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.

Longitudinal profile of an EM shower

Number of particle decreases after maximum

Cu

From Mauricio Barbi, TSI’07 lectures

Spring, 2009 Phys 521A 9

Electromagnetic Shower DevelopmentShower profileShower profile

From previous slide, one expects the longitudinal and transverse developments to scale with X0

Transverse development10 GeV electron

Longitudinal development10 GeV electron

RM less dependent on Z than X0:ZARZEZAX Mc 1,20

EGS4 calculationEGS4 calculation

From Mauricio Barbi, TSI’07 lectures

RM

Spring, 2009 Phys 521A 10

Electromagnetic Shower DevelopmentEnergy depositionEnergy deposition

The fate of a shower is to develop, reach a maximum, and then decrease in number of particles once E0 < Ec

Given that several processes compete for energy deposition at low energies, it is important to understand the fate of the particles in a shower.

Most of energy deposition is by low energy e±’s.

e± (< 4 MeV)

40%

60%

e± (< 1 MeV)

e± (>20 MeV)

Ionization dominates

EGS4 calculation

From Mauricio Barbi, TSI’07 lectures

Spring, 2009 Phys 521A 11

Shower images

• ICARUS, liquid argon drift chamber (measures ionization)

• Play around with an online simulator from Sven Menke:http://www.mppmu.mpg.de/~menke/elss/home.shtml