Physics and Engineering of Radiation Detection

2.5. Interaction of Electrons with Matter 131

0 5 10 15 20 25

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Figure 2.5.4: Typical shape of

the Landau distribution. The

most probable value for this dis-

tribution, corresponding to its

peak, is 5. Axes’ units are arbi-

trary.

whereNAis the Avogadro’s constant,Zis the atomic number of the target atom,
Ais its atomic mass, andxis the thickness of the material given in the second
expression inmg/cm^2.
Since Landau distribution is an asymmetric distribution therefore its most prob-
able value is different from its average value. The most probably value is simply the
value at which the distribution has a maximum. The average value, on the other
hand, is much more complicated to determine. The reason is that in order to deter-
mine the average value one generally has to cut the tail of the distribution at some
point, which may depend on some scheme or the personal bias. Since the average
value is rarely used in practice and does not even have much physical significance,
therefore we will concentrate here the most probable energy loss instead. The most
probable energy loss, as obtained from the Landau distribution, can be computed
from

(^) mp=ξ

[

ln

(

ξ

E

)

+0. 423

]

, (2.5.24)

which can also be written as

(^) mp=ξ

[

ln

(

2 mc^2 β^2 ξ

I^2 (1−β^2 )

)

−β^2 +0. 423

]

. (2.5.25)

Landau distribution is a skewed distribution with a long tail at the high energy
side (see Fig.2.5.4). The degree of its skewness increases with the decrease in the
thickness of the material. For very thin absorbers, the distribution no longer depicts
reality even for electrons and should be replaced by some other distribution. As a