Microscopic calculation of the β − decays of 151 Sm, 171 Tm, and 210 Pb with implications to detection of the cosmic neutrino background

The electron spectral shapes corresponding to the low- Q β− -decay transitions Sm151(5/2g.s.−)→Eu151(5/2g.s.+), Sm151(5/2g.s.−)→Eu151(7/21+), Tm171(1/2g.s.+)→Yb171(1/2g.s.−), Tm171(1/2g.s.+)→Yb171(3/21−), Pb210(0g.s.+)→Bi210(1g.s.−), and Pb210(0g.s.+)→Bi210(01−) have been computed using beta-decay theory with several refinements for these first-forbidden nonunique (ff-nu) β− transitions. These ff-nu β− transitions have non-trivial electron spectral shapes with transition nuclear matrix elements (NMEs) computed by using the microscopic Interacting Boson-Fermion Model (IBFM-2) for the decays of 151 Sm and 171 Tm, and the nuclear shell model (NSM) for the decay of 210 Pb. Within the respective Q windows, the computed ff-nu electron spectral shapes deviate maximally at sub-percent level from the universal allowed shape, except for the transition Pb210(0g.s.+)→Bi210(1g.s.−), where the maximal deviation is some 2.7%. This confirms that the so-called ξ approximation is fairly good for most of these low- Q β− transitions and thus the allowed shape is a rather good first approximation. Our computed spectral shapes could be of interest for experiments aiming to measure the cosmic neutrino background (C ν B), like the PTOLEMY experiment. We have also derived C ν B cross sections for the ground-state transitions of the considered nuclei at the β endpoint. Our findings indicate that more work on the atomic mismatch correction is needed in the future in order to extract reliable and precise C ν B cross sections for any nuclear target.