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Expressions are derived for the probability P₍, ₌ that a pulse initiated by n electrons (or holes) in a uniformly multiplying semiconductor diode will result in a total number of electrons (or holes) m, to give a gain m/n, and for the probability Q₍, ₌ that the gain will be m/n or greater. It is shown that the distributions are far from Gaussian. The gain distribution P₁, ₌ for a single photoelectron, for example, is shown to have a maximum value for m = 1 for any value of the average gain M=m/n. The derivations are valid for any electric field distribution and assume only that the hole ionization coefficient (E) can be approximated by the relation (E) =k (E), where (E) is the electron ionization coefficient and k is a constant. A method of determining an effective value of k, for cases where =k is not a good approximation, is presented. The results can be used to calculate the average gain and the mean square deviation from the average, giving results in agreement with previously published relations 1, 2. The implications of this theory on the use of avalanche diodes for low-level photodetection are discussed. It is shown that in the near infrared, cooled avalanche photodiodes can compare favorably with the best available photomultiplier when used either in a photon-counting mode, or for the reliable detection of low-level laser pulses.
R.J. McIntyre (Thu,) studied this question.