Physics and Engineering of Radiation Detection

2.5. Interaction of Electrons with Matter 125

Solution

The cut-off wavelength is given by

λmin =

hc

eV

=

(6. 634 × 10 −^34 )(2. 99 × 108 )

(1. 602 × 10 −^19 )(40× 103 )

=3. 095 × 10 −^11 m=30. 95 fm.

The energy corresponding to this wavelength is given by

Emax=

hv

λmin

=

(

6. 634 × 10 −^34

)(

2. 99 × 108

)

3. 095 × 10 −^11

=6. 4 × 10 −^15 J

=40keV. (2.5.4)

A.6 CherenkovRadiation......................

The concept of Cherenkov radiation has already been discussed earlier in the chapter.
We saw that an electron, being a very light particle, can emit Cherenkov radiation
when accelerated to high energies in a medium. Its energy should be so high that
its velocity becomes higher than the velocity of light in that medium.
The velocity of light in a medium of refractive indexnis given byc/n,wherecis
the velocity of light in vacuum. For Cherenkov radiation to be emitted, the velocity
of the charged particle traversing the medium must be greater than this velocity,
such as
vth≥

c

n

. (2.5.5)

Similarly the threshold energy is given by

Eth=γthm 0 c^2 , (2.5.6)

wherem 0 is the electron rest mass,cis the velocity of light in vacuum, andγthis
the relativistic factor defined as

γth =

[

1 −

vth^2

c^2

]− 1 / 2

=

[

1 −βth^2

]− 1 / 2

. (2.5.7)

(2.5.8)

Here the termβth=vth/cis an oft quoted condition for Cherenkov emission. From
the above equations we can also deduce that

γth=

n

√

n^2 − 1

(2.5.9)