Physics and Engineering of Radiation Detection

2.4. Interaction of Heavy Charged Particles with Matter 111

Here eis the unit electron charge,

meis the mass of electron,

Neis the electron number density,

qis the charge of the ion,

vis the velocity of the ion,

f(Z) is a function of the atomic numberZof the material, and

γis the relativistic factor given by

(

1 −v^2 /c^2

)− 1 / 2

.

Example:

Derive equation 2.4.8.

Solution:

Let us suppose that a heavy charged particle (such as an ion) is moving in

x-direction in the electric field of an electron and define an impact parameter

bas the perpendicular distance between the two particles (see figure 2.4.4).

The rationale behind this definition is the fact that the impulse experience

by the electron as the ion approaches it will be canceled by the impulse

delivered by the receding ion and consequently the only contribution left will

be perpendicular to the motion of the ion.

Supposing that the ion passes by the electron before it can move any sig-

nificant distance, we can calculate the momentum transferred to the electron

through the impulse delivered by the ion as

∆p=

∫∞

−∞

(

−eE ̄⊥

)

,

whereE ̄⊥is the perpendicular component of the electric field intensity given

relativistically by

E ̄⊥= qeγb

(b^2 +γ^2 v^2 t^2 )^3 /^2

.

Hereγ=1/sqrt

(

1 −v^2 /c^2

)

is the relativistic factor for the ion moving with

velocityv.

Integration of the above equation yields

∆p=

2 qe^2

bv

.

In terms of energy transferred to the electron ∆E=∆p^2 / 2 me,thiscanbe

written as

∆E=

2 q^2 e^4

mev^2 b^2

(2.4.9)