Determination of the neutrino mass by electron capture in ${}^{163}\mathrm{Ho}$ and the role of the three-hole states in ${}^{163}\mathrm{Dy}$

${}^{163}\mathrm{Ho}$ to ${}^{163}\mathrm{Dy}$ is, probably due to the small $Q$ value of about 2.5 keV, the best case to determine the neutrino mass by electron capture. The energy of the $Q$ value is distributed between the excitation of dysprosium (and the neglected small recoil of holmium) and the relativistic energy of the emitted neutrino including the rest mass. The reduction of the upper end of the deexcitation spectrum of dysprosium below the $Q$ values allows us to determine the neutrino mass. The excitation of dysprosium can be calculated in the sudden approximation of the overlap of the electron wave functions of holmium minus the captured electron and one-, two-, three-, and multiple hole excitations in dysprosium. Robertson [R. G. H. Robertson, Phys. Rev. C 91, 035504 (2015)] and Faessler and Simkovic [A. Faessler and F. Simkovic, Phys. Rev. C 91, 045505 (2015)] have calculated the influence of the two-hole states on the dysprosium spectrum. Here for the first time the influence of the three-hole states on the deexcitation bolometer spectrum of ${}^{163}$Dy is presented. The electron wave functions and the overlaps are calculated self-consistently in a fully relativistic and antisymmetrized Dirac-Hartree-Fock approach in holmium and in dysprosium. The electron orbitals in Dy are determined including the one-hole states in the self-consistent iteration. The influence of the three-hole states on the Dy deexcitation (by x rays and Auger electrons) can hardly be seen. The three-hole states seem not to be relevant for the determination of the electron neutrino mass.

Figure 1

Logarithmic (${log}_{10}$) three-hole Bolometer spectrum (1) with (3) and (13) for the sum of the one-, two-, and three-hole probabilities calculated in this work with the resonance energies and widths of Tables 1 and 2 with the assumed $Q$ value $Q=2.8$ keV for the bolometer energy between 0.0 and 2.8 keV. The ${log}_{10}$ coordinates of the ordinate have to be read as ${10}^{\text{ordinate}}$; e.g. the ordinate value $-2$ corresponds to ${10}^{-2}$. The theoretical spectra are normalized to the experimental $4s_{1/2}$ hole peak at 0.411 keV (see Fig. 4).

Figure 2

Logarithmic (${log}_{10}$) plot of the bolometer spectrum (1) including the one-hole, the one- plus two-hole, and the one- plus two- plus three-hole states. The $Q$ value is assumed to be $Q=2.8$ keV according to an ECHo measurement [16]. The resonance energies and widths are listed in Tables 1 and 2. The three-hole states can hardly be seen.

Figure 3

Comparison of the ${log}_{10}$ spectrum of the one-hole, the one- plus two-hole, and the one- plus two- plus three-hole states with the data of the ECHo Collaboration [13, 16, 24, 26]. The experimental data are binned every 2 eV (counts/2 eV). Some bins have no counts and thus the logarithmic value is $-\infty$. The effect of the three-hole states hardly shows up even in this logarithmic plot.

Figure 4

Comparison of the ${log}_{10}$ bolometer spectrum around the $N1={\left(4s_{1/2}\right)}^{-1}0.411\text{keV}$ (relative probability to the $3s_{1/2}$ state 24.4%) and $N2={\left(4p_{1/2}\right)}^{-1}0.333\text{keV}$ (relative probability 1.22%) resonances with the ECHo data [13, 16, 24, 26]. At 0.569 keV one sees the degenerate two-hole states: $N1={\left(4s_{1/2}\right)}^{-1}$, $N4={\left(4d_{3/2}\right)}^{-1}$ with the relative probability of 0.088 % and $N1={\left(4s_{1/2}\right)}^{-1}$, $N5={\left(4d_{5/2}\right)}^{-1}$ with the relative probability of 0.125% (total 0.213 %). The resonance energies and the widths are taken from the ECHo Collaboration data [24, 26]. The theory does not include the finite resolution of the measurement, which is about 10 eV FWHM. The spectra of the sum of the one- plus two-hole states and the sum of the one- plus two- plus three-hole states look identical, even in a logarithmic scale.