.
The Landau distribution for ionizing particles
It is well known that the BetheBloch formula describes the average energy loss of charged particles when travelling through matter, while the fluctuations of energy loss by ionization of a charged particle in a thin layer of matter was theoretically described by Landau in 1944 [1]. This description ends with a universal asymmetric probability density function.
Fig. 1 Landau
Distribution
It
resembles a Gaussian distribution with a
long upper tail, resulting from the small
number of individual collisions, each
with a small probability of transferring
comparatively large amounts of energy. This
energy is deposited by a subsequent cascade.
Its limit is the long tail, which
theoretically extends to infinite energies,
while the energy deposited by an incoming
particle cannot exceed its own energy.
The
convolution of two Landau distributions
results in another Landau distribution. This
property can be illustrated by the energy
loss of a particle traversing a layer of
thickness D or two subsequent layers of
thickness D/2, respectively. The overall
energy loss must be the same in both cases,
implying the convolution property mentioned
above.
The Landau
distribution has a finite area,
however, it is impossible to state a mean
value or moments of higher order. One
possible workaround is to cut the Landau
tail, which implies the loss of the
convolution property.
Protons, pions and other types of charged
particles, which are in most cases close to
MIPs, all
produce approximately Landaudistributed
spectra when traversing the matter. Several
approximations exist, the simplest way is to
use the Gaussian function, if the intention
is to fit at the most probable value
(peak) only.
The first and second momenta Φ (1,x) and Φ (2, x) of the density function truncated on the righthand tail can be defined through the general formula
It
is possible to use of the Landau function
for all the situation where k < 0.01 (where
k is the ratio of the mean energy loss and
the Maximum Transferable Energy) with the
assumptions that:
 The maximum
energy transfer is infinite
 The electron
binding energy in a collision is
negligible, in other words the electrons
involved in collisions are treated as
free and the distant collision is
ignored
 The particle
velocity remains approx the same>
 Taking only the first term of BetheBlock formula the mean energy loss is approximated to:
The Landau distribution is then given by:
r is an arbitrary real constant and the variable l is:
where C_{E} is the Euler constant. The function φ(λ) is a universal function that must be evaluatednumerically. A tabulation for various λ can be found on some article, (6) moreover a computer program has been developed and an implementation can be found in the Cern ROOT package. In this page is reported a translation of that function in Matlab code. The φ(λ) has a maximum for a λ of 0.229 and a full width at half maximum: W_{L}=4.02ξ. The energy loss corresponding to the maximum of the function f_{L}(x,Δ) is called the most probable energy loss ( Δ_{p}).
Earlier values for the constant 0.2000 were
0.37 (Landau, 1944) and 0.198 [Maccabee and
Papworth (1969), quoted by Sternheimer and
Peierls (1971); see, also, Ahlen (1980)].
The equation includes the density effect,
which had not been used by Landau.
For γ >> 100,
we get Δ_{p}=
ξ (
12.325 + ln(ξ/I)
)
If
we enter the I value, we obtain Δ_{p}(keV)
= t (0.1791 + 0.01782�lnt ) with t in
um.
For
small thickness the Landau fails to fit the
experimental energy loss distribution [2].
In fact for small thickness the Landau shows
a lower position of the peak with respect to
the energy loss distribution measured (Fig.
2(a)).
Fig. 2 (a) Energy Loss by
12 GeV Protons in 5.6 um of silicon with in
red the Landau contribution. (b) Landau
distribution for different values of β
It
can be understood qualitatively: for very
thin absorbers, Kshell electrons do not
contribute to the energy loss. Thus, the
effective thickness of the absorber can be
considered to be t_{e}=12/14 t. For
t = 10 um, t_{e}=8.57 um.
For
the Landau function, the full width at half
maximum, w depends on the absorber
thickness and is independent of particle
type and speed.
References

L. Landau, On the
Energy Loss of Fast Particles by
Ionization, J. Phys.
USSR 8 (1944) 201.
 Blunck and S. Leisegang, Zum Energieverlust schneller Elektronen in d�nnen Schichten, Z. Physik 128 (1950) 500.
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