Higher Order Corrections to the Allowed Beta Decay

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.