This paper is concerned with the theory of the probability distribution of the total number of electrons in the avalanche produced by the release of a single electron in the gas of a proportional counter. The disagreement between existing theory and experimental results is discussed and a new theory is proposed, based on the fact that fluctuations in the number of electrons at a given point in the avalanche are accompanied by fluctuations in the average electron energy. This aspect of the problem is incorporated directly into a simple one-dimensional model of the multiplication process, and the resulting distribution function has a mathematical foim in agreement with that observed experimentally.
The fluctuation in the number of electrons predicted by this theory is not constant, but is determined by a parameter which, for large mean values, is essentially the mean fraction of electrons in the avalanche having energies above the ionization energy of the counter gas. Limits on the variation of this parameter are obtained by calculations of the mean values using a particular two-dimensional model, in which electrons are divided into two classes according as they have energies above, or below, the ionization energy. The experimentally observed fluctuation lies within the predicted range and close to the lower limit; it is concluded that there is little scope for improvement in the resolution to be obtained from the conventional type of proportional counter.