Deviation from theξApproximation in First ForbiddenβDecay

The failure of the $\xi$ approximation in which only the first nonvanishing term is kept in the expansion in descending powers of the Coulomb energy factor is due to either the cancellation or selection rule effect. The latter may be due to $K$ or $j$ forbiddennesses. In order to distinguish these three possibilities experimentally, the two transitions, $3\left(\beta \right)2\left(\gamma \right)0\mathrm{}$ and $2\left(\beta \right)2\left(\gamma \right)0\mathrm{}$, are discussed. The present data on ${\mathrm{Sb}}^{124}$ and ${\mathrm{Rb}}^{86}$ are insufficient to permit drawing a definite conclusion. Similar arguments can be extended to other $\beta$ decays. To get more information, the general energy and angular dependences are given in convenient form for various observables, and are shown numerically for ${\mathrm{Sb}}^{124}$. Certain $\beta -\gamma$ correlation experiments, especially the $\beta$-circularly polarized $\gamma$ correlation and the transverse $\beta$ polarization, are proposed for a variety of special $\beta$ decays, e.g., ${\mathrm{Ga}}^{72}$, ${\mathrm{Y}}^{92}$, and so on. It is also concluded that the unique shape energy spectrum does not necessarily correspond to a unique forbidden transition. An example is ${\mathrm{Eu}}^{152}$. Measurements of $\beta -\gamma$ correlations are useful in order to decide this correspondence. Other $\beta$ decays, which may be characterized by the cancellation, are ${\mathrm{Ag}}^{111}$, ${\mathrm{Re}}^{186}$, and ${\mathrm{Tm}}^{170}$.