Introduction

One of the goals of modern nuclear physics is to understand the structures of nuclei far from the line of stability and at extremes of isospin. Thus far such studies have largely dealt with the light nuclei for which radioactive beams have been available at facilities such as TRIUMF/ISAC, NSCL, RIKEN, and CERN/ISOLDE. Early experiments have lead to the identification of novel structures of which the halo [1] is the best known after its identification in ¹¹Li. In light of projected experimental facilities using radioactive ion beams to extend such investigations to heavy nuclei, there have been advances in the theoretical study of neutron rich nuclei across the whole mass range (see refs. [1,2,3,4,5] for reviews). Studies of such nuclei have topical interest since nuclei far from stability play an important role in nucleosynthesis with short-lived species formed as part of the r- and rp- processes. The structures of those nuclei determine the rates at which the nucleon-capture reactions proceed against the interplay of weak decays and photo-disintegration and thus are key elements in the determination of the abundances of stable nuclei in the universe.

Of course, these exotic systems are of interest in their own right given that they may exhibit forms of nuclear matter quite different from that known with the stable isotopes. Of particular interest are the separate density distributions of proton and neutrons. As well as halos, heavier systems may lead to the identification of pronounced neutron skins [6] and/or to dramatic changes in the nuclear shell structure [7].

An important observable by which the model structures are tested is the ground state density. For stable nuclei, one usually seeks a measure of that from the elastic scattering of electrons which measures the charge and current densities from the longitudinal and transverse form factors, respectively. To complement that information, which primarily focuses on the proton density itself, one looks to the elastic scattering of nucleons from nuclei which measures the matter density. As the effective nucleon-nucleon (NN) interaction is dominated by the isoscalar ³S₁ channel, proton scattering predominantly measures the neutron density and vice versa. For radioactive nuclei, the only available measure of the ground state density comes from the scattering of nuclei from hydrogen which, in the inverse kinematics, corresponds to proton scattering. However, to gauge the validity of the structure model from which the density is obtained, a model of scattering which is able to predict the scattering observables and is sensitive to changes in the density is required.

For nuclei above the fp-shell, mean-field models of structure are at the forefront of current studies of the ground state densities. Usually those calculations are made in a relativistic Hartree or Hartree-Fock model [8,9,10,11] or a non-relativistic Skyrme-Hartree-Fock-Bogoliubov model [12,13,14]. This mean-field approach works as the ground state properties for heavy nuclei arise more generally from the bulk properties of the density as opposed to single particle properties. As one approaches the drip lines, however, surface properties become more important and, in concert, so do single particle densities.

A test case has been to use nucleon elastic scattering to determine/extract the neutron skin thickness in ²⁰⁸Pb. Karataglidis et al. [15] have shown that the differential cross sections from the elastic scattering of 200 MeV protons from ²⁰⁸Pb suggests a neutron skin thickness in the nucleus of $\sim 0.17$ fm. More importantly, that study also showed that such scattering analyzes could select between disparate model structures that had the same root mean square (rms) radii. By itself, the rms radius is not an adequate indicator of the validity of a model structure. To use the scattering data to differentiate between models of structure, a predictive theory of nucleon-nucleus (NA) scattering was needed. Just such has been developed in the past decade [16]. That theory is direct in that all quantities required are defined a priori with no a posteriori adjustment of results. With the nucleus viewed as a system of A nucleons, NA scattering is determined by an optical potential formed by a folding process. That folding requires use of realistic interactions of the projectile with each and every nucleon constituting the ground state target density. Such microscopic approaches defining the NA optical potential have been quite successful in predicting both angle-dependent and integral observables of elastic scattering [16,17]. In the coordinate space approach, and for analyzes that use the DWBA98 programs [18], the projectile-target nucleon interaction not only is complex but also it is energy dependent and non-local. We have used that program to calculate cross sections for the scattering of all the even mass isotopes ^{100 - 176}Sn from hydrogen at an energy of 200A MeV using optical potentials formed by folding realistic effective NNinteractions with the details of densities of those nuclei given by the mean-field models of their structure. With optical potentials so formed previously, differential cross sections and spin observables for proton scattering at 200 MeV specifically and from diverse targets ranging from ³He to ²³⁸U have been predicted and found to agree very well with data. The complex (non-local) optical potentials are formed by folding effective interactions determined from NN gmatrices [solutions of Brueckner-Bethe-Goldstone (BBG) equations for nuclear matter] found from realistic free NN forces. Details of the specifications of those effective NN interactions, of the folding process that gives the optical potential, and of the successful predictions of differential cross sections and analyzing powers from the scattering of protons at diverse energies and from diverse mass targets, have been summarized before [16] and so are not repeated herein.

In the next section, we outline the structure models used in our scattering calculations and present therein details of the proton and neutron matter distributions generated therefrom. In Section 3 we show the results found using those structures and an established effective NN force in nuclear matter in generating g-folding optical potentials for a wide range of Sn isotopes scattering from hydrogen. An energy of 200A MeV has been used. In Section 4 application of structure and scattering models is made for two cases ( ^118,120Sn) for which proton elastic scattering data are available. Conclusions are presented in Section 5.