The Pseudoscalar Interaction and the Beta Spectrum of RaE

Abstract

It is shown that the theory of forbidden beta transitions, as hitherto used, does not provide a correct treatment of the pseudoscalar interaction. This is demonstrated by the application of a canonical transformation to the Hamiltonian of interacting nucleons and leptons whereby all odd nuclear Dirac operators are eliminated from the theory. When this is done properly it is seen that the pseudoscalar interaction makes a contribution to the $\ensuremath{\beta}$-decay process only by virtue of the fact that the lepton covariants are not constant. The corresponding additional operators introduced in the other three interactions ($S$ is pure even) are examined and it is shown that, except in one case, these make trivial corrections which would not be observed in practice. The exception occurs in second and higher forbidden transitions wherein the spin change (tensor rank, strictly speaking) is lower than the forbiddenness order. By virtue of present knowledge of the beta interaction these must be regarded as small correction terms. The possibility of calculating all nuclear matrix elements using nonrelativistic wave functions based on some coupling model is discussed. The results presented here also show that wherever $P$ and $T$ interactions interfere (spin change zero, first-forbidden transitions) it will be possible to obtain the ratio of coupling coefficients ($\frac{{g}_{P}}{{g}_{T}}$ for example) by comparison with the observed spectral shapes. A method of reduction of the $\ensuremath{\beta}$-decay operators is described and it is pointed out that the same procedure is very convenient for obtaining the nuclear matrix elements. In this method the irreducible tensors are obtained automatically and the retardation expansion is a trivial operation performed at the end of the calculation rather than at the beginning. The correction factors for pseudoscalar ($P$) as well as $P\ensuremath{-}T$ and $P\ensuremath{-}A$ mixtures are obtained and the fact that the shape of the correction factor is strongly modified as compared to the customary result is observed. The correction factor is now more strongly $Z$ dependent and this would perhaps account for the appearance of the $P$ interaction only for heavy elements. The results of this investigation are applied to the RaE spectrum. No fit with the assumption of zero spin for RaE can be obtained. It is suggested that the spin of RaE is unity, and other evidence pointing in this direction is cited.