Microscopic calculation of the ${\beta }^{-}$ decays of ${}^{151}$Sm, ${}^{171}$Tm, and ${}^{210}$Pb with implications to detection of the cosmic neutrino background

Abstract

The electron spectral shapes corresponding to the low-$Q$ ${\beta }^{-}$-decay transitions ${}^{151}$Sm$(5/2_{g.s.}^{-})\to {}^{151}\text{Eu}(5/2_{g.s.}^{+})$, ${}^{151}$Sm$(5/2_{g.s.}^{-})\to {}^{151}\text{Eu}(7/2_1^{+})$, ${}^{171}$Tm$(1/2_{g.s.}^{+})\to {}^{171}\text{Yb}(1/2_{g.s.}^{-})$, ${}^{171}$Tm$(1/2_{g.s.}^{+})\to {}^{171}\text{Yb}(3/2_1^{-})$, ${}^{210}\text{Pb}(0_{g.s.}^{+})\to {}^{210}\text{Bi}(1_{g.s.}^{-})$, and ${}^{210}\text{Pb}(0_{g.s.}^{+})\to {}^{210}\text{Bi}(0_1^{-})$ have been computed using beta-decay theory with several refinements for these first-forbidden nonunique (ff-nu) ${\beta }^{-}$ transitions. These ff-nu ${\beta }^{-}$ 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 ${}^{210}\text{Pb}(0_{g.s.}^{+})\to {}^{210}\text{Bi}(1_{g.s.}^{-})$, where the maximal deviation is some 2.7$\%$. This confirms that the so-called $\xi$ approximation is fairly good for most of these low-$Q$ ${\beta }^{-}$ 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$\nu$B), like the PTOLEMY experiment. We have also derived C$\nu$B cross sections for the ground-state transitions of the considered nuclei at the $\beta$ 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$\nu$B cross sections for any nuclear target.