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

The proportionality between single charge-exchange reaction cross sections in the forward direction as found, for example from $(p,n)$ and $(^3$He,$t)$ and from $(n,p)$ and $(d,^2$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 ${}^{76}$Ge-${}^{76}$Se system that may undergo double beta-decay, we demonstrate that the proportionality is a relative good assumption for reactions changing a neutron into a proton, i.e. ${}^{76}$Ge$(p,n){}^{76}$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 putting 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 beta decay a single proton must be taken from an occupied SP state and a neutron placed in an unoccupied one. This second process often is Pauli forbidden in medium-heavy nuclei and only can be effected if the Fermi surface is smeared out. Such is the case for ${}^{76}$Se$(n,p){}^{76}$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 ($NN$) 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.