Determination of scattering phase shifts via the generalized unitarity theoremfor spin-orbit interactions

Abstract

The unitarity conditions upon the scattering amplitudes for the elastic scattering of spin- particles from spin-0 targets at energies below the first inelastic threshold transcribe to a set of coupled nonlinear integral equations for the phase functions of two helicity amplitudes and thence, by simple linkage, to the non-spin-flip and spin-flip scattering amplitudes. From the latter set, by Legendre integrations, one obtains the scattering phase shifts, ${\mathrm{\ensuremath{\delta}}}_{(\mathrm{l},\mathrm{}\mathrm{j}\mathrm{=}\mathrm{l}\ifmmode\pm\else\textpm\fi{}1\mathrm{/}2)}$. Input to the study are the differential cross section and the polarization, {(d\ensuremath{\sigma}/d\ensuremath{\Omega})(\ensuremath{\theta}),P(\ensuremath{\theta})}. An iterative method of solution based upon Frech\'et derivatives and with generalized cross validation (GCV) smoothing of the variations between iterates can give convergent, stable, and accurate results. Two test cases, the first built upon a model set of (small) phase-shift values and the second for an optical model calculation of 1-MeV neutrons scattered from an \ensuremath{\alpha} particle, have been used to demonstrate convergence and accuracy. There are natural ambiguities (fourfold, in fact) for the phase functions of the scattering amplitudes since data are invariant to complex conjugation of, or the Minami transform on, the phase shifts of the mirror data set {(d\ensuremath{\sigma}/d\ensuremath{\Omega})(\ensuremath{\theta}),-P(\ensuremath{\theta})}, as well as to the combined action of complex conjugation and Minami transformation of the phase shifts given by the initial solution. Those ambiguities are presented herein and are shown not to pose numerical problems in solution, provided the initial guesses are not near to the symmetry ``lines'' of the four solutions, and the GCV process is used to prevent branch flips occurring at scattering angles where the allowed solutions intersect.