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 , which is proportional to the ratio of mean energy loss to the maximum allowed energy transfer in a single collision with an atomic electron

/

= 2z2e4NAvZx me2c2A = 153.4 z2 2 Z Ax keV,

z	charge of the incident particle
NAv	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 :

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 > 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 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 /I₀, where I₀ 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 /I₀ » 1. In GEANT (see URL http://wwwinfo.cern.ch/asdoc/geant/geantall.html), the limit of Landau theory has been set at /I₀ = 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 /I₀ 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 1 The variable /I0 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.

2 Landau theory

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]:

f(, x)

=

1 ()

()	=	1_ 2i c - ic + i exp du      c > 0
	=	- - ' - 2 - ln ___ Emax
'	=	0.422784 . . . = 1 -
	=	0.577215 . . . (Euler's constant)
	=	average energy loss
	=	actual energy loss

2.1 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 3).
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[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 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:

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 Emax = . Now we can write[2]:

f

=

1 v

v	=	1_ 2i c - ic + i esds      c > 0
	=	exp   exp ,
	=	s ln + (s + 2) - e-s/,

E1(z)	=	zt-1e-tdt   (the exponential integral)
v	=

The 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).

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 > 10 the Vavilov distribution can be replaced by a Gaussian of the form :

f(, s) 1 __________ exp

mean	=
2	=	2 (1 - 2/2) = E max(1 - 2/2)

5 Urbán model

The 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.

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 E₁ and E₂. The particle--atom interaction will then be an excitation with energy loss E₁ or E₂, or an ionisation with an energy loss distributed according to a function g(E) ~ 1/E²:

(1)

The macroscopic cross-section for excitations (i = 1, 2) is

(2)

and the macroscopic cross-section for ionisation is

(3)

Emax is the GEANT cut for -production, or the maximum energy transfer minus mean ionisation energy, if it is smaller than this cut-off value. The following notation is used:

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

(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, and---in consequence---the cross-section can be considered constant along the path length. The energy loss due to the excitation is

(9)

where n₁ and n₂ are sampled from Poisson distribution as discussed above. The loss due to the ionisation can be generated from the distribution g(E) by the inverse transformation method:

u = F(E)	=	IEg(x)dx
E = F-1(u)	=	I_____ 1 - u Emax__ Emax+I		(10)
				(11)

(12)

5.1 Fast simulation for n₃ > 16

If the number of ionisation n₃ 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)

The mean energy loss and the standard deviation for this type of collision are

(14)

and

(15)

If the collision number is high , we assume that the number of the type A collisions can be calculated from a Gaussian distribution with the following mean value and standard deviation:

<nA>	=	n3P()		(16)
A2	=	n3P()(1 - P())		(17)

<EA>	=	nA<E()>		(18)
E, A2	=	nA2()		(19)

The collisions where the energy loss is in the interval B are sampled directly from

(20)

The total energy loss is the sum of these two types of collisions:

(21)

The approximation of equations ((16), (17), (18) and (19) can be used under the following conditions:

<nA> - c A	>	0		(22)
<nA> + c A	<	n3		(23)
<EA> - c E,A	>	0		(24)

(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 n₃. The maximum number of these collisions can be estimated as

(27)

From the previous expressions for <n_A> and _A one can derive the condition

(28)

The following values are obtained with c = 4:

n3	nB,max		n3	nB,max
16	16		200	29.63
20	17.78		500	31.01
50	24.24		1000	31.50
100	27.59			32.00

5.2 Special sampling for lower part of the spectrum

If the step length is very small (< 5 mm in gases, < 2-3 m in solids) the model gives 0 energy loss for some events. To avoid this, the probability of 0 energy loss is computed

(29)

If the probability is bigger than 0.01 a special sampling is done, taking into account the fact that in these cases the projectile interacts only with the outer electrons of the atom. An energy level E₀ = 10 eV is chosen to correspond to the outer electrons. The mean number of collisions can be calculated from

(30)

The number of collisions n is sampled from Poisson distribution. In the case of the thin layers, all the collisions are considered as ionisations and the energy loss is computed as

(31)

References

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