Introduction

When traversing ordinary matter, elementary particles undergo deflection from their originary trajectory due to the interaction with the atoms. This effect is rather large for charged particles, which are deflected by the electric field of nuclei and electrons via a large number of small elastic collisions. To simulate precisely the transport of particles in matter, it is important to provide a precise description of this effect. Considerable effort has gone over the years into the formulation of a theory of Coulomb multiple scattering. In GEANT we follow the theory originally formulated by Molière [] [] and then elaborated by Bethe []. A review of this theory can be found in [].

It should be noted that this is not the only theory which describes multiple Coulomb scattering, and a thorough analysis of the problem has been performed by Goudsmit and Saunderson [] [].

A complete account of the Molière theory is outside the scope of this manual. For more information the interested reader could refer to the works cited above and to the EGS4 manual [], from which the notation used in the present chapter has been derived.

The principal limitations of Molière theory are the following:

1. the angular deflection is small. Effectively this constraint provides the upper limit for the step-size;
2. the theory is a multiple scattering theory; that is, many atomic collisions participate in causing the incident particle to be deflected. In some implementations, this constraints provides a lower limit on the step size. In GEANT, when the number of scatters is below the limit of applicability of Molière single Coulomb scatterings are simulated, so that there is no effective minimum step size below which multiple scattering is switched off;
3. the theory applies only in semi-infinite homogeneous media. This constraint calls for a very careful handling of particles near to a medium's boundary;
4. there is no energy loss built into the theory. This again introduces the necessity to take small steps;

In Molière theory corrected for finite angle scattering ($sin\theta \ne \theta$ ) as described by Bethe [,], the angular distribution is given by:

$=$

r(η) &sp;ηdη,

where for $f$_r(η) we use the first three terms of Bethe's expansion:

$f$r(η)	$=$	$f$r(0)(η)+fr(1)(η)B-1+fr(2)(η)B-2
$\eta$	$=$	$\{\theta \chi$c B }

$\eta$ is called the reduced angle.