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Recent analysis of the viability of solid state-based relic neutrino detectors has revealed the fundamental necessity for the use of heavy, $A>100$, $\beta$-decayers as neutrino targets. Of all heavy isotopes, $^{171}$Tm and $^{151}$Sm stand out for their sufficiently low decay energies, reasonable half-life times and stable daughter nuclei. However, the crucial bit of information, that is the soft neutrino capture cross-section is missing for both isotopes. The main reason for that is a particular type of $\beta$-decay, which precludes a simple link between the isotope's half-life time and the neutrino capture rate. Here we propose an experimental method to bypass this difficulty and obtain the capture cross-section of a soft neutrino by a given isotope from the isotope's $\beta$-spectrum., Comment: 11 pages

Beta-spectrum of radioactive atoms was long ago predicted to bear an imprint of the Cosmic Neutrino Background (C$\nu$B). Over the years, it has been recognised that the best chance of achieving the signal-to-noise ratio required for the observation of this effect lies with solid-state designs. Here we bring to the fore a fundamental quantum limitation on the type of beta-decayer that can be used in such a design. We derive a simple usability criterion and show that $^3\rm H$, which is the most popular choice, fails to meet it. We provide a list of potentially suitable isotopes and discuss why their use in C$\nu$B detection requires further research.