Method

The photon energy is sampled according to the Seltzer and Berger bremsstrahlung spectrum []. Seltzer and Berger have calculated the spectra for materials with atomic numbers Z = 6,13,29,47,74,92 in the electron (kinetic) energy range 1 keV - 10 GeV. Their tabulated results have been used as input in a parametrising-fitting procedure. The functional form of the parameterisation for the quantity:

$S(x)\; =\; C\; k\; \{d\; \sigma d\; k\}$

can be written as $S(x)\; =$

$(1-a$hε)F1(δ) + bhε2F2(δ)	$T\; \ge 1\; MeV$
$1\; +\; a$lx + blx2

$.$

where:

$C$		$normalisation\; constant$	$k$		$photon\; energy$
$T,\; E$		$kinetic\; and\; total\; energy\; of\; the\; primary\; electron$	$x$	$=$	$\{kT\}$
$\epsilon$	$=$	$\{kE\}=\; x\; \{TE\}$

The $F$_i(δ) screening functions depend on the screening variable:

$\delta$	$=$	$\{136\; m$eZ1/3E}{ε1-ε}
$F$1(δ)	$=$	$F$0(42.392 - 7.796 δ+1.961 δ2- F)	$\delta \le 1$
$F$2(δ)	$=$	$F$0(41.734 - 6.484 δ+1.250 δ2- F)	$\delta \le 1$
$F$1(δ)	$=$	$F$2(δ) =F0(42.24 - 8.368 ln(δ+ 0.952) -F)	$\delta >\; 1$
$F$0	$=$	$\{142.392-F\}$
$F$	$=$	$4\; lnZ\; -\; 0.55\; (lnZ)2$

$a$_h,l and $b$_h,l are parameters to be fitted.

The `high energy' (T > 1 MeV) formula comes from the Coulomb-corrected, sceened Bethe-Heitler formula (see e.g. [,,]). However, there are two things in eq. () which make a difference:

1. $a$_h, b_h depend on T and on the atomic number Z ( in the case of the Bethe-Heitler spectrum $a$_h= 1 , $b$_h=0.75 );
2. the function F is not the same than that in the Bethe-Heitler cross-section, this function gives a better behaviour in the high frequency limit, i.e. when $k\; \to T$ ($x\; \to 1$ ).

The T and Z dependence of the parameters are described by the equations:

$a$h	$=$	$1\; +\; \{a$h1u}+{ah2u2}+{ah3u3}
$b$h	$=$	$0.75+\{b$h1u}+{bh2u2}+{bh3u3}
$a$l	$=$	$a$l0+ al1u + al2u2
$b$l	$=$	$b$l0+ bl1u + bl2u2
$with$
$u$	$=$	$ln(\; \{Tm$e})

the $a$_hi, b_hi, a_li, b_li parameters are polynomials of second order in the variable:

$v\; =\; [Z\; (Z+1)]1/3$

It can be seen relatively easily that for the limiting case , $a$_h→1, b_h→0.75 , so eq. () gives the Bethe-Heitler cross section.

There are altogether 36 linear parameter in the formulae , their values are given in GBREME. The parameterisation reproduces the Seltzer-Berger tables within a few % (2-3 % on average, the maximum error being less than 10-12 %), the tables, on the other hand, agree well with the experimental data and theoretical (low- and high-energy) results (less than 10 % below 50 MeV, less than 5 % above 50 MeV).

Apart from the normalisation the cross section differential in photon energy can be written as:

$\{d\; \sigma d\; k\}=\; \{1ln\{1x$_c}}{1x}g(x) = {1ln{1x_c}}{1x}{S(x)S_max}

where $x$_c= k_c/T , $k$_c is the photon cut-off energy below which the bremsstrahlung is treated as a continuous energy loss (this cut is BCUTE in the program). Using this decomposition of the cross section and two random numbers $r$₁ , $r$₂ uniformly distributed in $]0,1[$ , the sampling of x is done as follows:

1. sample x from

   $\{1ln\{1x$_c}}{1x}&sp;setting&sp;x = e^r₁lnx_c

2. calculate the rejection function $g(x)$ and:
   • if $r$₂> g(x) reject x and go back to 1;
   • if $r$₂≤g(x) accept x.

To apply the Migdal correction [] all it has to be done is to multiply the rejection function by the Migdal correction factor:

$C$_M(ε) ={1 + C₀/ ε_c²1 + C₀/ ε²}

where

$C$₀={nr₀λ²π}, &sp;ε_c= {k_cE}

n

electron density in the medium

$r$₀

classical electron radius

$\lambda$

reduced Compton wavelength of the electron.

After the successful sampling of $\epsilon$ , GBREME generates the polar angles of the radiated photon with respect to the parent electron's momentum. It is difficult to find in the literature simple formulas for this angle. For example the double differential cross section reported by Tsai [,] is the following:

$\{d\; \sigma dkd\; \Omega \}$	$=$	$\{2\; \alpha 2e2\pi k\; m4\}[\; \{2\epsilon -2(1+u2)2\}+\{12u2(1-\epsilon )(1+u2)4\}]Z(Z+1)\; .$
		$+\; .\; [\; \{2-2\epsilon -\epsilon 2(1+u2)2\}-\; \{4u2(1-\epsilon )(1+u2)4\}][\; X-2Z2f$c((αZ)2)]
$u$	$=$	$\{E\; \theta m\}$
$X$	$=$	$\int$tminm2(1+u2)2[ GZel(t) + GZin(t) ]{t-tmint2}dt
$G$Zel, in(t)		$atomic\; form\; factors$
$t$min	$=$	$[\; \{k\; m2(1+u2)2\; E\; (E-k)\}]2=\; [\; \{\epsilon m2(1+u2)2\; E\; (1-\epsilon )\}]2$

This distribution is complicated to sample, and it is anyway only an approximation to within few percent, if nothing else, due to the presence of the atomic form-factors. The angular dependence is contained in the variable $u\; =\; E\; \theta m-1$ . For a given value of u the dependence of the shape of the function on Z, E, $\epsilon =\; k/E$ is very weak. Thus, the distribution can be approximated by a function $f(u)\; =\; C\; (\; u\; e-au+\; d\; u\; e-3au)$

where

where E is in GeV. While this approximation is good at high energies, it becomes less accurate around few MeV. However in that region the ionisation losses dominate over the radiative losses.

The sampling of the function $f(u)$ can be done in the following way ($r$_i,&sp;i=1,2,3 are uniformly distributed random numbers in [0,1]):

1. Choose between $u\; e-au$ and $d\; u\; e-3au$ :

   $b\; =$

   $a$	1< 9/(9+d)
   $3a$	1≥9/(9+d)

   $.$

2. Sample $u\; e-bu$ :

   $u=-\{log(\; r$₂r₃) b}

3. check that:

   $u\; \le u$_max= {E πm}

   otherwise go back to 1.

The probability of failing in the last test is reported in table .

 

2c$P\; =\; \int$umax&inf;f(u) &sp;du
E (MeV)	P()
0.511	3.4
0.6	2.2
0.8	1.2
1.0	0.7
2.0	0.1

Table: Angular sampling efficiency

The function $f(u)$ can be used also to describe the angular distribution of the photon in $\mu$ bremsstrahlung and to describe the angular distribution in photon pair production.

The azimuthal angle, $\Phi$ , is generated isotropically. This information is used to calculate the momentum vector of the radiated photon, to transform it to the GEANT coordinate system and to store the result into common block /GCKING/. Also, the momentum of the parent electron is updated.

• Restrictions