DETERMINATION OF SCATTERING PHASE-SHIFTS VIA THE GENERALIZED UNITARITY THEOREM FOR SPIN-ORBIT INTERACTIONS

The unitarity conditions upon the scattering amplitudes for the elasti c scattering of spin-1/2 particles from spin-0 targets at energies bel ow the first inelastic threshold transcribe to a set of coupled nonlin ear integral equations for the phase functions of two helicity amplitu des and thence, by simple linkage, to the non-spin-Rip and spin-flip s cattering amplitudes. From the latter set, by Legendre integrations, o ne obtains the scattering phase shifts, delta((l, j=l+/-1/2)). Input t o the study are the differential cross section and the polarization, { (d sigma/d Omega)(theta),P(theta)}. An iterative method of solution ba sed upon Frechet derivatives and with generalized cross validation (GC V) smoothing of the variations between iterates can give convergent, s table, and accurate results. Two test cases, the first built upon a mo del set of (small) phase-shift values and the second for an optical mo del calculation of 1-MeV neutrons scattered from an cu particle, have been used to demonstrate convergence and accuracy. There are natural a mbiguities (fourfold, in fact) for the phase functions of the scatteri ng amplitudes since data are invariant to complex conjugation of, or t he Minami transform on, the phase shifts of the mirror data set {(d si gma/d Omega)(theta),-P(theta)}, as well as to the combined action of c omplex conjugation and Minami transformation of the phase shifts given by the initial solution. Those ambiguities are presented herein and a re shown not to pose numerical problems in solution, Provided the init ial guesses are not near to the symmetry ''lines'' of the four solutio ns, and the GCV process is used to prevent branch hips occurring at sc attering angles where the allowed solutions intersect.