Partial-Wave Scattering by Non-Spherically-Symmetric Potentials. I. General Theory of Elastic Scattering

A general method is developed to obtain the elastic-scattering cross section of an asymmetric potential $V\left(\mathrm{r}\right)=\Sigma l^{}{v}_l\mathrm{}\left(r\right)\mathrm{}{P}_l\left({\hat{\Omega }}_l·\hat{r}\right)$ where we have expanded the potential as a sum of multipoles and ${\hat{\Omega }}_l$ is a unit vector along the $2l$-pole axis. We assume that $v_l\mathrm{}\left(r\right)\mathrm{}$ is effectively zero beyond a cutoff radius $R$ and we solve the resulting set of coupled differential equations in terms of a scattering matrix S. A general expression for the scattering amplitude $f(\mathrm{k}, {\mathrm{k}}^{\prime })$ is derived in terms of the elements of S and $3j$ symbols. Finally, as an example, we apply the method to the problem of calculating the total and momentum-transfer cross sections for a randomly orientated set of axially symmetric centers.