It is well known that the Bethe-Bloch 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. In fact, if a particle does not stop in the sensor, the response varies around the distribution's peak, with a high probability of high signals.

Because of this tail, the average value is greater than the distribution's most probable value. The fluctuation around the maximum of this distribution becomes greater as the sensor becomes thinner. When designing the dynamic range of the readout circuit for these devices, this must be taken into account.

The Landau fluctuation is caused primarily by the rare but measurable occurrence of knock-on electrons, which gain enough energy from the interaction to become ionizing particles themselves. Because the direction of the knock-on electron is typically perpendicular to the direction of the incoming particle, it causes irregular charge clouds and degrades spatial resolution.

Fig. 1 Landau Distribution

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 Landau-distributed 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 right-hand 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:

*1 - The maximum energy
transfer is infinite*

*2 - 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*

*3 - The particle velocity
remains approx the same>*

*4 - Taking only the
first term of Bethe-Block 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

**in um.**

*t*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, K-shell 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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