Hyperspherical hidden crossing calculation of Ps formation in low-energy e$^+$-Na collisions

The hyperspherical hidden crossing method (HHCM) can provide important insight into scattering processes. Previously, we have used the HHCM to calculate the Ps(1s)-formation cross section in low-energy e$^+$-H [1] and e$^+$-Li [2] collisions. Here we apply the HHCM to low-energy e$^+$-Na collisions. We use the Peach model potential and treat e$^+$e$^-$Na$^+$ as an effective three-body system. We calculate the Ps(1s)-formation cross sections for $0 \le L \le 3$ and compare our results with a hyperspherical close-coupling calculation [3]. The HHCM provides an explanation for the small S-wave Ps(1s)-formation cross section. The S-wave St\"uckelberg phase is close to $\pi$ for the three collision systems due to destructive interference between the two amplitudes that correspond to different paths leading to Ps(1s) formation. \\[4pt] [1] S.~J.~Ward, J.~H.~Macek, and S.~Yu.~Ovchinnikov, Phys.~Rev.~A {\bf 59}, 4418 (1999); S.~J.~Ward and J.~Shertzer (unpublished).\\[0pt] [2] S.~J.~Ward and J.~Shertzer, Phys.~Rev.~A {\bf 68}, 032720 (2003); {\bf 75}, 012713 (2007).\\[0pt] [3] Anh-Thu Le, M.~W.~J.~Bromley and C.~D.~Lin, Phys.~Rev.~A {\bf 71}, 032714 (2005); Anh-Thu Le (private communication).