Evidence for the exchange effect in the $\beta$ decay of ${}^{241}$Pu

The exchange effect has been previously given as a possible explanation for a significant deviation from an allowed shape observed at low energy in the ${}^{241}$Pu $\beta$ spectrum. Calculations set out here confirm that this atomic effect explains a large part of this deviation. The equations needed to calculate the exchange effect are detailed, as well as the evaluation of the confluent hypergeometric function for complex arguments of large magnitudes. After a review of the possible other effects that could explain the remaining discrepancy at low energy, the screening correction using effective nuclear charges seems to be the best explanation. For radionuclides with high $Z$, this work has demonstrated the necessity to take into account the spatial variation of the nuclear charge experienced by the ejected electron to accurately correct for the screening effect.

Figure 1

Total exchange factor for the ${}^{241}$Pu $\beta$ decay and contributions of each orbital. The effective nuclear charges used are the relativistic ones from Harston and Pyper [7] (see Table ).

Figure 2

Comparison between the measured $\beta$ spectrum of ${}^{241}$Pu from Loidl et al. [1] and an allowed spectrum calculated with and without exchange effect. The effective nuclear charges used are the relativistic ones calculated for a neutral Pu atom (see Table ).

Figure 3

Residuals relative to the allowed spectrum for the measured spectrum of ${}^{241}$Pu from Loidl et al. [1], for our calculation including the exchange effect (red solid line), for the one from Harston and Pyper [7] (green long-dashed line), and for the latter one including the 7$s$ orbital (blue dot-dashed line). The details of the calculations are explain in the text.