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Selection Rules in Nuclear Radiation

Published online by Cambridge University Press:  24 October 2008

H. M. Taylor
H. M. Taylor
Affiliation:
Clare College
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Abstract

In the first part of this paper we discuss the radiation from a single charged particle moving in an arbitrary central field of force and obeying Dirac's equation. We consider the electric quadripole and magnetic dipole radiation as well as the electric dipole. We derive the selection rules for the magnetic dipole radiation and collect together for reference the corresponding selection rules for the electric dipole and quadripole radiations. In the second part we discuss the relative intensities of the various types of radiation, treating in detail the cases where the selection rules for magnetic dipole and electric quadripole are simultaneously satisfied. Finally we show that these results have an important bearing on the theory of internal conversion of γ-rays. The internal conversion of soft γ-rays occurs with such high probability that the theory is unable to account for the experimental results unless it is assumed that the radiation is largely magnetic dipole in character. On the other hand, Fisk and Taylor (loc. cit.) were unable to account for the presence of magnetic dipole radiation in appreciable amounts. We show that this is due to the fact that, of the two possible transitions (a and e of § 2) in which both magnetic dipole and electric quadripole radiation can be emitted, Fisk and Taylor considered only the second. In the case of the second, corresponding to a transition between two distinct terms, we show that Fisk and Taylor were correct in predicting a negligible amount of magnetic dipole radiation, but in the case of the first, corresponding to a transition between two levels of one multiplet term, we find that there is indeed a high percentage of magnetic dipole radiation.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 32 , Issue 2 , May 1936 , pp. 291 - 300
Copyright
Copyright © Cambridge Philosophical Society 1936

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References

† Fisk, and Taylor, , Proc. Roy. Soc. A, 146 (1934), 178CrossRefGoogle Scholar; Hulme, Mott, Oppenheimer and Taylor, Proc. Roy. Soc. A, to appear shortly; Stahel, Helvetica Phys. Acta.

‡ The theory of atomic spectra (1935), p. 85Google Scholar, equations (6)–(9).

§ Since we are developing the radiated potentials in powers of d/λ and retaining only the first two powers, it is of interest to note how well the condition λ»d is satisfied. For the emission of light from the extra-nuclear structure it is so well satisfied that, as is well known, we normally retain only the first power of d/λ. It is reasonably well satisfied for the case of X-ray emission, since for hard X-rays λ˜ A.u.while d ∼0·05 A.U. Finally it it well satisfied for soft γ-radiation, since for nuclei d ∼ 10–12cm., while even for the hardest γ-rays λ∼5 × 10–11cm.

† A discussion of the properties of dyadics is given in Weatherburn's Advanced vector analysis.

‡ Condon and Shortley, loc. cit. p. 59, equation (1) and paragraph (d). See also p. 43 for the definition of the quantity [A, B].

† Condon and Shortley, p. 63, equations (11).

‡ Ibid. p. 123, equations (8b) and p. 127.

† Condon and Shortley, pp. 127–8, equations (7) and (10).

† κ is the number called k by Darwin, (Proc. Roy. Soc. A, 118 (1928), 667)Google Scholar and is of course the analogue of the quantum number l of the spinless hydrogen problem: Dirac (Quantum mechanics, Second edition, p. 226) uses j for the quantum number which we call k. Since J is the symbol now generally accepted for the total angular momentum and j for the corresponding quantum number, we have thought it beat to follow Condon and Shortley in using the symbol K for the observable α(2ћ–2 L·S+1) and k for the corresponding quantum number. Weyl (Theory of groups and quantum mechanics, English translation, 1931) uses the same notation but, owing to a slip on p. 234 where he compares the vanishing instead of the non-vanishing component of his wave function with the non-relativistic wave function, he has the wrong sign for k in his correlation of k with the spectroscopic notation (p. 233).Darwin and Condon and Shortley give this correlation correctly.

† Special cases. j = 0 → j= 0 is forbidden for all types of radiation. j = ½ → j = ½ and j = 1 → j = 1 are forbidden for quadripole radiation.

† Loc. cit. To find explicit forms for Fand G we must of course know V, but our purpose can be achieved without the knowledge of explicit forms for Fand G.

‡ Taylor, and Mott, , Proc. Roy. Soc. A, 142 (1933), § 4, p. 227.Google Scholar Note that Taylor and Mott give the potentials only for large values of r, whereas we insert the extra terms in (qr)–1and (qr)–2 which make the potentials valid for all values of r. We follow Taylor and Mott's notation.

† Taylor, , Proc. Camb. Phil. Soc. 31 (1935), 411,CrossRefGoogle Scholar equations (3.06) et seq.

‡ Fisk and Taylor, loc. cit. equation (2.1).

† Taylor, loc. cit. equation (3.04).

† Hulme, Mott, Oppenheimer and Taylor, loc. cit.; Stahel, loc. cit.

† The Δι = 2 of Fisk and Taylor, p. 180, is our Δζ= 2.

§ The exceedingly small value which these authors obtain for the magnetic dipole intensity is due to the fact that, in their case, the quantities D and E of our equation (32) have opposite sign and are also fortuitously of almost the same magnitude.