The shape of the allowed beta spectrum and the directional correlations of the allowed beta ray and gamma or alpha ray have been investigated theoretically with an assumption of $\mathrm{VA}$. We take into account the Coulomb field due to the daughter nucleus, the finite de Broglie wavelength effect, and the contribution of the second forbidden matrix elements, $\mathfrak{M}\left(r^2\right)$, $\mathfrak{M}\left(\alpha ·\mathrm{r}\right)$, $\mathfrak{M}\left(\sigma r^2\right)$, $\mathfrak{M}\left(\right(\sigma ·\mathrm{r}\left)\mathrm{r}\right)$, $\mathfrak{M}\left({\gamma }_5\mathrm{r}\right)$, and $\mathfrak{M}(\alpha \times \mathrm{r})$, simultaneously. Relations between coordinate-type and momentum-type matrix elements are given in the nonrelativistic approximation. In this case, $\frac{\mathfrak{M}(\alpha \times \mathrm{r})}{\mathfrak{M}\left(\sigma \right)}=M^{-1\mathrm{}}$. The correction factors for the beta spectra of ${\mathrm{B}}^{12}$ and ${\mathrm{N}}^{12}$ as well as their ratio have almost no energy dependence, since several corrections cancel each other. On the other hand, this ratio varies by 12% over the whole spectrum, if we adopt $\frac{\mathfrak{M}(\alpha \times \mathrm{r})}{\mathfrak{M}\left(\sigma \right)}=M^{-1\mathrm{}}({\mu }_p-{\mu }_n)$ as given by Gell-Mann. The beta-alpha directional correlation of ${\mathrm{Li}}^8$ is discussed also.

• Received 8 September 1958

DOI:https://doi.org/10.1103/PhysRev.113.1584