Neutrino mass, electron capture, and the shake-off contributions

Electron capture can determine the electron neutrino mass, while the $\beta$ decay of tritium measures the electron antineutrino mass and the neutrinoless double $\beta$ decay observes the Majorana neutrino mass. In electron capture, e.g., ${}_{67}{}^{163}\mathrm{Ho}+e^{-}\to \stackrel{*}{{}_{66}{}^{163}\mathrm{Dy}}+{\nu }_e$, one can determine the electron neutrino mass from the upper end of the decay spectrum of the excited Dy, which is given by the $Q$ value minus the neutrino mass. The excitation of Dy is described by one, two, and even three hole excitations limited by the $Q$ value. These states decay by x-ray and Auger electron emissions. The total decay energy is measured in a bolometer. These excitations have been studied by Robertson and by Faessler et al. In addition the daughter atom Dy can also be excited by moving in the capture process one (or more) electrons into the continuum. The escape of these continuum electrons is automatically included in the experimental bolometer spectrum. Recently a method developed by Intemann and Pollock was used by DeRujula and Lusignoli for a rough estimate of this shake-off process for “$s$” wave electrons in capture on ${}^{163}\mathrm{Ho}$. The purpose of the present work is to give a more reliable description of “$s$” wave shake-off in electron capture on holmium. One uses the sudden approximation to calculate the spectrum of the decay of $\stackrel{*}{{}_{66}{}^{163}\mathrm{Dy}}$ after electron capture on ${}_{67}{}^{163}\mathrm{Ho}$. For that one needs very accurate atomic wave functions of Ho in its ground state and excited atomic wave functions of Dy including a description of the continuum electrons. DeRujula and Lusignoli use screened nonrelativistic Coulomb wave functions for the Ho electrons $3s$ and $4s$ and calculate the Dy* states by first-order perturbation theory based on Ho. In the present approach the wave functions of Ho and Dy* are determined self-consistently with the antisymmetrized relativistic Dirac-Hartree-Fock approach. The relativistic continuum electron wave functions for the ionized Dy* are obtained in the corresponding self-consistent Dirac-Hartree-Fock potential. The result of this improved approach is that shake-off can hardly be seen in the bolometer spectrum after electron capture in ${}^{163}\mathrm{Ho}$ and thus can probably not affect the determination of the electron neutrino mass.

Figure 1

Nonrelativistic Coulomb wave functions $3s$ (solid), $4s$ (dashed-dot-dot) in Ho with the effective charges chosen by DeRujula and Lusignoli [9] $Z_{\mathrm{eff}}\left(3s\right)=54.9$ and $Z_{\mathrm{eff}}\left(4s\right)=43.2$. In Ref. [9] the $s$-wave functions for Dy including the continuum are calculated according to [10] in first order with the perturbation (5). But for this figure the screened nonrelativistic “Coulomb” wave functions for ${}^{163}\mathrm{Dy}$ are calculated exactly [11] and not with first order perturbation relative to Ho as in Ref. [9], for the potential (6) with a $3s$ or $4s$ hole in Dy. The charge of the nucleus is reduced by one from Ho to Dy, but the electron hole in Dy in the orbit $3s$ (dashed-dot) or $4s$ (dashed) increases for the outer electrons the charge again effectively by one. The change from Ho to Dy is almost not visible in the wave function of this figure. The hole state has in Dy the effect to smear out the positive charge of the nucleus compared to Ho and increases by that effectively the nuclear radius. Thus the effect is to shift the functions in Dy slightly to the right to larger radial distances. This small change of the wave function from Ho to Dy restricts in the sudden approximation using the Vatai approximation [7, 8] the probability for two-hole states including also shake-off to the very small values 0.00271 for $3s$ and 0.00607 (see Table 2) for $4s$ in column five of Table 2. These small probabilities for two-hole states including shake-off agrees qualitatively with the more reliable self-consistent relativistic Dirac-Hartree-Fock [12] result of column three in Table 2. This shows, that shake-off can have only a very minor effect on the bolometer spectrum of the Dy decay after capture in Ho.

Figure 2

Large amplitudes $P\left(\vec{r}\right)$ for $1s,2s,2p_{1/2},2p_{3/2}$, and $3s$ Dy electrons normalized DHF wave functions in atomic units for the the radial distance (Bohr radii) and ${\left(\mathrm{atomic}\mathrm{units}\right)}^{-1/2}$ for the wave functions.

Figure 3

Self-consistent DHF potential [dotted; dimension: 1/(length = a.u.)] and analytic approximation (solid) with a = 3.4. [see Eqs. (15) and (17).]

Figure 4

Radial distance $r$ times the self-consistent potential [dimensionless] for the ground state of the Dy atom with 66 electrons (dashed), $r$ times the self-consistent potential with a hole in $3s$ and 65 electrons (solid), $r$ times the self-consistent potential with a hole in $4s$ and 65 electrons (dotted), and $r$ times the analytical fit to the ground state potential (17) (dashed-dot lines).

Figure 5

Large amplitude $P\left(r\right)$ of the s wave at $E=4$ [Hartree] $\equiv 108.8$ [eV] before (solid) and after Schmidt orthogonalization (dashed).

Figure 6

Wave function $P\left(r\right)$ (dashed-dot line) and $Q\left(r\right)$ (solid line) Eq. (16) in the Dy continuum at $50\left[\mathrm{Hartree}\right]=1.36$ keV. The continuum wave functions $P$ and $Q$ are normalized to $\int dr[P{\left(r\right)}^2+Q{\left(r\right)}^2]=2\pi \delta (k-k^{\prime })$. The $4s$ bound state in Ho is normalized to unity [dashed for $P\left(r\right)$ and dotted for $Q\left(r\right)]$: $\int dr[P{\left(r\right)}^2+Q{\left(r\right)}^2]=1$. The overlap $\langle ns,\mathrm{Ho}|\mathrm{E},s,\mathrm{Dy}\rangle$ squared is proportional to the shake-off process as a function of energy. The continuum wave functions $P_E$ and $Q_E$ are dimensionless, while the bound states $P_{4s}\left(r\right)$ and $Q_{4s}\left(r\right)$ are in atomic units $\left[{(\mathrm{a}.\mathrm{u}.)}^{-1/2}\right]$.

Figure 7

Theoretical results in arbitrary units of the sum of the one- and two-hole de-excitations compared to the sum of the one-, two-hole, and the shake-off de-excitation as measured by the bolometer spectrum (7). The arbitrary units are adjusted to the experimental $N1, 4s_{1/2}$ one-hole peak. The nature of the one hole states are indicated. The two-hole peaks are by about two orders of magnitudes smaller than the one hole peaks. Shake-off can almost not been seen.

Figure 8

Experimental and theoretical results of the sum of the one- and two-hole de-excitations (dashed line) and the sum of the one-, two-hole, and the shake-off de-excitation (solid line) for the bolometer spectrum (7). The experimental data are from the ECHo collaboration [4, 21]. The two theoretical spectra are adjusted to experiment at the $N1,4s_{1/2}$ peak. The nature of the one-hole states are indicated. The two-hole peaks are by about two orders of magnitudes smaller than the one-hole peaks. The shake off contributions can hardly been seen in this scale. Some bins contain no experimental counts, thus the ${log}_{10}$ for these experimental values are minus infinity. To fit the 1931 experimental points for the bolometer energy of 0.0 to 2.8 keV, the theoretical spectrum of 200 mesh points had to be interpolated to the data points for this figure. Figure 7 contains the 200 original theoretical results without interpolation for the bolometer spectrum over $E_c$ between 0.0 and 2.8 keV. The interpolation is normally very good (compare Figs. 7 and 8) but difficult at some sharp minima and maxima.

Figure 9

Shake-off contributions for different two-hole excitations in Dy normalized for the experimental bolometer spectrum (see Fig. 8) to the $N1, 4s_{1/2}$ peak. Increasing the energy $E_c$ of the bolometer spectrum the $Q$ value = 2.8 keV is first used to excite the two-hole state. So the shake-off contribution for the bolometer spectrum starts as function of $E_c$ with the two-hole binding energy. Energy conservation yields an upper limit of $Q=2.8$ keV for the bolometer spectrum. To integrate over the continuum energy of the shake-off electron (7) we divided the interval $\langle 0.0,2.8\rangle$ keV into 417 mesh points. From the left to the right with increasing bolometer energy $E_c$ the start of the different shake-off contributions are indicated in the figure. The energy difference between $Q$ and $E_c$ is carried away by the neutrino, which cannot contribute to the bolometer spectrum.

Figure 10

Theoretical results of the sum of all $s$-waves shake-off contributions. The steep rise on the left of the maxima is connected with the binding energy of the two-hole states (see Table 3), which contribute to the excitation of Dy. The rise is softened by the width of these states (Table 3). The shake-off continuum contribution has to be added on top of these two-hole contributions. The energy difference between $Q=2.8$ keV and $E_c$ is carried away by the neutrino and does not show in the bolometer spectrum. The theoretical results are adjusted to the experiment [4, 21] at the $N1,4s_{1/2}$ peak in Fig. 8.