Relativistic Effect and $\alpha -^{40}$Ca and $\alpha -^{48}$Ca Scattering

Past optical model calculations of elastic differential scattering cross sections of $\alpha $- particles incident upon $^{40}$Ca and $^{48}$Ca have been examined to illustrate the need for relativistic calculations at 1.37 GeV. Khoa et al. and Nakano et al$^{1}$. have constructed potentials that match the experimental data very well in the forward angles using Woods-Saxon and Mexican hat types of potentials using the non-relativistic Schroedinger equation. However, when the potentials are used in the relativistic Schroedinger equation (RSE) to produce scattering cross sections, the fits are no longer that good. At 1.37 GeV, the $\alpha $-particle momentum calculated relativistically differs significantly from the one determined from the non-relativistic energy-momentum relation. The new $\alpha -^{40}$Ca and $\alpha -^{48}$Ca potentials determined using RSE, differ from those of Khoa et al$^{1}$. and Nakano$^{2}$. et al. $^{1}$D.T. Khoa, M. Nakano et al., Phys. Rev. C. 65, 024611, 2002 $^{2}$ M. Nakano et al., Phys. Rev. C. 40, 1323, 1989.