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

TL;DR

This study performs microscopic calculations of electron spectral shapes for specific low-Q beta-minus decays, with implications for detecting the cosmic neutrino background, confirming the validity of the allowed shape approximation in most cases.

Contribution

The paper provides detailed spectral shape calculations for forbidden beta decays using advanced nuclear models, aiding future cosmic neutrino background detection efforts.

Findings

Spectral shapes deviate less than 1% from the allowed shape in most cases.

The transition $^{210}$Pb to $^{210}$Bi shows a 2.7% deviation, indicating the $\xi$ approximation is generally valid.

Derived C$ u$B cross sections highlight the need for improved atomic mismatch corrections.

Abstract

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.