Neutron-Proton Scattering with Spin-Orbit Coupling. II. Variational Formulation and Effective Range Theory

The variational methods of Schwinger et al. are used to give variationally correct expressions for the three independent real parameters in the scattering matrix (δ_Jα, δ_Jβ, and εJ) for the "mixed" states, such as ³S₁+³D₁, of the neutron-proton system with spin-orbit coupling. The relation between these variational principles and the effective range theory is discussed briefly. The effective range expansions are then derived nonvariationally. The two leading terms are given for k⁵cotδ_1β and for k^-2tanε₁. The three leading terms are given for kcotδ_1α.