PENELOPE 2003 - OECD Nuclear Energy Agency

A.1. Two-body reactions 211
Clearly, by symmetry, we can obtain a corresponding expression for cos θ 4 by interchanging
the indices 3 and 4
cos θ 4 = m2 3 c4 − m 2 1 c4 − m 2 2 c4 − m 2 4 c4 + 2W 1 (W 4 − W 2 ) + 2W 2 W 4
2 (W 2 1 − m 2 1c 4 ) 1/2 (W 2 4 − m 2 4c 4 ) 1/2 . (A.16)
x
2
3
θ 3
p 1
p 3
p 4
z
1
4
θ 4
Figure A.1: Kinematics of two-body reactions.
The different two-body reactions found in Monte Carlo simulation of coupled electron-photon
transport can be characterized by a single parameter, namely the energy
of one of the particles that result from the reaction. The energy of the second particle
is determined by energy conservation. Eqs. (A.15) and (A.16) then fix the polar angles,
θ 3 and θ 4 , of the final directions. Explicitly, we have
• Binary collisions of electrons and positrons with free electrons at rest.
Projectile: Electron or positron m 1 = m e , W 1 = E + m e c 2 .
Target: Electron m 2 = m e , W 2 = m e c 2 .
Scattered particle: m 3 = m e , W 3 = E − W + m e c 2 .
Recoil electron: m 4 = m e , W 4 = W + m e c 2 .
( E − W
cos θ 3 =
E
(
W
cos θ 4 =
E
E + 2m e c 2
E − W + 2m e c 2 ) 1/2
,
E + 2m e c 2
W + 2m e c 2 ) 1/2
.
(A.17)
(A.18)
• Compton scattering of photons by free electrons at rest.
Projectile: Photon m 1 = 0, W 1 = E ≡ κ m e c 2 .
Target: Electron m 2 = m e , W 2 = m e c 2 .
Scattered photon: m 3 = 0, W 3 ≡ τE.
Recoil electron: m 4 = m e , W 4 = m e c 2 + (1 − τ)E.