
Geant4 Physics Reference Manual 
Computing the occurrence of a process
Cross sections per atom and mean free path values are tabulated during initialisation.
and it is independent of the material traversed. If n_{r} is
a random variable denoting the number of mean free paths from a given
point until the point of interaction, it can be shown that n_{r} has the
distribution function
The short description given above is the differential approach of the particle transport , which is used in most of the simulation codes( e.g. [GEANT3],[EGS4]).
In this approach besides the other (discrete) processes the continuous energy loss imposes a limit on the stepsize, too. The reason of this is the energy dependence of the cross sections. It is assumed in this approach that the cross sections of the particles are constant during a step , i.e. the step size should be so small that the relative difference of the cross sections at the beginning of the step and at the end can be small enough.In principle one has to use very small steps in order to have an accurate simulation , but the computing time increases if the stepsize decreases. As a good compromise the stepsize is limited in GEANT4 by the requirement that the stopping range of the particle can decrease by not more than 20 % during the step. This condition works fine for a particle of kinetic energy > 0.5 MeV  1. MeV , but for low energy it gives very short step sizes. To cure this problem a lower limitation on the stepsize is also introduced. The lower limit of the stepsize is the cut in range parameter of the program.
There is another disadvantage of this usual differential algorithm , which can be a serious problem in the case of certain processes. If the interaction process has a cross section with narrow peaks in it , in this approach the particle can easily skip over these peaks.(This can happen in the case of a number of hadronic processes.)
To overcome these shortcomings the integral approach has been implemented in GEANT4 as an option .
In this algorithm the integral in eq. 1.1 is really computed for every process and another equation is used instead of eq. 1.4 when is updated. This section gives a short overview on the formulae used in this approach.
The change in the number of mean free path after a small step can be written as
This formula can be rewritten in the following form
where
T  kinetic energy of the particle 
inverse of the wellknown quantity . 
After these preliminaires the number of mean free paths (eq. 1.1)
can be rewritten as
or
where has been denoted by f() . Eqs. 1.7 and 1.8 are the basic equations of the integral approach . The meaning of the integrals on the right hand sides of the equations is the number of mean free paths for a given process between the initial state of energy T and the stopping of the particle ( energy 0).
The steps of the algorithm using the integral approach are the following:
The proper and lab. time of the particle should be updated after each step. This update is done differently in the two approaches.
In the differential approach the following formula is used to update the particle time in the laboratory system
Using eq. 1.14 the expression for
is
The update of the proper time is performed similarly in the two approaches.