Interaction of alpha particles in the lead region near the Coulomb barrier

The ($\alpha ,n$) cross section on ${}_{}{}^{208}{}_{}{}^{}\mathrm{Pb}$ and ${}_{}{}^{209}{}_{}{}^{}\mathrm{Bi}$ has been measured between 16 and 24 MeV by observing the $\alpha$ activity of the residual nuclei ${}_{}{}^{211}{}_{}{}^{}\mathrm{Po}$ and ${}_{}{}^{212}{}_{}{}^{}\mathrm{At}$. The relative yields to both the ground and metastable states of each residual nucleus were obtained and new half-life determinations have been made for these states. The absolute yield was measured at 20 and 22 MeV to normalize the relative data. Below the ($\alpha ,2n$) threshold, $\sigma (\alpha ,n)$ constitutes most of the total reaction cross section, and the ($\alpha ,n$) data, together with accurate elastic scattering measurements at 19, 20, and 22 MeV are reproduced well by a six-parameter optical model. The fits are sensitive to the reaction cross-section data, and serve to define the magnitude and shape of the real potential at a radial distance of about 11 fm. The real potential in this region dominates the interaction at the present energies. The imaginary potential is relatively small; it is not well determined and serves only to absorb those waves which penetrate the real potential. The real potential obtained is interpreted in terms of an $\alpha$-folding model using a free $\alpha$-nucleon two-body force. The potential near the peak of the Coulomb barrier is determined primarily by the target matter density near 9 fm, where it is approximately 1% of its value at the center of the nucleus. Implications concerning the relative distributions of target neutron and protons are discussed.