Validity ranges for the different models

The limits of the validity for the various models are somewhat arbitrary within a certain range of material densities and particle energies. To help determine these limits, some characteristic parameters are used:

$\xi$

typical energy loss of a particle in a given layer of material. Its value is:

$\xi =\; 2\; \pi r$_e²m_ec²N_Av( {Z_incβ})²{Z ρtA}= 0.1535 ( {Z_incβ})²{Z ρtA}&sp;(MeV)

$E$_Max

maximum transferable energy in a single collision:

$E$_Max= {2 m_eβ²γ²1 +2 γ{m_eM}+( {m_eM}) ²}

where M is the mass of the incoming particle. Note that for incoming electrons this value has to be divided by 2 due to the impossibility distinguishing the two electrons in the final state.

I

typical electron binding energy. This is a value which characterises the average energy levels of the atomic electrons. In GEANT it is parametrised as:

average energy loss by the particle in a layer of thickness t according to the GEANT energy loss tables.

$\xi /E$_Max

(hereafter $\kappa$ ) relative importance of high energy transfer collisions in the ionisation process

(hereafter $\Delta$ ) estimation of the number of collisions with energy close to the ionisation energy.

In GEANT two variables control the model used to describe the energy loss fluctuations:

ILOSS

fluctuation model:

0

no energy loss;

1

$\delta$ -rays are produced above the threshold, reduced fluctuations from $\delta$ -rays below the threshold are added to the energy loss;

2

no $\delta$ -rays are produced, complete fluctuations are calculated;

3

same as 1;

4

no fluctuations;

ISTRA

energy loss model for thin layer (see below):

0

Urbán model;

1

PAI model;

2

ASHO model for ,(not yet available) PAI model otherwise;

1. large number of low-energy collisions: $\Delta \ge 50$

   if $\delta$ -rays generation is requested ( ILOSS = 1 or 3 and IDRAY = 1) the Urbán model is used. If $\delta$ -rays are not produced ( ILOSS = 2 and IDRAY = 0) we distinguish three regions:

   1. very few energy transfers close to the maximum: $\kappa \le 10-2$

      the Landau distribution is used;

   2. few energy transfers close to the maximum:

      the Vavilov distribution is used;

   3. many energy transfers close to the maximum: $\kappa >\; 10$

      the Gauss distribution is used;

2. small number of low-energy collisions:

   in this region the same model is used for $1\; \le ILOSS\le 3$ irrespective of the value of IDRAY. The model used depends on the value of ISTRA.

   1. Urbán model if ISTRA = 0, PAI model otherwise.

   2. Urbán model if ISTRA = 0, PAI model if ISTRA =1 and ASHO model if ISTRA =2.

   3. Urbán model if ISTRA = 0, PAI model otherwise.

   4. $\Delta \le 0.01$

      Urbán model if ISTRA =0, PAI model otherwise.