Spring, 2009 Phys 521A 1 Interactions of Photons Interactions of Photons Pair Production An electron-positron pair can be created when (and only when) a photon passes by the Coulomb field of a nucleus or atomic electron this is needed for conservation of momentum. Threshold energy for pair production at E = 2mc 2 near a nucleus. E = 4mc 2 near an atomic electron Pair production is the dominant photon interaction process at high energies. Cross- section from production in nuclear field is dominant. First cross-section calculations made by Bethe and Heitler using Born Interactions of Particles with Matter nucleu e e nucleus γ Z e + e - + e - e + + e - + e - + nucleus e + + e - + nucleus From Mauricio Barbi, TSI’07 lectures
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Spring, 2009Phys 521A1 Interactions of Photons Pair Production An electron-positron pair can be created when (and only when) a photon passes by the Coulomb.
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Spring, 2009 Phys 521A 1
Interactions of PhotonsInteractions of Photons
Pair Production An electron-positron pair can be created when (and only when) a photon passes by the Coulomb field of a nucleus or atomic electron this is needed for conservation of momentum.
Threshold energy for pair production at E = 2mc2 near a nucleus. E = 4mc2 near an atomic
electron
Pair production is the dominant photon interaction process at high energies. Cross- section from production in nuclear field is dominant.
First cross-section calculations made by Bethe and Heitler using Born approximation (1934).
Interactions of Particles with Matter
nucleus eenucleusγ
Z
e+
e-
+ e- e+ + e- + e-
+ nucleus e+ + e- + nucleus
From Mauricio Barbi, TSI’07 lectures
Spring, 2009 Phys 521A 2
Interactions of Particles with Matter
Interactions of PhotonsInteractions of Photons
Pair Production
Photon pair conversion
probability (attenuation length is 9/7 X0)
Cross-section independent ofphoton energy (once well abovethreshold), ~ Z2
P=54%
09
7
1 X
X
eP
http://pdg.lbl.gov
From Mauricio Barbi, TSI’07 lectures
31
22 183ln
9
74
ZαZrσ e
npair
Spring, 2009 Phys 521A 3
Photon absorbtion lengthsInteractions of PhotonsInteractions of PhotonsPhoton attenuation length for different elemental absorbers versus photon energy
http://pdg.lbl.gov
Here λ = 9/7 X0
Spring, 2009 Phys 521A 4
Interactions of Particles with MatterSummary of the basic EM interactionsSummary of the basic EM interactions
e+ / e-
Ionisation
Bremsstrahlung
P.e. effect
Comp. effect
Pair production
E
E
E
E
E
g
dE
/dx
dE
/dx
s
s
s
Z
Z(Z+1)
Z5
Z
Z(Z+1)
From Mauricio Barbi, TSI’07 lectures
Spring, 2009 Phys 521A 5
Electromagnetic showers
• Cascade of pair production and bremsstrahlung is known as an electromagnetic shower
• number of low-energy photons (or electrons) produced is proportional to initial energy of electron or gamma
• Energy collected in each of e± and γ is also proportional to initial energy
Spring, 2009 Phys 521A 6
Electromagnetic Shower DevelopmentA simple shower modelA simple shower model
Shower development:
Start with an electron with E0 >> Ec
After 1X0 : 1 e- and 1 , each with E0/2
After 2X0 : 2 e-, 1 e+ and 1 , each with E0/4.. After tX0 :
Maximum number of particles reached at E = Ec
[ X0 ]t
tt
EE(t)
eN(t)
2
2
0
2ln
Number of particles increases exponentially with t equal number of e+, e-,
E
E)EN(E
EE)Et(
0
0
2ln
12ln
ln Depth at which the energy of a shower particle equals some value E’ Number of particles in the shower with energy > E’
ct
c
EEeN
EEt
02ln
max
0max
max
2ln
ln
From Mauricio Barbi, TSI’07 lectures
Spring, 2009 Phys 521A 7
Electromagnetic showers
• Radiation length X0 used to characterize longitudinal shower development
• Peaks at depth of ~7 X0
• Transverse shower size due to multiple Coulomb scattering; scales with Moliere radius (radius of cylinder containing 90% of shower energy)
RM = X0Es/Ec
where Es = me√4π/α ~ 21 MeVand Ec is the critical energy
• Two dimensionless variables: t=x/X0 and y=E/Ec govern shower development
Spring, 2009 Phys 521A 8
Electromagnetic Shower DevelopmentA simple shower modelA simple shower model
Simulation of the energy deposit in copper as a function of the shower depth for incident electrons shows the logarithmic dependence of tmax with E.
EGS4* (electron-gamma shower simulation)
*EGS4 is a Monte Carlo code for doing simulations of the transport of electrons and photons in arbitrary geometries.