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

Electron capture can determine the electron neutrino mass, while the β decay of tritium measures the electron antineutrino mass and the neutrinoless double β decay observes the Majorana neutrino mass. In electron capture, e.g., ₆₇¹⁶³Ho+e^-→*₆₆¹⁶³Dy+νₑ, 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 ¹⁶³Ho. 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 *₆₆¹⁶³Dy after electron capture on ₆₇¹⁶³Ho. 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 ¹⁶³Ho and thus can probably not affect the determination of the electron neutrino mass.