Physics and Engineering of Radiation Detection

2.4. Interaction of Heavy Charged Particles with Matter 113

x

electron

ion

b Figure 2.4.4: Definition of impact param-

eter for a charged particle (such as an ion)

moving in the electric field of an electron.

Later on Bethe and Bloch derived another expression for the stopping power
using quantum mechanics.
[
−

dE

dx

]

Bethe−Bloch

=

4 πNAre^2 mec^2 ρZq^2

Aβ^2

[

ln

(

Wmax

I

)

−β^2

]

(2.4.10)

Here NA=6. 022 × 1023 mole−^1 is the Avogadro’s number;

re=2. 818 × 10 −^15 mis the classical radius of the electron;

me=9. 109 × 10 −^31 kgis the rest mass of the electron;

qis the electrical charge of the ion in units of unit electrical charge;

ρis the density of the medium;

Ais the mass number of the medium;

Iis the ionization potential of the medium;

βis a correction factor. It is generally calculated from the

relation:β=

[

1 −E 0 +EE/A^0 i

] 1 / 2

whereE 0 = 931. 5 MeV

is the rest mass energy per nucleon andEis the energy

of the incident particle having mass numberAi;and

Wmaxis the maximum energy transferred in the encounter.

It can be calculated from:Wmax=2mec^2 β^2 /(1−β^2 ).

The factor 4πNAre^2 mec^2 is constant and therefore can be permanently substituted
in the above formula, which then becomes
[
−

dE

dx

]

Bethe−Bloch

=

4. 8938 × 10 −^18 ρZq^2

Aβ^2

[

ln

(

Wmax

I

)

−β^2

]

Jm−^1. (2.4.11)

Although the units ofJm−^1 are in standard MKS system, however in literature the
stopping power is generally mentioned in units ofMeV cm−^1. Therefore it is more