Pion-nucleus inelastic scattering: Reaction contributions and nuclear spin determinations

Formulas for pion-nucleus inelastic scattering are presented in a form that may suggest experiments to isolate various contributions to the reaction, including $S$-wave, $P$-wave spin and nonspin flip, and effects of nucleon Fermi motion. Adopting a form of the distorted wave impulse appoximation, we obtain an expression for inelastic cross sections that clearly separate the pion laboratory energy $E$, three-momentum transfer $q$, and scattering angle $\theta$ dependences. The result is similar to the separation of longitudinal and transverse form factors in inelastic electron scattering. By varying the energy of the incident pion, but working at fixed $q$, one can determine whether a given nuclear excitation has natural or unnatural parity. By working at fixed $\theta$, and varying $E$ and thus $q$, one can isolate different reaction contributions—spin, scalar, and "convection current." We also discuss the potential usefulness of studying the energy dependence of angle-integrated differential cross sections at fixed energy loss. The predictions of our formulas are in good agreement with recent data on natural and unnatural parity excitations in ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}(\pi ,{\pi }^{\prime }){}_{}{}^{12}{}_{}{}^{}\mathrm{C}_{}^{*}{}_{}{}^{}$. Thus, this approach may be useful in analyzing future data in which the final nuclear spin is uncertain. Future experiments with selective $q$, $E$, and $\theta$ variations to separate nuclear structure from reaction-mechanism uncertainities are suggested.