Application of the Faddeev-Watson expansion to thermal collisions of Rydberg atoms with neutral particles

The Faddeev-Watson expansion (FWE) for the $T$ operator is applied to the study of thermal collisions between Rydberg atom and neutral atom. These collisions are considered as a three-body problem (the perturber, the Rydberg electron, and its parent core) and it is assumed, as already done in most theoretical works dealing with Rydberg-atom-atom collisions, that the core-perturber interaction can be neglected. Then the evaluation of the FWE first- and second-order terms is made tractable by using an appropriate separable potential for the Rydberg-electron-perturber interaction. The evaluation of the second-order term allows us to estimate the importance of taking into account explicitly the Rydberg-electron-core interaction in the expression of the (three-body) $T$ operator for the thermal collisions considered. Detailed calculations for the process $\mathrm{Rb}(n, l=0)+\mathrm{He}\to \mathrm{Rb}(n^{\prime },l^{\prime })+\mathrm{He}$ are presented and discussed. The FWE second-order term has been evaluated for the first time by taking the (two-body) $t$ operator associated with the Rydberg atom (valence electron plus parent core) as the Coulomb potential. The contribution of the FWE second-order term to the scattering amplitude decreases as $n$ increases and is found especially significant when both the momentum transfers involved in the collision are large and the values of $l$ and $l^{\prime }$ are small.