Semiclassical derivation of a local optical potential for heavy-ion elastic scattering

A semiclassical method to determine the contribution to the optical potential in the elastic channel due to the coupling to other processes taking place in heavy-ion collisions is developed. An application is made to the case of Coulomb excitation. The lowest-order term of our potential is shown to be identical to the potential derived by Baltz et al.     

Nuclear matter approach to the heavy-ion optical potential

An energy-dependent local potential for heavy-ion (HI) scattering is derived from Reid's softcore interaction using the Brueckner theory. The Bethe-Goldstone equation in momentum space is first solved with the outgoing boundary condition for two colliding systems of nuclear matter with the relative momentum K_r per nucleon. The Fermi distribution is assumed to consist of two spheres without double counting of their intersection separated by the relative momentum K_r. The angle-averaged Pauli projection function is given in the form of a one-dimensional integral. Secondly the optical potential for HI scattering is evaluated using the energy-density formalism. The energy density is calculated for two limiting cases: (i) the sudden approximation in which the spatial distribution of the two HI is described by an antisymmetrized cluster wave function, and (ii) the adiabatic limit represented by an antisymmetrized two-centre wave function. The complex HI potential is defined in terms of the energy density from nuclear matter so that both volume elements in the finite and the infinite systems have the same nucleon and kinetic energy density. This method is applied to the ¹⁶O + ¹⁶O, ⁴⁰Ca + ¹⁶O, and ⁴⁰Ca + ⁴⁰Ca potentials. The calculated results are compared with phenomenological potentials. Though in principle our approach can generate an imaginary part for the HI potential, the magnitude is too small. Reasons and possible improvements of this point are discussed.     

On the nuclear matter approach to the heavy-ion optical potential

A simple theory of the heavy-ion optical potential oV, based on the local density approximation, is presented. The colliding ions are described locally as two slabs of nuclear matter. The real part of the energy density of the two slabs is derived from the properties of nuclear matter, and for the imaginary part the “frivolous model” is applied. Results for oV in the case of two slabs are presented and compared with results of other calculations. Arguments are given in favour of using the frivolous model in the optical potential and the VUU calculations for heavy-ion collisions.     

Global optical model potential describing 12C-nucleus elastic scattering

We construct a new global optical model potential to describe the elastic scattering of ¹²C. The experimental data of elastic-scattering angular distributions and total reaction cross sections for targets from ²⁴Mg to ²⁰⁹Bi are considered below 200 MeV within the framework of the optical model. The results calculated using the derived global optical potential are then compared with the existing experimental data. The reliability of the global optical potential is further tested by predicting the elastic scattering data out of the mass and energy ranges, within which the global potential parameters are determined, and reasonable results are also obtained.     

The optical potential for the elastic heavy ion scattering

For the elastic scattering of heavy ions a microscopic model is developed which uses shell model wave functions which are localised at a certain distance both in coordinate and momentum space. The antisymmetrization is treated exactly. Using oscillator functions and the Brink-Boeker force the energy dependent real part of the optical potential for O₁₆- O₁₆ scattering is numerically determined. The results are compared with the analysis of experiments.     

The optical model for the elastic deuteron scattering

An optical model for the elastic scattering of deuterons by nuclei is derived from first principles. The influence of the Pauli principle is investigated in nuclear matter and found to be small. Surface effects due to the finite size of the deuteron are studied in terms of a perturbation expansion. The first order term agrees well with the 100 MeVfit of the real potential depth to the experimental data, the second order term is shown to be two orders of magnitude smaller.     

Estimation of nuclear effects in heavy-ion elastic scattering

The elastic scattering of heavy ions interacting by a central complex nuclear potential is calculated in first-order perturbation theory. The differential cross section can be expressed by simple analytical formulas for a Yukawa-type potential as well as for a Woods-Saxon potential. This leads to a very easy estimate of the nuclear effects, if the energy of the projectile is in the neighbourhood of the Coulomb barrier. It is shown that the derived expressions are quite accurate compared to a full numerical solution of the Schrödinger equation, as long as the elastic cross section deviates less than about 50 % from the pure Rutherford cross section.     

Strong coupling effects in near-barrier heavy-ion elastic scattering

Accurate elastic scattering angular distribution data measured at bombarding energies just above the Coulomb barrier have shapes that can markedly differ from or be the same as the expected classical Fresnel scattering pattern depending on the structure of the projectile, the target or both. Examples are given such as ¹⁸O + ¹⁸⁴W and ¹⁶O + ^{148, 152}Sm, where the expected rise above Rutherford scattering due to Coulomb-nuclear interference is damped by coupling to the target excited states, and the extreme case of ¹¹Li scattering, where coupling to the ⁹Li + n + n continuum leads to an elastic scattering shape that cannot be reproduced by any standard optical model parameter set. An early indication that the projectile structure can modify the elastic scattering angular distribution was the large vector analyzing powers observed in polarised ⁶Li scattering. The recent availability of high-quality ⁶He, ¹¹Li and ¹¹Be data provides further examples of the influence that coupling effects can have on elastic scattering. Conditions for strong projectile-target coupling effects are presented with special emphasis on the importance of the beam-target charge combination being large enough to bring about the strong coupling effects. Several measurements are proposed that can lead to further understanding of strong coupling effects by both inelastic excitation and nucleon transfer on near-barrier elastic scattering. A final note on the anomalous nature of ⁸B elastic scattering is presented as it possesses a more or less normal Fresnel scattering shape whereas one would a priori not expect this due to the very low breakup threshold of ⁸B . The special nature of ¹¹Li is presented as it is predicted that no matter how far above the Coulomb barrier the elastic scattering is measured, its shape will not appear as Fresnel like whereas the elastic scattering of all other loosely bound nuclei studied to date should eventually do so as the incident energy is increased, making both ⁸B and ¹¹Li truly “exotic”.     

Ab-initio optical model theory of elastic electron-atom scattering

We analyze the elastic scattering of fast electrons by atoms, using a second order optical potential obtained from first principles. Our results are in excellent agreement with recent absolute experimental data for electron-helium and electron-neon collisions.     

Description of elastic heavy-ion scattering within the Glauber-Sitenko approach

Analytic expressions for the amplitudes of elastic nucleus-nucleus scattering for various collision regimes are derived within the Glauber-Sitenko approach. The procedure used to do this employs an extended optical potential of the Woods-Saxon type and takes into account the deflection of trajectories by a strong Coulomb field. A comparison of the analytically calculated cross sections with numerical results and experimental data shows that the approach in question can be successfully used in the energy range from 10 to 100 MeV per nucleon. It is demonstrated that, for a preset potential, it is possible to find angular ranges dominated by specific patterns of scattering, such as classical or rainbow scattering and Fresnel or Fraunhofer diffraction.     

Nuclear matter approach to the heavy-ion optical potential: (II). Imaginary part

The heavy-ion optical potentials are constructed in a nuclear matter approach, for the ¹⁶O + ¹⁶O, ⁴⁰Ca + ¹⁶O and ⁴⁰Ca + ⁴⁰Ca elastic scattering at the incident energies per nucleon E^lab/A ? 45 MeV. The energy density formalism is employed assuming that the complex energy density of colliding heavy ions is a functional of the nucleon density ?(r), the intrinsic kinetic energy density τ⁽²⁾(r) and the average momentum of relative motion per nucleon K_r(≦ 1.5 fm^?1). The complex energy density is numerically evaluated for the two units of colliding nuclear matter with the same values of ρ, τ⁽²⁾ and K_r. The Bethe-Goldstone equation is solved for the corresponding Fermi distribution in momentum space using the Reid soft-core interaction. The “self-consistent” single-particle potential for unoccupied states which is continuous at the Fermi surface plays a crucial role to produce the imaginary part. It is found that the calculated optical potentials become more attractive and absorptive with increasing incident energy. The elastic scattering and the reaction cross sections are in fair agreement with the experimental data.     

Equivalent bare optical potentials in heavy-ion inelastic scattering

The equivalent bare optical potentials have been calculated for the inelastic scattering of ¹⁶O and ¹³C by ⁴⁰Ca at 60 and 68 MeV, respectively. The potentials obtained are quite consistent with those found phenomenologically by coupled-channels calculations. The shape of the bare potential is interpreted by showing the significant contribution of the nuclear—Coulomb cross term.     

Investigation of optical model parameters for deuteron elastic scattering from 40Ar

Differential cross sections, vector and tensor analysing powers of deuterons elastically scattered by ⁴⁰Ar at 9.0, 10.75 and 12.0 MeV have been measured. The results have been analysed in terms of an optical potential containing complex central, spin-orbit and tensor terms. It is found that not only could all five observables be fitted very well but the parameters found also give excellent predictions for the excitation function of the cross section and T₂₀ at 175°, as well as for the 15 MeV vector analysing power data.     

On the nuclear polarization potential for heavy-ion scattering

The real polarization potential ΔV due to transfer reactions is studied using a microscopic semiclassical formalism. It is found that ΔV is typically of the same order of magnitude as the corresponding absorptive potential W. Different types of possible energy dependences for ΔV and W are explored qualitatively. Specific calculations for ¹⁶O + ²⁰⁸Pb and ¹⁶O + ⁶⁰Ni scattering show features similar to those which have been deduced empirically.     

Connection between elastic scattering and inclusivek T 2 distribution

A connection between the elastic scattering and the inclusive one-particlek _T ² distribution is pointed out in the context of thes channel unitarity. One of the implications of this connection is that the slope of thek _T ² distribution atk _T ² =0 is about a factor two larger than the slope of the elastic scattering att=0.     

Spin-orbit effects in the optical model analysis of deuteron and proton elastic scattering

Spin-orbit effects on the differential cross-sections for the deuteron and proton elastic scattering are considered. Relations between the parameters of the optical potentials of the deuteron and its constituent nucleons are discussed. The variation of the deuteron and proton potential depths with energy is also considered.     

The optical theorem and forward glory effects in heavy-ion scattering

A detailed discussion of the optical theorem for heavy-ion scattering is given. It is pointed out that a careful application of this theorem to light heavy-ion systems may yield information about the nuclear interaction at distances corresponding to forward glory trajectories. Applications to several cases are presented.