Positronium formation and threshold behavior in positron-sodium collisions at low energies

We investigate the elastic and inelastic scattering of positrons by sodium atoms in both the ground state, Na($3s$), and excited states, Na*($3p$, $4s$, $3d$), using the hyperspherical coordinate method with a model potential to represent the atomic core. The threshold behavior of positronium (Ps) formation cross sections is analyzed as the positron impact energy $E$ approaches zero. Within this framework, we derive a generalized expression for partial-wave Ps-formation cross sections at low positron energies, which applies to both ground-state and excited-state sodium targets. Our results confirm that the total threshold behavior follows the expected power-law dependence: $\sigma_{\text{Ps}} \propto E^{-1/2}$ for exothermic reactions and $\sigma_{\text{Ps}} \propto E^{a}$ for endothermic reactions, where $a > 0$. Furthermore, we find that Ps-formation cross sections for positron scattering from excited Na states are significantly larger than those from the ground state in the low-energy region. A notable enhancement of Gailitis-Damburg oscillations is observed above the Ps($n=2$) threshold, which may account for the increase observed in experimental data. Incorporating contributions from excited sodium targets improves agreement with experimental results and may help resolve discrepancies between theoretical predictions and measurements.