Phase-Shift Analysis of Single-Channel Reactions

The calculation of phase shifts from experimental data has usually involved "matching" of differential cross-section data at a number of angles equal to the number of assumed nonzero phase shifts, followed by minimization of the errors at other angles. It is possible to derive $n$ simultaneous equations for $n$ assumed phase shifts by calculating $n$ "moments" of a suitable expression for the differential cross section. We have derived such a theoretical expression for the problem of single-channel elastic scattering of charged particles with channel spin ½, and performed the integrations. For each particular experimental problem, the cross-section data are to be integrated numerically. Then $n$ simultaneous equations are obtained which may be solved for $n$ phase shifts. As an example of this method, we have analyzed proton-alpha scattering data at an incident energy of 5.78 Mev and obtained values for five assumed phase shifts.