The electron spectral shapes corresponding to the low-$Q$ $\beta^-$-decay transitions $^{ 151}$Sm$(5/2^-_{ \rm g.s.})\to\,^{ 151}\textrm{ Eu}(5/2^+_{ \rm g.s.})$, $^{ 151}$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 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 $^{ 210}\textrm{ Pb}(0^+_{ \rm g.s.})\to\,^{ 210}\textrm{ Bi}(1^-_{ \rm g.s.})$, 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.
Categories: nucl-th