114 Chapter 2. Interaction of Radiation with Matter
convenient to write the above equation in the form
[
−
dE
dx]
Bethe−Bloch=
0. 30548 ρZq^2
Aβ^2[
ln(
Wmax
I)
−β^2]
MeV cm−^1. (2.4.12)One of the difficult parameters to evaluate in the above expression is the ionization
potentialI of the medium. For this a number of empirical formulas have been
proposed, such as
I =12Z+7,Z < 13
I =9. 76 Z+5. 58 Z−^0.^19 ,Z≥ 13 (2.4.13)This equation has been corrected for two factors that become significant at very
high and moderately low energies. One is the shielding of distant electrons because
of the polarization of electrons by the electric field of the moving ion. This effect
depends of the electron density and becomes more and more important as the energy
of incident particle increases. The second correction term applies at lower energies
and depends on the orbital velocities of the electrons. Both of these correction terms
are subtractive and generally represented by the symbolsδandCrespectively.
The modern form of the Bethe-Bloch formula for stopping power after applying
the above corrections is given by
[
−
dE
dx]
Bethe−Bloch=
4 πNAre^2 mec^2 ρZq^2
Aβ^2[
ln(
Wmax
I)
−β^2 −δ
2−
C
Z
]
(2.4.14)
Bethe-Bloch formula can also be written in terms of mass stopping power, which
is simply the stopping power as defined by 2.4.10 or 2.4.14 divided by the density
of the medium.
[
−
1
ρdE
dx]
Bethe−Bloch=
4 πNAr^2 emec^2 Zq^2
Aβ^2[
ln(
Wmax
I)
−β^2 −δ
2−
C
Z
]
(2.4.15)
It should be noted that the above expression for mass stopping power deals with
a medium with unique atomic number and hence is valid for a pure element only.
In case of a compound or a mixture of more than one element, we can use the so
calledBragg-Kleeman ruleto calculate the total mass stopping power.
[
1
ρdE
dx]
total=
∑ni=1[
wi
ρi(
dE
dx)
i]
(2.4.16)
Herewiandρiare the fraction by mass of elementiin the mixture and its density
respectively.
The Bragg-Kleeman rule can also be applied to compute stopping power of a
compound material using
[
dE
dx]
total=
∑ni=1wi(
dE
dx)
i. (2.4.17)
If we substitute the expressions for stopping power and mass stopping power in
the above relations, we can find expressions for average ionization potential,