THE EGS5 CODE SYSTEM

Transformation to the laboratory system In the scattering system used here, k ⃗ 0 and ⃗e 0 are
in the direction of ⃗e z and ⃗e x , respectively, as shown in Figure 2.23, whereas k ⃗ 0 and ⃗e 0 may be in an
arbitrary direction in the laboratory system. The scattering and laboratory systems are connected
via three rotations, which are calculated from the direction of k ⃗ 0 and ⃗e 0 in the laboratory system.
Using these three rotations, ⃗ k and ⃗e are transformed from the scattering system to the laboratory
system.
Here, we describe the relation of the laboratory system, which is used in the default EGS4
simulation, to the scattering system used in Compton and Rayleigh scattering routines for linearly
polarized photons.
The laboratory system, in which k ⃗ 0 and ⃗e 0 are in arbitrary directions, and the scattering system,
in which k ⃗ 0 is parallel to ⃗e z and ⃗e 0 is parallel to ⃗e x , is transformed to each other by three rotations.
⃗e z and ⃗e x are unit vectors parallel to the z- and x-axes. Two rotations are necessary to make
⃗k 0 ‖ ⃗e z . These rotations were described by Cashwell and Everett [42]. The default version of EGS4
already treats these rotations. Using Cashwell’s notation, this A −1 matrix is
⎛
⎞
uw vw
A −1 ρ ρ
− ρ
⎜
=
−v u ⎟
⎝ ρ ρ
0 ⎠ . (2.441)
u v w
Here, ρ = √ 1 − w 2 . This matrix is written with an inverse sign, since A is mainly used for a
transformation from the scattering system to the laboratory system. It is clear that
⎛ ⎞ ⎛ ⎞
u 0
A −1 ⎝ v ⎠ = ⎝ 0 ⎠ . (2.442)
w 1
In the laboratory system, ⃗e 0 ⊥ ⃗ k 0 ; rotation by A −1 does not change this relation. As A −1 makes
⃗k 0 ‖ ⃗e z , A −1 moves ⃗e 0 onto the x-y plane. In Figure 2.25, ⃗ k 0 and ⃗e 0 after two rotations by the A −1
matrix is shown. Another rotation by an angle (−ω) along the z-axis is necessary to make ⃗e 0 ‖ ⃗e x
. The cos ω and sin ω are calculated using
⎛ ⎞
cos ω
A −1 ⃗e 0 = ⎝ sin ω ⎠ . (2.443)
0
By these three rotations, ⃗ k 0 and ⃗e 0 in the laboratory system are transferred to those in the scattering
system. The scattered photon propagation vector ( ⃗ k) and the polarization vector (⃗e) are transferred
from the scattering system to the laboratory system by an inverse of these three rotations after
Compton or Rayleigh scattering.
The relationship of ⃗ k 0 , ⃗ k, ⃗e 0 and ⃗e in laboratory system and those in scatter system are:
⃗k 0 (lab) = A · B · ⃗k 0 (scatter),
⃗e 0 (lab) = A · B · ⃗e 0 (scatter),
⃗ k (lab) = A · B · ⃗ k (scatter), (2.444)
⃗e (lab) = A · B · ⃗e (scatter),
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