Effect of final-state interactions in allowed β decays. I. General formalism

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Abstract
In this work the β decay of a nucleus hosted in an atomic or a molecular system is considered. A formalism is derived that allows the calculation of the changes in the β spectrum that are caused by the Coulombic final-state interaction of the β electron (or positron) with all molecular electrons as well as nuclei. Explicit formulas are derived for the matrix elements occurring in a perturbational treatment up to the pure first-order correction term, and different approximations are discussed. The result obtained in this work is especially applicable to cases where the decaying nucleus is part of a molecular system. In contrast to previous works a complete partial-wave analysis of the interaction operator has been performed, and for the first time explicit results are derived for all partial waves. The total symmetric contribution agrees (for atoms) with previous results and confirms the early work of Rose. However, the nonspherical contributions are of the same order of magnitude as the total symmetric contribution and have therefore to be taken into account, if the Rose correction is applied. This can be done in a very simple way, since the correction turns out to be (approximately) proportional to the number of electrons but otherwise completely independent of the atomic or molecular system under consideration. In the case of molecular systems also the contribution of the spectator nuclei has to be considered, since this contribution is again of the same order of magnitude as the correction term given by Rose. It is shown that the interaction with the decaying nucleus (that leads to the occurrence of the Fermi function) is very well approximated within the present perturbational approach, where it is treated in the same way as the interaction with the other particles.