Improved description of one- and two-hole excitations after electron capture in ${}^{163}\mathrm{Ho}$ and the determination of the neutrino mass

The atomic pair ${}^{163}\mathrm{Ho}$ and ${}^{163}\mathrm{Dy}$, because of its small $Q$ value of about 2.5 keV, seems to be the best pair to use to determine the neutrino mass by electron capture. The bolometer spectrum measures the full deexcitation energy of dysprosium (by x rays and Auger electrons plus the recoil of holmium, which can be neglected). The spectrum has an upper limit given by the $Q$ value minus the neutrino mass. Till now this spectrum has been calculated in dysprosium allowing excitations with $3s_{1/2}, 3p_{1/2}, 4s_{1/2}, 4p_{1/2}, 5s_{1/2}$, and $5p_{1/2}$ (and $6s_{1/2}$) holes only. Robertson [R. G. H. Robertson, arXiv:1411.2906v1] also recently calculated the spectrum with two-electron-hole excitations in Dy. He took the probability for the excitation for the second electron hole from the work of Carlson and Nestor [T. A. Carlson, C. W. Nestor, T. C. Tucker, and F. B. Malik, Phys. Rev. 169, 27 (1968); T. A. Carlson and C. W. Nestor, Phys. Rev. A 8, 2887 (1973)] for $Z=54$ xenon. The neutrino mass must finally be obtained by a simultaneous fit of the $Q$ value, together with the properties of the relevant resonances, and the neutrino mass to the the upper end of the spectrum. Under the assumption that only one resonance (independent of its nature: one-hole, two-hole, multihole, or of other origin) near the $Q$ value determines the upper end of the spectrum and that the profile of this leading state is Lorentzian, one has to fit simultaneously four parameters (neutrino mass, strength, distance of the leading resonance to the $Q$ value, and width). If more than one resonance is of comparable importance for the upper end of the spectrum, it might be difficult or even impossible to extract the neutrino mass reliably. Compared to the work of Robertson this work includes the following improvements. (i) The two-hole probabilities are calculated in the Dirac-Hartree-Fock (DHF) approach for holmium and dysprosium but not for xenon. (ii) In calculating the probability for the second electron hole in dysprosium the $n{s}_{1/2}$ or $n{p}_{1/2}$ ($n\ge 3$) one-hole states are included self-consistently in the DHF iteration. (iii) Because dysprosium has $Z=66$ electrons and xenon only has $Z=54$ electrons, one has at least eight additional two-hole states that do not exist in xenon and thus their probabilities have not been calculated by Carlson and Nestor and have not been included by Robertson. They are included here. (iv) For the probabilities of the one-hole states, which determine the main structure of the spectrum, the overlap and exchange corrections are taken into account. (v) In solving the DHF electron wave functions the finite size of the nuclear charge distribution is included. (vi) The nuclear matrix elements for electron capture integrate the charge of the captured electron over the nucleus with the weight $\psi {\left(r\right)}_{e,n,\ell ,j}^2{r}^2$. Thus, for the capture probability the value ${\psi }_{e,n,\ell ,j}^2\left(R\right)R^2$ is taken at the nuclear radius and not the value ${\psi }_{e,n,\ell ,j}^2(r=0.0)$ at $r=0.0$, which has the weight $r^2$ zero. (vii) The formulas are derived in second quantization including automatically the antisymmetrization.

Figure 1

${\mathrm{Logarithmic}}_{10}$ (with basis 10) bolometer spectra (4) and (21) for the one- and two-hole probabilities calculated in this work (dashed line) and with the parameters of Robertson (solid line) [7] with the assumed $Q$ value = 2.8 keV for the bolometer energy between 0.0 and 2.8 keV. Apart from the excitation of one-hole states in ${}^{163}\mathrm{Dy}$ after electron capture in ${}^{163}\mathrm{Ho}$, the excitations of the two-hole states in ${}^{163}\mathrm{Dy}$ are also included. The ${\mathrm{logarithmic}}_{10}$ plot stresses the effect of the two-hole states. In a linear plot (see Fig. 2) the two-hole states are hardly to be seen. The values at the ordinate have to be read as ${10}^{\mathrm{ordinate}}$. So “$-2$” is ${10}^{-2}$.

Figure 2

Experimental (solid line) and theoretical (dashed line) linear bolometer spectra (4) and (21) for electron capture in ${}^{163}\mathrm{Ho}$ to ${}^{163}\mathrm{Dy}$ with the $Q$ value = 2.8 keV [see Eq. (5)] including the one- and two-hole probabilities calculated in this work and compared to the Heidelberg experiment of Ranitzsch et al. [28] and Gastaldo et al. [29] for the total bolometer energy from 0.02 to 2.80 keV. The theoretical spectrum is normalized to the data of the $N1$ ($4s_{1/2}$) peak at 0.42 keV. The three small experimental lines between 1.2 and 1.6 keV originate from small admixtures of ${}^{144}\mathrm{Pm}$. The experimental counts are binned in 2-eV intervals. Due to background subtraction the number of counts in some bins can be negative. In some regions of the energy the number of counts per bin is zero or one. Due to negative and zero counts in a bin it is not possible to plot the data logarithmically as in Fig. 1, which would show the small effects of the two-hole excitations in Dy better. The theoretical spectrum is calculated for zero neutrino mass and with the excitation and the width of the one-hole states of the ECHo Collaboration [28, 29] listed in Table 4.

Figure 3

The upper 10 eV of the bolometer spectrum from 2.790 to the $Q$ value = 2.800 keV for neutrino masses $m_{\nu }=0.0$ eV (dashed line) and $m_{\nu }=2$ eV (solid line).

Figure 4

The upper 10 eV of the bolometer spectrum from 2.345 keV to the assumed $Q$ value = 2.355 keV for neutrino masses $m_{\nu }=0.0$ eV (dashed line) and $m_{\nu }=2$ eV (solid line). The two-hole state $3s_{1/2}, 4p_{3/2}$ at $E_c=2.350$ keV in ${}^{163}\mathrm{Dy}$ is just below the assumed $Q$ value = 2.355 keV within the width $\Gamma =13.2$ eV. In a simultaneous fit of the neutrino mass and the $Q$ value, also the position, the width, and the strength of the resonance state must also be included. A finite neutrino mass produces at the upper end of the spectrum a special fingerprint, which cannot be produced by a resonance state. Thus one can hope, that even in this situation the upper end of the spectrum shows the fingerprint of a finite neutrino mass.

Figure 5

The upper 20 eV of the bolometer spectrum in ${}^{163}\mathrm{Dy}$ for a neutrino with mass $m_{\nu }=0$ eV (dashed line) and an electron neutrino, which is a mixture of two mass eigenstates with $m_{\nu }=2$ eV and $m_{\nu }=10$ eV (solid line) for the assumed $Q$ value = 2.8 keV. The mixing coefficients are adopted from Ref. [30] assuming only mixing of two neutrinos with probabilities of $U_{e,1}^2\left(2\mathrm{eV}\right)=0.636$ and $U_{e,2}^2\left(10\mathrm{eV}\right)=0.364$. The neutrino with $m_{\nu }=0$ eV is assumed to have $U_{e,1}^2=1.0$ and $U_{e,2}^2=0.0$. The onset of the heavy neutrino of $m_{\nu }=10$ eV at $(Q-10)\mathrm{eV}=2.790$ keV can be seen in the theoretical spectrum. Finite experimental energy resolution will at least partially smear out this effect.