Method

In the Molière theory the average number of Coulomb scatters for a charged particle in a step is expressed by the parameter $\Omega$₀ (see [PHYS325]). When $\Omega$₀≤20 , the Molière theory is not applicable any more, even if it has been noted [] that it still gives reasonable results down to its mathematical limit $\Omega$₀= e . The range ₀≤20 is called the plural scattering regime. An interesting study of this regime can be found in [].

In GEANT, when $\Omega$₀≤20 , a direct simulation method is used for the scattering angle. The number of scatters is distributed according to a Poissonian law with average $\Omega =\; k\; \Omega$₀= e^2γ-1Ω₀≈1.167 Ω₀ with $\gamma =\; 0.57721...$ Euler's constant. Using the customary notations for the probability distribution function for small angle ($sin\theta \approx \theta$ ) single scattering, we can write:

$f(\theta )\; \theta d\theta =\; \{d\; \sigma \theta d\theta \}\{1\sigma \}\theta d\theta$

where $\sigma$ is the cross section for single elastic scattering. We use as cross section the one reported by Molière [] []:

$\{d\; \sigma \theta d\theta \}=\; 2\; \pi (\; \{2\; Z\; Z$_ince²p v}) ²{1(θ²+ χ_α²)²}

This is the classical Rutherford cross section corrected by the screening angle $\chi$_α . This angle is described by Molière as a correction to the Born approximation used to derive the quantum mechanical form of the Rutherford cross section. We have then:

$\sigma =\; \int$₀^&inf;{d σθdθ}θdθ= 2 π( {2 Z Z_ince²p v}) ²∫₀^&inf;{θd θ(θ²+ χ_α²)²}=π( {2 Z Z_ince²p v}) ²{1χ_α²}

and so equation () becomes:

$f(\theta )\; \theta d\theta =\; \chi$_α²{2 θd θ(θ²+ χ_α²)²}= { 2 Θd Θ(1+Θ²)²}

where we have set $\Theta =\; \theta /\; \chi$_α . To sample from this distribution we calculate the inverse of the distribution function:

$\eta =\; \int$₀^Θ{2 t d t(1+t²)²}= 1 - {11+Θ²}&sp;⇒&sp;Θ= {11-η}-1

where $\eta$ is a number uniformly distributed between 0 and 1. If we observe that also $1-\eta$ is uniformly distributed between 0 and 1 and we remember the definition of $\Theta$ , we obtain:

$\theta =\chi$_α²({1η}-1 )

To calculate $\chi$_α² we observe that:

$\Omega$0	$=$	$\{\chi$c2k χα2}
$\chi$α2	$=$	$\{\chi$c2k Ω0}={χcc2Zinc2t/E2β4k bcZinc2t /β2}={χcc2k bcp2c2}

where we have used the notations of [PHYS325] and $k\; =\; 1.167$ .

PHYS330