
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
dünnen Schichten,
Z. Physik 128
(1950) 500.
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