Consistent calculation of the screening and exchange effects in allowed ${\beta }^{-}$ transitions

The atomic exchange effect has previously been demonstrated to have a great influence at low energy on the ${}^{241}\mathrm{Pu}$ ${\beta }^{-}$ transition. The screening effect has been given as a possible explanation for a remaining discrepancy. Improved calculations have been made to consistently evaluate these two atomic effects, compared here to the recent high-precision measurements of ${}^{241}\mathrm{Pu}$ and ${}^{63}\mathrm{Ni}$ $\beta$ spectra. In this paper a screening correction has been defined to account for the spatial extension of the electron wave functions. Excellent overall agreement of about 1% from 500 eV to the end-point energy has been obtained for both $\beta$ spectra, which demonstrates that a rather simple $\beta$ decay model for allowed transitions, including atomic effects within an independent-particle model, is sufficient to describe well the current most precise measurements.

Figure 1

Total exchange factor ${\eta }^{\mathrm{tot}}$ for the ${}^{63}\mathrm{Ni}$ decay, with the contribution of each orbital ${\eta }^{ns}$.

Figure 2

Measured ${}^{63}\mathrm{Ni}$ $\beta$ spectrum compared to analytical calculation (green), numerical calculation with screening (red), and numerical calculation with screening and exchange (blue). Normalization to the data was done by integration from 500 eV to $E_{\max }$. Standardized residuals between the data and the most complete calculation are also given. A zoom of the low-energy region is also given.

Figure 3

Total exchange factor ${\eta }^{\mathrm{tot}}$ for the ${}^{241}\mathrm{Pu}$ decay, with the contribution of each orbital ${\eta }^{ns}$.

Figure 4

Measured ${}^{241}\mathrm{Pu}$ $\beta$ spectrum compared to analytical calculation (green), numerical calculation with screening (red), and numerical calculation with screening and exchange (blue). Normalization to the data was done by integration from 500 eV to $E_{\max }$. Standardized residuals between the data and the most complete calculation are also given. A zoom of the low-energy region is also given.