114
H. J. Hilke and W. Riegler
Edge Effects
Near edges, the electric field is reduced over distances similar to the gap between
the electrode planes. It can be recovered largely by additional field shaping lines on
the edges [46].
Space Charge
Due to the low velocity of the positive ions (falling off as 1/r from >1mm/μs at
r = a), space charge will build up at high particle fluxes and lower the avalanche
amplification. In drift tubes the voltage drop due to the space charge from a given
particle flux is proportional to the third power of the tube radius. A smaller radius
thus improves the rate capability drastically.
4.2.3.4 Statistical Fluctuations of the Amplification
In the proportional regime, the amplification A is simply defined by A = n/n T and
one assumes that each of the n T initial electrons produces on average the same A
ion pairs. We define P(n) as the probability to produce n electrons in the individual
avalanche with mean A and variance σ 2 If n T > > 1 and if all avalanches develop
independently, it follows from the central limit theorem that the distribution function
F(n) for the sum of the n T avalanches approaches a Gaussian with mean n = n T A
and variance S 2 = n T σ 2 , independent of the actual P(n).
On the other hand, for detection of single or a few electrons, knowledge of the
individual P(n) is required.
For a parallel plate geometry calculations [47] agree well with measurements
[48]. The distributions found theoretically [49] and experimentally [50] for the
strong inhomogeneous field around a thin wire also look similar and approach Polya
distributions (Fig. 4.14).
For these distributions
(σ A / < A >)
2
= f, with f ≤ 1.
(4.51)
The limiting case f = 1 is an exponential distribution (Yule-Furry law)
P (A) = (1/ < A >) exp (−A/ < A >) .
(4.52)
Experimental results point to f = 0.6 − 1.0. Measurements with laser tracks
[19], indicate that the r.m.s. width σ A of a single-electron avalanche is close to the
mean, as it is for the Polya distribution with f = l. An approximately exponential
distribution for single-electron avalanches is also reported in [28, 48].
H. J. Hilke and W. Riegler
Edge Effects
Near edges, the electric field is reduced over distances similar to the gap between
the electrode planes. It can be recovered largely by additional field shaping lines on
the edges [46].
Space Charge
Due to the low velocity of the positive ions (falling off as 1/r from >1mm/μs at
r = a), space charge will build up at high particle fluxes and lower the avalanche
amplification. In drift tubes the voltage drop due to the space charge from a given
particle flux is proportional to the third power of the tube radius. A smaller radius
thus improves the rate capability drastically.
4.2.3.4 Statistical Fluctuations of the Amplification
In the proportional regime, the amplification A is simply defined by A = n/n T and
one assumes that each of the n T initial electrons produces on average the same A
ion pairs. We define P(n) as the probability to produce n electrons in the individual
avalanche with mean A and variance σ 2 If n T > > 1 and if all avalanches develop
independently, it follows from the central limit theorem that the distribution function
F(n) for the sum of the n T avalanches approaches a Gaussian with mean n = n T A
and variance S 2 = n T σ 2 , independent of the actual P(n).
On the other hand, for detection of single or a few electrons, knowledge of the
individual P(n) is required.
For a parallel plate geometry calculations [47] agree well with measurements
[48]. The distributions found theoretically [49] and experimentally [50] for the
strong inhomogeneous field around a thin wire also look similar and approach Polya
distributions (Fig. 4.14).
For these distributions
(σ A / < A >)
2
= f, with f ≤ 1.
(4.51)
The limiting case f = 1 is an exponential distribution (Yule-Furry law)
P (A) = (1/ < A >) exp (−A/ < A >) .
(4.52)
Experimental results point to f = 0.6 − 1.0. Measurements with laser tracks
[19], indicate that the r.m.s. width σ A of a single-electron avalanche is close to the
mean, as it is for the Polya distribution with f = l. An approximately exponential
distribution for single-electron avalanches is also reported in [28, 48].