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 , which is proportional to the ratio of mean energy loss to the maximum allowed energy transfer in a single collision with an atomic electron
z  charge of the incident particle 
N_{Av}  Avogadro's number 
Z  atomic number of the material 
A  atomic weight of the material 
density 

x  thickness of the material 
measures the contribution of the collisions with energy transfer close to Emax. For a given absorber, tends towards large values if x is large and/or if is small. Likewise, tends towards zero if x is small and/or if approaches 1.
The value of distinguishes two regimes which occur in the description of ionisation fluctuations :
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 nonrelativistic particles and is described by the inequality > 10 (i.e. when the mean energy loss in the absorber is greater than the maximum energy transfer in a single collision).
The relevant inequalities and distributions are 0.01 < < 10, Vavilov distribution, and < 0.01, Landau distribution.
An additional regime is defined by the contribution of the collisions with low energy transfer which can be estimated with the relation /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 /I_{0} » 1. In GEANT (see URL http://wwwinfo.cern.ch/asdoc/geant/geantall.html), the limit of Landau theory has been set at /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 1 shows the behaviour of /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.

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.
For a particle of mass m_{x} traversing a thickness of material x, the Landau probability distribution may be written in terms of the universal Landau function () as[1]:
where()  =  1_ 2i _{ c  i}^{c + i} exp du c > 0 
=    '  ^{2}  ln ___ Emax  
'  =  0.422784 . . . = 1  
=  0.577215 . . . (Euler's constant)  
=  average energy loss  
=  actual energy loss 
The Landau formalism makes two restrictive assumptions :
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:
This is realised introducing a _{max}() such that if only values of < _{max} are accepted, the average value of the distribution is .
A parametric fit to the universal Landau distribution has been performed, with following result:
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 Emax = . Now we can write[2]:
where andThe Vavilov parameters are simply related to the Landau parameter by _{L} = _{v}/  ln . It can be shown that as 0, the distribution of the variable _{L} approaches that of Landau. For < 0.01 the two distributions are already practically identical. Contrary to what many textbooks report, the Vavilov distribution does not approximate the Landau distribution for small , but rather the distribution of _{L} defined above tends to the distribution of the true from the Landau density function. Thus the routine GVAVIV samples the variable _{L} rather than _{v}. For > 10 the Vavilov distribution tends to a Gaussian distribution (see next section).
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 > 10 the Vavilov distribution can be replaced by a Gaussian of the form :
thus implyingThe method for computing restricted energy losses with 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.

It is assumed that the atoms have only two energy levels with binding energy E_{1} and E_{2}. The particleatom interaction will then be an excitation with energy loss E_{1} or E_{2}, or an ionisation with an energy loss distributed according to a function g(E) ~ 1/E^{2}:
 (1) 
The macroscopic crosssection for excitations (i = 1, 2) is
 (2) 
 (3) 
r, C  parameters of the model 
E_{i}  atomic energy levels 
I  mean ionisation energy 
f_{i}  oscillator strengths 
The model has the parameters f_{i} , E_{i} , C and r (0 < r < 1). The oscillator strengths f_{i} and the atomic level energies E_{i} should satisfy the constraints
The parameter C can be defined with the help of the mean energy loss dE/dx in the following way: The numbers of collisions (n_{i} , i = 1,2 for the excitation and 3 for the ionisation) follow the Poisson distribution with a mean number <n_{i}>. In a step x the mean number of collisions is
 (6) 
 (7) 
 (8) 
The following values have been chosen in GEANT for the other parameters:
The energy loss is computed with the assumption that the step length (or the relative energy loss) is small, andin consequencethe crosssection can be considered constant along the path length. The energy loss due to the excitation is
 (9) 
 (12) 
If the number of ionisation n_{3} is bigger than 16, a faster sampling method can be used. The possible energy loss interval is divided in two parts: one in which the number of collisions is large and the sampling can be done from a Gaussian distribution and the other in which the energy loss is sampled for each collision. Let us call the former interval [I, I] the interval A, and the latter [I, Emax] the interval B. lies between 1 and Emax/I. A collision with a loss in the interval A happens with the probability
 (13) 
 (14) 
 (15) 
The collisions where the energy loss is in the interval B are sampled directly from
 (20) 
 (21) 
The approximation of equations ((16), (17), (18) and (19) can be used under the following conditions:
where c > 4. From the equations (13), (16) and (18) and from the conditions (22) and (23) the following limits can be derived:
 (25) 
 (26) 
The number of collisions with energy loss in the interval B (the number of interactions which has to be simulated directly) increases slowly with the total number of collisions n_{3}. The maximum number of these collisions can be estimated as
 (27) 
 (28) 
n_{3}  n_{B,max}  n_{3}  n_{B,max} 

16  16  200  29.63 

20  17.78  500  31.01 

50  24.24  1000  31.50 

100  27.59  32.00 
If the step length is very small (< 5 mm in gases, < 23 m in solids) the model gives 0 energy loss for some events. To avoid this, the probability of 0 energy loss is computed
 (29) 
 (30) 
 (31) 
[1] L.Landau. On the Energy Loss of Fast Particles by Ionisation. Originally published in J. Phys., 8:201, 1944. Reprinted in D.ter Haar, Editor, L.D.Landau, Collected papers , page 417. Pergamon Press, Oxford, 1965.
[2] B.Schorr. Programs for the Landau and the Vavilov distributions and the corresponding random numbers. Comp. Phys. Comm., 7:216, 1974.
[3] S.M.Seltzer and M.J.Berger. Energy loss straggling of protons and mesons. In Studies in Penetration of Charged Particles in Matter , Nuclear Science Series 39, Nat. Academy of Sciences, Washington DC, 1964.
[4] R.Talman. On the statistics of particle identification using ionization. Nucl. Inst. Meth., 159:189, 1979.
[5] P.V.Vavilov. Ionisation losses of high energy heavy particles. Soviet Physics JETP , 5:749, 1957.