Calculation of the reduced angle

In formula () above $\chi$_c the critical scattering angle defined as: $\chi$_c²= {4 πe⁴Z_sρN_AvZ_inc²tW c²p²β²}= χ_cc²Z_inc²{tE²β⁴}

with

The variable $\chi$_c is factorised into two parts. The first part, $\chi$_cc , is a constant of the medium for a given incident particle. The second part depends on the incident particle energy and on the path length in the medium.

$\chi$_cc²= {4 πe⁴Z_sρN_AvW}= {4 πc⁴r₀²m_e²Z_sρN_AvW}≈(0.39612 &sp;10^-3)²Z_s{ρW}[ &sp;GeV²&sp;cm^-1]

The variable CHC or CHCMOL stored in Q(JPROB+25) corresponds to $\chi$_cc . B is defined by the equation $B-lnB\; =ln\Omega$₀,

where $\Omega$₀ can be interpreted as the number of collisions in the step:

$\Omega$₀={ χ_c²e^2γ-1χ_α²}= b_cZ_inc²{tβ²}

Here $\gamma$ is Euler's constant ($\gamma =0.57721...$ ) and $\chi$_α the atomic electron screening angle. For a single element this is given by:

$\chi$_α²= ( {λ₀2 πr_TF}) ²[ 1.13 + 3.76 ( {αZ Z_incβ}) ²]

where

$\{\lambda$02 π}	$=$
$r$TF	$=$	$0.885\; a$0Z-{13}&sp;Thomas-Fermiradius of the atom
$a$0	$=$	$\{$2m e2}&sp;Bohr's radius
$\alpha$	$=$

so that we have

$\chi$_α²= {m²e⁴1.13p²²(0.885)²}Z^{23}[ 1 + 3.34 ( {αZ Z_incβ}) ²].

For a compound/mixture the following rule holds:

$ln(\chi$α2)	$=$	$\{\sum$i=1NniZi(Zi+1) ln( χαi2)∑i=1NniZi(Zi+1)}= {∑i=1N{piAi}Zi(Zi+1) ln( χαi2) ∑i=1N{piAi}Zi(Zi+1)}
	$=$	$\{\sum$i=1N{piAi}Zi(Zi+1) ln{m2e41.13p22(0.885)2}Zi{23}[ 1 + 3.34 ( {αZiZincβ}) 2]∑i=1N{piAi}Zi(Zi+1)}
	$=$	$ln(\; \{m$2e41.13p22(0.885)2}) -{∑i=1N{piAi}Zi(Zi+1) lnZi-{23}∑i=1N{piAi}Zi(Zi+1)}
		$+\; \{\sum$i=1N{piAi}Zi(Zi+1) [ 1 + 3.34 ( {αZiZincβ}) 2]∑i=1N{piAi}Zi(Zi+1)}

To understand the transformations in the above formulae, the following should be noted. Let $n$_i be the number of atoms of type i in a compound/mixture, that is the number of moles of element i in a mole of material, and $p$_i the proportion by weight of that element. We have the following relation:

$p$_i= {n_iA_i∑_j=1^Nn_jA_j}&sp;thatis&sp;n_i= ( ∑_j=1^Nn_jA_j) {p_iA_i}= W {p_iA_i}

so that we can simplify expressions in the following way:

$\{\sum$_i=1^Nn_i...W}= ∑_i=1^N(p_i/A_i) ...

If now we set

$Z$s	$=$	$\sum$i = 1NniZi(Zi+1)	$=$	$W\; \sum$i = 1N{piAi}Zi(Zi+1)	$=$	$W\; Z$s'
$Z$E	$=$	$\sum$i = 1NniZi(Zi+ 1 )lnZi-2/3	$=$	$W\sum \; \{p$iAi}Zi(Zi+ 1 )lnZi-2/3	$=$	$W\; Z$E'
$Z$x	$=$	$\sum$i = 1NniZi(Zi+ 1 ) ln[ 1+3.34( {αZiZincβ})2]
	$=$	$W\; \sum$i = 1N{piAi}Zi(Zi+ 1 ) ln[ 1+3.34( {αZiZincβ})2]			$=$	$W\; Z$x'

we can write:

$ln(\chi$α2)	$=$	$ln(\; \{m$2e41.13p22(0.885)2})+ {Zx-ZEZs}
$\chi$α2	$=$	$\{m$2e41.13p22(0.885)2}e(Zx- ZE)/Zs= {m2e41.13p22(0.885)2}e(Zx' - ZE' )/Zs'

and finally:

$\Omega$0	$=$	$\{\; \chi$c2e2γ-1χα2}
	$=$	4ZsρNAvZinc2tW c2p2β2}&sp;{p22(0.885)21.13 m2e4}e-(Zx' - ZE' )/Zs'
	$=$	$b$cZinc2{tβ2}
$b$c	$=$	$\{4\; \pi N$Av2m2c2}{(0.885)21.13 x1.167}ρZs' e(ZE' - Zx' )/Zs'≈6702.33 ρZs'e(ZE' - Zx' )/Zs'

The quantity $b$_c is calculated during initialisation by the routine GMOLI setting $\beta =1$ and $Z$_inc=1 . This variable is called OMC or OMCMOL and it is stored in Q(JPROB+21). It has, via the atomic electron screening angle $\chi$_α² , a small dependence on $\beta$ and $Z$_inc in the term $Z$_x . The quantity $b$_cρZ_s'

is re-evaluated during tracking by the routine GMOLIO, called by GMULTS, only when necessary.

The dependence of $Z$_x on $\beta$ and $Z$_inc is via a term of the form: $1+3.34(\alpha Z\; Z$_incβ^-1)² , so it is not necessary to recalculate it as long as $1\; +\; 3.34(\alpha Z\; Z$_incβ^-1)²≈1 + 3.34(αZ )² that is:

$3.34\; (Z\; \alpha )$2[ ( {Zincβ})2-1 ]	$\ll$	$1$	$\Rightarrow$	$Z$2{Zinc2- β2β2}	$\ll$	$3.34\; \alpha$-2≈5500
$Z$2( Zinc2- β2)	$\le$	$50\; \beta$2

For mixtures/compounds this condition should be checked for every component and this would be unacceptable from the point of view of computer time. So we simply make the condition more restrictive multiplying the first term by the number of elements in the mixture/compound.