Extraction of neutron densities from elastic proton scattering byPb206,207,208at 650 MeV

The method for extraction of neutron and matter densities from data on cross sections and spin observables for elastic p-nucleus scattering is presented. The effective nuclear potential was treated to the first order with respect to the t-matrix and density distributions. To eliminate uncertainties associated with the t-matrix, the latter has been determined directly from p${\mathrm{-}}^{40}$Ca scattering data, where the proton and neutron densities are known sufficiently well. The medium modifications in the t- matrix have been included by means of density dependent terms added to the free t matrix. The neutron densities to be found were expressed as a Hartree-Fock (HF) distribution plus a small correction expanded in a Fourier-Bessel series with inclusion of only low-frequency components determined by the data. High-frequency components are presented by the HF term. Expressions for the physical observables were linearized in terms of the expansion coefficients, the latter being determined through an iterative procedure from the ${\mathrm{\chi }}^2$ criterium. The method enables one to obtain statistical fluctuations in those Fourier components as they are generated by experimental error bars. The normalization and model uncertainties have been also estimated. The extracted densities ${\mathrm{of}}^{206,207,208}$Pb and their differences were compared with the HF results calculated by us with the Skyrme interaction. On the whole, pretty good agreement has been achieved.