The Effect of the Finite deBroglie Wavelength in the Theory of Beta-Decay

An accurate numerical evaluation of the $Z$ and energy dependent parts of the correction factors for allowed and forbidden beta-transitions has been carried out. The range of parameters considered is: 88 values of $Z$ (from 10 to 96 inclusive in steps of 2 and for positrons ($Z<\mathrm{}0\mathrm{}$) as well as electrons ($Z>\mathrm{}0\mathrm{}$)); 33 values of $p$, the electron momentum, in mc units ($0.1<~p<~25$) for $\mathrm{}\left|Z\right|\mathrm{}$ through 24 and 31 values of $p$ (0.1≤ p≤ 15) for $\mathrm{}\left|Z\right|>24\mathrm{}$; four values of $j\left(\mathrm{the}\mathrm{electron}\mathrm{momentum}\right(\frac{1}{2}<~j<~\frac{7}{2})$. For each value of $Z$, $p$, and $j$ the six wave-function combinations ($L_{\nu }, M_{\nu }, N_{\nu }, P_{\nu }, Q_{\nu }, R_{\nu }$) needed for the general mixed beta-interaction are obtained. For the given range of $j$, analysis of beta-spectra can be carried out through third forbidden transitions. The results given here show that the theoretically allowed spectra should be modified by an energy-dependent factor. For not-too-low maximum energies, this modification in the shape of the allowed beta-spectrum may be observable with presently attainable precision.