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The electron spectral shapes corresponding to the low-$Q$ $\beta^-$-decaytransitions $^{151}$Sm$(5/2^-_{\rm g.s.})\to\,^{151}\textrm{Eu}(5/2^+_{\rmg.s.})$, $^{151}$Sm$(5/2^-_{\rm g.s.})\to\,^{151}\textrm{Eu}(7/2^+_{1})$,$^{171}$Tm$(1/2^+_{\rm g.s.})\to\,^{171}\textrm{Yb}(1/2^-_{\rm g.s.})$,$^{171}$Tm$(1/2^+_{\rm g.s.})\to\,^{171}\textrm{Yb}(3/2^-_{1})$,$^{210}\textrm{Pb}(0^+_{\rm g.s.})\to\,^{210}\textrm{Bi}(1^-_{\rm g.s.})$, and$^{210}\textrm{Pb}(0^+_{\rm g.s.})\to\,^{210}\textrm{Bi}(0^-_{1})$ have beencomputed using beta-decay theory with several refinements for thesefirst-forbidden nonunique (ff-nu) $\beta^-$ transitions. These ff-nu $\beta^-$transitions have non-trivial electron spectral shapes with transition nuclearmatrix elements (NMEs) computed by using the microscopic InteractingBoson-Fermion Model (IBFM-2) for the decays of $^{151}$Sm and $^{171}$Tm, andthe nuclear shell model (NSM) for the decay of $^{210}$Pb. Within therespective $Q$ windows, the computed ff-nu electron spectral shapes deviatemaximally at sub-percent level from the universal allowed shape, except for thetransition $^{210}\textrm{Pb}(0^+_{\rm g.s.})\to\,^{210}\textrm{Bi}(1^-_{\rmg.s.})$, where the maximal deviation is some 2.7$\%$. This confirms that theso-called $\xi$ approximation is fairly good for most of these low-$Q$$\beta^-$ transitions and thus the allowed shape is a rather good firstapproximation. Our computed spectral shapes could be of interest forexperiments aiming to measure the cosmic neutrino background (C$\nu$B), likethe PTOLEMY experiment. We have also derived C$\nu$B cross sections for theground-state transitions of the considered nuclei at the $\beta$ endpoint. Ourfindings indicate that more work on the atomic mismatch correction is needed inthe future in order to extract reliable and precise C$\nu$B cross sections forany nuclear target.

physics letters b

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