Detectors for High-Energy (Heavy-Ion) Experiments

Lecture Week:
Detectors for High-Energy (Heavy-Ion)
Experiments
2.-5.April 2007
Hans Rudolf Schmidt, GSI Darmstadt
1
Outline of Lectures
I.
Introduction
History
Basic Interactions
II.
IIIa.
IIIb.
IV.
Gaseous Detectors
Semiconductors
Calorimeters
High Energy Experiments
CMS
ALICE
2
Literature
•
K. Kleinknecht
–
•
C. Grupen
–
•
Semiconductor Radiation Detectors (Springer)
W. Blum & L. Rolandi
–
•
Instrumentation in High Energy Physics (World Scientific)
G. Lutz
–
•
Experimental Techniques in High Energy Physics (World Scientific)
F. Sauli
–
•
Instrumentation in elementary particle physics (World Scientific)
J. Ferbel
–
•
Radiation Detection and Measurement (Wiley)
C. W. Fabjan & J.E. Pilcher
–
•
Review of Particle Physics (Phys. Rev. D)
G.F. Knoll
–
•
Techniques for Nuclear and Particle Physics Experiments (Springer)
NN
–
•
The Physics of Particle Detectores (Cambridge)
W.R. Leo
–
•
Teilchendetektoren (BI Wissenschaftsverlag)
Dan Green
–
•
Particle Detectors (Cambridge)
Particle Detection with Drift Chambers (Springer)
L. Rossi, P. Fischer, T.Rohe, N. Wermes
–
Pixel Detectors
3
Internet Sources
•
•
•
•
•
•
•
Particle Detector BriefBook: http://physics.web.cern.ch/Physics/ParticleDetector/BriefBook/
Data Analysis BriefBook: http://rkb.home.cern.ch/rkb/titleA.html
Particle Date Group: http://pdg.web.cern.ch/pdg/pdg.html
Fermilab: http://quarknet.fnal.gov/run2/biblio.shtml
SLAC: http://www2.slac.stanford.edu/vvc/
CERN Academic Training: http://cdsweb.cern.ch/
CERN Summer Student Lectures: http://agenda.cern.ch/
4
Introduction - History
detector technology – the driving force for
new discoveries.....
5
A Photon Detector
The „oldest“ photon detector (built many
million times)
Properties:
• high sensitivity for photons
• high position resolution
• <100 µm
• 8 MegaPixel/DIN A4 page
• high dynamic range (1:1014)
• select energy (wave length)
• date rate ~10 Hz (incl. online processing)
Retina
6
Photon Detectors
1885: W.C. Röntgen discovers „X-Rays“
Detection of X-Rays with photo paper,
i.e., AgBr/AgCl + energy -> metallic silver (blackened)
Properties of the „detector“:
+ very good position resolution
+ good dynamic range
no timing information
no „online“ data taking
Gas-Electron-Multiplier (1990)
7
The Discovery of Radioactivity & Cosmic Rays
•
1896 – Becquerel discovers, - by chance - , that the
certain minerals blacken photographic emulsions
•
But: electroscopes (principle of the dosimeters)
discharge also (slowly) without radioactive material
Experiments „showed“, that the radiation was in the
“environment”, but did not origin from the sun or the sky
– Proof: no significant reduction of the radiation in
underground caves
– first historic misinterpretation (many more to come)
of data by „background events“: accumulation of
radioactive radon gas in caves causes radiation
puzzle: radiation cannot be shielded by even very thick
lead plates
•
•
8
The Discovery of Cosmic Rays– Victor Hess
(1912/13)
•
•
•
•
•
Question: How far one has to leave the earth‘s,
to reduced the intensity of the environmental
radiation?
surprising result: the radiation increases with
increasing height (x2)!
no reduction with (partial) solar eclipse
interpretation of Hess: radiation originates from
the universe („cosmic rays“)
Nobel Price 1936
hot-air balloon flight (max. height 5 km)
detector: electroscope
9
The Electroscope
• two thin foils are repelled from each other by the charge
• ionizing radiation discharge the electroscope
• the time needed for the discharge is proportional to the intensity of the
radiation
• NB: the discharging has already been observed in 1875 by Coulomb
during an experiment on electrical forces
• integrating counter
-
- +
-
do-it-yourself construction
10
First Visualizations of Cosmic Rays (1932)
cloud chamber (Wilson, 1910)
Nobel Price 1927
11
Cloud Chamber (Wilson, 1910)
• water vapor volume A at its saturation point
• adiabatic expansion produces super-saturated water
vapor
• ionizing particles form condensation seeds
• similar: condensation trail air planes
TPC – 2000
• very first photograph of particle tracks
radioactive source
streamer chamber – 1986
12
Build your own Cloud Chamber:
The Idiot’s Guide to How to View Cosmic Ray Tracks From the Comfort of Your
Own Home. Kuro5hin.org, 2002. Available:
http://www.kuro5hin.org/story/2002/12/27/15929/126
Foland, Andrew. How to Build a Cloud Chamber. Cornell University.
Available:
http://w4.lns.cornell.edu/~adf4/cloud.htm
13
Discovery of the Positron
•
•
C. Anderson (Nobel Price 1936)
•
⇒
ionization density ⇒ Z=1
curvature in mag. field ⇒ positive
charge, momentum measurement
energy loss in 6 mm lead (together
with know momentum and Z) ⇒
m<20me ( ⇒ no proton)
electron with positive charge , later
named positron
6 mm Pb
cloud chamber picture of positron
14
Geiger-Müller-Counter
signal, which can be
amplified
electronically
H. Geiger
1908: Hans Geiger develops the Counter (later on improved together with Erwin Müller)
⇒ electronic single event detection
1910: exp. demonstration of Rutherford scattering
Rutherford: Nobel Prize in Chemistry 1908
Ernest Rutherford and Hans Geiger
with apparatus for counting alpha
particles, Manchester, 1912
15
Bubble Chamber
Donald A Glaser (Nobel Price 1960)
• a liquid (e.g. liquid hydrogen) is cooled at 5-6 bar until completely bubble-free
• sudden pressure drop brings a liquid to sub-cooled boiling, i.e. the temperature of the liquid is above its
boiling temperature
• it is now sensitive (~ms) for ionizing particles
• due to the energy loss (ionization) of the particles vapor bubbles are formed along the particle's track
• the tracks are recorded with stereo cameras
16
Bubble Chamber: Discovery of the Ω-
BNL (USA)
K − + p → Ω− + K 0 + K +
↓
Ξ 0 + π−
existence of the Ω-:
confirmation of the
SU3 symmetry
↓
Λ 0 + γ1 + γ 2
↓
p + π−
17
BEBC (CERN)
BEBC (Big European Bubble
chamber)
- 35 m3 liquid hydrogen
- in operation until 1983
CERN Poster
18
Discovery of the Vector Bosons (CERN)
Nobel Price 1984: C. Rubia, S.v.d.Meer
technological improvement of accelerators had a significant part in the discovery...
development of the stochastic beam cooling enables the
storage of a sufficiently large number of anti-protons in a
small phase space volume
SPS → SppS
this allowed the change from the SpS to the SppS-collider;
only in the cms of pp enough energy for the production of
W+/-, Z0 available
luminosity: 5 x 1027 s-1cm-2
19
Discovery of the Vector Bosons (CERN)
ν (invisible)
UA1 (1981)
p+p → W
+ /−
+X
W + / − → e+ / − + ν
mW=80.6 GeV
e
Wire Chamber
ECAL
HCAL
Muon Chamber
20
Wire Chamber
1992: Nobel Price for Georges Charpak (CERN) for...
... his invention and development of particle detectors, in particular the multiwire proportional
chamber, a breakthrough in the technique for exploring the innermost parts of matter'
e.g..: ALICE @ LHC → readout the central detectors (TPC, TDR) are based on MWPC
21
Prerequisites for new Discoveries in Particle
Physics
1)
open questions (next slides)
2)
a new, powerful accelerator (this would be another lecture)
nonetheless, a few facts about the LHC:
•
biggest cryogenic plant on earth, 700’000 liters of liquid helium (1.9 K)
•
1232 dipole magnets (8.5 T/m, 15 m length) – internal forces 165 tons
•
energy/Pb-ion: 0.2 mJ (macroscopic energy)
•
LHC - Challenges in Accelerator Physics: http://lhc.web.cern.ch/lhc/general/acphys.htm
3)
major advances in detector technology
•
this lecture
22
Open Questions of High Energy Physics
•
Is there a unified theory, which describes gravitation together with the other three forces??
•
What is the origin of mass? ⇒ LHC
•
In how many space-time dimensions do we live ? ⇒ LHC
•
Are the elementary particles fundamental or do they have structure?
•
Why is the charge of the electron equal to the one of the proton but has an opposite sign?
•
Why are there exactly three generations of quarks and leptons?
•
Why there is an excess of matter over anti-matter in the universe?
•
Are protons stable?
•
What is the nature of Dark Matter, which dominates our universe? ⇒ LHC
•
Are there new states of matter at extreme densities and temperatures? ⇒ LHC
•
Are neutrinos massive? (ok)
•
Dark energy?
23
What is Mass?
Why have the fundamental particles in the universe different masses?
Why is the electron 350.000 lighter then the top quark?
These questions are not answered in the standard model!
However: 40 years ago Peter Higgs proposes a mechanism, which gives mass to
elementary particles: the so-called Higgs-field penetrates the empty space
completely; particles gain mass through interaction with the exchange particle of
the Higgs-Field, the Higgs Boson.
Analogy: Apparent increase of the momentum of inertia (=mass) when
accelerating in a viscous medium...
c.f: The Waldegrave Higgs Challenge (http://hepwww.ph.qmw.ac.uk/epp/higgs.html)
24
from: D. Froidevaux, CERN, CERN Summer Student Lectures
Higgs boson has been with us
for several decades as:
1. a theoretical concept,
2. a scalar field linked to the vacuum,
3. the dark corner of the Standard
Model,
4. an incarnation of the Communist
Party, since it controls the masses P.W. Higgs, Phys. Lett. 12 (1964) 132
(L. Alvarez-Gaumé in lectures for Up to now, only
CERN summer school in Alushta), unambiguous example of
5. a painful part of the first chapter
of our Ph. D. thesis
observed Higgs
(apologies to ALEPH
collab.)
25
How to discover the Higgs-Boson?
first: production in collision at
√s=14 TeV
Assumption: mH > 650 GeV
pp→H
↓
ZZ→l+ l- j j
requires:
• tracking
• lepton identification
• hadron jet identification
• high count rate capability
CMS
26
How to discover the Higgs-Boson?
Higgs
4µ
27
How to discover the Higgs-Boson?
Higgs
4µ
+30 min. bias events
28
Technological Challenges
In the LHC protons come in bunches (about 1011 protons/bunch), which collide every 25 ns. n each
in each bunch-bunch crossing, i.e., every 25 ns, about 25 minimum bias events are produced (ca. 1000 particles), Which
overlap („pile-up“) rare, interesting events.
s
25 n
Detectors have to be fast („fast response“), otherwise they integrate over to many bunch crossings. Typical response
time: 20-50 ns.
The readout electronics has operate at this frequency (⇒ very challenging readout electronics)
LHC detectors have to be highly granular (many detector cells), to minimize the probability, that interesting events
coincide with „min bias“ type particles (cost)
Radiation Damage, in particular of the front-end electronics and the detectors close to the vertex
29
Puzzle
Fi nd 4 straight tracks.
30
Answer
1. Measure precisely the
transverse momentum
2. cut on tracks with pT< 2
GeV
31
Susy (Super Symmetry)
•
Super Symmetry postulates that each
particle type has a massive shadow
particle
32
Super Partners
33
Decay of SUSY-Particles
q~ , g~
heavy → complicated decay
q
g~
q~
χ02
χ01
q
Z
decay pattern: leptons, jets and much (!) missing energy (because of
LSP = lightest super symmetric particle)
34
km
1
≈
quark-hadron
transition
35
Challenge: Extreme Multiplicities
Simulation Pb+Pb @ LHC at 5.5 TeV/A
Measurement Au+Au @ RHIC at 200 GeV/A
60° < ϑ < 62°!
36
Challenge: Extreme Multiplicities
Simulation Pb+Pb @ LHC at 5.5 TeV/A
Measurement Au+Au @ RHIC at 200 GeV/A
full acceptance
37
LHC Detectors - Atlas
38
LHC Detectors - Atlas
39
LHC Detectors - Atlas
40
LHC Detectors - Atlas
41
LHC Detectors - Atlas
42
LHC Detectors - Atlas
43
Introduction - Basic Interactions
• interaction of charge particles and photons
with matter
44
Energy loss of charged particles
Processes at the passage of charged particles in matter:
1) inelastic collisions with atomic electrons
2)
3)
4)
5)
6)
elastic scattering (Rutherford scattering) with nuclei
nuclear reactions
emission of Cherenkov-radiation
emission of transition radiation
Bremsstrahlung
The dominant process responsible for the energy loss in matter is (1); (4)-(6) are rare, nuclear
reactions (3) play a role only at very high energies; Rutherford scattering is responsible for change
in direction of the particles („multiple scattering“)
The energy loss per collision of a heavy particle with an electron is small, nonetheless the total energy
loss is large because of the high frequency of the collisions (σ≈107-108 barn).
example: A Proton of 10 MeV has a range in copper of only 0.25 mm!
Sum of the differential energy loss dE/dx ⇒ Bethe-Bloch relation
45
Bethe-Bloch Relation
Energy loss dE/dx of heavy particles (M >> me) in matter leads to ionization. The average
energy loss via electro-magnetic interaction is described by the quantum mechanical
correct Bethe-Bloch relation:
⎡
⎢
⎢
⎢
2 ⎢
⎢
⎢⎣
⎤
2 γ 2β2
⎥
2m
c
δ
Z
z
⎥
2
2
2
e
− dE = 4πN A re m e c
− β − ⎥⎥
ln
dx
Aβ
I0
2⎥
2
⎥⎦
main parameter
of Bethe-Bloch
absorber properties (Z, A, mean excitation potential I0,
screening- and density correction δ)
dE/dx
+
-
+
- +
+ -
by polarization
particle properties (z, β)
46
Classical Derivation (Bohr)
Assumptions:
a) a heavy particle (M>>me) passes at a free electron at distance b
b) the particle does not change its path
Calculation of the momentum transfer to the electron:
∆pe = ∫ Fdt = e ∫ E⊥ dt = e
1
E ⊥ dx mit dx = β cdt
βc ∫
Integration of the electrical field along an infinitely long cylinder gives:
∫E
⊥
2πb dx = 4πze
2ze2
⇒ ∆pe =
bβ c
∆p2e
2z 2e 4
⇔ ∆E(b) =
=
2me meβ2c 2b2
Integration of the distance (impact parameter b) gives:
dE
−
=
dx
bmax
∫
bmin
b
max
∆p2e
2z 2 e4
1
4πz 2 e4 Z bmin
Ne 2πb db =
Ne ∫ 2 2πb db ∼
ln
2me
meβ2c 2 bmin
b
meβ2c 2 A bmax
47
Properties of Bethe-Bloch (I)
Valid for velocities β >> αz (α = fine
structure constant 1/137), i.e., β >>
velocity of shell electrons.
For small momenta (p < 1-2 GeV/c)
follows dE/dx ~ 1/β2, together with
Ekin=0.5 m (βc)2 follows the important
relation for particle identification :
∆E ~ mz2 1 ⇔ ∆E ⋅ E ~ mz2
E
For very high energies (> TeV) additional energy loss comes from
Bremsstrahlung.
48
Energy Loss of Electrons And Positrons
For electrons and positrons radiation losses (Bremsstrahlung) become important::
⎛ dE ⎞
⎛ dE ⎞
⎛ dE ⎞
⎜ dx ⎟ = ⎜ dx ⎟ + ⎜ dx ⎟
⎝
⎠ total ⎝
⎠rad ⎝
⎠coll
Bremsstrahlung is negligible for heavy particles:
σrad
σ
1
∼ 2 ⇒ e
σµ
m
2
rad
2
⎛ m ⎞ ⎛ 105.6 ⎞
=⎜ µ ⎟ =⎜
≈ 4.2 × 10 4
⎟
⎝ me ⎠ ⎝ 0.511 ⎠
material (Z) and energy dependence is complex, because of impact parameter dependant screening
effects (screening of charge Z by atomic electrons)
dE
Z2 2
183
−
= 4αN0
re E ln 1 3
dx rad
A
Z
(assumption of complete screening)
49
Properties of Bethe-Bloch (II)
Broad minimum at βγ≈4. Relativistic
particles (β≈1) with this energy loss are
called minimum ionizing. Minimum
ionizing particles cannot be separated.
In gases (Z/A ≈ 0.5) we have:
−
dE
MeV
|min ≈ 2
dx
g / cm2
Remark: Typically, the length scale dx is given
as g/cm2; dx =ρds (ds in cm, ρ in g/cm2), since
the energy loss per unit surface mass density in
material independent.
50
Properties of Bethe-Bloch (III)
Above γ>4 slow, so-called
„relativistic rise“ of dE/dx; functional
dependence is like 2 ln(γ).
Reason: Coulomb field of the
particles is Lorentz-contracted; the
cross section for ionization and
excitation is increased. The energy
loss saturates at large γ due to
screening effects (Fermi plateau)
51
PID in the Relativistic Rise
particle separation:
NS.D. ( π;K) =
dE dx( π) − dE dx(K)
σ(dE dx)
52
Properties of Bethe-Bloch (V)
cancer therapy with heavy ions (GSI)
The Bethe-Bloch graph shows that the energy loss increases
with decreasing energy ⇒ Bragg-Peak
Bragg
53
Energy Loss of Charged Particles (Summary)
<27 MeV
27 MeV < E < 1000 MeV
1 GeV <E <300 GeV
> 300 GeV
e
Ionization
Bremsstrahlung
Bremsstrahlung
Bremsstrahlung
µ
Ionization
Ionization
Ionization
Bremsstrahlung
π
Ionization
Ionization
Nuclear Reactions
Nuclear Reactions
Range of a 1 TeV particle in iron:
e: <50 cm
π: <1 m
µ: 264 m
54
Photo Absorption Model I
An exact, unified derivation (within the photo absorption model; W.W.M. Allision
und J.G.H. Cobb, Ann. Rev. Nucl. Sci. 30 (1980) 253) of the emission of a photon
at the passage of a charged particle through matter is given by:
dσ
α σ γ (E)
1
α σ γ (E) ⎛ 2mc 2β2 ⎞ α 1
=
ln
+ 2
ln ⎜
⎟+ 2
12
2
2
dE β2 π EZ
EZ
E
π
β
⎤
⎡
⎝
⎠ β πE
2
4 2
⎢⎣ 1− β ε1 + β ε2 ⎥⎦
(
)
Bethe-Bloch
E
∫
0
σ γ (E′)
Z
⎛
⎞
α
1 ⎜ 2 ε1 ⎟
ŹdE′ + 2
β − 2 ⎟θ
β π ZN c ⎜⎜
ε ⎟⎠
⎝
δ-Electrons
Cherenkov
55
Photo Absorption Model II
The optical behavior of the medium is
characterized by the complex dielectric
constant ε=ε1+iε2
optical domain (Eγ<I): σγ=0
index of refraction
Im ε = k
absorption parameter
⎛
⎞
1 ⎜ 2 ε1 ⎟
α
dσ
β − 2 ⎟θ
=
dE β 2 π ZN c ⎜⎜
ε ⎟⎠
⎝
dσ
≥ 0 für
dE
ionization domain (2eV <Eγ<5 keV)
Re ε = ε1 = n
β ≥1
ε = ε1 + iε2 = ε1
ε = 1/ n
(Cherenkov-threshold)
ε = ε1 + iε 2 complex with ε 2 > 1 und ε1 < 1
βCerencokov > 1
Exchange of virtual photons with excitation or
ionization of atoms
56
Cerenkov Radiation I
57
Cerenkov-Radiation (classical) II
Cherenkov is emitted, if a charged particle passes through a dielectric with a velocity β>βthreshold=1/n
(n= index of refraction).
cos θc =
1
; n = n(λ ) ≥ 1
nβ
βSchwelle =
1
→ θc ≈ 0
n
Cerenkov-threshold
1
n
max. angle (β=1)
θmax = arccos
58
Cerenkov Radiation (classical) III
Vparticle<<C
symmetric polarization
no dipole field
no radiation
Vparticle>C/n
asymmetric polarization
time-dependant dipole field
radiation
59
Cherenkov-Radiation V
Number of photons emitted per unit length and wave length interval
d2N 2πz2α ⎛
1 ⎞ 2πz2α
2
=
1
−
=
sin
θc
⎜
⎟
2
2 2
2
dxdλ
λ ⎝ βn ⎠
λ
λ=
c hc
=
υ E
d2N
z2α 2
=
sin θc = cons tan t
dxdE
c
60
Transition Radiation I
X-ray range (Eγ > 5 keV)
ε = ε1 + iε 2 complex with ε 2 = 1 und ε1 < 1
βCerencokov > 1
Experimentally one observes X-rays (i.e., real photons) of very low intensity! The radiation
occurs when a charged particle penetrates through matter with discontinuities in the index of
refraction n (n2=ε1).
simple picture
D = ε(t)E
The particle form with its mirror charge in the medium a time-dependant dipole ⇒ emission of
electro-magnetic radiation
61
Transition Radiation II
transition radiation as „sub threshold“ Cerenkov radiation
uncertainty relation ( δθc ⋅ δk c ∼ ) allows variance of the
emission angle θc.
Coherent emission for ∆L<λ~1/k, thus diffraction pattern
θc =
1
of width
nβ
∆θ'c ∼
λ
∆L
(∆θ'c → 0 for ∆L
λ)
Emission of Cerenkov light passing thin foils for
β<
⎛
1
1 ⎞
= βthresh ⎜ cos θc =
⎟
n
nβ ⎠
⎝
∆L
possible (sub-threshold)!
θc
⇒ Emission of TR is collinear (θc≈0°)
radiated energy/transition:
W=
1
α ωp ⋅ γ
3
only extremely energetic e+/- emit TR of detectable intensity (→PID)
62
Transition Radiation: Properties
yield per transition ~ α = 1/137, i.e. one needs about 100 transitions for one single
TR photon!
typical setup of a TR-Detectors:
self absorption!
important for applications: TR-yield saturates
with γ (Cerenkov saturates with β)
Reasonable yield/threshold from γ=1000 on.
Emission co-linear: Θ~1/γ
63
Interaction of Photons with Matter
Processes at the passage of photons(X- and γ-radiation) through matter:
1) Photo-electric effect
2) Compton-scattering
3) Pair Production
4) Nuclear reactions (γ,n) (rare)
Qualitative difference to charged particles:
a) Photons are much more penetrating than charged particles, since their cross section is much
smaller;
b) Photon beams suffer intensity loss, not energy loss (total absorption or scattering). For the passage
through a layer of matter of thickness x of mass thickness X=ρx one has:
( ρ) X
I(x) = I0 e−µx = I0 e− µ
µ=linear absorption coefficient
µ/ρ=mass absorption coefficient
we have:
µ = σN0ρ / A ⇔ µ / ρ = N0 ⋅ σ / A
σ = σPhoto + σCompton + σPaar
64
Photo Effect
Absorption of a photon by an atomic shell electron with subsequent emission of the electron. The
energy of the electron is:
E = hν − EB
EB=binding energy (or „work function“).
The photo electric effect can, for reason of momentum conservation, only happen on a bound electron.
energy and Z-dependence:
a) for non-relativistic energies above the K-shell
(EK< hν<<mec2) we have (ε=hν/mec2):
σPhoto =
1
32π
2 α 4r e2 Z5 7
3
ε 2
b) for hν>>mec2
σPhoto = 4π r e2 α 4 Z5
1
ε
65
Compton Effect
Compton-scattering is the scattering on a free electron. In the relativistic limit (ε>>1)
also bound electron are considered as free. The quantum-mechanical evaluation
gives:
⎧⎪⎛ 1 + ε ⎞ ⎡ 2 (1 + ε ) 1
⎤ 1
1 + 3ε ⎫⎪
σc = 2πr ⎨⎜ 2 ⎟ ⎢
− ln (1 + 2ε ) ⎥ + ln (1 + 2ε ) −
2⎬
1
2
ε
+
ε
ε
⎝
⎠
(1 + 2ε ) ⎭⎪
⎣
⎦ 2ε
⎪⎩
2
e
(Klein-Nishina relation, from QED)
no implicit Z-dependence
log. energy dependence
Rayleigh/Thompsen scattering: classical
limit (ε<<1)
66
Pair Production
Eγ>1.022 MeV
Momentum conservation requires 3. body
because of screening effects the calculation of the cross
section for pair production is not „straight forward“.
approximation for small Z and ε>>1:
109 ⎞
⎛7
σp = re2 4αZ2 ⎜ ln 2ε −
54 ⎟⎠
⎝9
for large ε:
⎛ 7 183 1 ⎞
σp = re2 4αZ2 ⎜ ln 1 3 −
54 ⎟⎠
⎝9 Z
67
Photons and Matter: Summary
energy
range
(MeV)
energy
dependence
Z-dependence
photo effect
<10
ε-1-ε-3
Z5
Compton
1-10
~e-ε
(Z, number of
electrons)
pair
>1
~log ε constant
Z2
σtotal (E γ ,Z) = σphoto (Z,E γ ) + Zσcompton (E γ ) + σpair (Z,E γ )
68