Physics and Engineering of Radiation Detection

164 Chapter 3. Gas Filled Detectors

The above equation is true only for a uniform electric field. In an non-uniform
field, the Townsend coefficient becomes a function ofx. In that case the multipli-
cation factor for an electron that drifts from pointr 1 tor 2 can be calculated from

M=exp

[∫r 2

r 1

α(x)dx

]

. (3.3.5)

Hence if we want to compute the multiplication factor, we must know the spatial
profile of the first Townsend coefficient. Although it is quite challenging to determine
this profile analytically, it has been shown that the reduced Townsend coefficient has
a dependence on the reduced electric field intensity, that is

α

P

=f

(

E

P

)

, (3.3.6)

whereEis the electric field intensity andP is the gas pressure. Several authors
have reported different forms of the first Townsend coefficient but a commonly used
expression is the one originally proposed by Korff (19). It is given by

α

P

=Aexp

(

−

BP

E

)

, (3.3.7)

where the parametersAandBdepend on the gas and the electric field intensity.
These parameters have been experimentally determined for a number of gases (see
Table.3.3.1).
Another simple expression forαthat has been reported in literature is based on
the intuition that sinceαis inversely related to the mean free path of electrons in
a gas it should therefore be directly related to the molecular densityNmof the gas
and the energyξof the electrons. This argument leads to the expression

α=DαNmξ (3.3.8)

where the proportionality constantDαhas been experimentally determined for sev-
eral gases (see Table.3.3.1).

Table 3.3.1: Experimentally determined values of parameters appearing in equations
3.3.7 and 3.3.8 (37).

Gas A(cm−^1 Torr−^1 ) B(Vcm−^1 Torr) Dα(× 10 −^17 cm^2 V−^1 )

He 3 34 0.11

Ne 4 100 0.14

Ar 14 180 1.81

An interesting aspect of avalanche is its geometric progression, which assumes
the shape of a liquid drop because of the large difference between the drift velocities