4 Gaseous Detectors
113
where q s is the charge per cm. Therefore,
A = exp
q s α(E) dE/
2πε 0 E
2
.
(4.47)
Two approximations in particular have been used to describe practical cases.
The early Korff model [43] uses the parameterization
α/p = A exp (−Bp/E) ,
(4.48)
with empirical constants A and B depending on the gas.
In the Diethorn approximation [44], α is assumed to be proportional to E. One
then obtains for a proportional tube with wire radius a and tube radius b
ln A = (ln 2/ ln (b/a)) (V /ΔV ) ln
V / (ln (b/a) aE min ) ,
(4.49)
where the two parameters E mm and V are obtained from measurements ofA
at various voltages and gas pressures. E mm is the minimum E field to start the
avalanche and eV the average energy required to produce one more electron. E mm
is defined for a density p 0 at STP. For another density E mm (p) = E mm (p 0 )(p/p 0 ). A
list for E mm and V for various gases is given, e.g., in [19]. Reasonable agreement
with the experimental data is obtained; discrepancies show up at high A.
4.2.3.3 Dependence of Amplification on Various Factors
The gas amplification depends on many operational and geometrical parameters.
Some examples are:
Gas Density
The Diethom approximation gives
dA/A = − (ln 2/ ln (b/a)) (V /Δ∇) (dp/p) →= (5-8) dp/p
(4.50)
typically.
Geometrical Imperfections
The effects will obviously depend on the geometry and the operation details. An
early publication [45] gives analytic estimates of the effects of wire displacements
and variations in wire diameter. In a typical geometry dA/A~2.5 dr/r, where r is the
wire radius; dA/A~9gap/gap.
113
where q s is the charge per cm. Therefore,
A = exp
q s α(E) dE/
2πε 0 E
2
.
(4.47)
Two approximations in particular have been used to describe practical cases.
The early Korff model [43] uses the parameterization
α/p = A exp (−Bp/E) ,
(4.48)
with empirical constants A and B depending on the gas.
In the Diethorn approximation [44], α is assumed to be proportional to E. One
then obtains for a proportional tube with wire radius a and tube radius b
ln A = (ln 2/ ln (b/a)) (V /ΔV ) ln
V / (ln (b/a) aE min ) ,
(4.49)
where the two parameters E mm and V are obtained from measurements ofA
at various voltages and gas pressures. E mm is the minimum E field to start the
avalanche and eV the average energy required to produce one more electron. E mm
is defined for a density p 0 at STP. For another density E mm (p) = E mm (p 0 )(p/p 0 ). A
list for E mm and V for various gases is given, e.g., in [19]. Reasonable agreement
with the experimental data is obtained; discrepancies show up at high A.
4.2.3.3 Dependence of Amplification on Various Factors
The gas amplification depends on many operational and geometrical parameters.
Some examples are:
Gas Density
The Diethom approximation gives
dA/A = − (ln 2/ ln (b/a)) (V /Δ∇) (dp/p) →= (5-8) dp/p
(4.50)
typically.
Geometrical Imperfections
The effects will obviously depend on the geometry and the operation details. An
early publication [45] gives analytic estimates of the effects of wire displacements
and variations in wire diameter. In a typical geometry dA/A~2.5 dr/r, where r is the
wire radius; dA/A~9gap/gap.