Physics and Engineering of Radiation Detection

3.5. Proportional Counters 185

Ec

rc

Ea

Eb

a r b

Electric Field Intensity

Cathode

Anode

Avalanche Region

Figure 3.5.2: Depiction of

avalanche region around the an-

ode wire, the critical radial dis-

tancerc, and critical electric field

intensityEcin a cylindrical pro-

portional counter. In reality the

avalanche region is very close to

the anode wire.

there will not be any avalanche. The integral in relation 3.3.5 can then be evaluated
from the surface of the anode wireatorc. Hence we have

M =exp

∫rc

a

[

DαNmV 0

rln(b/a)

] 1 / 2

dr

=exp

[

2

(

DαNmV 0 a

ln(b/a)

) 1 / 2 (√

rc

a

− 1

)]

. (3.5.4)

The ratiorc/ain the above expression can also be expressed in terms of applied
voltageV 0 and the threshold voltageVt. The threshold voltage is defined as the
voltage applied at the anode below which there will not be any avalanche. Hence it
can be evaluated by substitutingr=aandE=Ecin equation 3.5.1.

Vt=aEcln

(

b

a

)

(3.5.5)