132 Chapter 2. Interaction of Radiation with Matter
matter of fact, in real practice, the use of Landau distribution is somewhat limited
due to one or more of the following reasons.
The Landau distribution is valid only if the maximum energy loss in a single
collision is much larger than the typical energy loss. In the actual formalism
of the distribution, it is assumed that the maximum energy transfer can be
infinite.The Landau theory assumes that the typical energy loss is much larger than
the binding energy of the innermost electron such that the electrons may essen-
tially be considered free. This condition is not really fulfilled by most gaseous
detectors, in which the average energy loss can be a fewkeV,whichcanbe
lower than the binding energy of the most tightly bound electrons of the gas
atoms.It assumes that the velocity of the particle is constant, meaning that the de-
crease in particle’s velocity is insignificantly small.If the Landau distribution is integrated, the result is an infinite value.Landau distribution is difficult to handle numerically since the energy loss
computed from it depends on the step size used in the computations.Another distribution describing the energy straggling phenomenon is the so called
Vavilov distribution. With increase in the thickness of the material, the tail of the
Landau distribution becomes smaller and the distribution approaches the Vavilov
distribution. Because of this some authors prefer to call the Vavilov distribution a
more general form of the Landau distribution. However, since one can not approx-
imate the Landau distribution from the Vavilov distribution, we do not encourage
thereadertomakethisassumption.
A major problem with Vavilov distribution is its difficult analytic form requiring
huge numerical computation. Its use is therefore only warranted in situations where
highly accurate results are needed and speed in computations is not an issue.
For general radiation measurements, we encourage the reader to concentrate on
the Landau distribution as it gives acceptable results without computational diffi-
culties.
2.5.D RangeofElectrons........................
As opposed to heavy charged particles, the range of electrons is very difficult to treat
mathematically. The primary reason for the difficulty lies in the higher large-angle
scattering probability of electrons due to their extremely low mass as compared to
the heavy charged particles. However it has been found that the bulk properties of an
electron beam can be characterized by relatively simple relations. The attenuation of
an electron beam, for example has been seen to follow an approximately exponential
curve given by
N=N 0 e−μx. (2.5.26)
HereNrepresents the number of electrons transmitted through a thicknessxof
the material. μis the absorption coefficient of the material for the electrons and
is also a function of the electron energy. For electrons having a continuous energy