Semiclassical Calculation of ElasticH+-He Differential Scattering Cross Sections

Abstract

Elastic differential cross sections for proton-helium scattering are calculated in the semi-classical approximation for two assumed interaction potentials. Both potentials are of the form $V(r;A,B,C)=\left(\frac{2}{r}\right)e^{\frac{-r}{A}}[1+\frac{r}{A}+\frac{1}{2}{r}^2(\frac{1}{A^2}-\frac{U}{B}\left)\right]-U{[1+\frac{r}{B}+{\left(\frac{r}{C}\right)}^2+\frac{2U{r}^4}{\alpha }]}^{-1\mathrm{}},$ where $U$ is the difference between the ground-state energies of He and ${\mathrm{Li}}^{+}$ (4.373 11 hartree) and $\alpha$ is the polarizability of He (1.3835 ${\mathrm{bohr}}^3$). The first potential $V_M\equiv V(r;0.423,0.483,0.441\mathrm{})\mathrm{}$ fits the He-${\mathrm{H}}^{+}$ ground-state energies which Michels has calculated. The second, $V_W\equiv V(r;0.442,0.505,0.451\mathrm{})\mathrm{}$, is similar to $V_M$ except that its minimum is decreased by 10% to agree with the value obtained by Wolniewicz. The cross sections for these two potentials are shown for protons incident at energies $T$ of 7, 19, 58, and 116 eV in the laboratory frame and for scattering angles, at each energy, out to the rainbow angle ${\theta }_R$. ${\theta }_R$ is given in center-of-mass coordinates by the expression ${\theta }_{R}T\cong 0.1$ rad hartree. As the collision energy decreases, the cross sections develop oscillatory structure not present in the classical cross sections. This structure and the rainbow angle are sensitive to the choice of potential, which suggests that measurements of ${\mathrm{H}}^{+}$-He cross sections may be used to test the suitability of, e.g., the Born-Oppenheimer potential for scattering phenomena. It is also suggested that many-body calculations of these cross sections would allow, by comparison with the present results, an evaluation of the potential scattering model.