Method

Due to the statistical nature of ionisation energy loss, large fluctuations can occur in the amount of energy deposited by a particle traversing an absorber element. As recently reviewed [], continuous processes such as multiple scattering and energy loss play a relevant role in the longitudinal and lateral development of electromagnetic and hadronic showers, and in the case of sampling calorimeters the measured resolution can be significantly affected by such fluctuations in their active layers. The description of ionisation fluctuations is characterised by the significance parameter $\kappa$ , which is proportional to the ratio of mean energy loss to the maximum allowed energy transfer in a single collision with an atomic electron

$\kappa =\{\xi E$_max}

$E$_max

is the maximum transferable energy in a single collision with an atomic electron.

$E$_max={2 m_eβ²γ²1 + 2γm_e/m_x+ ( m_e/m_x)²},

where $\gamma =\; E/m$_x , E is energy and $m$_x the mass of the incident particle, $\beta 2=\; 1\; -\; 1/\gamma 2$ and $m$_e is the electron mass. $\xi$ comes from the Rutherford scattering cross section and is defined as:

$\xi =\; \{\; 2\pi z2e4N$AvZ ρδx meβ2c2A}= 153.4 {z2β2}{ZA}ρδx &sp;keV,

where

$z$	charge of the incident particle
$N$Av	Avogadro's number
$Z$	atomic number of the material
$A$	atomic weight of the material
$\rho$	density
$\delta x$	thickness of the material

$\kappa$ measures the contribution of the collisions with energy transfer close to $E$_max . For a given absorber, $\kappa$ tends towards large values if $\delta x$ is large and/or if $\beta$ is small. Likewise, $\kappa$ tends towards zero if $\delta x$

is small and/or if $\beta$

approaches 1.

The value of $\kappa$ distinguishes two regimes which occur in the description of ionisation fluctuations :

1. A large number of collisions involving the loss of all or most of the incident particle energy during the traversal of an absorber.

   As the total energy transfer is composed of a multitude of small energy losses, we can apply the central limit theorem and describe the fluctuations by a Gaussian distribution. This case is applicable to non-relativistic particles and is described by the inequality $\kappa >\; 10$

   (i.e. when the mean energy loss in the absorber is greater than the maximum energy transfer in a single collision).

2. Particles traversing thin counters and incident electrons under any conditions.

   The relevant inequalities and distributions are , Vavilov distribution, and , Landau distribution.

An additional regime is defined by the contribution of the collisions with low energy transfer which can be estimated with the relation $\xi /I$₀ , where $I$₀

is the mean ionisation potential of the atom. Landau theory assumes that the number of these collisions is high, and consequently, it has a restriction $\xi /I$₀≫1 . In GEANT, the limit of Landau theory has been set at $\xi /I$₀= 50 . Below this limit special models taking into account the atomic structure of the material are used. This is important in thin layers and gaseous materials. Figure shows the behaviour of $\xi /I$₀

as a function of the layer thickness for an electron of 100 keV and 1 GeV of kinetic energy in Argon, Silicon and Uranium.

Figure: The variable $\xi /I$₀ can be used to measure the validity range of the Landau theory. It depends on the type and energy of the particle, Z, A and the ionisation potential of the material and the layer thickness.

In the following sections, the different theories and models for the energy loss fluctuation are described. First, the Landau theory and its limitations are discussed, and then, the Vavilov and Gaussian straggling functions and the methods in the thin layers and gaseous materials are presented.

• Landau theory
• Vavilov theory
• Gaussian Theory
• Urbán model