128 Chapter 2. Interaction of Radiation with Matter
The collisional component not only includes the inelastic impact ionization process
but also the other scattering mechanisms we discussed earlier, such as Moeller and
Bhabha scattering. The analytic forms of the collisional and radiative components
of the total stopping power for electrons are given by
[
−
dE
dx]
collisional=
2 πZe^4 ρ
mev^2[
ln(
mev^2 E
2 I^2 (1−β^2 ))
−
ln 2(
2
√
1 −β^2 −1+β^2)
+
(
1 −β^2)
+
1
8
(
1 −
√
1 −β^2) 2 ]
,
(2.5.15)
and [
−dE
dx]
radiative=
Z(Z+1)e^4 ρE
137 m^2 ec^4[
4ln(
2 E
mec^2)
−
4
3
]
. (2.5.16)
From these two equations we can deduce that the rate of energy loss of an electron
through the collisional and radiative processes can be approximately expressed as
Scollisional ∝ ln(E)and
Sradiative ∝ E.This implies that the losses due to radiative effects such as Bremsstrahlung increase
more rapidly than the losses due to collisional effects such as ionization. This can
also be seen from Fig.2.5.3, where the two effects have been plotted for electrons
traveling through a slab of copper. The energy at which these two types of losses
become equal is called thecritical energy. A number of attempts have been made
to develop a simple relation for this critical energy, the most notable of which is the
one that uses the approximate ratio of equations 2.5.15 and 2.5.16 given by
Sradiative
Scollisional≈
(Z+1.2)E
800
, (2.5.17)
whereE is inMeV. From this equation we can find the critical energyEcby
equating the two types of loss rates. Hence
Ec≈800
Z+1. 2
MeV. (2.5.18)This definition was originally given by Berger and Seltzer (6). Although this is a
widely used and quoted definition of the critical energy but there are other definitions
as well that work equally well for most materials. For example, Rossi (48) has
defined critical energy as the point where the ionization loss per unit radiation length
becomes equal to the electron energy. This definition is also graphically depicted in
Fig.2.5.3.
Equation 2.5.18 does not give accurate results for all types of matter because it
does not take into account the state of the matter, that is, it gives same results
whether the matter is in solid, liquid, or gaseous state. A much better approach is
to parameterize the curve
Ec≈
a
Z+bMeV, (2.5.19)