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The Shot Effect for Showers

  • J. M. Whittaker (a1)

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1. A well-known theorem on probability may be stated as follows. Let λdt be the chance that a certain event will happen in time dt, so that λT is the average number of occurrences of the event in an interval T. If we take a large number of repetitions of the interval T the actual numbers of occurrences will fluctuate about the mean λT, being say λT + x′, λT + x″,…. Then the theorem states that

The ejection of an α-particle by a radioactive substance is an event of the type contemplated and the theorem was applied to this physical problem by von Schweidler in 1905. Much similar work, both theoretical and experimental, has been done at intervals since then*.

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* For the theory see especially Campbell, N. R., Proc. Camb. Phil. Soc. 15 (1909), 117 and 310; Bateman, H., Phil. Mag. 20 (1910), 704. For recent experimental work and numerous references see Moullin, E. B., Proc. Roy. Soc. 147 (1934), 107.

* Cf. Ince, E. L., Ordinary differential equations (Longmans, 1927), p. 408.

* The discussion of Campbell's theorem by Rowland, E. N. (Proc. Camb. Phil. Soc. 32 (1936), 580, §§ 1–6) seems to me mistaken. If I understand him rightly, Rowland seeks to prove the theorem on the hypothesis that it is most probable that, in the mean for the set, there are Ndr, events in an interval (Tv, Tv + dTv), and he calculates and for this distribution, ignoring the fluctuations altogether. Now the existence of the shot effect is entirely due to the fact that there are fluctuations, and that the mean square fluctuation is not negligible. Rowland's method should therefore lead to the conclusion that there is no shot effect at all, i.e. that = 0. I think that his method will in fact give this result, if correctly worked out. In his equation (22) he separates a sum Σr, s into a sum , (with r = s omitted) and a sum Σr (over r = s), and the latter terms give the shot effect. He gives no reason for using (i.e. for omitting the terms r = s when forming the mean value), and if, as I believe, it should be replaced by Σr, s, the extra terms which give the shot effect disappear.

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