Elastic and inelastic scattering of 100 MeV protons from the even-even titanium isotopes

The differential cross sections for elastic and inelastic scattering to the first excited (${2}^{+}$) states of 100 MeV protons from $^{46,48,50}\mathrm{Ti}$ were measured. The angular ranges covered were 9\ifmmode^\circ\else\textdegree\fi{} to 110\ifmmode^\circ\else\textdegree\fi{} for $^{46,50}\mathrm{Ti}$ and 9\ifmmode^\circ\else\textdegree\fi{} to 168\ifmmode^\circ\else\textdegree\fi{} for $^{48}\mathrm{Ti}$. The elastic scattering data were analyzed with a local optical model potential. The quality of fits to the data covering the smaller angular range ( 110\ifmmode^\circ\else\textdegree\fi{}) is good for all three isotopes, with $\frac{{\ensuremath{\chi}}^{2}}{N}\ensuremath{\sim}1.5$; whereas the fit to the large angle data of $^{48}\mathrm{Ti} (\ensuremath{\lesssim}{168}^{\ensuremath{\circ}})$ is not as satisfactory ($\frac{{\ensuremath{\chi}}^{2}}{N}\ensuremath{\sim}3.75$). Various modifications to the Woods-Saxon shape of the real central potential showed no real improvement. Nevertheless, the large angle data appear to favor a slightly sharper falloff in the real central potential near the surface, such as the Woods-Saxon-squared potential. Systematics of the phenomenological optical model potential of 100 MeV protons were studied. The derived volume integrals and rms radii seem to follow the usual $A$ dependence. The isospin dependence of the real potential was also extracted. The mean free paths computed directly from the imaginary potential strength have large uncertainties due to the insensitivity of the data to the central region of the imaginary potential. Distorted-wave Born approximation and coupled-channels Born approximation calculations for the collective excitation of the target nuclei to the first ${2}^{+}$ states provided deformation parameters, ${B}_{2}$, in agreement with those from other proton and alpha scattering data and electromagnetic measurements.