Landau theory

  For a particle of mass $m$_x traversing a thickness of material $\delta x$ , the Landau probability distribution may be written in terms of the universal Landau [], [] function $\phi (\lambda )$ as:

$f(\; \epsilon ,\; \delta x\; )$

$=$

$\{1\xi \}\phi (\; \lambda )$

where

$\phi (\lambda )$	$=$	$\{12\; \pi i\}\int$c-i&inf;c+i&inf;exp( u lnu + λu ) du &sp;c ≥0
$\lambda$	$=$	$\{\epsilon -\epsilon \xi \}-\; \gamma \text{'}\; -\; \beta 2-\; ln\{\xi E$max}
$\gamma \text{'}$	$=$	$0.422784...=\; 1\; -\; \gamma$
$\gamma$	$=$	$0.577215...(Euler\text{'}s\; constant)$
$\epsilon$	$=$	$average\; energy\; loss$
$\epsilon$	$=$	$actual\; energy\; loss$

Restrictions

The Landau formalism makes two restrictive assumptions :

1. The typical energy loss is small compared to the maximum energy loss in a single collision. This restriction is removed in the Vavilov theory (see section ).
2. The typical energy loss in the absorber should be large compared to the binding energy of the most tightly bound electron. For gaseous detectors, typical energy losses are a few keV which is comparable to the binding energies of the inner electrons. In such cases a more sophisticated approach which accounts for atomic energy levels (see for instance Talman []) is necessary to accurately simulate data distributions. In GEANT, a parameterised model by L. Urbán is used (see section ).

In addition, the average value of the Landau distribution is infinite. Summing the Landau fluctuation obtained to the average energy from the $dE/dx$ tables, we obtain a value which is larger than the one coming from the table. The probability to sample a large value is small, so it takes a large number of steps (extractions) for the average fluctuation to be significantly larger than zero. This introduces a dependence of the energy loss on the step size which can affect calculations.

A solution to this has been to introduce a limit on the value of the variable sampled by the Landau distribution [], in order to keep the average fluctuation to 0. The value obtained from the GLANDO routine is:

$\delta dE/dx\; =\; \epsilon -\epsilon =\; \xi (\; \lambda -\; \gamma \text{'}+\beta 2+ln\{\xi E$_max})

In order for this to have average 0, we must impose that:

$\lambda =\; -\gamma \text{'}\; -\; \beta 2-ln\{\xi E$_max}

This is realised introducing a $\lambda$_max(λ) such that if only values of $\lambda \le \lambda$_max are accepted, the average value of the distribution is $\lambda$ .

A parametric fit to the universal Landau distribution has been performed, with following result:

$\lambda$_max= 0.60715 + 1.1934λ+(0.67794+0.052382λ)exp(0.94753+0.74442λ)

only values smaller than $\lambda$_max are accepted, otherwise the distribution is resampled.