SetCuts: conversion from range to kinetic energy

Charged particles

Starting from a particle cut given in stopping range, the method constructs a vector holding the cut values in kinetic energy for every material. The main steps of the algorithm are the same for all the charged particles, the only difference is that different energy loss formulae are used for electrons, positrons and heavy charged particles (muon,pion,proton,...).

In the case of charged hadrons this procedure is applied for the proton and antiproton only, the cut values in kinetic energy for other hadrons are computed using the proton and antiproton energy loss and range tables.

General scheme

  1. An energy loss table is created and filled for all the elements in the element table.
  2. For every material in the material table the following steps are performed:
    1. a range vector is constructed using the energy loss table and formulae for the low energy part of the calculations,
    2. the conversion from stopping range to kinetic energy is performed and the corresponding element of the KineticEnergyCuts vector is set (It contains the particle cut value in kinetic energy for the actual material.),
    3. the range vector is deleted.
  3. The energy loss table is deleted at the end of the process.

Energy loss formula for heavy charged particles

The energy loss of the particle is calculated from a simplified Bethe-Bloch formula if the kinetic energy of the particle is above the value:
\begin{displaymath} T_{lim}=2 \,MeV \times \left( \frac { \mbox{particle mass}}{\mbox{proton mass}} \right) \end{displaymath}

The simplified here means that the low energy shell correction term and the high energy Sternheimer density correction term have been omitted. Below the energy value $T_{lim}$ a simple parametrized energy loss formula is used to compute the loss, which reproduces the energy loss values of the stopping power tables fairly well. The main reason of using a parametrized formula for low energy is that the Bethe-Bloch breaks down at low energy. The formula has the following form :

\begin{displaymath} \frac{dE}{dx} = \left \{ \begin{array}{ll} a*\sqrt{\frac{T... ... & \mbox{for $T \in [T_0, \, T_{lim}]$} \end{array} \right. \end{displaymath} 


      where : 		 M = particle mass, 


T = kinetic energy, 

		 $ T_0=0.1 \, MeV \times Z^{1/3} \times \mbox{ M/(proton mass)} $


Z = atomic number.
the paramaters $a,b,c$ have been chosen in such a way that $dE/dx$ is a continuous function of T at $T=T_{lim}$ and $T=T_0$ , and $dE/dx$ reaches its maximum at the correct T value.

Energy loss of electrons and positrons

The Berger-Seltzer energy loss formula has been used for T $>$ 10 keV to compute the energy loss comes from the ionization . This formula plays the role of the Bethe-Bloch for electrons ( see e.g. the GEANT3 manual).Below 10 keV the simple c/(T/mass of electron) parametrization has been used, where c can be determined from the requirement of continuity at T = 10 keV. As in the case of electrons the radiation loss is important even at relative low (few MeV ) energy, a second term has been added to the energy loss formula which accounts for the radiation losses (losses due to bremsstrahlung). This second term is an empirical parametrized formula. For positrons a different formula is used to calculate the ionization loss, the term accounting for the radiation losses is the same than that for electrons.

Range calculation

The stopping range is defined as :

\begin{displaymath}R(T)= \int_0^T \frac{1}{(dE/dx)} \, dE \end{displaymath}

The integration has been done analytically for the low energy part and numerically above an energy limit.

Photons

Starting from a particle cut given in absorption length, the method constructs a vector holding the cut values in kinetic energy for every material. The main steps of the algorithm are the following:
  1. A cross section table is created and filled for all the elements in the element table.
  2. For every material in the material table the following steps are performed:
    1. an absorption length vector is constructed using the cross section table
    2. the conversion from absorption length to kinetic energy is performed and the corresponding element of the KineticEnergyCuts vector is set (It contains the particle cut value in kinetic energy for the actual material.),
    3. the absorption length vector is deleted.
  3. The cross section table is deleted at the end of the process.

Cross section formula for elements

An approximate empirical formula is used to compute the absorption cross section of a photon in an element. The absorption cross section means here the sum of the cross sections of the gamma conversion, Compton scattering and photoelectric effect. These processes are the 'destructive' processes for photons , they destroy the photon or decrease its energy. (The coherent or Rayleigh scattering changes the direction of the gamma only, its cross section is not included in the absorption cross section.)

Absorption length vector

The AbsorptionLength vector is calculated for every material as:
AbsorptionLength = 5/(macroscopic absorption cross section)
The factor 5 comes from the requirement that the probability of having no 'destructive' interaction should be small, hence

\begin{eqnarray*} \exp(-\mbox{{\tt AbsorptionLength * MacroscopicCrossSection}}) &=&\exp(-5) \\ &=& 6.7 \times 10^{-3} \end{eqnarray*}



Meaningful cuts in absorption length

The photon cross section for a material has a minimum at a certain kinetic energy $T_{min}$. The AbsorptionLength has a maximum at $T=T_{min}$, the value of the maximal AbsorptionLength is the biggest "meaningful" cut in absorption length . If the cut given by the user is bigger than this maximum, a warning is printed and the cut in kinetic energy is set to the maximum gamma energy (i.e. all the photons will be killed in the material).

back