First-Forbidden Beta-Decay Matrix Elements

The general theory of first-forbidden beta-transitions involves seven nuclear matrix elements in the nonrelativistic approximation. In principle the number of independent parameters can be reduced to four with the help of the relation $(W_f-W_i)\left(f\right|X\left|i\right)=\left(f\right|[H_0,X]\left|i\right)+\left(f\right|[H_c,X]\left|i\right)+\left(f\right|[H_{\nu },X]\left|i\right).$ Here $W$ is an energy eigenvalue, $X$ a coordinate type first-forbidden operator, $H_0$ the free particle Hamiltonian, $H_c$ the Coulomb interaction, and $H_{\nu }$ the specifically nuclear interaction. The commutators of $H_0$ and $H_c$ with $X$ are easily evaluated, but [$H_{\nu }$, $X$] presents difficulties. However, the matrix elements of [$H_{\nu }$, $X$] can be estimated by general physical arguments based on the semi-empirical energy surface and the validity of shell model considerations. The explicit form of $H_{\nu }$ is not required a fortunate circumstance since $H_{\nu }$ for complex nuclei is at present essentially unknown. These calculations determine the common proportionality factor $\Lambda$ in the relations $\{\frac{1}{2}\Lambda \alpha Z\int \frac{\mathbf{r}}{R},=-i\int \mathbf{\alpha },\}\{\frac{1}{2}\Lambda \alpha Z\int \frac{\mathbf{\sigma }·\mathbf{r}}{R},=-i\int {\gamma }_5,\}\{\frac{1}{2}\Lambda \alpha Z\int \mathbf{\sigma }\times \frac{\mathbf{r}}{R},=-\int \beta \mathbf{\alpha }.\}$