Theoretical Half-Lives of Forbiddenβ-Transitions

Fermi's theory of $\beta$-decay is extended to the "nth forbidden" approximation. Precise formulas for the distribution in energy of the emitted $\beta$-rays are derived for arbitrarily charged nuclei, according to the five possible invariant forms of interaction, the so-called scalar, polar vector, tensor, axial vector, and pseudo-scalar interactions, respectively. The nuclear matrix elements of the transitions, made up of the components of certain irreducible tensors, are constructed. The selection rules appropriate to these matrix elements are given in Table II. The magnitudes of the nuclear matrix elements are estimated by a simple averaging process depending only on the directional properties of the tensors from which they are constructed and the order of magnitude of the tensor components. Theoretical half-lives of the "forbidden" $\beta$-decays of RaE, ${\mathrm{P}}^{32}$, ${\mathrm{K}}^{40}$, and ${\mathrm{Rb}}^{87}$ are calculated by numerical integration of the energy dependent electron emission probabilities. Upon comparing with the experimental determinations of the half-lives, the most satisfactory agreement seems to be obtained with the tensor form of interaction. The evidence in favor of Gamow-Teller selection rules is somewhat inconclusive for the case of the ${\mathrm{K}}^{40}$ decay because of the uncertainty of the experimental determination of the maximum electron energy.