Method

For a complete account of the Monte Carlo methods used the interested user is referred to the publications of Butcher and Messel [], Messel and Crawford [] and Ford and Nelson []. Only the basic formalism is outlined here.

The quantum mechanical Klein-Nishina differential cross-section is:

$\Phi (E,E\text{'})\; =\{X$₀n πr₀²m_eE²}[{1ε}+ε][1 - {εsin²θ1+ε²}]

where,

$E$	energy of the incident photon
$E\text{'}$	energy of the scattered photon
$\epsilon$	$E\text{'}/E$
$m$e	electron mass
$n$	electron density
$r$0	classical electron radius
$X$0	radiation length

Assuming an elastic collision, the scattering angle $\theta$ is defined by the Compton formula:

$E\text{'}\; =\; E\; \{m$_e m_e+ E(1-cosθ)}

Using the combined ``composition and rejection'' Monte Carlo methods described in chapter PHYS211, we may set:

$f(\epsilon )$	$=$	$[\{1\epsilon \}+\epsilon ]=\; \sum$i=12αifi(E)	$2lfor$	$\epsilon$0> ε> 1
$g(\epsilon )$	$=$	$[\; 1\; -\; \{\epsilon sin2\theta 1+\epsilon 2\}]$	$3lrejection\; function$
$\alpha$1	$=$	$\{1ln(1/\epsilon$0)}	$\alpha$2	$=$	$\{12\}(1-\epsilon$02)
$f$1(ε)	$=$	$\{1\epsilon ln(1/\epsilon$0)}	$f$2(ε)	$=$	$\{2\epsilon 1-\epsilon 2\}$

The value of $\epsilon$ corresponding to the minimum photon energy (backward scattering) is given by:

$\epsilon$0

$=$

$\{11+2E/m$e}

Given a set of random numbers $r$_i uniformly distributed in [0,1], the sampling procedure for $\epsilon$ is the following:

1. decide which element of the $f(\epsilon )$ distribution to sample from. Let $\alpha$_T= (α₁+α₂)r₀ . If $\alpha$₁≥α_T

   select $f$₁(ε) , otherwise select $f$₂(ε) ;

2. sample $\epsilon$ from the distributions corresponding to $f$₁ or $f$₂ . For $f$₁ this is simply achieved by:

   $\epsilon =\; \epsilon$₀e₁^αr₁

   For $f$₂ , we change variables and use:

   $\epsilon \text{'}\; =$

   $max(r$2,r3)	$for$	$E/m\; \ge (E/m+1)r$4
   $r$5	$2lfor\; all\; other\; cases$

   $.$

   Then, $\epsilon =\; \epsilon$₀+(1-ε₀)ε' ;

3. calculate $sin2\theta =\; max(0,t(2-t))$

   where $t=m$_e(1-ε)/E'

4. test the rejection function, if $r$₆≤g(ε) accept $\epsilon$ , otherwise return to step 1.

After the successful sampling of $\epsilon$ , GCOMP generates the polar angles of the scattered photon with respect to the direction of the parent photon. The azimuthal angle, $\phi$ , is generated isotropically and $\theta$ is as defined above. The momentum vector of the scattered photon is then calculated according to kinematic considerations. Both vectors are then transformed into the GEANT coordinate system.