Re-enactment of Rutherford's and Geiger's counting of alpha-particles

Introduction
and
Background

While the average number of particles from a steady source is nearly constant, when a large number is counted, the number appearing in a given short interval is subject to wide fluctuations. These variations are especially noticeable when only a few scintillations appear per minute. For example, during a considerable interval it may happen that no particle appears; then follows a group of particles in rapid succession; then an occasional a particle, and so on.
Important to settle whether these variations in distribution are in agreement with the laws of probability. i.e., whether the distribution of a particles on an average is that to be anticipated if the particles are expelled at random both in regard to space and time. It might be conceived, for example, that the emission of a particle might precipitate the disintegration of neighbouring atoms.

Methods and
Experimental
Arrangement

Because it would be to long and tedious to do so using the electric method, it was simpler, if not quite so accurate, to count the particles by the scintillation method. Source small disk coated with polonium, placed inside an exhausted tube, closed at one end by a zinc sulphide screen. Scintillations counted in the usual way by means of a microscope on an area of about one sq. mm. of screen During the 5 days, to correct for the decay, polonium was moved daily closer to the screen in order that the average number of alpha particles impinging on the screen should be nearly constant. Scintillations recorded on a chronograph tape by closing an electric circuit by hand at the instant of each scintillation. Time-marks at intervals of one half-minute were also automatically recorded on the same tape.

The (theoretical) distribution of alpha particles according to the law of probability was kindly worked out for us by Mr. Bateman. The mathematical theory is appended as a note to this paper. Mr. Bateman has shown that if (in modern notation) μ be the true average number of particles for any given interval falling on the screen from a constant source, the probability that y alpha particles are observed in the same interval is given by (1/y!) × (μ to the power y) × exp(- μ). y is here a whole number, which may have all positive values from 0 to ∞. The value of μ is determined by counting a large number of scintillations and dividing by the number of intervals involved. The probability for y alpha particles in the given interval can then at once be calculated from the theory.

Results

The following table contains the results of an examination of the groups of alpha particles occurring in 1/8 minute interval. For convenience the tape was measured up in four parts, the results of which are given separately in horizontal columns I. to IV. For example, combining the four horizontal columns, it is seen that out of 2608 intervals containing 10,097 particles, the number of times that 3 alpha particles were observed was 525. The number calculated from the equation was the same, viz. 525.

On the whole, theory and experiment are in excellent accord. The relation between theory and experiment is shown in fig. 1 for the results given in Table I., where the o represent observed points and the broken line the theoretical curve.
The results also been analysed for 1/4 minute intervals, in two ways, which give two different sets of numbers. For example, let A, B, C, D, E represent the number of particles observed in successive 1/8 minute intervals. One set of results, given in Table A, is obtained by adding A+B, C+D, &c. ; the other ...

Bottom line

We may consequently conclude that the distribution of particles in time is in agreement with the laws of probability and that the alpha particles are emitted at random. As far as the experiments have gone, there is no evidence that the variation in number of a particles from interval to interval is greater than would be expected in a random distribution.

Apart from their bearing on radioactive problems, these results are of interest as an example of a method of testing the laws of probability by observing the variations in quantities involved in a spontaneous material process.

Appendix

Let λdt be the chance that an alpha particle hits the screen in a small interval of time dt [...] Now let W_y(t) denote the chance that y alpha particles hit the screen in an interval of time t.   [ ... ]  

{We surmise that Bateman used the letter W for probability because Schweidler - and Abbe earlier -- writing in German, used it as short for wahrscheinlichkeit, the German word for probability.}

Hence

      W_y+1(t+dt) = (1 - λdt)W_y+1(t) + λdtW_y(t).

Proceeding to the limit, we have

      dW_y+1/dt = λ(W_y - W_y+1).

[...]

Putting y = 0, 1, 2 .. in succession we have a system of differential equations [...]. The equations may be solved by multiplying each of them by exp(λt) and integrating. Since W₀(0) = 1, W_y(0), we have in succession :

      W₀ = exp(-λt),

      W₁ = (λt) × exp(-λt),

      W₂ = (1/2!) × [ λt to power 2] × exp(-λt),

and so on. Finally, we get

      W_y = (1/y!) × [ λt to power y] × exp(-λt),

The average number of alpha particles which strike the screen in the interval t is λt. Putting this equal to μ, we see that the chance that y particles strike the screen in this interval is

      W_y = (1/y!) × [ μ to power y] × exp(-μ).     

Snow

Goodness of Fit of Rutherford's 1-parameter, and some multi-parameter (Pearson) 'ideal frequency curves'.

Marsden&Barratt 1

Marsden&Barratt 2

The Probability Distribution of the Time Intervals of alpha Particles with Application to the Number of alpha Particles emitted by 1. Uranium: 2. Thorium and Actinium.
''It would, therefore, seem preferable in many respects to test the application of the probability laws to actual time intervals between successive alpha particles'''.

(Following 7-sec. lead-in) ten 1-minute segments,
in 8 colour-coded divisions of 1/8 min. each

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