THE EGS5 CODE SYSTEM

are then given by
P 1 = (E 1 , 0, 0, p 1 ) (2.161)
P 2 = (m, 0, 0, 0) (2.162)
P 3 = (E 3 , p 3 sin θ 3 , 0, p 3 cos θ 3 ) (2.163)
P 4 = (E 4 , −p 4 sin θ 4 , 0, p 4 cos θ 4 ) . (2.164)
If now we want to find the scattering angle θ 3 , assuming we have determined all energy and
momenta, we solve Equation 2.160 for P 4 and take its invariant square to get
P 2 4 = P 2 1 + P 2 2 + P 2 3 − 2(P 1 + P 2 ) · P 3 + 2P 1 · P 2 . (2.165)
Making use of the relation P 2
i
= m 2 i
and Equations 2.161 and 2.164, we obtain
and for cos θ 3 we arrive at
m 2 4 = m2 1 + m2 + m 2 3 + 2[E 1m − (E 1 + m)E 3 + p 1 p 3 cos θ 3 ] , (2.166)
cos θ 3 = m2 4 − m2 1 − m2 − m 2 3 + 2(E 1 + m)E 3 − 2E 1 m
2p 1 p 3
. (2.167)
Clearly, by symmetry, the above equation is still true if we interchange the indices 3 and 4 to obtain
a relation for cos θ 4 . Thus, we see that the scattering angles are uniquely determined by the final
energies. Azimuthal angles are uniformly distributed, provided, of course, that the two particles
have the opposite azimuth.
In the sections that follow we shall focus on the computation of the differential and total cross
sections for these four processes, and we will provide the sampling methods used to determine the
secondary energies. We shall also reduce Equation 2.167 to the specific reactions and derive the
expressions used in EGS to determine the cosines of the scattering angles. In most cases, we get
the sines of the scattering angles using the formulas
sin θ 3 = √ 1 − cos 2 θ 3 (2.168)
sin θ 4 = − √ 1 − cos 2 θ 4 . (2.169)
The reason for making sin θ 4 negative is that this effectively achieves the opposite azimuth within
the frame work of the EGS routine UPHI, as discussed later. Note that we drop the subscripts on
the scattering angles for simplicity in the discussions below.
2.9 Compton Scattering
The differential and total Compton scattering cross sections are given by formulas originally due
to Klein and Nishina [90]:
d˘Σ Compt (˘k 0 )
= X 0n πr0 2m
[( )/
]
C1
d˘k
˘k 0
2 E + C 2 E + C 3 + E
(2.170)
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