Elastic scattering of alpha particles near the Coulomb barrier and matter distribution of medium and heavy nuclei

The elastic scattering of $\alpha$ particles near 180° was measured in the vicinity of the Coulomb barrier for ${}_{}{}^{110,112,114,116}{}_{}{}^{}\mathrm{Cd}$, ${}_{}{}^{112,114,116,118,120,122,124}{}_{}{}^{}\mathrm{Sn}$, ${}_{}{}^{122,124,126,128,130}{}_{}{}^{}\mathrm{Te}$, ${}_{}{}^{144,148,150,152}{}_{}{}^{}\mathrm{Sm}$, and ${}_{}{}^{204,206,208}{}_{}{}^{}\mathrm{Pb}$. An optical-model analysis using Woods-Saxon potentials shows that the usual parameters of the real part of the potential $V$, $R_{\mathrm{opt}}$ and $a$ must obey the relationship $V\exp \left[\frac{(R_{\mathrm{opt}}-R_{0.2})}{a}\right]=0.2\mathrm{}$ MeV in order to fit the data. The $\alpha$-nucleus distance $R_{0.2}$ at which the nuclear potential depth is -0.2 MeV can then be determined for each nucleus within ± 0.03 fm. An analysis in terms of a folding model was performed for ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ and ${}_{}{}^{124}{}_{}{}^{}\mathrm{Te}$. For the class of potentials thus obtained, it is the $\alpha$-nucleus distance at 0.5 MeV depth rather than at 0.2 MeV that appears to be best determined. The same analysis determines the radius $R_{\mathrm{FD}}=0.002\mathrm{}$ at which the nucleon density is 2 × ${10}^{-3\mathrm{}}$ nucleon/${\mathrm{fm}}^3$. The value of $R_{\mathrm{FD}}$ is found to depend mostly on the $\alpha$-nucleon effective interaction used, and very little on the functional form of the density distribution. Further evidence is presented in favor of the Gaussian interaction $-U_0\exp (-K^2{r}^2)$ with $U_0=127\mathrm{}$ MeV and $K=0.6\mathrm{}$ ${\mathrm{fm}}^{-1\mathrm{}}$, which has been proposed by Sumner and which leads to the probable value $R_{\mathrm{FD}}=R_{0.2}-(3.06\mathrm{}\pm 0.03)$ fm. Other interactions are not excluded, however, and considering those proposed so far in the literature leads to $R_{\mathrm{FD}}=R_{0.2}-(3.11\mathrm{}\pm 0.14)$ fm. The average variation of $R_{\mathrm{FD}}$ with mass number is found to be $〈R_{\mathrm{FD}}〉=(1.355\mathrm{}{A}^{\frac{1}{3}}+0.87)$ fm for spherical nuclei. The rate of variation of $〈R_{\mathrm{FD}}〉$ with mass number is found to be in good agreement with the droplet model predictions, which is taken as an evidence that the surface thickness of spherical nuclei is practically constant from Sn to Pb.