184 Chapter 3. Gas Filled Detectors
whereMis the multiplication factor, which for typical chambers lies between 10^3 -
104. Since the output signal is proportional to the total number of charges, therefore
it is evident that such a chamber can amplify the signal considerably.
A detrimental effect of increasing the high voltage in proportional counters is
the build up of space charge around the anode wire due to slower motion of heavy
positive ions. The higher the voltage, the more ions are produced, which move
much slower than electrons. The result is the screening of the electrodes and the
consequent decrement in the effective electric field intensity inside the active volume.
When this happens, the proportionality between the deposited energy and the pulse
height can no longer be guaranteed. The proportional counters are therefore always
operated below the onset of this region oflimited proportionality.
3.5.A MultiplicationFactor.......................
Determination of the multiplication factor is central to the operation of a propor-
tional counter. We saw earlier in this chapter that for a uniform field the multipli-
cation factor can be obtained from the relation 3.3.4
M=eαx,whereαis the first Townsend coefficient. In a non-uniform field in which the
Townsend coefficient has a spatial dependence, the relation 3.3.5, that is
M=exp[∫
α(x)dx]
,
should be used instead. To evaluate this integral we need the spatial profile (more
specifically radial profile for a cylindrical geometry) ofα. We start with the simple
relation 3.3.8 betweenαand the average energy gained by the electron between
collisionsξ
α=DαNmξ.
Since the electron is drifting in the influence of the electric field intensityE, therefore
the energy it gains while traversing the mean free pathλ=1/αcan be written as
ξ = Eλ=E
α. (3.5.2)
Substituting this in the above expression forαgives
α=(DαNmE)^1 /^2. (3.5.3)Now we are ready to evaluate the multiplication factor using the relation 3.3.5. But
before we do that we should first decide on the limits to the integral in that relation.
We know that the initiation of the avalanche depends on the electric field strength.
Therefore there must be a critical value of the field below which the avalanche will
not occur. Let us represent this critical electric field intensity byEcand the radial
distance from the center of the cylinder at which the field has this strength byrc.
What we have done here is to essentially defined a volume around the anode wire
inside which the avalanche will take place (see Fig.3.5.2). Outside of this volume