Computed total $\beta$-electron spectra for decays of Pb and Bi in the ${}^{220,222}\mathrm{Rn}$ radioactive chains

The radon radioactivity is an unavoidable background in present and future underground experiments attempting to detect the neutrinoless double $\beta$ decay, WIMP-nucleus interactions in direct dark-matter searches, etc. In particular, the ${}^{220,222}\mathrm{Rn}$ radioactive chains lead to ${\beta }^{-}$ decays of ${}^{212,214}\mathrm{Pb}$ and ${}^{212,214}\mathrm{Bi}$, notorious backgrounds in the mentioned experiments. In this paper, we compute the total $\beta$-electron spectral shapes of these decays by including next-to-leading-order terms and other correction factors in the $\beta$ spectral shape. The studied ${\beta }^{-}$ decays involve strong allowed and first-forbidden $\beta$ transitions, the nonunique first-forbidden transitions being nuclear-structure dependent through the numerous involved nuclear matrix elements (NMEs). We compute these NMEs by using the nuclear shell model with the khpe Hamiltonian. This Hamiltonian renders a very nice description of the level energies of the daughter nuclei ${}^{212,214}\mathrm{Bi}$ and ${}^{212,214}\mathrm{Po}$ of the mentioned mother nuclei. We adopt experimental endpoint energies and engage the small relativistic NMEs (sNME), to accurately describe the measured branching ratios, a necessary prerequisite for a precise description of the total $\beta$ spectra. We also discuss the uncertainties of our computed spectra and hope that these computations will be of help for present and future rare-decays and dark-matter experiments.

Figure 1

Computed level schemes for the ${\beta }^{-}$-decay daughters ${}^{212}\mathrm{Bi}, {}^{214}\mathrm{Bi}, {}^{212}\mathrm{Po}$, and ${}^{214}\mathrm{Po}$ using the Hamiltonian khpe. A comparison with the available data is performed with the parentheses denoting uncertainty in parity and spin-parity assignments. The evaluated data are gathered from [3].

Figure 2

Value of the enhancement factor ${\varepsilon }_{\mathrm{MEC}}$ as a function of $g_{\mathrm{A}}^{\mathrm{eff}}$ for $0^{+}$ to $0^{-}$ decays of experimental branchings of $81.5\left(1\right)\%$ and $44.5\left(7\right)\%$ [3] in ${}^{212}\mathrm{Pb}$ and ${}^{214}\mathrm{Pb}$, respectively. Only solutions matching the experimental branchings within 0.1% relative error are plotted. The dashed lines point to our current choice for ${}^{214}\mathrm{Pb}$. The grey dots present the results when uncertainties in the $Q$ value, branching, and total half-life are considered for the ${}^{212}\mathrm{Pb}$ decay. The green dot with its error bars corresponds to the previous analysis of Warburton [24], however, our analysis is not directly comparable to it since in [24] many more transitions than our two were considered.

Figure 3

Computed total $\beta$ spectra and their dependencies on the choice of the sNME. The crossed-blue curves are those constructed by adopting the value closer to the CVC value for the sNME of an individual transition. The gray-hatched regions denote the span of the curves obtained through the $N_{\mathrm{tot}}$ (displayed at the top-right corner of each panel) different combinations of the two possible values of sNMEs listed in Table 2. The dotted-red curve in (b) uses the value of the sNME further away from its CVC value for the important ${}^{212}\mathrm{Bi}\to {}^{212}\mathrm{Po}\left(\text{g.s.}\right)$ transition, the gray-hatched region being constructed as for the crossed-blue curve. In (a), an error budget, considering the uncertainties of the crossed-blue curve in the $Q$ value, branchings, and total half-life, is presented by the red-dashed region. These spectral shapes are available from the authors at request. For more information see the text.

Figure 4

$\beta$ spectral shapes of the two transitions that are mostly affected by the choice between the two possible values of the sNME in Table 2. These are the transitions from ${}^{212}\mathrm{Bi}$ to the ground state (left panel) and the second excited $2^{+}$ state (right panel) in ${}^{212}\mathrm{Po}$. The crossed-blue (dotted-red) curve represents the choice closer (farther) from the CVC value of the sNME.