Charge-exchange reaction cross sections and the Gamow-Teller strength for double β decay

Abstract

The proportionality between single charge-exchange reaction cross sections in the forward direction as found, for example, from $(p,n)$ and $(^{3}\mathrm{He}$,$t)$ and from $(n,p)$ and $(d,^{2}\mathrm{He}$) reactions, and the Gamow-Teller (GT) strength into the same final nuclear states has been studied and/or assumed often in the past. Using the most physically justified theory we have at our disposal and for the specific example of the $^{76}\mathrm{Ge}$-$^{76}\mathrm{Se}$ system that may undergo double \ensuremath{\beta} decay, we demonstrate that the proportionality is a relatively good assumption for reactions changing a neutron into a proton, i.e., $^{76}\mathrm{Ge}(p,n)^{76}\mathrm{As}$. In this channel, the main contribution to the GT strengths comes from the removal of a neutron from an occupied single-particle (SP) state and the placement of a proton into an unoccupied SP state having either the same state quantum numbers or those of the spin-orbit partner. In contrast to this, in the second leg of the double \ensuremath{\beta} decay, a single proton must be taken from an occupied SP state and a neutron placed into an unoccupied one. This second process often is Pauli forbidden in medium-heavy nuclei and can only be effected if the Fermi surface is smeared out. Such is the case for $^{76}\mathrm{Se}(n,p)^{76}\mathrm{As}$. Our results suggest that one may not always assume a proportionality between the forward-angle cross sections of the charge-exchange reactions and the GT strength in any such medium-heavy nuclei. The discrepancy originates from a pronounced effect of the radial dependence of the nucleon-nucleon ($\mathit{NN}$) interaction in connection with the Pauli principle on the cross sections in the $(n,p)$ reaction channel. Such a radial dependence is completely absent in the GT transition operator.