Physics and Engineering of Radiation Detection

2.5. Interaction of Electrons with Matter 133

spectrum, it depends on their endpoint energy. Following the analogy of attenuation
of photons in matter, here also we can define a path length or absorber thickness as

t=

1

μ

, (2.5.27)

the interpretation of which can be understood by substitutingx=tin the exponen-
tial relation above.

N = N 0 e−^1

⇒No. of electrons absorbed = N 0 −N=N 0 (1−e−^1 )

=0. 63 N 0

This implies thattis the thickness of the material which absorbs about 63% of
the electrons of a certain energy. In Fig.2.5.5 we have plottedNversusxon both
linear and semilogarithmic scales. Since the behavior of electrons is not perfectly
logarithmic, therefore if one performs an experiment to measure the variation of
electron intensity with respect to the thickness of the material, a perfect straight
line on the semilogarithmic scale is not obtained. Such a curve is known asabsorption
curve. Absorption curves for specific materials are routinely obtained to determine
the range of electrons in the material.
An experimental setup to obtain the absorption curve for a material simply con-
sists of slabs of the material of varying thickness, a known source of electrons, and
a radiation detector. The electrons from the source are allowed to pass through
the material. The number of electrons transmitted through the material are then
counted through the detector. This process is repeated for various thicknesses of
the material. A plot of thickness versus log of number of counts gives the required
absorption curve. The obtained curve looks similar to the second curve shown in
Fig.2.5.5. However, since the simple exponential attenuation of electrons does not
strictly hold for most of the materials, therefore the variation is not as linear as
shown in the figure. The curve is actually seen to bend down at larger thicknesses.
The determination of range from a perfectly linear variation is very simple as it can
be done by extrapolating the line to the background level (see Fig.2.5.6). However
if the curve shows a curvature with increasing thickness then the end point is gen-
erally taken as the range. Sometimes the experiment is not performed till the end,
such as, till the detector stops seeing any electrons. In such a case the curve can
be extrapolated as shown in Fig.2.5.6. It is interesting to note that the range as
obtained from areal curveactually corresponds to the endpoint energy of electrons
from a radioactive source.
Although the best way to determine the range of electrons in a material is to
perform an experiment as described above, however it may not be always practical
to do so. Fortunately enough, the range of electrons in any material can be fairly
accurately determined from the following simple formulae given by Katz and Penfold
(29).

Rspe[kgm−^2 ]=

⎧

⎪⎨

⎪⎩

4. 12 E^1.^265 −^0 .0954 ln(E) for 10keV < E≤ 2. 5 MeV

5. 30 E− 1. 06 forE> 2. 5 MeV

(2.5.28)