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On the theory of successive radioactive transformations

Published online by Cambridge University Press:  24 October 2008

W. F. Sedgwick
W. F. Sedgwick
Affiliation:
Trinity College, Cambridge
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Extract

13. The main object of this paper is to supply simple algebraical proofs of the general formulae for the number of particles, at any time t, of the rth product in a series of successive radioactive transformations. This is done both for ‘Case 1’, where the matter is initially all of one kind (paragraphs 3 and 4), including a stable end-product (paragraph 5), and for ‘Case 2’, where the successive products are initially all in existence and in equilibrium (paragraph 6). In paragraph 7 it is remarked that there exists a relation between the amounts of the end-product in ‘Case 1’, supposed to be the (r + 1)th product, and of the preceding rth product in ‘Case 2’, for the same time t, and the necessity of this relation is explained on general grounds. In paragraph 9 numerical values are given for the particular example (paragraph 8) considered by Rutherford in his Newer alchemy (pp. 11, 12), whilst paragraphs 10 and 11 deal with the degree of accuracy attained, and show the need for close accuracy in the experimental data.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 38 , Issue 3 , July 1942 , pp. 280 - 289
Copyright
Copyright © Cambridge Philosophical Society 1942

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References

* Or chap. xi of Rutherford's earlier book on Radioactive substances and their radiations, 1913.

* It is assumed througthout that no two of the λ's are exactly equal. If any two were equal, terms of the form te −λt would be introduced, and the solution would take a different form. But it is exceedingly improbable that for any radioactive process any two of the λ's would be exactly equal, though they might be equal within the limits of experimental accuracy, and then further consideration would be necessary.

* The factor λ4 has been accidentally omitted in Bateman's paper (p. 426, loc. cit.) from the denominator of the coefficient of e −λ4t in n4 (≡S, Case 2), and this omission has been carried into Rutherford, Chadwick and Ellis's book, p. 14, value of ‘d’ in (13), and into Rutherford's earlier book of 1913, p. 424, whilst in each of these last two references the numerator of ‘a’ should be λ2λ3 instead of λ1λ2.

* In the table on p. 25 of Radiations from radioactive substances, λ6 (radium D) is incorrectly given as 1·37 × 10−9, corresponding to an earlier figure for the half-period, since increased by about 50%.