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.

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 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 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, 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

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