186 Chapter 3. Gas Filled Detectors
Also, if we substituter=rc in equation 3.5.1, we will get an expression for the
critical field intensityEc,thatis
Ec=V 0
rcln(b/a). (3.5.6)
By combining the above two relations we get
rc
a=
V 0
Vt. (3.5.7)
The good thing about this expression is that the fraction on the right hand side
can be easily determined sinceVtis simply the voltage at which the avalanche
multiplication begins. In other wordsVtcorresponds to the onset of proportional
region. Substituting this ratio in equation 3.5.4 gives
M=exp[
2
(
DαNmV 0 a
ln(b/a)) 1 / 2 (√
V 0
Vt− 1
)]
. (3.5.8)
This expression has been shown to be in good agreement with experimental results
up to the moderate values ofM(on the order of 10^4 ). At very high electric fields,
the initial approximation forαwe used in this derivation breaks down and therefore
can not be used. However general proportional counters are operated such that the
multiplication factor falls within the applicability range of this expression.
The above expression can also be written in terms of capacitance per unit length
C=
2 π 0
ln(b/a).
Hence it can be shown that
M=exp[
2
(
DαNmCV 0 a
2 π 0) 1 / 2 (√
V 0
Vt− 1
)]
. (3.5.9)
Either equation 3.5.8 or 3.5.9 can be used to determine the multiplication factor
for a cylindrical chamber at a certain voltage. As stated earlier, the values ob-
tained from these expressions are good up to the usual range of applied voltages for
proportional counters (37).
Example:
Plot the dependence of multiplication factor for a cylindrical proportional
counter filled with argon under standard temperature and pressure on the
applied voltage of up to 1000 V. The active volume of the counter has
adiameterof6cmand the anode wire has a radius of 10μm.Takethe
threshold voltage to be 500V. The weight density of argon under standard
conditions is 1. 784 kg/m^3.Solution:
The given parameters area=10−^3 cm,b=3cm,andVt= 500V.