Higher Partial Waves in Positron-Hydrogen Scattering

Abstract

The minimum-principle (MP) formulation of scattering theory is applied to the study of the scattering of positrons by atomic hydrogen for a number of incident energies between 0 and 6.8 eV, the threshold for pickup. Rigorous lower bounds, ${\ensuremath{\eta}}_{1}$ (MP) and ${\ensuremath{\eta}}_{2}$ (MP), are obtained on the $p$-wave $L=1$ and $d$-wave $L=2$ phase shifts, ${\ensuremath{\eta}}_{1}$ and ${\ensuremath{\eta}}_{2}$. The method requires the exact numerical solution of the scattering wave function and of the Green's function of the static (one-body) equation, and the orthogonalization of the trial function $Q{\ensuremath{\Psi}}_{t}$ to the ground-state wave function; at that stage the method proceeds as in a standard variational calculation. $Q{\ensuremath{\Psi}}_{t}$ includes contributions from hydrogenic states with angular momenta ${l}_{1}=0,1,2, \mathrm{and} 3$ for $L=1$, and ${l}_{1}=0,1, \mathrm{and} 2$ for $L=2$; the associated radial functions contain arbitrary variational parameters, as do the radial functions associated with the positron. Contributions from each ${l}_{1}$ to the total ${\ensuremath{\eta}}_{L}$ (MP), and the effects of the coupling of different ${l}_{1}$, are studied, and extrapolation estimates of ${\ensuremath{\eta}}_{1}$ and ${\ensuremath{\eta}}_{2}$ are obtained. The rigorous (variational) bounds and the estimates based on extrapolation are compared with close-coupling and other previous estimates and, at very low energies, with predictions based on long-range polarization effects.