Pick-up and inelastic scattering contributions to the proton optical potential

The contributions from couplings with the available inelastic (p, p′) and pick-up (p, d) channels to the elastic scattering of 30 MeV protons from ⁴⁰Ca, ⁴⁸Ca and ²⁰⁸Pb were studied. The equivalent non-local potentials were constructed and attempts were made to find local equivalents to them. These potentials are dependent upon angular momentum. Inelastic scattering produces an attractive contribution to the real potential but pick-up results in a much larger repulsive term. The main features of the observed scattering from ²⁰⁸Pb can be reproduced by coupling to 24 channels provided a supplementary absorptive term is included. The explicit couplings by themselves can produce 90 % of the observed absorption cross section but only about one-half of the empirical absorptive potential strength.     

Elastic scattering of 1-GeV protons by nuclei

The calculated elastic scattering of 1.04-GeV protons by ²⁰⁸Pb, ⁵⁸Ni, and ⁴⁰Ca is compared with experiment. The effects of the spin-orbit potential, Pauli and short-range correlations, variations in the ratio of the real to the imaginary part of the nucleon-nucleon scattering amplitude at 0° are found to have similar effects. Further experiments are required before these effects can be disentangled. Changes of the neutron density compared to the proton density can be distinguished. The proton polarization by elastic scattering by ²⁰⁸Pb and ⁴⁰Ca is calculated omitting the effects of correlations.     

DWBA approach to inelastic scattering at medium energies

Medium energy proton elastic and inelastic scattering to states of ⁵⁸Ni and ²⁰⁸Pb, and ⁴He elastic and inelastic scattering to states of ⁴⁰Ca, are analyzed using the partial wave approach, by solving the Schrödinger equation with relativistic kinematics and using the distorted wave Born approximation. Our results can be compared with results of several previous analyses of the nucleon inelastic data using the Glauber approximation. Our calculations are absolute, using nuclear collective parameters obtained from a survey of a large number of low-energy analyses of inelastic scattering of electrons, nucleons and nuclei from ⁴⁰Ca, ⁵⁸Ni and ²⁰⁸Pb.     

Multiple diffractive analysis of elastic and inelastic scattering by 1 GeV protons on 12C, 58Ni,and 208Pb

Recent elastic and inelastic scattering experiments of 1 GeV protons on ¹²C, ⁵⁸Ni, and ²⁰⁸Pb have been analyzed using multiple diffractive scattering theory.Elastic angular distributions have been calculated, including pair correlations in many-particle densities. There is evidence from the Ni and Pb data for the presence of correlations, but some evaluated correlation corrections do not produce detailed agreement.It is shown that the Glauber series for coherent inelastic scattering can be approximately summed to yield an eikonal form of the DWIA. In particular the smaller angle parts of the inelastic distributions are well described in this approximation.The expression for the summed incoherent inelastic cross section is considered as a sum rule. It is found that for ¹²C the excitation of the three lowest excited states exhaust ? 25 % of the sum rule up to ?t ? 1 fm^?2. For larger t quasi-elastic knock-out largely exhausts the sum rule.     

Nonrelativistic description of nucleon-nucleus scattering

Within the three-dimensional semiclassical approximation, an analytic expression is obtained for the amplitude of proton-nucleus scattering at intermediate energies of incident protons. The method for deriving this amplitude is based on the use of the high-energy approximation with distorted waves. In view of the short-range character of proton-nucleon interaction, the process of proton-nucleus scattering is represented as a series of single scattering events occurring on each individual nucleon. With the aid of the proposed mathematical formalism, a recursion relation is derived that makes it possible to express the nuclear form factor obtained within the distorted-wave method in terms of the sum of an infinite Born series. Parameters that characterize the distributions of protons and neutrons in the spherical nuclei ⁴⁰Ca, ⁴⁸Ca, ⁹⁰Zr, and ²⁰⁸Pb and which include the width of the surface layer of nucleons and the root-meansquare radii of the proton-, neutron-, and nucleon-density distributions are determined from an analysis of the measured cross sections for the elastic scattering of 1-GeV protons, a modified Fermi function being employed for the nucleon-density distribution.     

Angular distribution of 1.04 GeV protons scattered by 12C, 58Ni, 208Pb

Angular distribution measurements for elastic and inelastic scattering of 1.04 GeV proton by ¹²C, ⁵⁸Ni and ²⁰⁸Pb with an energy resolution of the order of 100 keV are presented.     

Relativistic DWBA calculations for polarization transfer in proton inelastic scattering

The matrix element for inelastic scattering of protons leading to the excitation of collective states in even-even nuclei is calculated in the framework of Dirac phenomenology using a DWBA approach. The deformed parts of the Lorentz scalar and four-vector optical potentials serve as the transition operators. The results are compared with the recently measured polarization transfer coefficients on ⁴⁰Ca and ²⁰⁸Pb at 500 MeV. The agreement with experiment is good.     

The excitation of collective nuclear states by high energy particles

The excitation of collective nuclear states by high energy particles is considered within the framework of the Glauber theory. The approach is based on the adiabatic approximation and expansion of the scattering amplitude in powers of the nonsphericity parameter. The formulae for the excitation cross sections of the rotational and one- and two-phonon vibrational states, as well as the elastic scattering cross section with the collective motion included, are obtained. The effect of nucleon correlations in both elastic and inelastic cross sections is also studied. The theoretical predictions are compared to new data on 1 GeV proton scattering by ⁵⁸Ni, ²⁰⁸Pb, ¹²C, ⁴⁰Ca and ³⁹K. A comparison with electron scattering data is simultaneously carried out. The agreement with the experimental data is generally good.     

Nuclear matter sizes and isoscalar octupole transition rates of204,206,208Pb from 104 MeV α-particle scattering

Differential cross sections for elastic and inelastic scattering of 104 MeVα-particles from^204,206,208Pb were measured with high angular accuracy. The experimental results were analysed on the basis of a semimicroscopic folding model in order to determine the matter distributions at the nuclear surface and the transition densities. Using phenomenological parametrizations of the densities the analyses of elastic scattering yield rms radii of 〈r ²〉^1/2=5.55±0.06 (²⁰⁴Pb), 5.57±0.06 (²⁰⁶Pb), 5.63±0.05 (²⁰⁸Pb) fm. Various sensitivities affecting the results were studied. From the inelastic cross sections for the 3 ₁ ^? -states octupole transition probabilities and transition radii were derived by using different methods proposed in literature.     

Proton-nucleus elastic scattering at 1 GeV in A = 48 nuclei

The elastic scattering differential cross section for p-⁴⁸Ca and p-⁴⁸Ti at 1 GeV are discussed. Calculations based on self-consistent proton and neutron densities fail to reproduce the experimental data. As shown by simple arguments, the discrepancy may be related to incorrect nuclear surface slopes.     

Ground state correlations and electron scattering

The effect of random phase ground state correlations on charge densities and elastic electron scattering cross sections is studied for²⁰⁸Pb and⁴⁰Ca. Modifications of the charge densities relative to densities obtained by Hartree-Fock calculations with a density dependent force are studied using different RPA wave functions. The differences with respect to the HF charge distributions are discussed.     

Optical model potential for neutrons in the resonance region

A method is described for determining the strength of the complex central optical potential directly from neutron total cross sections measured below the threshold for inelastic scattering. Real and imaginary strengths deduced for ²⁰⁸Pb are consistent with higher-energy results and are shown to provide a good fit to above-threshold data.     

The contribution of nuclear inelastic excitation to the optical potential for heavy ions

Using the plane-wave approximation we derive analytical expressions for both the real and imaginary parts of the polarization potential arising from nuclear inelastic scattering. These potentials and the resulting elastic and inelastic cross sections are compared with exact coupledchannel calculations for ¹³C on ⁴⁰Ca at 68 MeV. The agreement, for the most part, is good. We also briefly discuss the numerical non-local potentials for this system and the imaginary polarization potential for ¹⁶O on ²⁰⁸Pb at 104 MeV.     

Distorted wave theory of scattering of nucleons with intermediate energies on nuclei

In the context of nonrelativistic theory in the distorted wave approximation, a three-dimensional form of analytical expression for the differential cross section of scattering of nucleons with intermediate energies on atomic nuclei is derived. In the context of this theory, the main parameters of elastic scattering of protons with incident energy of 1 GeV on the ²⁰⁸Pb nucleus are determined. For inelastic scattering of protons with nuclear surface vibrations, giant multipole resonances in the excited nucleus are investigated for the collective nucleus model. The energy losses of the scattered proton are calculated together with the energies of giant dipole and quadrupole resonances and nuclear surface vibration energy. This allows the deformation parameter of the excited nucleus to be calculated.     

Proton elastic scattering by nuclei: Fresnel and two-body correlation effects

In this paper we deal with the effects of Fresnel diffraction in the elastic scattering of 1 GeV protons by nuclei. Explicit expressions for the optical potential in⁵⁸Ni and²⁰⁸Pb are given, showing numerically that Fresnel effects and two-body correlations are of little importance in the elastic scattering of protons by nuclei.     

Elastic differential cross sections and analyzing powers for p+40,42,44,48Ca at 0.8 GeV

Differential cross sections and analyzing powers for the elastic scattering of 800 MeV polarized protons from ^40,42,44,48Ca are reported. A first-order, spin-dependent KMT optical potential analysis is presented from which the rms radii of the neutron densities are deduced. A comparison of these results with other determinations and with various theoretical predictions is given.     

Elastic and the 2+ inelastic scattering of high energy protons on12C and the alpha-particle model

The alpha-particle model has been applied to calculate the elastic and the 2⁺ inelastic angular distribution of 1 GeV protons on¹²C within the framework of Glauber multiple scattering theory. The results are compared with the recent experimental data and with some previous calculations. The model gives a fairly good account of the inelastic data and its predictions for the elastic scattering are essentially the same as those of the shell model.