Geant4 Physics Reference Manual

Computing the occurrence of a process

Particle transport using the differential approach

The interaction length

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In a simple material the number of atoms per volume is:

$$n = \frac{\mathcal{N}\rho}{A}$$

where:

$$\begin{eqnarray*} \mathcal{N} & & \mbox{Avogadro's number} \\ \rho & & \mbox{density of the medium} \\ A & & \mbox{mass of mole} \end{eqnarray*}$$

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In a compound material the number of atoms of Element elm per volume is:

$$n_{elm} = \frac{\mathcal{N}\rho w_{elm}}{A_{elm}}$$

where:

$$\begin{eqnarray*} \mathcal{N} & & \mbox{Avogadro's number} \\ \rho & & \mbox{... ...nt elm}\\ A_{elm} & & \mbox{mass of mole of the Element elm} \end{eqnarray*}$$

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The mean free path, $\lambda$, of a process can be given in terms of its total cross section.

$$\lambda(E) \equiv \frac{1}{\Sigma (E)} = \frac{1}{\sum_{elm}{\lbrack n_{elm} \sigma(Z_{elm},E)\rbrack}}$$

where $\sigma(Z,E)$ is the total cross section per atom of the process and $\sum_{elm}$ runs over all Elements the material is made of.

Cross sections per atom and mean free path values are tabulated during initialisation.

Determination of the interaction point

The mean free path of a particle for a given process, $\lambda$, depends on the medium and cannot be used directly to sample the probability of an interaction in a heterogeneous detector. The number of mean free paths which a particle travels is:

$$n_\lambda =\int \frac{dx}{\lambda(x)}$$ (1.1)

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

$$P( n_r < n_\lambda ) = 1-e^{-n_\lambda}$$ (1.2)

The total number of mean free paths the particle travels before the interaction point, $n_\lambda$, is sampled at the beginning of the trajectory as:

$$n_\lambda = -\log \left ( \eta \right )$$ (1.3)

where $\eta$ is a random number uniformly distributed in the range (0,1). $n_\lambda$ is updated after each step $\Delta x$ according the formula:

$$n'_\lambda=n_\lambda -\frac{\Delta x }{\lambda(x)}$$ (1.4)

until the step originating from $s(x) = n_\lambda \lambda(x)$ is the shortest and this triggers the specific process.

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 .

Particle transport with integral approach

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 $n_\lambda$ 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

$$\Delta n_\lambda = \frac{\Delta x}{\lambda (x)} .$$ (1.5)

This formula can be rewritten in the following form

$$\Delta n_\lambda = \frac{dx}{dT}\cdot \frac{1}{\lambda (T)} \cdot dT ,$$ (1.6)

where

T	kinetic energy of the particle
$\frac{dx}{dT} = \frac{1}{\frac{dT}{dx}}$	inverse of the well-known quantity $\frac{dE}{dx}$ .

After these preliminaires the number of mean free paths (eq. 1.1) can be rewritten as

$$n_\lambda (T) = \int_{0}^{T}\frac{d\tau}{f(\tau)\cdot \lambda (\tau)}$$ (1.7)

or

$$n_\lambda (T) = \int_{0}^{T}\frac{\sigma (\tau ) \cdot d\tau}{f(\tau) }$$ (1.8)

where $\frac{dE}{dx}$ 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 number of mean free paths the particle travels to the interaction point , $\Delta n_\lambda$ is sampled using eq. 1.3
• initial value of the quantity $n_\lambda (T_0)$ is computed
• the (possible) final energy T is calculated from
  

  $$\Delta n_\lambda = n_\lambda (T_0) - n_\lambda (T)$$ (1.9)

  
  

• the step limitation originating from the process is
  

  steplimit = r(T₀) - r(T) (1.10)

  
  
  , where the function r(T) gives the range of the particle for a kinetic energy value T.

If the interaction occurred at the end of the step was some other process or the step was limited by the geometry (at medium boundary) , the number of mean free paths $\Delta n_\lambda$ should be updated. This update can be easily done using the function $n_\lambda (T)$

$$\Delta n_{\lambda,updated} = \Delta n_\lambda - [n_\lambda (T_0)- n_\lambda (T) ]$$ (1.11)

Updating of the proper and laboratory time of the particle

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

$$\Delta t_{lab} = \frac{\Delta x}{\frac{v_0 + v}{2}}$$ (1.12)

, where

$$\begin{array}{ll} \Delta x & \mbox{step travelled by the par... ...box{particle velocity at the end of the step .} \\ \end{array}$$

This expression is a good approximation for use in the differential approach , because in this scheme the velocity is not allowed to change too much during the step.This is not the case in the integral approach, the kinetic energy and the velocity of the particle can change a lot here, therefore the update of the laboratory time is done by using the integral expression

$$t = \int_{0}^{s} \frac{dx}{v(x)}$$ (1.13)

which can be written as

$$t = \int_{0}^{T} \frac{f(\tau ) \cdot d\tau }{v(\tau )}.$$ (1.14)

In the eqs. 1.13 and 1.14

$$\begin{array}{ll} t & \mbox{time} \\ v & \mbox{velocity of ... ... \\ T & \mbox{kinetic energy} \\ f(T) & dE/dx \\ \end{array}$$

Using eq. 1.14 the expression for $\Delta t_{lab}$ is

$$\Delta t_{lab} = t(T_0)-t(T) .$$ (1.15)

The update of the proper time is performed similarly in the two approaches.

Status of this document

9.10.98 created by L. Urbán.