The interaction length

Let $\sigma (E,Z,A)$ be the total microscopic cross section for a given interaction. The mean free path, $\lambda$ , for a particle to interact is given by: $\lambda =\; \{1\Sigma \}$

where $\Sigma$ is the macroscopic cross-section in $cm-1$ . This quantity is given for an element by:

$\Sigma =\; \{N$_Avρσ(E,Z,A)A}

and for a compound or a mixture by:

$\Sigma =\; \{N$_Avρ∑_in_iσ(E,Z_i,A_i)∑_in_iA_i}= N_Avρ∑_i{p_iA_i}&sp;σ(E,Z_i,A_i)

$N$Av	Avogadro's number ($6.02486\; x1023$ )
$Z$	atomic number
$A$	atomic weight
$\rho$	density
$\sigma$	total cross-section for the reaction
$n$i	proportion by number of the $ith$ element in the material
$p$i	$=n$iAi/ ∑jnjAj , proportion by weight of the $ith$ element in the material

For electromagnetic processes which depend linearly on the atomic number Z we can write:

$\Sigma (E)$	$=$	$N$Avρ∑i{piAi}&sp;σ(E,Zi) =NAvρ∑i{piAi}&sp;Zif(E)
	$=$	$N$Avρf(E) ∑i{piAi}&sp;Zi= NAvρf(E) Zeff
$Z$eff	$=$	$\sum$i{piAi}&sp;Zi

the value above of $Z$_eff is calculated by GPROBI. This mean free path is tabulated at initialisation time as a function of the kinetic energy of the particle, or, for hadronic interactions, it is calculated at tracking time.

Cross sections are tabulated in the energy range defined as: $ELOW(1)\le E\; \le ELOW(NEK1)$

in NEK1 bins. These values can be redefined by the data record RANGE. Default values are $ELOW(1)=\; 10\; keV$ , $ELOW(NEK1)=\; 10\; TeV$ and $NEKBIN=\; NEK1-1\; =\; 90$ . NEKBIN cannot be bigger than 199. The array ELOW is in the common /GCMULO/.

Numerically, if we measure the microscopic cross section in b where $1b=10-24cm-2$ , we can express the macroscopic cross section as:

$\Sigma [cm-1]$	$=$
	$=$

which is the formula mostly used in GEANT.