Finite-Range Effects in the Distorted-Wave Impulse Approximation

The usual (zero-range) distorted-wave, impulse approximation applied to the inelastic scattering of high-energy protons by nuclei ignores the averaging of the two-nucleon $t$ matrix over the range of momentum transfers introduced by the distorted waves. The effects of including this averaging (the finite-range calculation) are investigated here at 50, 100, and 150 MeV for quadrupole transitions in ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ and ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ using a much simplified form of the two-nucleon $t$ matrix. It is found that the finite-range effects are not negligible even at the highest energy; in particular, the 150-MeV cross section for ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ is reduced by a factor of two in the forward direction, while the peak cross section for ${}_{}{}^{40}{}_{}{}^{}\mathrm{Ca}$ is reduced by 20% at this energy. An attempt is also made to fit the data on the $2^{+}$ level of ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$ at 45.5 MeV using the finite-range calculation, and it is found that the strength of the transition predicted by the impulse approximation (using the simplified $t$ matrix) is too small by a factor of more than three. Exchange effects were ignored in the finite-range calculations.