158
0159
0160
CONTRIBUTION TO S MATRIX FROM THE VARIOUS LEVELS
73
72
76
75
77
MT=MCC(1)
0 0 74 7 = 1 , MT
C=CH( T )
MODE=MD(TI
MK=8 F ( C)
ML=0L(MK)
MC=0C(MKI
DO 73 M=1 , MC
I F ( P C ( M K , M ) . E O . C ) GO TO 72
CONTINUE
DO 7 5 LA=1, ML
CTEM=( 0 . 0 0 , 0 . 0 0 )
DO 7 6 C $ = l , M C
CTEM=CTEM+GAM(PLCIMK,LA,C6 I ) * W K ( P C C ( MK, C$, MI I
CONTINUE
M1=PL( MK, LA)
S L ( L A ) = M E ( M I , M 0 0 E ) *FAC( MODE) / ( EL( M l ) - E ) * C T E M
CTM=(0.,0.)
NA2=0.
DO 77 LA=1 , ML
NA2=NA2+CABS( S L ( L A ) I
CTM=C TM+ SL ( LA I
S ( TI=CTM
NA1=CABS(CTMI
I F I N A 2 . L T . l . E - 5 0 ) GO TO 74
KEY=NA1/ NA2
I F I E S T E P . N E . O . ) GO TO 74
I F I N A 1 . L T . l . E - 1 0 ) GO TO 82
T E M= 5 7 . 3 * A I MA G ( C L 0 G( C T M) )
I F 1 T E M . L T . 0 . I TEM=360. +TEM
-465-
82 0 0 78 LA=1, ML
A R ( L A ) = C A B S ( S L ( L A ) 1 * 1 0 0 . / NA2
MP H ( L A ) = 1 . E 1 0
I F ( A R I L A ) . L r . l . E - 1 0 ) GO TO 78
MPH(LA)=57.3*A1MAG(CL0G(SL(LA) ) 1 + 0 . 5
I F ( M P H I L A ) . L T . O ) MPH( LA) =360+MPH( LA>
7 8 CONTINUE
ML2=2*ML
ML 2=ML/ 2
WRITE( 6 , 8 0 I T , K E Y , N A 1 , T E M , ( A R ( L A ) , M P H ( L A I , L A = 1 , M L 2 )
ML2=ML2+1
WRITE! 6 , 81 MARI LA) ,MPH(LA) , LA=ML2, ML)
8 0 FORMAT ( 1 4 , < KEY = ' , F 4 . 3 , F 1 1 . 7 , • ( ' , F 5 . 1 , ' ) • , 7 ( F 7 . 1 , ' ( • , 13 , • ) •
81 FORMAT( 3 I X , 7 ( F 7 . 1 , • ( • , 1 3 , • ) • ) )
7 4 CONTINUE
o o o
0161
0162
0163
0164
0165
0166
0167
0168
0169
0170
0171
0172
0173
0174
0175
0176
0177
0178
0179
0180
0181
0182
0183
0184
0185
0186
0187
0188
0189
COMPUTE CROSS-SECTION
o
MREL=0
0 0 5 0 MM=1, 5
TEM1=0.
TEM2 =0 •
MTT=MCC(MM)
0 0 51 MP=1 ,MTT
MREL=MREL+l
T=TT(MREL) /256
T$=TT(MREL)-256*T
T E M = R E A L ( S l T ) * C O N J G ( S ( T $ ) ) * W ( MR E L ) )
TEM1= TEM1 + T EM
51 TEM2 =TEM2 + ABS <TEM)
CSC(MM)=TEM1
50 AB(MH)= TEM2
0190
0191
0192
0193
0194
0195
non
A(1)=35.69*CSC<1)/(E-15.96)
A l l 1 = 1 0 0 0 0 . * A ( 1)
0 0 52 M l = 2 , 5
52 A ( M 1 ) = C S C ( M 1 ) / C S C ( 1 )
WRITE(6,53)E,EP,A
5 3 FORMAT! 1 E = • , F 5 . 2 , 5 X , ' EP = •, P 5 . 2 , 5 X , • * * * • , F B . 2 , •
1
10X,'A ='.4F 10.3)
0196
0197
0198
0199
0200
FIND RELATIVE CONTRIBUTIONS OF TRANSITION PAIRS
55
0201
0202
0203
0204
0205
0206
0207
0208
0209
90
0210
0211
0212
0213
0214
0215
0216
0217
0218
0219
0220
0221
0222
58
57
I F ( F L A G ( 6 ) . O R . E S T E P . E Q . 0 . ) GO TO 5 5
GO TO 54
MREL=0
DO 5 6 MM=1 , 5
I F ( A B ( M M ) , G T . 1 . E - 2 0 ) GO TO 90
MREL=MREL+MCC(MM)
GO TO 56
M1=0
T EM1 =0 . 002 * AB( MM)
CC=8 LANK
MTT =MCC( MM)
DO 5 7 MP=1 ,MTT
MREL=MREL+1
T=TT( MREL1 / 2 5 6
T$=TT(MREL)-256*T
TEM=REAL( S(T)*CONJG( S( T$) )*W(MREL))
I F ( A B S ( T E M ) . L T . T E M 1 ) GO TO 57
M1=M1+1
I F ( M l . L E . 1 0 0 ) GO TO 5 8
CC=AST
GO TO 57
TU(M1)=256*T+T*
ST( Ml ) =TEM
CONTINUE
0223
0224
0225
0226
0227
0228
0229
0230
02 31
59 M2=7
I F ( M l . L T . M 2 )M2=M1
DO 6 0 M3=1, M2
DO 60 M4=M3,M1
I F ( ABS( S T ( M 3 ) ) . G T . ABS( S T ( M4) ) )
TEM=ST(M4)
S T ( M 4 ) = S T (M3)
S T ( M3) = T EM
TS2 = TU(M4)
TUIM4 >=TU(M3)
TUIM3) =TS2
6 0 CONTINUE
02 32
0233
0234
0235
0 0 61 M3 = 1 , M2
M Y ( M3 ) = T U ( M3 ) / 2 5 6
MZ(M3)=TU(M3)-256*MY(M3)
61 S T ( M 3 ) = S T ( M 3 ) / A B ( M M ) * 1 0 0 .
GO TO 6 0
MICROBARNS' ,
-4G6-
,
0236
M 3= M M -1
0237
0238
0239
0240
0241
0242
0243
0244
WRI TE! 6 , 6 2 I C C , M 3 , ( M Y I M 4 ) , M Z ( M4) , ST< M4 ) , M 4 = 1 , M 2 )
62 FORMAT!• ' , A 1 , I 7 , 7 ( *
( 1, 1 2 , • , • , I 2 , ’ ) • , F 5 . 1 ,• # • ))
56 CONTINUE
WR1TE( 6 , 6 3 I
6 3 FORMAT! 1 1 )
54 CONTINUE
GO TO 2
END
C
0001
SUBROUTINE C 1 6 ( A , B , C , N , TOL, * I
C
C
C
C
C
C
C
C
C
C
C
000 2
0003
0004
0005
0006
0007
0008
0009
0010
0011
0012
0013
0014
INVERSION OF GENERAL C0MPLEX*16 MATRIX
METHOO IS GAUSSIANELIMINATION
CHECKS THE ACCURACY OF ANSWERS
A
ORIGINAL N*N MATRIX
C
INVERSE OF A
B
IS AN N*N WORK AREA(ORIGINAL MATRIX
A IS
NOTOESTROYEO)
TOL
IS THE MAXIMUM TOLERANCE ALLOWED
ONTHEPRODUCT
A*C
IF TOLERANCE IS NOT MET, AN ERROR RETURN OCCURS AND
TOL
WILL CONTAIN THE ERROR WHICH CAUSED THE REJECTION.
C0MPLEX*16 A ( 1 ) , B ( 1 ) , C (1 I , O N E / ( 1 . 0 0 , 0 . 0 0 ) / , Z E R O / ( 0 . 0 0 , 0 . D O ) /
C0MPLEX*16 DIA
I F I N . G E . 2 ) GO TO 1
C(1)=0NE/A(II
RETURN
1 N2=N*N
0 0 2 1= 1 , N2
B ( I ) =A( I I
2 C ( I ) =ZERO
J=1
00 3 1 = 1 , N
C ( J)=ONE
3 J=J+N+1
C
0015
15 DO 4
1=1,N
C
0016
0017
0018
0019
0020
0021
0022
0023
0024
0025
0026
0027
0028
0029
DIA = B( ( 1 - 1 ) * N + I )
0 0 8 K=1, N
L=(K-1)*N+I
B(L)=B(L)/DIA
8 C ( L ) =C ( L I / O IA
00 9 J=1,N
I F ( J . E Q . I ) GO TO 9
0IA=B((I-1)*N+J)
00 9 K=1 , N
L=(K-1)*N+J
B(L)=B(L>-B(L+I-J>*DIA
C(L)=C(Ll-C<L+I-J)*DIA
9 CONTINUE
4 CONTINUE
C
"'X
\
I
0030
0031
0032
0033
0034
0035
0036
0037
0038
0039
0040
0041
0042
0043
AC=0.
DO 1 0 1 = 1 , N
0 0 10 J = 1 , N
DIA=ZERO
DO 11 K=1, N
11 0 1 A = D I A + A ( I + I K - l ) * N ) * C ( K + ( J - l ) * N )
IF(I,EO.J)DIA=DIA-ONE
1 0 AC = AC+CDABS ( DI A )
I F ( A C —T OL 1 1 2 . 1 2 , 1 3
12 TOL=AC
RETURN
13 TOL=AC
RETURN1
END
ROW
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0001
COEFFICIENT,
A.B.C
IS
TOP
ROW,
X,Y,Z
IS
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— «
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FUNCTION
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-4G8C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
c
C
C
C
C
0001
THIS FUNCTION HAS MULTIPLE ENTRIES AND IS USEDTO COMPUTE THE
LINEAR
SUBSCRIPTS OF ELEMENTS OF AN ARRAY IN CUMPACT STORAGE.
THESE ARRAYS CONSIST OF BLOCKS ALONG THE DIAGONAL (NOT NECESSARILY
SQUARE), THE REST BEING ZERO.
ONLY THE NON-ZERO BLOCKS ARE STORED
AND THIS SUBROUTINE FINOS THE POSITION IN STORAGE FROM THE BLOCK
NUMBER ANO THE ROW AND COLUMN NUMBERS WITHIN THIS BLOCK OF A GIVEN
ELEMENT.
THE FUNCTION MUST BE I NI T I A L I Z E D BEFORE USE BY CALLING THE ENTRY
RELOC WITH K EQUAL TO THE TOTAL NUMBER OF BLOCKS (NOT MORE THAN 12 )
THE DIMENSIONS OF THE K BLOCKS MUST BE SUPPLIED IN COMMON, IN ARRAYS
OL ANO DC. ALL ELEMENTS OF DL MUST BE EVEN (SEE
BELOW)
THE ENTRY RELOC IS UNSUCCESSFUL AND RETURNS THE INTEGER - 1 IF THE
DIMENSIONS GIVEN IN DL ARE NOT EVEN OR IF THEREARE MODE THAN
12
BLOCKS. OTHERWISE THE INTEGER 1 I S RETURNED.
ONE DIMENSIONAL ARRAYS
P L ( K , I ) IS POSITION OF ELEMENT I OF BLOCK K WHEN BLOCK DIMENSIONS
ARE GIVEN BY DL.
PC(K ,I)
IS POSI TI ON OF ELEMENT I OF BLOCK K WHEN DIMENSIONS ARE
GIVEN BY DC.
TWO-DIMENSIONAL ARRAYS
PC C( K, I , JI
IS POSITION OF ELEMENT OF ROW 1 AND COLUMNJ IN BLOCK K
FOR MATRIX WITH ROW DIMENSIONS AND COLUMN DIMENSIONS GIVEN 8 Y DC.
PLCI K, I , J ) IS POSI TI ON OF ELEMENT OF ROW I ANO COLUMN J IN BLOCK K,
FOR MATRIX WITH ROW DIMENSIONS GIVEN BY DL ANO COLUMN DIM
GIVEN BY DC.
PLL( K , I , J )
I S MORE COMPLEX.
I ANO J ARE STILL THE ROW AND
COLUMN NUMBERS OF THE ELEMENT WITHIN BLOCK K BUT THE
D L ( K I = D L ( K) BLOCK K HAS BEEN SPLI T INTO 4 EQUAL SUB-BLOCKS
AND ONLY THE TWO SUB-BLOCKS ALONG THEDIAGONAL
WERE STORED,
THE REST BEING ASSUMED ZERO
t h i s i s c o n v e n i e n t f o r s t o r i n g w a v e f u n c t i o n s when t i s a
GOOD QUANTUM NUMBER.
ONE BLOCK THEN REFERS TO A S P I N PARITY COMBINATION, ANO IT I S SPLI T INTO
TWONON-ZERO
0 1 SCONNECTEO PARTS, T=0 ANO
T=l.
INTEGER FUNCTION RELOC( K)
C
0002
0003
0004
I MPLI CI T I N T E G E R ! A - Z )
COMMON / P A R A M/ A ( 7 I , D L ( 1 2 ) , 0 C ( 1 2 )
INTEGER RL ( 12 ) / 0 / , R C ( 12 ) / O / , P.LL ( 12 ) / O / , R C C ( 12 ) / O / , RLC < 12 I / O / ,
1
0LL(12),DCC(12),DLC(12)
C
0005
0006
0007
0008
0009
DO 3 L = 1 , K
3 I F ( D L ( L ) / 2 * 2 . N E . 0 L ( L ) ) GO TO 4
I F ( K . L E . 1 2 ) GO TO
1
4 RELOC = - 1
RETURN
C
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
0020
0021
1 0 0 2 L= l , K
0LL(L)=DL(L)*DL(L)/2
DCC( L) =DC (L )* OC (L)
2 OLC<L)=DC( L) *OL( L)
0 0 5 L=2 , K
RL(L)=RL(L-1)+DL(L-l)
RC( L ) = R C ( L - 1 ) + 0 C ( L - l )
RLL(L)=RLL(L-1)+0LL(L-1)
RCC(L)=RCC(L-lI+OCC(L-l)
5 RLC(L)=RLC(L-1)+DLC(L-1)
REL0C = 1
RETURN
C
0022
0023
0024
ENTRY P L ( K , I )
PL=RL(K)+I
RETURN
C
0025
0026
0027
ENTRY P C ( K , I )
PC=RC( KI + I
RETURN
C
0028
0029
0030
0031
0032
0033
0034
'
ENTRY P L L ( K , I , J )
D=DL(K)/2
I F ( I . G T . D ) GO TO 6
PLL=RLL(K )+I+0#(J-l)
RETURN
6 PLL=RLL(K)+0*J+I-0*2
RETURN
C
0035
0036
0037
ENTRY P C C I K , I , J »
PCC=RCC(K)+I+DC(K)#(J-1)
RETURN
C
0038
0039
0040
ENTRY P LCI K, I , J )
PLC=RLC(K)+I+0L(K)*CJ-1)
RETURN
C
0041
END
-4G 9 -
C
0001
0002
C
C
C
C
C
C
C
C
C
C
C
C
C
C
COULOMB FUNCTIONS PROGRAM
SUITABLE FOR PROTONS ON CARBON
THE REGULAR FUNCTIONS ARE COMPUTED FROM RECURSION RELATIONS AND NOR
MALI SAT I ON CONDITIONS.
THE IRREGULAR FUNCTIONS OF ORDER 0 AND 1 ARE OBTAINED FROM EIGHT
TERMS IN THE ASYMPTOTIC EXPANSION. WHEN RHO I S LESS THAN 2 . 5 ,
THE ASYMPTOTIC EXPANSION IS USED AT RHO=2. 5 AND THE DIFFERENTIAL
EQUATION INTEGRATED TOWARDS ZERO.
THE SUBROUTINE COULD BE USED FOR HEAVIER NUCLEI PROVIDED THE ASYMPT
EXP STARTING VALUE IS INCREASED. THE RESULTS WILL NOT NECESSARILY
AGREE WITH BLOCH ANO BREI T•S TABLES WHICH CONTAIN SOME INACCURACIES.
THE REMAINING IRREGULAR FUNCTIONS AND THE DERIVATIVES ARE COMPUTED
FROM RECURSION RELATIONS.
SUBROUTINE C0 U L 2 ( ETA, RHO, F , G , F P , G P I
C
C
C
C
C
C
C
C
ETA AND RHO ARE INPUT ARGUMENTS
ALL OTHER MUST BE DIMENSIONED 4 IN CALLING PROGRAM.
RETURNS F , G COULOMB FUNCTIONS AND FP. FG THEIR DERIVATIVES
FOR L=0 TO L = 3 .
MESSI AH' S DEFI NITI ON (APPENDIX ALBERT MESSIAH MFCANIQUE QUANTIQUEI
REAL*4 P I / 3 . 1 4 1 5 9 2 6 5 / , C F ( 2 1 I , D ( 2 1 I , F ( 4 I » G ( 4 ) , F P ( 4 ) , G P ( 4 )
REAL* 8 D E T A , D R H 0 , R K , S F , S G , A K , B K , F K , G K , F K 1 , G K 1
REAL* 8 0 G1, DG2
COMPLEX* 8 CETA. AETA, ACC
REAL*4 C ( 1 5 1 / 1 . 0 , 0 . 5 7 7 2 1 5 6 6 , - 0 . 6 5 5 8 7 8 0 7 , - . 0 4 2 0 0 2 6 4 , 0 . 1 6 6 5 3 8 6 1 ,
1
-.04219773,-.00962197,0.00721894,-.00116517,-.00021524,
1
0.0 C 012 8 0 5 , - . 0 0 0 0 2 0 1 3 , - . 0 0 0 0 0 1 2 5 , 0 . 0 0 0 0 0 1 1 3 , - . 0 0 0 0 0 0 2 1 /
LOGICAL FLAG
0003
0004
0005
0006
0007
C
0008
0009
FLAG=. TRUE.
I F ( RHO.GT. 2 . 5 ) FLAG=. FALSE.
RHOK=RHO
I F ( FLAG 1R H0 = 2 . 5
0010
0011
0012
C
C
C
ASYMPTOTIC EXPANSION
00 9 1=1,2
L= I - 1
CETA= CMPL X( 0 . , ET A)
AETA=CET A
ACC=(0 . , 0 . )
DO 4 K= 1 » 1 5
ACC=ACC+C( K) *AETA
4 AETA=AETA*CETA
SIGMA=ATAN(AIMAGI A C C ) / R E A L ( A C C ) )
I F I RE A HAC C I . L T . O . ) S I GMA = S IGMA+PI
SI GMA=- SI GMA+RHO- ETA*ALOG( 2. *RHO)
IF(L.EQ.0)THETA=SIGMA+PI/2.
I F ( L . E Q . 1 ) T H E T A = SIGMA+AT AN( ETA)
0013
0014
0015
0016
0017
0018
0019
0020
0021
0022
0023
0024
c
002 5
0026
0027
0028
0029
0030
0031
0032
0033
0034
0035
0036
0037
0038
0039
0040
0041
0042
0043
5F=1.
FK = 1.
SG=0 •
GK=0.
DETA=ETA
DRHO= RHO
DO 6 J = 1 , 6
K= J —1
KK=2*K+1
RK=( KK+1) *DRH0
AK=KK*DETA/RK
B K = ( L * ( L + l ) - K * ( K+ l I + DETA* DET AI / RK
FK1=AK*FK-BK*GK
GK1=AK*GK+BK*FK
SF=SF+FK1
SG=SG+GK1
FK=FK1
GK = GK 1
6 CONTINUE
C
0044
0045
0046
0047
Ft I ) =SG*C0S(THETA> + SF*SI N( THETA)
G< I ) = S F * C O S ( T H E T A ) - S G * S I N( T H E T A )
9 CONTINUE
I F ( . N O T . F L A G ) GO TO 5 0
C
C
C
0048
0049
0050
0051
PROCEED TO INTEGRATE
SPAN=RHO-RHOK
STEP =SPAN
NSTEP=1
DO 5 2 1 = 1 , 1 0
INWARDS FOR G ( l )
ANO G ( 2 ) .
-470-
0052
0053
0054
0055
0056
0057
0058
0059
0060
0061
0062
0063
0064
0065
0066
0067
0068
0069
0070
STEP=STEP/2.
NSTEP=2*NSTEP
I F I S T E P . L T . 0 . 0 0 5 ) GO TO 53
52 CONTINUE
53 0G1=G(1)
DG2 = G( 2 )
DO 54 I = 1, NS TEP
RHO=RHOK+(NSTEP—1 + 1 )*STEP
Z l = l »/ KHO+ETA
Z 2 = S QR T ( 1 . + E T A * E T A )
GP 1 = Z 1 * DG1 - Z 2 * DG2
GP2=Z2*DG1-Z1*0G2
Z1=2.*ETA/RH0-1.
Z2=STEP*STEP/2.
DGl =-GPl =>STEP+( Z1=>Z2 + 1 . >*DG1
5 4 DG2=- GP2*STEP + ( ( Z l + 2 . / R H 0 / R H 0 ) * Z 2 + l . ) *DG2
G( 1 ) = DG1
G ( 2 ) = DG2
RHO=RHOK
COMPUTE REGULAR SOLUTION F ( L )
0071
0072
0073
0074
0075
0076
0077
50 C F ( 2 1 ) = 0 .
CF(20)=1.
0 0 11 1 = 1 , 1 9
L= 2 0 - l
XL = L+1
C F ( L ) = ( 2 * L + 1) *<ETA+< L * X L / R H 0 ) ) * C F ( L + l >1
L*S0RT( XL*XL+ETA*ETA)*CF(L+2)
11
C F ( L ) = C F ( L ) / X L / S O R T ( L* L+ETA*ETA)
NORMALIZE
D<1)=0.
0078
0079
0080
DO 12 L = l , 2 0
12 0 ( L + l ) = 0 ( L ) + A T A N ( E T A / L ) + P I / 2 .
0081
0082
0083
0084
0085
0086
0087
0088
TPE=2.*PI*ETA
SUC=0 .
SUS=0.
DO 13 1 = 1 , 2 1
XL=2. * 1 - 1 .
SUS=SUS+XL*SIN(0(I) ) *CF( I )
13 SUC = SUC + XL*COS( 0 ( I ) ) * C F ( I I
X =( EXP(TPE) -1. )/ RH0/ RH0 /TPE*(SUS*SUS+SUC*SUC>
0089
0090
00 91
X= SOR T( X >
DO 1 4 1 = 1 , 4
1 4 C F ( I ) =CF( I ) / X
0092
0093
0094
0095
0096
009 7
0098
0099
19 DO 30 1 = 1 , 4
30 F ( I ) = C F < I )
Z1=S0RT(4.+ETA*ETA)
Z2 =SQRT( l . + E T A * E T A )
G (3 1 = ( 3 . * ( ETA+2./RH0)*G( 2 ) - 2 . * Z 2 * G ( I ) ) / Z l
G ( 4 ) = ( 2 . 5 * ( E T A + 6 . / R H 0 ) # G ( 3 ) - 1 . 5 * Z l * G ( 2 ) ) / S ORT ( 9 . + ET A* ET A)
FP(1)=(1./RH0+ETA)*F(1)-Z2*F(2)
GP(1)=(1./RH0+ETA)*G(1)-Z2*G<2>
0 0 31 L = 1 , 3
Z1 = SQRT( L*L + E T A*E T A )
Z2 = L*L/ RH0+ETA
F P IL + l) = ( Z 1 *F (L )-Z 2 *F IL + 1 ) )/L
31 G P ( L + 1 ) = ( Z 1 * G ( L ) - Z 2 * G ( L + 1 ) ) / L
COMPUTE REMAINING G AND DERIVATIVES OF F ANO G
0100
0101
0102
0103
0104
0105
0106
^\
V
V
)
RETURN
END
-471-
0001
FUNCTI0N C 3 J ( A , B , C , X , Y , Z )
C
0002
REAL*8 F ( 5 0 » , P, DS QRT
LOGICAL F I R S T / . F A L S E . /
INTEGER L C9 J
OOOOOOOOOOO
0003
0004
0005
0006
0007
0008
0009
3J SYMBOL SUBROUTINE. THE INPUT ARGUMENTS ARE ASSUMED TO SATISFY
THE TRIANGULARITY CONDITIONS. THE MAGNETIC QUANTUM NUMBERS TO BE
LESS THAN OR EQUAL TO THE THEIR RESPECTIVE ANGULAR MOMENTA IN
ABSOLUTE VALUE AND THE ARGUMENTS ARE ASSUMED EXACT (NO ROUND UFF
ERRORS PRESENT) .
I T IS NOT ASSUMED THAT X+Y+Z IS 0
THE SUBROUTINE IS LIMITED TO A+B+C . L E . 48
A,B,C
IS THE TOP ROW, X , Y , Z
IF ( X+Y+Z I 7 , 6 , 7
7 C3J=0.
RETURN
6 I F I F I R S T J GO TO 1
FIRS T = . TRUE.
F (l)=l
00 2 1=2,5 0
2 F ( I )= ( 1 - 1 ) * F ( 1 - 1 1
1 L ( 1 1=A+B- C
L(2)=A-B+C
L ( 3) = - A+B+C
L ( 4 )=A+X
L(5)=A-X
L ( 6 )=B+Y
L ( 7 1=B- Y
L ( 8 )=C+Z
L ( 9 I =C—Z
0010
0011
0012
0013
0014
0015
0016
0017
0018
0019
O
0020
0021
0 022
P= l .
DO 3 1 = 1 , 9
3 P = P * F ( L ( I 1+ 1 )
J = A+B+C
C3J = DSQRT ( P / F I J + 2 ) )
o
0023
0024
0025
0026
0027
0028
0029
0030
0031
0032
0033
0034
0035
0036
0037
0038
0039
0040
00 41
0042
0043
0044
0045
0046
0047
0048
L ( 1 ) =A+B- C
L ( 2 1= 0
L(3)=A-X
L(4)=C-B+X
L ( 5 ) =B+Y
L(6)=C-A-Y
K1=-MIN0(0,L( 4 ) , L ( 6 ) )
K2 =MI NO ( L ( 1 ) , L ( 3 ) , L ( 5 ) >
S=0.
DO 4 K=K1, K2
CS = 1.
M=K
00 5 1=1,6
M=-M
5 CS = C S / F ( L ( I I + M+ l )
I F ( M / 2 * 2 . N E . M ) CS=-CS
4 S = S+CS
N= A - B - Z
C3J=C3J*S
I F ( N / 2 * 2 . N E . N ) C3J=- C3 J
RETURN
END
0001
OOO
SUBROUTINE T R I A L ( A , B , C , * I
0002
000 3
0004
0005
0006
CHECK OF TRIANGULAR RULE
I F ( A . G T •B+CI
IF(a.GT.C+A)
IFIC.GT.A+B)
RETURN
END
RETURN 1
RETURN 1
RETURN 1
THE BOTTOM (MAGNETIC Q . N . ) .