- Department of Theoretical Physics UMCS

Transcription

Nuclear Mean-Field Hamiltonians and
Their Spectroscopic Predictive Power
Jerzy DUDEK
Department of Subatomic Research, CNRS/IN2 P3
and
University of Strasbourg, F-67037 Strasbourg, FRANCE
25th September 2009
Jerzy DUDEK, University of Strasbourg, France
Nuclear Hamiltonians and Spectroscopic Predictive Power
COLLABORATORS:
Karolina RYBAK, DRS Strasbourg
Bartlomiej SZPAK, IFJ Kraków
Marie-Geneviève PORQUET, CSNSM Orsay
Hervé MOLIQUE, DRS Strasbourg
Bogdan FORNAL, IFJ Kraków
1. Posing the Problem of Statistical Insignificance
Part I
Mean-Field Hamiltonian - New Ways of Thinking
About this Particular Project: What? and Why?
• We assume that the Nuclear Mean-Field Theory will remain
one of the fundamental nuclear-physics tools for a while
• Existing developments, especially the Relativistic Mean-Field
(explicit link with particle physics) and Density Functionals
(Kohn-Sham theorem, its potentialities in nuclear physics) do
represent the best we can and should do at present
• Possibilities of extrapolations to unknown, extreme isospin
areas remain, using the mildest formulation, unsatisfactory
• Possibilities of extrapolations to unknown, extreme isospin
areas remain, using the mildest formulation, unsatisfactory
• Questions of statistical significance of the theory modeling
the most basic questions asked in other domains of research
are seldom - if ever - posed. This will be our subject today.
About the Goals and New Technical Means
Subjects behind the Title and Exotic-Nuclei Physics
• Shells, Spherical Gaps, Shell Evolution with Isospin etc.
• Nuclear Central, Spin-Orbit and Tensor Mean-Fields
• The Concept and Properties of Theoretical Errors etc.
• The Concept and Properties of Theoretical Errors etc.
• The Concept of Predictive Power of the Nuclear M-F
Hamiltonians, Mathematical Criteria and Precision etc.
1.1 Inconsequences of the Past & How to Do It Better
1.1 Constructing Experimental Single-Particle Energies
Introduction: Defining the Physics Problem
Consider a mean-field Hamiltonian: RMF, HF, Phenomenological ...
Hmf = Hmf (r̂, p̂, ŝ; {p});
{p} → parameters
After laborious constructions of Hmf , we often get terribly exhausted
and forget that: Parameter determination is a noble, mathematically
sophisticated procedure based on the statistical theories often more
involved than the physical problems under study!
Hmf = Hmf (r̂, p̂, ŝ; {p});
{p} → parameters
Hmf = Hmf (r̂, p̂, ŝ; {p});
{p} → parameters
• In their introduction to the chapter ‘Modeling of Data’, the authors
of ‘Numerical Recipes” (p. 651), observe with sarcasme:
”Unfortunately, many practitioners of parameter estimation never proceed
beyond determining the numerical values of the parameter fit. They deem
a fit acceptable if a graph of data and model ‘l o o k s g o o d’. This
approach is known as chi-by-the-eye. Luckily, its practitioners get what
they deserve” [i.e.: what is ment is a nonsense]
Experimental SP Levels - Traditional Treatment
The single-particle levels of doubly-magic spherical nuclei can be
used to extract Hamiltonian parameters
P
Ĥ = k λk ĥk
Z
Usually the Hamiltonian
parameters are fitted ...
132
90
40
16
Ca
56
Ni
48
Ca
208
Pb
Sn
Zr
O
N
Experimental SP Levels - Traditional Treatment
The single-particle levels of doubly-magic spherical nuclei can be
used to extract Hamiltonian parameters
P
Ĥ = k λk ĥk
Z
Usually the Hamiltonian
parameters are fitted ...
132
90
40
16
Ca
56
Ni
48
Ca
208
Pb
Sn
Zr
... using the chi−by−the−eye
"principle"
O
N
Experimental SP Levels as Uncertainty Distributions
Suppose, we know the theoretical levels for a number of pre-studied
nuclei - together with the theoretical uncertainties
.
Z
208
132
48
Pb
Sn
Ca
N
.
Suppose, we wish to study a so far unknown nucleus (Z , N) by using
the information coming from the known fits (known Hamiltonians)
.
Z
208
132
48
Pb
Sn
(Z,N)
Ca
N
.
From one of the nuclei we obtain the theoretical predictions of the
single particle energies - together with the uncertainty distributions
.
Z
Pb
208
132
48
Pb
Sn
(Z,N)
Ca
N
.
... and similarly from the other one we obtain theoretical predictions
of the single particle energies - with their uncertainty distributions
.
Z
Sn
208
132
48
Pb
Sn
(Z,N)
Ca
N
.
If the results are consistent within uncertainties, the predictions can
be considered ’valid within statistics’, otherwise: improve exp. input
.
Z
Extrapolated from:
Pb
208
132
48
Sn
Pb
Sn
(Z,N)
Ca
N
.
Theory, Experiment, Experimental Data Bases
Even a theorist can produce an experimental
single-particle energy on his or her computer
Theory, Experiment, Experimental Data Bases
Even a theorist can produce an experimental
single-particle energy on his or her computer
Illustrative examples follow →
Example of an Analysis: Neutron States in
91
Zr
• We consider the reaction
90 Zr
91
Zr
(~d, p) 91 Zr , with polarised d-beam
90 Zr
91
Zr
• We observe peaks in the proton spectrum corresponding to each
|jmi state populated in 91 Zr
90 Zr
91
Zr
• Angular distributions from the measured proton spectrum allow
to deduce the associated proton orbital momenta ` - and through
conservation - possible orbital momenta of the odd neutron in 91 Zr
90 Zr
91
Zr
• From polarisation of the deuterons we deduce the orientation of
intrinsic spins s with respect to orbital momentum ` → we deduce j
90 Zr
91
Zr
• From polarisation of the deuterons we deduce the orientation of
intrinsic spins s with respect to orbital momentum ` → we deduce j
• To summarise: Such an experiment provides ` ↔ parity, j and,
with an extra effort, the spectroscopic factors
Using NNDC Data to Obtain Neutron States in
91
Zr
91
Zr
• From Audi et al., Nucl. Phys. A729 (2003) 345, we find that:
Neutron separation from
91 Zr:
Sn = 7194.5 keV
91
Zr
91 Zr:
• In NNDC, we select 91 Zr and next
the known information about the
Sn = 7194.5 keV
90 Zr
(pol d,p), and we obtain
Orbitals above N=50: νd5/2 , νs1/2 , νg7/2 , νd3/2 , νh11/2
91
Zr
91 Zr:
• In NNDC, we select 91 Zr and next
the known information about the
Sn = 7194.5 keV
90 Zr
(pol d,p), and we obtain
Orbitals above N=50: νd5/2 , νs1/2 , νg7/2 , νd3/2 , νh11/2
• The first example - the data for states 5/2+ :
E1 = 0 keV with S1 = 1.09 and E2 = 1244 keV with S2 = 0.032
hE i = (E1 × S1 + E2 × S2 )/(S1 + S2 ) = 42 keV
Ed5/2 = −Sn + hEi = (−7194.5 + 43) keV = −7152.5 keV
91
Zr
• Another example: the data for states 3/2+ :
E1
E2
E3
E4
E2
hEi =
P
i
= 2042
= 2871
= 3083
= 3290
= 3681
keV
keV
keV
keV
keV
with
with
with
with
with
P Ei × Si /
i Si
S1
S1
S1
S1
S1
= 0.78
= 0.08
= 0.16
= 0.22
= 0.16
→→→→ hE i = 2592 keV
Ed3/2 = −Sn + hE i = (−7194.5 + 2592) keV = −4602.5 keV
Experimental Single Particle States: Comments
1. Paradoxally, the so-called experimental mean-field single-particle
levels are highly complicated, model-dependent objects
2. They depend, among others, on the spectroscopic factors, defined
in the presence of simplifying assumptions in the reaction theory; the
latter may complicate the self-control through the sum rule tests
3. Many levels need to be measured (including collectivity) to extract ‘reliably’ one, average energy, interpreted as exp. s. p. energy
4. To obtain reliable results, attention should be payed to measuring
all the j π states that contribute significantly to the average
4. To obtain reliable results, attention should be payed to measuring
all the j π states that contribute significantly to the average
5. Vibration-coupling corrections can be measured and one should
aim at subtracting them (alternatively: include in the theory)!
Experimental Single Particle States: Comments (II)
1. Probably the most powerful way of obtaining ’spherical levels’
is to use the information about the axially-deformed odd-A nuclei
(‘band-heads’); our primary goal is to combine the ’spherical’ and
’deformed’ level information
2. This is by the way how the ’Universal’ Woods-Saxon parameters
were fitted
were fitted
3. There are many ways of loosing the predicitve power, the way we
defined it, of the mean-field hamiltonians: one of the most powerful
ways of doing so is to use experimental nuclear masses...
were fitted
3. There are many ways of loosing the predicitve power, the way we
defined it, of the mean-field hamiltonians: one of the most powerful
ways of doing so is to use experimental nuclear masses...
4. ... at least the way it has been practiced so far ...
Experimental levels represent, from both quantum-mechanical and
experimental points of view an ensemble of probability distributions
.
Single Particle Energies
(Experimental, Schematic)
|jlm> 1
|jlm> 2
|jlm> 3
Uncertainty
Distributions
.
These uncertainties propagate to the parameters-solutions p1 , p2 ...
implying that the latter turn into distributions
.
Implied Uncertainties
of Adjusted Parameters
|jlm> 1
|jlm> 2
|jlm> 3
Uncertainty
Distributions
.
These uncertainties propagate to the parameters-solutions p1 , p2 ...
implying that the latter turn into distributions
.
|jlm> 1
p1
p2
|jlm> 2
|jlm> 3
p
3
Uncertainty
Distributions
.
As it turns out, even small uncertainties on some experimental levels
may cause very large uncertainties on the adjusted parameters ...
.
|jlm> 1
p1
p2
|jlm> 2
|jlm> 3
p
3
Uncertainty
Distributions
.
... while at the same time big uncertainties on other levels have a
small impact: as in life, all are equal but some more than the others
.
|jlm> 1
|jlm> 2
p1
|jlm> 3
p2
p
3
Uncertainty
Distributions
.
Conclusion: the quality of adjustement depends strongly on the
quality of the data implying the existence of theoretical error bars
.
|jlm> 1
p1
p2
|jlm> 2
p1
|jlm> 3
p
3
p2
p
3
Uncertainty
Distributions
.
Inverse Problem and Nuclear Mean-Field Theory
• Starting from a limited experimental data set {eexp
µ } we wish
to obtain the information about all of the single particle levels
µ } we wish
• In Applied Mathematics this is called the ‘Inverse Problem’
µ } we wish
• The mathematical formulation is the same as e.g. when optimizing the position of mines from limited number of drilling
µ } we wish
• The mathematical formulation is the same as e.g. when optimizing the position of mines from limited number of drilling
• The goal of these mathematical theories is to optimize the
chances of meaningful predictions or: provide us with the full
‘statistical significance of predicted result’
2. Single-Particle Levels: Theory vs. Experiment
Part II
Exact Techniques Involving Mean-Field
Hamiltonians - Instructive New Tools
2.1 Posing the Problem
2.2 Modeling Single-Particle Energies
About this Particular Project: What? and How?
• Here we limit our considerations to the simplest but realistic
mean field approaches and focus on the single-particle spectra
• We introduce the notion of Exact Reference Hamiltonians decribing the present experimental status with full precision
• Having introduced the exact-model Hamiltonians we study
them using the standard tools such as the ‘input noise’, error
distributions etc. that propagate to the final result
• Having introduced the exact-model Hamiltonians we study
them using the standard tools such as the ‘input noise’, error
distributions etc. that propagate to the final result
• We arrive at the final-result certitude distributions: In other
words, if the mathematics is done correctly, we not only give
the theoretical result but also provide its statistical significance
Selection of Nuclei and Test Model-Hamiltonians
• We will be interested in extracting today’s single-nucleon energies
in the so-called well-known doubly-magic spherical-nuclei:
16 O, 40 Ca, 48 Ca, 56 Ni, 90 Zr, 132 Sn
and
208 Pb
16 O, 40 Ca, 48 Ca, 56 Ni, 90 Zr, 132 Sn
and
208 Pb
• We wish to have the Hmf capable of reproducing the single-particle
level energies exactly (”Exact Reference Hamiltonians”)
16 O, 40 Ca, 48 Ca, 56 Ni, 90 Zr, 132 Sn
and
208 Pb
• The only simple Hamiltonian that we know of, which reproduces
the single-particle spectra exactly, is the Woods-Saxon Hamiltonian
ws
ws
Hws
mf = T + Vcent + Vso
16 O, 40 Ca, 48 Ca, 56 Ni, 90 Zr, 132 Sn
and
208 Pb
• The only simple Hamiltonian that we know of, which reproduces
the single-particle spectra exactly, is the Woods-Saxon Hamiltonian
ws
ws
Hws
mf = T + Vcent + Vso
• The simple Woods-Saxon mean-field has certain advantages:
clearly defined roles of the size, depth and surface-diffuseness
Examples of High-Precision Woods-Saxon Mean-Fields
• Illustrating the results of the standard χ2 -fit in
90 Zr:
Z=40 N= 50 protons S2= 0.01 Rrms=4.16
theor
exper
1g7/2
1g7/2
2d5/2
2d5/2
1g9/2
1g9/2
2p1/2
2p1/2
2p3/2
2p3/2
1f5/2
1f5/2
0
Energy [MeV]
-2
-4
-6
-8
-10
V0=-58.18 r0=1.25 a0=0.79 Vso= 20.33 rso=1.21 aso=0.37
2
-12
• The quality of the result is such that the graphical illustration is
insufficient to show the discrepancies !!!
High Precision of the Fits: ∼ 1 keV!
• There are several high-precision examples produced with the WoodsSaxon Hamiltonian: The case of 90 Zr
No.
Ecalc
1.
2.
3.
4.
5.
6.
-10.089
-9.861
-8.348
-5.150
-1.300
0.399
Eexp
-10.090
-9.860
-8.350
-5.150
-1.300
0.400
Level
Err.(th-exp)
1f5/2
2p3/2
2p1/2
1g9/2
2d5/2
1g7/2
0.0007
-0.0007
0.0018
0.0000
-0.0001
-0.0006
• With as many as 6 parameters, Woods-Saxon Hamiltonian reproduces 6 experimental energies with δe = 2 keV maximum error!
• We arbitrarily call it Exact Reference Hamiltonian (‘Zr-exact’)
Partial Conclusion at This Stage of the Discussion:
Let us emphasize:
Although we study an Exact Reference Hamiltonian,
in reality we aim at the realistic fully microscopic
mean-field theories
Examples of High-Precision Woods-Saxon Mean-Fields
• Illustrating the results of the standard χ2 -fit in 16 O; single-particle
mean-field energies corrected for the Wigner-type contributions[1]
Z= 8 N= 8 neutrons S2= 0.03 Rrms=2.59
exper
1d3/2
Energy [MeV]
0
-2
2s1/2
2s1/2
-4
1d5/2
1d5/2
-6
-8
-10
-12
1p1/2
1p1/2
-14
-16
-18
-20
1p3/2
1p3/2
2
Z= 8 N= 8 protons S = 0.00 Rrms=2.56
theor
6
4
2
0
exper
1d3/2
1d3/2
2s1/2
2s1/2
1d5/2
1d5/2
1p1/2
1p1/2
1p3/2
1p3/2
-2
-4
-6
-8
-10
-12
-14
-16
-18
V0=-59.48 r0=1.15 a0=0.69 Vso= 18.02 rso=0.79 aso=0.39
theor
Energy [MeV]
1d3/2
V0=-54.99 r0=1.22 a0=0.70 Vso= 36.75 rso=0.45 aso=0.95
2
• The quality of the result is such that the graphical illustration is
again insufficient to show the fit inaccuracies !!!
[1] For details about Wigner corrections, cf. R. R. Chasman, PRL 99, 082501 (2007)
Unprecedented Precision of the Fits: 100 eV!
• The six-parameter Woods-Saxon Hamiltonian,
No.
Ecalc
1.
2.
3.
4.
5.
-15.300
-9.000
-0.600
-0.100
4.400
Eexp
-15.300
-9.000
-0.600
-0.100
4.400
16 O,
neutrons:
Level
Err.(th-exp)
1p3/2
1p1/2
1d5/2
2s1/2
1d3/2
-0.0001
-0.0001
0.0000
0.0000
0.0001
• The results of this type definitely need an interpretation:
Is this property limited to one single nucleus? Not at all!
Is this a good? or rather a bad news? - We will see soon
No.
Ecalc
1.
2.
3.
4.
5.
-15.300
-9.000
-0.600
-0.100
4.400
Eexp
-15.300
-9.000
-0.600
-0.100
4.400
16 O,
neutrons:
Level
Err.(th-exp)
1p3/2
1p1/2
1d5/2
2s1/2
1d3/2
-0.0001
-0.0001
0.0000
0.0000
0.0001
No.
Ecalc
1.
2.
3.
4.
5.
-15.300
-9.000
-0.600
-0.100
4.400
Eexp
-15.300
-9.000
-0.600
-0.100
4.400
16 O,
neutrons:
Level
Err.(th-exp)
1p3/2
1p1/2
1d5/2
2s1/2
1d3/2
-0.0001
-0.0001
0.0000
0.0000
0.0001
No.
Ecalc
1.
2.
3.
4.
5.
-15.300
-9.000
-0.600
-0.100
4.400
Eexp
-15.300
-9.000
-0.600
-0.100
4.400
16 O,
neutrons:
Level
Err.(th-exp)
1p3/2
1p1/2
1d5/2
2s1/2
1d3/2
-0.0001
-0.0001
0.0000
0.0000
0.0001
High Precision of the Fits: 10−1 keV!
• Another nucleus with the standard Woods-Saxon Hamiltonian:
The case of 132 Sn
No.
Ecalc
1.
2.
3.
4.
5.
-16.010
-15.710
-9.680
-8.720
-7.240
Eexp
-16.010
-15.710
-9.680
-8.720
-7.240
Level
Err.(th-exp)
2p1/2
1g9/2
1g7/2
2d5/2
2d3/2
-0.0004
-0.0000
0.0000
0.0000
0.0001
• With as much as Z=50 protons, the Woods-Saxon Hamiltonian
reproduces 5 experimental energies with δe = 0.4 keV max. error!
An Example with the Presence of High-j Orbitals
• The case of
208 Pb
No.
1.
2.
3.
4.
5.
6.
7.
Ecalc
Eexp
Level
Err.(th-exp)
-9.338
-8.519
-7.757
-3.791
-2.534
-1.835
-0.372
-9.350
-8.350
-8.100
-3.800
-2.490
-1.830
-0.400
1h11/2
2d3/2
3s1/2
1h9/2
2f7/2
1i13/2
2f5/2
0.0119
-0.1686
0.3428
0.0090
-0.0438
-0.0045
0.0284
• As we will see immediately, we are beginning to learn something:
the levels which are the most rigid are not the high-j but rather low-`
orbitals. By arbitrarily removing 3s1/2 and 2d3/2 from the fit ...
... Back to High Precision of ∼100 eV
• Indeed, by removing two lowest-` orbitals, 3s1/2 and 2d3/2 , we are
again down to the precision of 200 electron-volts !!!
No.
1.
2.
3.
4.
5.
Ecalc
Eexp
Level
Err.(th-exp)
-9.350
-3.800
-2.490
-1.830
-0.400
-9.350
-3.800
-2.490
-1.830
-0.400
1h11/2
1h9/2
2f7/2
1i13/2
2f5/2
0.0002
0.0000
0.0000
-0.0002
0.0000
• ... and all that for the nucleus as heavy as
208 Pb
Quite Often Good Is Better than Bad, and so:
• At first glance we should be happy:
If the Hamiltonian gives exactly the experimentally measured
single-particle states, it gives most likely the other, not yet measured (at least close lying levels) at the nearly right positions!
Before planing a new measurement - it is certainly interesting
to consult the prediction of the Exact Reference Hamiltonian!!
It becomes a useful tool to study many-body matrix-elements
within e.g. shell-model traditional techniques - it generates the
wave-functions that are ‘exact’ desired input to other codes
It becomes a useful tool to study many-body matrix-elements
within e.g. shell-model traditional techniques - it generates the
wave-functions that are ‘exact’ desired input to other codes
• But on the other hand, the Woods-Saxon Hamiltonian is too
simple to contain the truth, all the truth and the only truth
about nuclear Mean-Field! Overparametrized Hamiltonians?
3. Instability of Hamiltonians: Mathematics
Part III
Hamiltonians and Parametric Correlations:
Mathematical Background
3.1 A Lesson in Applied Mathematics
3.2 Mathematics of Confidence Intervals
What Is this Section about?
• If parameters are correlated [e.g. pi = f(pk , p` , . . .)]:
- The confidence intervals tend to infinity
- The problem becomes ill-conditioned (looses stability)
- No extrapolations e.g. to exotic nuclei are possible
• The problem is that one never knows these correlations - unless
one studies them specifically. This problem is serious: ignoring it
renders extrapolations to exotic nuclear ranges a priori impossible
• The problem is that one never knows these correlations - unless
one studies them specifically. This problem is serious: ignoring it
renders extrapolations to exotic nuclear ranges a priori impossible
• Here we introduce the formalism to treat the problem
Introduction: Choice of the Experimental Input
• Consider a mean-field Hamiltonian:
Hmf = Hmf (r̂, p̂, ŝ; {p});
{p} → parameters
Hmf = Hmf (r̂, p̂, ŝ; {p});
{p} → parameters
• In a search for ‘an as adequate approach as possible a search for
coupling constants’ we need some guidelines. Our choice:
The mean-field Hamiltonian should first of all describe optimally
the mean field degrees of freedom
{ϕν , eν } in Hmf ϕν (r, {p}) = e ν (p) ϕ ν (r, {p})
Hmf = Hmf (r̂, p̂, ŝ; {p});
{p} → parameters
• In a search for ‘an as adequate approach as possible a search for
coupling constants’ we need some guidelines. Our choice:
The mean-field Hamiltonian should first of all describe optimally
the mean field degrees of freedom
{ϕν , eν } in Hmf ϕν (r, {p}) = e ν (p) ϕ ν (r, {p})
• In an Appendix we explain at length why not the nuclear masses...
Introduction: χ2 -Problem and Its Linearisation
• Parameter adjustement usually corresponds to a minimisation of:
χ2 (p) =
1
m−n
Pm
j=1 Wj
[ejexp − ejth (p)]2 →
∂χ2
∂pk
= 0, k = 1 . . . n
where: m - number of data points; n - number of model parameters
• Parameter adjustement usually corresponds to a minimisation of:
χ2 (p) =
1
m−n
Pm
j=1 Wj
[ejexp − ejth (p)]2 →
∂χ2
∂pk
= 0, k = 1 . . . n
where: m - number of data points; n - number of model parameters
• Introduce linearisation procedure [after simplification ejth (p) → fj ]
n X
∂fj fj (p) ≈ fj (p0 ) +
(pi − p0,i )
∂pi p=p0
i=1
Jjk ≡
q
Wj
χ2 (p) =
∂fj ∂pk 1
m−n
and
bj =
p=p0
Pm Pn
j=1
i=1
q
Wj [ejexp − fj (p0 )]
Jji · (pi − p0,i ) − bj
2
χ2 -Problem Using Applied-Mathematics’ Jargon
• One may easily show that within the linearized representation
∂χ2
∂pi
= 0 → (JT J) · (p − p0 ) = JT b
∂χ2
∂pi
= 0 → (JT J) · (p − p0 ) = JT b
• In Applied Mathematics we usually change wording and notation:
{p} ↔ x ↔ ‘causes’ and {e} ↔ b ↔ ‘effects’ ↔ A · x = b
From the measured effects represented by ‘b’ we extract information
about the causes ‘x’ by inverting the matrix: A → x = A−1 · b
∂χ2
∂pi
= 0 → (JT J) · (p − p0 ) = JT b
• In Applied Mathematics we usually change wording and notation:
{p} ↔ x ↔ ‘causes’ and {e} ↔ b ↔ ‘effects’ ↔ A · x = b
From the measured effects represented by ‘b’ we extract information
about the causes ‘x’ by inverting the matrix: A → x = A−1 · b
• Some mathematical details: J ≡ A ∈ Rm×n ; x ∈ Rn , and b ∈ Rm
A Powerful Tool: Singular-Value Decomposition
• The problems with instabilities (i.e. ill-conditioning) can be easily
illustrated using the so-called Singular-Value Decomposition of A:
A = U · D · VT with U ∈ Rm×m , V ∈ Rn×n , D ∈ Rm×n
where diagonal matrix has a form D = diag{d1 , d2 , . . . dmin(m,n) }
|
{z
}
decreasing order
• The problems with instabilities (i.e. ill-conditioning) can be easily
illustrated using the so-called Singular-Value Decomposition of A:
A = U · D · VT with U ∈ Rm×m , V ∈ Rn×n , D ∈ Rm×n
where diagonal matrix has a form D = diag{d1 , d2 , . . . dmin(m,n) }
|
{z
}
decreasing order
• Formally (but also in practice), the solution ‘x’ is expressed as
x = AT b; AT = V · DT · UT
where
DT = diag d11 ,
1
,
d2
...
1
; 0, 0,
dp
... 0
Ill-Conditioned Problems: Qualitative Illustration
• An academic 2 × 2 problem: A has eigenvalues: d1 and d2 ∼
1
10 d1
1
• An academic 2 × 2 problem: A has eigenvalues: d1 and d2 ∼ 10
d1
• Introduce errorless data (b∗ ) and uncertain (noisy) data: b1 & b2
x − space
b − space
2
Ax=b
data points
with noise
2
x 1= A b 1
b1
x 2= A b 2
−1
b2
x=A b
errorless
data b *
1
x =Ab
*
*
1
1
• An academic 2 × 2 problem: A has eigenvalues: d1 and d2 ∼ 10
d1
• Introduce errorless data (b∗ ) and uncertain (noisy) data: b1 & b2
x − space
b − space
2
Ax=b
data points
with noise
2
x 1= A b 1
b1
x 2= A b 2
−1
b2
x=A b
errorless
data b *
x =Ab
*
*
1
1
Left: Red circle represents points equally distant from ‘noisless’ data b∗ .
Right: Purple oval represents the image of the circle through Ax=b. One
may show that instability is directly dependent on the condition number
cond(A) ≡
d1
dr
- the bigger the condition number the more ‘ill-conditioned’ the problem
Fitting, Implied Errors and Confidence Intervals
• Let us come back to the underlying χ2 minimum condition:
∂χ2
∂pj
→ (JT J)(p − p0 ) = JT b
∂χ2
∂pj
→ (JT J)(p − p0 ) = JT b
• Using the Singular-Value Decomposition written down explicitly
P
T
Jik = r`=1 Ui` d` V`k
one shows that (JT J)−1 = V d12 VT
∂χ2
∂pj
→ (JT J)(p − p0 ) = JT b
P
T
• Independently one derives the expression for the correlation matrix
h(pi − hpi i) · (pj − hpj i)i = χ2 (p) t2α/2,m−n (JT J)−1
ij
∂χ2
∂pj
→ (JT J)(p − p0 ) = JT b
P
T
• Independently one derives the expression for the correlation matrix
h(pi − hpi i) · (pj − hpj i)i = χ2 (p) t2α/2,m−n (JT J)−1
ij
• If one or more dk → 0 then (JT J)−1 tends to infinity and
generally, the confidence intervals of all parameters diverge
A Kind of Summary and a Historical Analogy
If the confidence intervals diverge we loose unique answers,
but on the other hand we are confident of our errors
A similar problem has been encountered,
according to Umberto Eco, about 1327 (”Il nome della rosa”)
”So you don’t have unique answers to your questions?”
”Adson, if I had, I would teach theology in Paris”
”Do they always have a right answer in Paris”
”Never”, said William,
”Never”, said William,
”but there they are quite confident of their errors”.
4. Parametric Correlations within a Hamiltonian
Part IV
Physical Background of Parametric Correlations:
Realistic Illustrations
• Heurestic arguments: We know that in nuclear physics:
- Nuclear volume is constant, deformation independent
- ... and grows proportionally to the particle number A
- Surface-diffuseness is independent of anything (shape...)
• The above quantities do not enter the Hamiltonian directly. They
are reproduced through external conditions on the r.m.s, on the difussivity, on the saturation properties...
• The above quantities do not enter the Hamiltonian directly. They
are reproduced through external conditions on the r.m.s, on the difussivity, on the saturation properties...
• This imposes correlations among the parameters and
below we illustrate these features using the simplest, intuitive Hamiltonian: the spherical Woods-Saxon
4.1 Extracting Central Potential
4.2 Extracting Spin-Orbit Potential
Can We Obtain the Whole Hmf out of a Few Levels?
• We introduce the exact modelling with features close to the experimental ones
• We define the ‘experimental spectrum’ by setting the result of the
calculations with a certain set of realistic parameters
• The advantage of such an exercise consists in the fact that we
know the solution of the fitting (χ2 minimisation) procedure exactly
• What is the minimum information necessary in order to recover
the full/complete result i.e. all the (known) parameters of the whole
Hamiltonian?
• We begin with a central Woods-Saxon potential (no spin-orbit)...
for simplicity
Hamiltonian?
for simplicity
Hamiltonian?
for simplicity
Hamiltonian?
for simplicity
Hamiltonian?
for simplicity
Will it be sufficient to use four ‘experimental’
levels to recover the central potential?
Limited Experimental Input → May Be Too Limited!
constant parameter: A0CENT=0.69
1.4
1.3
1.2
1.1
Behaviour of the χ Function
208
82 Pb
9
8
7
2
6
5
4
6
4
2
1.0
0.9
-70
χ[MeV]
10
18 6
1
14
12
10
8
6
4
3
2
8
10
-65
1
-60
-55 -50 -45 -40
Depth Parameter Vo
-35
-30
0
Experimental levels: central interaction with Universal WS parameters
V0CENT=-55.95 R0CENT=1.21 A0CENT=0.69
Radius Parameter ro
1.5
Experimental Neutron Energy Levels: 3s1/2 1h11/2 2f5/2 3p1/2 Window Size: Emin =-15.87 MeV Emax =-8.35 MeV
Four levels are used; they lie close to the Fermi level. Clearly the radius
and potential depth parameters a very strongly correlated [cond(A) → ∞]
Will it help to use six levels?
Limited Experimental Input → May Be Too Limited
constant parameter: A0CENT=0.69
1.4
1.3
1.2
1.1
208
82 Pb
9
8
7
2
6
5
4
6
4
2
1.0
0.9
-70
χ[MeV]
10
18 6
1
14
12
10
8
6
4
3
8
-65
2
10
-60
1
-55 -50 -45 -40
Depth Parameter Vo
-35
-30
0
Experimental levels: central interaction with Universal WS parameters
Radius Parameter ro
1.5
Experimental Neutron Energy Levels: 1g9/2 2d5/2 3s1/2 1h11/2 2f5/2 3p1/2 Window Size: Emin =-20.35 MeV Emax =-8.35 MeV
Still six levels close to the typical nucleonic binding are used. Despite slight
improvements, the confidence intervals risk to be very large.
First Conclusions from Tests of This Type (Part I)
• Flat valleys on the χ2 -plots are a convenient graphical tool
to express instabilities (large confidence intervals)
• Because of these uncertainties, knowing ∼6 experimental
levels (a typical situation) will be in-sufficient to recover the
parameters with any significant precision
• Conversely: It may be relatively ”easy” to fit six levels with
high numerical precision and still large parametric uncertainty.
• Recall: We need to fight parametric uncertainties since we
wish to extrapolate to exotic nuclear ranges!
First Conclusions from Tests of This Type (Part II)
Our discussion brings us to the notion
of iso-spectral (or approximate iso-spectral)
parametrisations of the nuclear Hamiltonians:
”The same spectrum can be obtained using
many different, yet spectroscopically
equivalent parametrisations”
First Conclusions from Tests of This Type (Part II)
Our discussion brings us to the notion
of iso-spectral (or approximate iso-spectral)
parametrisations of the nuclear Hamiltonians:
”The same spectrum can be obtained using
many different, yet spectroscopically
equivalent parametrisations”
Analogous Tests for the Spin-Orbit Potential
• We ”define the experimental spectrum” by setting the result of
the calculations with a certain standard set of parameters - this time
with the spin-orbit term
• As before the main advantage of such an exercise consists in the
fact that we know the exact solution of the fitting (χ2 minimisation)
procedure exactly
• What is the minimum information necessary to recover the result?
• What are expected parametric uncertainties of the result?
• Can a unique information be recovered at all?
procedure exactly
procedure exactly
procedure exactly
Limited Experimental Input: How Little is Sufficient?
constant parameter: A0SORB=0.52
1.2
1.5
0.8
10
1.6
1.4
0.3
1.0
208
82 Pb
1.8
0.3
0.7
1.2
1.0
0.8
0.6
0.4
1.1
15
20
25
30
35
Spin-orbit Depth ParameterVo
0.2
40
0.0
Experimental levels: Universal WS parameters (central + spin-orbit interaction)
V0SORB=23.00 R0SORB=1.14 A0SORB=0.52
2.0
1.4
0.7
0.6
χ[MeV]
1.9
Spin-orbit Radius Parameterro
Experimental Neutron Energy Levels: 1s1/2 1p3/2 1d3/2 2s1/2 1f5/2 2p1/2 Window Size: Emin =-41.686 MeV Emax =-23.424 MeV
We know already that too few levels offer no good prospects - we start
with six very lowest levels. Observation: no way to distiguish between the
spin-orbit strength in the interval from 15 to 40 !
We will gradually increase the energy of the six-level
window to approach the nucleon binding region and
thus simulate the present-day experimental situation
χ[MeV]
2.0
0.7
1.8
1.6
1.2
1.4
0.3
1.2
1.0
1.0
0.8
1.1
1.5 1.9
0.8
0.6
0.4
0.6
10
208
82 Pb
0.7
15
20
25
30
35
0.2
40
0.0
1.4
1.1
2.3
Experimental Neutron Energy Levels: 1p3/2 1d3/2 2s1/2 1f5/2 2p1/2 1g9/2 Window Size: Emin =-37.676 MeV Emax =-22.09 MeV
Increasing the energy of the six levels helps localising the spin-orbit strength
only very slowly!
χ[MeV]
2.0
0.7
1.8
1.6
1.2
1.4
0.3
1.2
1.0
1.0
0.8
1.1
0.6
10
208
82 Pb
0.8
0.7
15
20
25
30
35
0.6
1.5
40
0.4
0.2
0.0
1.4
1.1
1.9
Experimental Neutron Energy Levels: 1d3/2 2s1/2 1f5/2 2p1/2 1g9/2 2d5/2 Window Size: Emin =-31.742 MeV Emax =-17.732 MeV
only very slowly...
1.4
1.2
χ[MeV]
2.0
1.1
1.8
1.6
0.3
0.7
1.4
1.1
1.5
1.0
1.2
1.9
1.0
0.8
0.8
0.6
10
208
82 Pb
0.6
0.4
0.7
1.1
15
20
25
30
35
0.2
40
0.0
1.5
2.3
Experimental Neutron Energy Levels: 2s1/2 1f5/2 2p1/2 1g9/2 2d5/2 1h11/2 Window Size: Emin =-30.7 MeV Emax =-15.916 MeV
only very slowly...
1.2
1.8
0.7
10
1.5
1.0
0.8
1.1
0.6
0.4
1.1
1.5
1.9
15
20
25
30
35
1.4
1.2
1.9
0.8
208
82 Pb
1.6
0.3
1.0
0.6
2.0
0.2
40
0.0
1.4
χ[MeV]
1.5
1.9
2.3
Experimental Neutron Energy Levels: 2p1/2 1g9/2 2d5/2 1h11/2 3s1/2 1i13/2 Window Size: Emin =-23.424 MeV Emax =-9.303 MeV
only very slowly...
10
208
82 Pb
1.4
1.2
1.9
1.0
0.6
1.6
0.3
1.0
0.8
1.5
1.1
1.9
0.6
0.4
0.2
1.5
15
20
25
30
35
40
0.0
1.8
1.2
0.8
2.0
2.3
1.4
χ[MeV]
1.5
1.9
0.7
Experimental Neutron Energy Levels: 1g9/2 2d5/2 1h11/2 3s1/2 1i13/2 2f5/2 Window Size: Emin =-22.09 MeV Emax =-8.461 MeV
only very slowly...
0.7
1.8
1.0
1.6
0.3
1.5
1.4
1.0
1.2
0.8
0.8
0.6
10
208
82 Pb
0.6
1.5
1.9
15
20
25
30
35
0.4
0.2
40
0.0
1.2
2.0
1.1
2.3
1.4
χ[MeV]
1.5
1.9
1.9
Experimental Neutron Energy Levels: 2d5/2 1h11/2 3s1/2 1i13/2 2f5/2 3p1/2 Window Size: Emin =-17.732 MeV Emax =-7.839 MeV
only very slowly...
1.2
2.0
1.8
1.1
1.6
0.3
1.4
0.7
1.2
1.0
1.5
1.0
0.8
0.8
0.6
10
208
82 Pb
0.6
0.4
1.5
1.9
1.9
15
20
25
30
35
0.2
40
0.0
1.4
χ[MeV]
1.5
1.9
1.9
Experimental Neutron Energy Levels: 1h11/2 3s1/2 1i13/2 2f5/2 3p1/2 2g9/2 Window Size: Emin =-15.916 MeV Emax =-3.785 MeV
only very slowly... Attention: Second solution is coming !
1.8
0.7
1.6
1.4
1.0
10
208
82 Pb
1.0
0.8
0.3
0.8
0.6
1.2
0.6
1.5
1.9
15
20
25
30
35
0.4
0.2
1.9
40
0.0
2.0
1.1
1.5
1.2
χ[MeV]
1.9
1.4
1.9
1.5
Experimental Neutron Energy Levels: 3s1/2 1i13/2 2f5/2 3p1/2 2g9/2 4s1/2 Window Size: Emin =-15.875 MeV Emax =-1.402 MeV
... and here we discover the existence of two solutions - we call them
compact and non-compact.
What Can We Conclude from the This Set of Tests?
• First of all, the fitted spin-orbit strength may vary widely
from one doubly-magic nucleus to another - there exists a
considerable softness in χ2 dependence on λso
• Secondly - there are correlations between the parameters:
two relatively distant sets of parameters may be, what we call
spectroscopically equivalent (‘iso-spectroscopic’)
• We discover a possibility of double-valued solutions giving
rise to compact and non-compact spin-orbit parametrisations
• We confirm the presence of iso-spectral lines also in the
space of the spin-orbit potential parameters
5. Self-Consistent Limitation of Redundancies
Part V
Overparametrized Nuclear Hamiltonians: Why?
How to Deal with the Problem?
5.1 Density-Dependent Spin-Orbit (Minimum Coupling)
5.2 Beyond the Minimum Coupling Hypothesis
First Step: A More Microscopic Formulation
• We suggest to replace a too neat W-S spin-orbit parametrisation
by using the density gradient
so
Vws
∼
1d
r dr

λ
1 + exp[(r − Ro )/ao ]
ff
↔
so
Vws
∼
λ0 dρ
r dr
• The nucleonic density can be seen as describing the interaction
source: in systems with short range interactions, on the average,
the higher the density (gradient) - the more chance to interact.
• Similarly, in the relativistic approach
so
Vrel
∼
1d
[S(r) − V(r)]
r dr
with S and V expressed by the densities of (source) mesons.
so
Vws
∼
1d
r dr

λ
1 + exp[(r − Ro )/ao ]
ff
↔
so
Vws
∼
λ0 dρ
r dr
so
Vrel
∼
1d
[S(r) − V(r)]
r dr
so
Vws
∼
1d
r dr

λ
1 + exp[(r − Ro )/ao ]
ff
↔
so
Vws
∼
λ0 dρ
r dr
so
Vrel
∼
1d
[S(r) − V(r)]
r dr
Half-Phenomenological Microscopic Hamiltonian
• Consider a given nucleus with the density ρ and a series of neighbouring nuclei with extra occupied orbitals j1 ↔ ρj1 , j2 ↔ ρj2 , etc.
We expect that the density-dependent spin-orbit potentials
Vso ∼
dρ
,
dr
dρj1
,
dr
dρj2
dr
...
account much better for these extra orbitals than just a flat WS
potential introduced long ago for numerical simplicity
• Therefore we will test the following Hartree-Fock type hypothesis
Vπso =
λππ dρπ
r dr
+
λπν dρν
r dr
and
Vνso =
and
Vπc =
λνπ dρπ
r dr
+
λνν dρν
r dr
with the central WS potentials
Vπc =
Voπ
1+exp[(r −Roπ )/aoπ ]
Voν
1+exp[(r −Roν )/aoν ]
Half-Phenomenological Microscopic Hamiltonian
• Consider a given nucleus with the density ρ and a series of neighbouring nuclei with extra occupied orbitals j1 ↔ ρj1 , j2 ↔ ρj2 , etc.
We expect that the density-dependent spin-orbit potentials
Vso ∼
dρ
,
dr
dρj1
,
dr
dρj2
dr
...
account much better for these extra orbitals than just a flat WS
potential introduced long ago for numerical simplicity
• Therefore we will test the following Hartree-Fock type hypothesis
Vπso =
λππ dρπ
r dr
+
λπν dρν
r dr
and
Vνso =
and
Vπc =
λνπ dρπ
r dr
+
λνν dρν
r dr
with the central WS potentials
Vπc =
Voπ
1+exp[(r −Roπ )/aoπ ]
Voν
1+exp[(r −Roν )/aoν ]
A New Self-consistent Microscopic Formulation
• We will try to test two strategies: First of all, the central WoodsSaxon potentials seem to be very robust when ‘moving’ from one
corner of the Periodic Table to another - as independent tests show
• On the other hand we will modify the HF idea of self-consistency
from the usual variational context to the spectroscopic context:
X
H(ρ)ψn = en ψn → ρ =
ψ ∗ ψ → H(ρ)ψn = en ψn . . .
We will iterate to obtain the self-consistency that in this context we
call ‘auto-reproduction’ - it is not a result of energy minimisation!
• In other words: If at ith iteration the spectrum is {ein } and
at i + 1st → {ei+1
n }, we stop iterating when
|ei+1
− ein | < ,
n
∀n
X
ψ ∗ ψ → H(ρ)ψn = en ψn . . .
at i + 1st → {ei+1
|ei+1
− ein | < ,
n
∀n
X
ψ ∗ ψ → H(ρ)ψn = en ψn . . .
at i + 1st → {ei+1
|ei+1
− ein | < ,
n
∀n
Self-consistent Formulation: Minimum Coupling
• For the first tests we apply what we call a minimum coupling
hypothesis
df
λππ = λπν = λνπ = λνν = λ
• We adjust parameters of the central potential together with |λ|
• In other words: We will reduce the number of spin-orbit
parameters from 6 (3 per particle kind: λ, roso and aso ) to one!
hypothesis
df
hypothesis
df
Comparing with Experimental Results: Example
16
O
Single particle proton and neutron states in 16 O - recall that until
recently the fit-errors varied between 2-3 MeV!
theor
Z= 8 N= 8 neutrons S2= 5.95 Rrms=2.57
4
exper
2
Energy [MeV]
0
1d3/2
2s1/2
2s1/2
1d5/2
1d5/2
-2
-4
-6
-8
-10
1p1/2
1p1/2
1p3/2
1p3/2
-12
-14
-16
-4
exper
1d3/2
2s1/2
2s1/2
1d5/2
1d5/2
1p1/2
1p1/2
1p3/2
1p3/2
-6
-8
-10
-12
-14
-16
-18
-18
theor
1d3/2
0
-2
Energy [MeV]
4
2
1d3/2
V0=-63.24 r0=1.11 a0=0.75 λso= 1.21
6
V0=-58.34 r0=1.17 a0=0.76 λso= 1.21
Z= 8 N= 8 protons S2= 7.38 Rrms=2.54
8
-20
-22
This result corresponds to just one λ-parameter fit instead of 6
Comparing with Experimental Results: Example
Similarly the single particle proton and neutron states in
2
1f7/2
1f7/2
-2
-4
1d3/2
1d3/2
-8
2s1/2
exper
1f5/2
1f5/2
-4
2p1/2
2p1/2
-6
2p3/2
2p3/2
1f7/2
1f7/2
1d3/2
1d3/2
2s1/2
2s1/2
-2
-8
-10
-12
-14
-16
2s1/2
-10
V0=-57.53 r0=1.20 a0=0.79 λso= 1.24
2p3/2
Energy [MeV]
2p3/2
theor
0
2p1/2
V0=-58.10 r0=1.19 a0=0.80 λso= 1.24
2p1/2
exper
0
Energy [MeV]
40 Ca
2
theor
-6
Ca
Z=20 N= 20 neutrons S2= 1.18 Rrms=3.22
Z=20 N= 20 protons S = 0.91 Rrms=3.31
4
2
40
-18
-20
This result corresponds to just one (λ) parameter fit instead of 6
Limitation of Parametric Redundancy: Conclusions (I)
1. One can fully appreciate the physics success realising that:
We have replaced six adjustable parameters by a single one
Flat-bottom W-S functions are replaced by nucleon densities
that carry more physical information through various j-orbitals
2. This result implies that:
The new, simple and in a way natural notion of self-consistency
works in a powerful manner
The new, simple and in a way natural notion of self-consistency
works in a powerful manner
Most importantly, Fits show that the density fluctuations are
needed for the gradients in the realistic spin-orbit terms!
Following Messages are intended for Mature
Audiences
How About Contemporary Skyrme-Type Functionals?
• In their comprehensive study Carlsson, Dobaczewski and
Kortelainen introduce nuclear density functionals up to the
sixth order (the standard Skyrme is of the second order)
• Their total energy density reads
X
0 I0 ,n0 L0 v0 J0
m0 I0 ,n0 L0 v0 J0
H(~r) =
Cm
r),
mI,nLvJ,Q TmI,nLvJ,Q (~
m0 I0 ,n0 L0 v0 J0
mI,nLvJ,Q
0 0
0 0 0 0
I ,n L v J
where Cm
are coupling constants
mI,nLvJ,Q
• Their total energy density reads
X
0 I0 ,n0 L0 v0 J0
m0 I0 ,n0 L0 v0 J0
H(~r) =
Cm
r),
mI,nLvJ,Q TmI,nLvJ,Q (~
m0 I0 ,n0 L0 v0 J0
mI,nLvJ,Q
0 0
0 0 0 0
I ,n L v J
where Cm
are coupling constants
mI,nLvJ,Q
• It is instructive to think about the extentions of the EDF
based approaches in terms of increasing number of coupling
constants and the preceding illustrations...
Table: Numbers of terms of different orders in the EDF up to N3 LO.
Numbers of terms depending on the time-even and time-odd densities
are given separately. The last two columns give numbers of terms when
the Galilean or gauge invariance is assumed, respectively. To take into
account both isospin channels, the numbers of terms should be multiplied
by a factor of two.
order
0
2
4
6
3
N LO
T-even
1
8
53
250
312
T-odd
1
10
61
274
346
Total
2
18
114
524
658
Galilean
2
12
45
129
188
Gauge
2
12
29
54
97
Deviating from the Minimum Coupling Hypothesis
4.0
S2n
100
55
35
90
80
2.0
70
np
λpn
so = λso
15
60
0.0
-2.0
-4.0
1.0
208
82 Pb
50
15
35
40
30
20
55
10
75
1.5
2.0
2.5
3.0
3.5
nn
λpp
so = λso
4.0
4.5
5.0
0
Proton Central Potential Parameters: VOCENT=-61.989 R0CENT=1.2176 A0CENT=0.6376
Neutron Central Potential Parameters:VOCENT=-41.560 R0CENT=1.3316 A0CENT=0.7481
Weights:WEIRAD=1.00, WEWIWEI=1.00, WEIMAX=10.00
The χ2n (neutron) test letting λππ =λνν and λπν =λνπ separate
The two variables manifest a linear correlation
Deviating from the Minimum Coupling Hypothesis
The χ2p (proton) test letting λππ =λνν and λπν =λνπ separate
100
35
4.0
90
80
np
λpn
so = λso
2.0
0.0
70
60
15
15
50
40
-2.0
30
20
35
-4.0
1.0
208
82 Pb
1.5
2.0
10
2.5
3.0
3.5
nn
λpp
so = λso
4.0
4.5
5.0
0
Proton Central Potential Parameters: VOCENT=-61.989 R0CENT=1.2176 A0CENT=0.6376
Neutron Central Potential Parameters:VOCENT=-41.560 R0CENT=1.3316 A0CENT=0.7481
Weights:WEIRAD=1.00, WEWIWEI=1.00, WEIMAX=10.00
S2p
The two variables manifest a linear dependence as well, but not quite
the same as in the previous case
Summary and Conclusions from This Part
1. We have introduced a self-consistent Woods-Saxon Hamiltonian - this is a reference tool half-way between the pure
phenomenological and the microscopic formulations
2. In this Hamiltonian we reduce the number of spin-orbit
constants from 6 to 1 and thus show that the ‘usual’ WS
modelisation is over-parametrised
3. We believe at this stage that the new reference Hamiltonian
is very well equipped for the extrapolations into exotic isospins:
The central potential is robust against extrapolations
The spin-orbit part is self-determining (consistent)
6. Uncertainties: Exact Modeling
Part VI
Uncertainties Caused by Experimental Input:
Exact Modelling
6.1 Impact from One-Level
6.2 Schrödinger Transform
• Let us calculate {eµ }-levels for a given parameter set
• We can treat them ‘as experimental’; by trying to reproduce them through fitting we know an exact solution!
• Extra advantage: we may introduce the notion of
‘noise’ and start varying any level which we wish to study
• Extra advantage: we may introduce the notion of
‘noise’ and start varying any level which we wish to study
• We obtain the repsonse of all the levels to a ‘noise’ the standard source of information about uncertainties
Propagation of Experimental-Level Uncertainties
• Consider a single particle spectrum {eνo } ↔ Hϕoν = eνo ϕoν obtained
with a certain set of parameters {p}o ;
• Define an ‘experimental’ set of levels {eνexp } ≡ {eνo }. Repeating
the minimisation procedure will reproduce those {eνo } exactly;
• Chose one level, say eκo ∈ {eνo }, and arbitrarily modify its position,
eκo → eκ ≡ (eκo − e) with, say e ∈ [−2, +2] MeV;
then refit the χ2 -test → all other levels will move to new positions
• Collect these new positions: they are functions eν = eν (eκ ), below
referred to as ‘error response functions’ → see illustrations
Error Response Functions to 3p1/2 -Orbital
Change in Fitted Energy [MeV]
0.8
0.6
2g7/2
1i13/2
3p1/2
1h9/2
3p3/2
0.4
1g7/2
2f5/2
0.2
1g9/2
2f7/2
0.0
2g9/2
2g9/2
-0.2
2f7/2
1g9/2
-0.4
2f5/2
1g7/2
3p3/2
1h9/2
-0.6
-0.8
208
82 Pb
2g7/2
-2.0
1i13/2
-1.0
0.0
1.0
2.0
Relative Change of Experimental Energy [MeV]
If the experimental 3p1/2 level varies in [-2,+2] MeV window around its actual positions,
all other levels fixed, refitting gives relative variations shown; the effect is small
Error Response Functions to 3p3/2 -Orbital
0.8
0.6
2g9/2
1h9/2
3p3/2
1i13/2
3p1/2
0.4
1g7/2
2f7/2
0.2
1g9/2
2g7/2
0.0
2g7/2
2f5/2
-0.2
2f5/2
1g9/2
-0.4
2f7/2
1g7/2
3p1/2
1i13/2
-0.6
-0.8
208
82 Pb
2g9/2
-2.0
1h9/2
-1.0
0.0
1.0
2.0
If the experimental 3p3/2 level varies in [-2,+2] MeV window around its actual positions,
all other levels fixed, refitting gives relative variations shown; the effect increases
Error Response Functions to 2f5/2 -Orbital
0.8
0.6
2g9/2
2g7/2
2f5/2
1i13/2
3p1/2
0.4
2f7/2
3p3/2
0.2
1g7/2
1h9/2
0.0
3p3/2
1g9/2
-0.2
1h9/2
1g7/2
-0.4
1g9/2
2f7/2
3p1/2
2g9/2
-0.6
-0.8
208
82 Pb
2g7/2
-2.0
1i13/2
-1.0
0.0
1.0
2.0
The experimental 2f5/2 level varies in [-2,+2] MeV window. Observe a ‘paradox’: to be
precise on 2g7/2 we better take care of spectroscopic factors while measuring 2f5/2
Error Response Functions to 2g9/2 -Orbital
0.8
0.6
1g9/2
3p3/2
2g9/2
2g7/2
2f7/2
0.4
2f5/2
3p1/2
0.2
1g7/2
2g7/2
0.0
1h9/2
1i13/2
-0.2
1i13/2
1g7/2
-0.4
3p1/2
2f5/2
2f7/2
1h9/2
-0.6
-0.8
208
82 Pb
3p3/2
-2.0
1g9/2
-1.0
0.0
1.0
2.0
To determine precisely through parameter-fitting the energies of 3p3/2 , 2f7/2 ... the
right position of 2g9/2 auxiliary analysis must be done particularly carefully (associated
spectroscopic factors precise, particle vibration subtracted, pairing effect subtracted)
Inverse Representation for 2g9/2 -Orbital
0.8
0.6
1g9/2
2f7/2
2g9/2
2f5/2
3p3/2
0.4
1h9/2
1i13/2
0.2
1g7/2
1g7/2
0.0
3p1/2
3p1/2
-0.2
2g7/2
2f5/2
-0.4
1i13/2
1h9/2
3p3/2
2g7/2
-0.6
-0.8
208
82 Pb
1g9/2
2f7/2
-2.0
-1.0
0.0
1.0
2.0
Attention: The figure may look similar but it contains a totally opposite information:
All the curves represent the 2g9/2 -level - this is how the fitting will modify 2g9/2 if we
vary the indicated levels
Conclusions from the Error Response-Function Tests
• Observe precise indications as to ‘which levels influence which’
what allows to discuss the experimental strategies precisely
• The low-` orbitals (such as 3p1/2 , 3p3/2 ) have relatively small
impact on the error-response functions ...
• ... while some pairs of orbitals couple very strongly
• The highest-` orbitals do not couple in the strongest way;
Schrödinger Transform: Definition and Properties
• Given experimental level ν with the average position ēν and its
uncertainty distribution Dν (e − ēν )
• We refit the whole spectrum {eκ → eκ (e − ēν )} each time varying
e ∈ [ēν − ∆, ēν + ∆]
• In this way we obtain a new set of distributions for each level eκ
viz. Dκ (e − ēν ): we call them Schrödinger transforms
• Below are typical illustrations; the Schrödinger transforms do not
need to be similar to the original experimental uncertainty distribution ...
Schrödinger Transform of the Level 3p1/2
Probability Distribution 3p1/2
[arbitrary units]
5.0
1i13/2
4.0
3.0
2.0
1.0
0.0
208
82 Pb
-0.4
-0.2
0.0
0.2
0.4
We assume Gaussian distribution for the initial ‘experimental’ level i13/2
and obtain the Schrödinger transform for 3p1/2 , integral normalised.
[arbitrary units]
6.0
1i13/2
5.0
4.0
3.0
2.0
1.0
0.0
208
82 Pb
-0.4
-0.2
0.0
0.2
0.4
Schrödinger Transform of the Level 2f5/2
Probability Distribution 2f5/2
[arbitrary units]
20.0
1i13/2
15.0
10.0
5.0
0.0
208
82 Pb
-0.4
-0.2
0.0
0.2
0.4
and obtain the Schrödinger transform for 2f5/2 , integral normalised.
[arbitrary units]
450.0
1i13/2
400.0
350.0
300.0
250.0
200.0
150.0
100.0
50.0
0.0
208
82 Pb
-0.4
-0.2
0.0
0.2
0.4
[arbitrary units]
3.0
2g7/2
2.5
2.0
1.5
1.0
0.5
0.0
208
82 Pb
-0.4
-0.2
0.0
0.2
0.4
We assume Gaussian distribution for the initial ‘experimental’ level 2g7/2
[arbitrary units]
10.0
2g7/2
8.0
6.0
4.0
2.0
0.0
208
82 Pb
-0.4
-0.2
0.0
0.2
0.4
[arbitrary units]
2.5
2g7/2
2.0
1.5
1.0
0.5
0.0
208
82 Pb
-0.4
-0.2
0.0
0.2
0.4
[arbitrary units]
25.0
2g7/2
20.0
15.0
10.0
5.0
0.0
208
82 Pb
-0.4
-0.2
0.0
0.2
0.4
Experimental Single Particle States (Comments II)
1. For the Z=N nuclei the extra coherent effects (Wigner energy)
should be extracted (R. Chasman, M.G. Porquet, our group...)
2. The pairing, often considered negligible in the double-magic nuclei (by the way under in-acceptable arguments) should be taken
into account (see next transparency)
• Our project has started, in collaboration with experimentalists by,
re-evaluation of all the experimental results on the single particle
levels according to the documented present status of the data
• Some details are given in the Annexe B - not presented here
Comment: Exact Treatment of Pairing - Example
40
Ca
Even though the BCS collapses, the coherent effects of pairing are there;
Table shows the exact results1 in function of the pairing constant G up to
the BCS collapse limit Gcr ∼ 0.65. By definition: ∆Ecoll ≡ δeph − δe exact .
G (MeV)
0.00
0.10
0.20
0.30
0.40
0.50
0.60
1
∆Ecoll (MeV)
0.000
0.111
0.248
0.423
0.655
0.969
1.402
∆Ecoll /E (%)
0.000
2.575
5.754
9.814
15.197
22.483
32.529
Collectivity (%)
0.00
0.12
0.54
1.36
2.74
4.86
7.93
H. Molique and JD, Phys. Rev. C56 (1997) 1795
• Because of instrumental and other uncertainties on the spectroscopic factors, extracted average energies have experimental errors
• Because of the approximations in the reaction theory there are
additional methodological uncertainties
• If the coupling with vibrations are not taken into account there
are additional coupling related uncertainties
• Because of the residual interactions there are additional dynamical
uncertainties
• It turns out that the ‘experimental single particle energies’ take
the form of random or statistical variables: in the best case they are
characterised by their average and a certain probability distribution
uncertainties
uncertainties
uncertainties

- Department of Theoretical Physics UMCS

Transcription

Similar documents

Session C14-Preparing Your Students for Career in Energy

A Terrain of Legends reviewed in

El núcleo de las células

targeting in practise: solutions with data protection

Fisika Inti Nuclear Physics

our wedding brochure

MSc Programme Energy Technologies ENTECH

SAP Predictice Analytics Roadmap

CONTENTS - Debate Central