The theory of the shot effect. II

Abstract

Summary of the previous paperThe previous paper dealt first with the shot effect without secondary emission. We considered a stream of electrons arriving at random, with a fixed probable density N, at the anode of the resistance-capacity coupled first valve of a linear amplifier. This does not allow for the effect whereby the arrival of electrons alters the anode potential, and consequently the probability of arrival of succeeding electrons. The valve must be working with what is conventionally called an infinite internal resistance, and without space-charge effects. Each electron arriving at time tr is supposed to produce at time t an output effect f(t − tr) from the amplifier. We considered the hypotheses, first, that electrons may, with specific probabilities, have lives of any length on the anode system, during which they add − ε/C to the anode potential, and alternatively that each electron adds to the anode potential of the valve whose anode to earth capacity is C and whose feed resistance is R. These hypotheses determine the form of f(t − tr) when the behaviour of the main part of the amplifier is known. Since the amplifier is linear, the effects of various electrons are additive, so that the total output is ∂(t) = ∑rf(t − tr).

Spontaneous fluctuations

“…Of a rather different character, however, are the irregular fluctuations in a flowing electric current which can arise from the atomicity of electricity itself. This type of fluctuation, the shot effect, was recognized in photoelectric and associated studies by von Schweidler (1905Schweidler ( , 1910Schweidler ( , 1913 and Campbell (1909Campbell ( , 1910 and in the field of thermionic emission in a classic paper by Schottky (1918). I n the case of the thermionic valve a certain amount of confusion has arisen, .essentially because the electron emission is there provided from a thermal source.…”

“…* These theorems may be generalized (e.g. Rowland 1936Rowland , 1937 to allow for a possible statistical distribution in the type of event resulting from an individual stimulus. Domb (vide in&) and Rivlin (1945) have also published extensions of the theorems to cover higher order moments (e.g.…”

“…It is now clear (e.g. North et al 1940-1941, Schottky and Spenke 1937 that in the fluctuation problem the reduction effect arises from correlation of the fluctuation in the number of electrons with energies very close to that of the minimum with the fluctuations in electrons of other energy-classes. Similarly, when altering the anode potential, the variation in (the mean) current is provided solely by electrons with energies close to the minimum.…”

The shot effect with space charge

“…Now k is of the order 10 7 . If therefore L is, say, 10~6, the exponential is negligible and the modulus of the ratio of the first term to the second is If we consider only fluctuations | (p 1 -<fi 0 | which are not larger than the average value <f> 0 , it is therefore sufficient to retain the first term in (11), so that…”

The fluctuation theorem. (Shot effect)

“…Actually, of course, (6 -6) 2 is determined, not arithmetically, but from the steady reading of a rectifier with smoothing elements. But this departure from the strict conditions of (1) is again justified by experiments that show that the procedure adopted always leads to the same result as arithmetical calculation.…”