Simulation of Energy Loss Straggling
Maria Physicist
August 7, 1998
1 Introduction
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.
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
[MML math],
which is proportional to the ratio of mean energy loss to the maximum
allowed energy transfer in a single collision with an atomic electron
[MML math]
[MML math] is the
maximum transferable energy in a single collision with an atomic electron.
[MML math] where
[MML math],
[MML math] is energy and
[MML math] the mass of the
incident particle, [MML math]
and [MML math] is the
electron mass. [MML math]
comes from the Rutherford scattering cross section and is defined as:
[MML math]
where
[MML math]
measures the contribution of the collisions with energy transfer close to
[MML math]. For a given absorber,
[MML math] tends towards large
values if [MML math] is large
and/or if [MML math] is small.
Likewise, [MML math] tends
towards zero if [MML math] is
small and/or if [MML math]
approaches 1.
The value of [MML math]
distinguishes two regimes which occur in the description of ionisation fluctuations
:
- 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 parti-
cles and is described by the inequality [MML math]
(i.e. when the mean energy loss in the absorber is greater than the maxi-
mum energy transfer in a single collision).
- Particles traversing thin counters and incident electrons under any
conditions.
The relevant inequalities and distributions are [MML math],
Vavilov distribution, and [MML math],
Landau distribution.
An additional regime is defined by the contribution of the collisions
with low energy transfer which can be estimated with the relation
[MML math],
where [MML math]
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
[MML math]. In GEANT (see
URL http://wwwinfo.cern.ch/asdoc/geant/geantall.html), the limit of Landau theory has
been set at [MML math].
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 1 shows the behaviour
of [MML math] as
a function of the layer thickness for an electron of 100 keV and 1 GeV of kinetic
energy in Argon, Silicon and Uranium.
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.
2 Landau theory
For a particle of mass [MML math] traversing
a thickness of material [MML math],
the Landau probability distribution may be written in terms of the universal Landau
function [MML math]
as[1]:
[MML math]
where
[MML math]
2.1 Restrictions
The Landau formalism makes two restrictive assumptions :
- 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 3).
- 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[4] is necessary to accurately
simulate data distributions. In GEANT, a parameterised model by L.
Urbán is used (see section 5).
In addition, the average value of the Landau distribution is infinite.
Summing the Landau fluctuation obtained to the average energy from the
[MML math]
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:
[MML math]
In order for this to have average 0, we must impose that:
[MML math]
This is realised introducing a [MML math]
such that if only values of [MML math]
are accepted, the average value of the distribution is
[MML math].
A parametric fit to the universal Landau distribution has been performed, with following
result: [MML math] only values
smaller than [MML math]
are accepted, otherwise the distribution is resampled.
3 Vavilov theory
Vavilov[5] derived a more accurate straggling distribution by introducing the
kinematic limit on the maximum transferable energy in a single collision, rather than
using [MML math].
Now we can write[2]:
[MML math]
where
[MML math]
and
[MML math]
The Vavilov parameters are simply related to the Landau parameter by
[MML math]. It can be shown that
as [MML math], the distribution of
the variable [MML math] approaches
that of Landau. For [MML math]
the two distributions are already practically identical. Contrary to what many textbooks
report, the Vavilov distribution does not approximate the Landau distribution for small
[MML math], but rather the
distribution of [MML math]
defined above tends to the distribution of the true
[MML math] from
the Landau density function. Thus the routine GVAVIV samples the variable
[MML math] rather
than [MML math].
For [MML math]
the Vavilov distribution tends to a Gaussian distribution (see next section).
4 Gaussian Theory
Various conflicting forms have been proposed for Gaussian straggling functions, but most of
these appear to have little theoretical or experimental basis. However, it has been shown[3]
that for [MML math]
the Vavilov distribution can be replaced by a Gaussian of the form :
[MML math]
thus implying
[MML math]
5 Urbán model
The method for computing restricted energy losses with
[MML math]-ray
production above given threshold energy in GEANT is a Monte Carlo method that
can be used for thin layers. It is fast and it can be used for any thickness of
a medium. Approaching the limit of the validity of Landau's theory, the
loss distribution approaches smoothly the Landau form as shown in Figure
2.
Figure 2 | Energy loss distribution for a 3 GeV electron in Argon as given by
standard GEANT. The width of the layers is given in centimeters. |
|
It is assumed that the atoms have only two energy levels with binding energy
[MML math] and
[MML math].
The particle-atom interaction will then be an excitation with energy loss
[MML math] or
[MML math], or
an ionisation with an energy loss distributed according to a function
[MML math]:
[MML math]
The macroscopic cross-section for excitations
( [MML math]) is
[MML math]and
the macroscopic cross-section for ionisation is
[MML math] [MML math]
is the GEANT cut for [MML math]-production,
or the maximum energy transfer minus mean ionisation energy, if it is smaller than
this cut-off value. The following notation is used: