3.3. Regions of Operation of Gas Filled Detectors 167
To understand the breakdown quantitatively, let us write the equation for the
multiplication factorMunder steady state condition of discharge. It can be shown
that when the discharge becomes independent of the ionization in the gas, the equa-
tion 3.3.4 should be replaced by (27)
M=
eαx
1 −γ(eαx−1), (3.3.9)
whereαandγ are the first and second Townsend coefficients respectively. The
singularity in the above equation represents the breakdown (the value at which the
current becomes infinite, at least theoretically), which occurs if the bias voltage is
increased to very high values. The mathematical condition to start and sustain
breakdown can therefore be written as
1 −γ(eαx−1) = 0⇒γ =1
eαx− 1. (3.3.10)
Hereγis thecriticalvalue of the coefficient at which the breakdown starts. Fur-
thermore as long as this condition remains fulfilled, the breakdown is sustained.
Note that this value depends not only the first Townsend coefficient but also on
the positionx. As stated earlier, in most radiation detectors under normal oper-
ating conditions the second Townsend coefficient remains below 0.1, that is, the
probability that a breakdown will occur is less than 10%.
Since the first Townsend coefficient in the above expression depends on the electric
field intensity and gas pressure therefore we will see if we can derive an expression
for the breakdown voltage. For this we first write the above relation as
αx=ln(
1+
1
γ)
.
Substitution of equation 3.3.7 in this expression gives
(AP x)exp(
−
BP
E
)
=ln(
1+
1
γ)
⇒E =
BP
ln [AP x/ln (1 + 1/γ)]. (3.3.11)
Let us now suppose that we have a parallel plate geometry in which the electrodes
are separated by a distancex=d. To write the above equation in terms of voltage,
we note that for such a geometryE=V/d. Hence the breakdown voltageVbreakis
given by
Vbreak=BPd
ln [AP d/ln (1 + 1/γ)]. (3.3.12)
This useful relation tells us that for a given gas the voltage at which the breakdown
occurs depends on the product of pressure and electrode separation (that is,Pd). It
is generally known asPaschen’s lawand a curve drawn from this equation between
PdandVbreakis referred to asPaschen curve. Fig.3.3.3 shows such a curve for
helium with arbitrarily chosenγ=0.1. The point of minima in such a curve is