Electron capture in 163Ho, overlap plus exchange corrections and neutrino mass

Holmium 163 offers perhaps the best chance to determine the neutrino mass by electron capture (EC). This contribution treats the EC in (163)Holmium completely relativistic for the overlap and exchange corrections and the description of the bolometer spectrum. The theoretical expressions are derived consistently in second quantization with the help of Wick's theorem assuming single Slater determinants for the initial Ho and the final Dy atoms with holes in the final ns(1/2) and np(1/2) states. One needs no hand waving arguments to derive the exchange terms. It seems, that for the first time the multiplicity of electrons in the orbital overlaps are included in the numerical treatment. Electron capture e(-) + p -> n + nu(e) is proportional to the probability to find the captured electron in the parent atom at the nucleus. Non-relativistically this is only possible for ns(1/2) electron states. Relativistically also p(1/2) electrons have a probability due to the lower part of the relativistic electron spinor, which does not disappear at the origin. Moreover relativistic effects increase by contraction the electron probability at the nucleus. Capture from other states are suppressed. However they can be allowed with smaller intensity due to finite nuclear size. These probabilities are at least three orders smaller than the EC from 3s(1/2) and 3p(1/2) states. The purpose of this work is to give a consistent relativistic formulation and treatment of the overlap and exchange corrections for EC in Ho-163(67) to excited atomic states in Dy-163(66) and to show the influence of the different configurations in the final Dy states. The overlap and exchange corrections are essential for the calorimetric spectrum of the de-excitation of the hole states in dysprosium. The slope of the upper end of the spectrum, which contains the information on the electron neutrino mass, is different. In addition the effect of the finite energy resolution on the spectrum and on the determination of the neutrino mass is studied. The neutrino mass must finally be determined by maximum likelihood methods to fit the theoretical spectra at the upper end near the Q value varying the neutrino mass, the Q-value and probably also the energy resolution, because at the moment Q and the energy resolution are not known to the accuracy needed.