Polarization potentials in heavy-ion scattering

A polarization potential is defined in terms of the Feshbach projection operator formalism to represent the effect upon the elastic channel of the coupling to non-elastic channels in heavy-ion scattering. The polarization potential represents coupling to specific surface degrees of freedom of the particular reaction considered and it is contrasted to the complementary global approaches for the volume potential such as the folding model and the proximity potential. The coupled channels method is used both as a source of exact model solutions for comparison with the various approximate potential forms and also as a numerical means of constructing trivially equivalent local potentials. The imaginary Coulomb polarization potential is due in lowest order to quadrupole coupling to the lowest collective 2⁺ state of a nucleus. It is considered in detail since it provides the insight of closed analytical forms in various approximations. Multistep coupling to higher states, energy loss and off-energy shell effects are also considered analytically. The real Coulomb polarization potential due to the virtual excitation of multipole giant resonances, and the polarization potential arising from relativistic corrections, are investigated in detail. Polarization potential components due to nuclear coupling are investigated numerically. Analytical cross section approaches are contrasted with the polarization potential approach and with coupled channels.     

Factorization in relativistic heavy-ion scattering

We have calculated the total cross sections for relativistic nucleus-nucleus scattering in the Glauber theory and conclude that there will be no factorization, due to the short-range nature of nucleon-nucleon interaction as compared to the sizes of the colliding nuclei.     

Deformation effects in heavy-ion scattering

Many experiments on the scattering of heavy ions confirm that the elastic cross sections display characteristics of Fresnel diffraction. Recent experiments on the scattering of very heavy ions, however, show marked deviations from the expected Fresnel shapes and an analysis of these experiments using the “quarter-point-recipe” of the Fresnel cross section yields a larger radius for ²⁰⁸Pb than for ²³²Th. It is shown that the deviations may be described in terms of the ground state deformations of the nuclei involved. Taking the deformations into account removes the above anomaly in the radii. The actual fit to the experimental cross sections is not very good and suggests that the real part of the nuclear potential may have an appreciable effect on the orbits of nuclei which are elastically scattered below the Coulomb barrier.     

Parity dependence in heavy-ion scattering

A range is defined for the effects of parity dependence in heavy-ion scattering. This range is shown to be related to the terms of the antisymmetrizer which exchange the largest number of nucleons between both nuclei. A simple formula, derived in the two-center harmonic oscillator model, gives an upper bound for the parity range. A criterion is proposed to determine whether a parity-dependent real part should be used in heavy-ion optical potentials. The most important parity effects should be expected in scattering between nuclei with neighbouring masses.     

Resonance propagation in heavy-ion scattering

The formalism developed earlier by us for the propagation of a resonance in the nuclear medium in proton-nucleus collisions has been modified to the case of vector boson production in heavy-ion collisions. The formalism includes coherently the contribution to the observed di-lepton production from the decay of a vector boson inside as well as outside the nuclear medium. The medium modification of the boson is incorporated through an energy dependent optical potential. The calculated invariant ρ mass distributions are presented for the ρ-meson production using optical potentials estimated within the VDM and the resonance model. The shift in the invariant mass distribution is found to be small. To achieve the mass shift (of about 200 MeV towards lower mass) as indicated in the high energy heavy-ion collision experiments, an unusually strong optical potential of about −120 MeV is required. We also observe that, for not so heavy nuclear systems and/or for fast moving resonances, the shape, magnitude and peak position of the invariant mass distribution is substantially different if the contributions from the resonance decay inside and outside are summed-up at the amplitude level (coherently) or at the cross section level (incoherently).     

Interfering transfer processes in heavy-ion reactions

Heavy-ion reactions in which two different transfer processes may interfere are analyzed. Angular distributions of the reactions ¹⁴C(¹⁶O, ¹⁷O)¹³C and ¹⁴C(¹⁶O, ¹⁸O)¹²C were measured at incident energies of 20, 25 and 30 MeV. The strong oscillations observed at the Coulomb barrier together with a backward rise at higher energies are taken as evidence for the superposition of two competing transfer reactions. DWBA calculations for the two single transfer processes were performed using the fixed-range approximation, and the two transition amplitudes were summed coherently. The experimental angular distributions are well reproduced. The DWBA also explains the disappearance of the interference structures for higher transferred angular momenta l. Data on the reaction ¹¹B(¹⁶O, ¹⁵N)¹²C measured earlier are included in the analysis in order to show the systematic dependence on l-values.     

Electron-positron scattering matrix in supercritical heavy-ion collisions

The scattering matrix describing the electron-positron excitation during a heavy-ion collision is discussed. It is shown that spontaneous positron emission in the presence of a long nuclear delay time precisely corresponds to the formal resonance condition of Scharf.     

A new interaction for heavy-ion scattering

A new effective nucleon-nucleon interaction gives the real part of heavy-ion optical potential tails correctly when used in a folding model. Exchange effects are discussed.     

Quantal description of inelastic heavy-ion scattering

A model Schrödinger equation for scattering with energy loss is discussed. The equation is linear and is closely related to the coupled-channels approach to elastic and inelastic scattering. A possible parametrization of the model for applications to heavy-ion scattering is considered.     

An interaction potential for heavy-ion scattering

The long-range part of the nucleus-nucleus interaction is taken to be given by folding the density distribution of one nucleus with the real part of the single-nucleon optical potential of the other. Analytic approximations are derived for the folded potential and its derivative in the case where the density distribution and single-nucleon optical potential have Saxon-Woods form factors of equal surface thickness. The approximations are generalised to the case of different surface thicknesses and are compared with a previous parametrisation due to Broglia and Winther. The variation with mass number of the central density of the Saxon-Woods matter distribution required to obtain the correct normalisation is shown to be large and an expression for the variation is given. Some calculations are performed on various elastic scattering data using the “quarter-point recipe” of Frahn's diffraction theory. The parameters required to fit the quarter-points of the elastic cross sections are shown to be consistent with their accepted values. It is shown, however, that the quarter-point recipe leads to a larger radius for ²⁰⁸Pb than for ²³²Th. The positions and heights of the pure Coulomb barrier (L = 0) are evaluated for various nuclei. The barrier radii are found to be sufficiently large to suggest that an interaction of the folded type should be reasonable in this region.     

Nuclear excitation effects on heavy-ion scattering

The detailed coupled-channel analysis, including a ground state rotational coupling up to the 4⁺ state, for ¹⁶O scattering from ¹⁵²Sm at the incident-ion energy of 72 MeV is performed. It is found that nuclear excitation plays an important role in the reproduction of details of the observed experimental data, and that the extracted nuclear deformation lengths from this analysis agree with those from light-ion experiments.     

On semi-quantal approximations to heavy-ion scattering

The coupled-channel equations for heavy-ion scattering are approximately solved in a closed form, in the context of semi-quantal approach. Our solutions are shown to contain dynamic polarization potentials (arising from two and/or multi-step processes) in a natural way. A closed form treatment, of the effects of dynamic polarization by Coulomb excitation, on the elastic scattering of deformed heavy-ions is also presented. As an example, we compare our results for quadrupole Coulomb excitation of¹⁸⁴W ions by¹⁸O ions at 90 MeV, with those obtained from optical model treatments.     

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.     

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.     

Quantal theory of Coulomb absorption in heavy-ion scattering

The component in the heavy-ion optical potential due to the Coulomb coupling to inelastic channels has been calculated using the on-energy-shell approximation for the intermediate-channel Green's functions. Closed expressions were derived for the Coulomb polarization potential representing coupling to all orders in the K = 0 rotational band. As a test of this general aproach for coupling to higher states, elastic-scattering calculations were performed with a truncated expression which included reorientation in the 2⁺ state and coupling to the 4⁺ state to all orders. Comparison with coupled-channels calculations indicate the increasing importance of off-shell effects with increasing coupling strength. An analytical estimate of offshell effects is presented. Limits on the range of validity of the optical-potential approach are determined.     

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”.     

A test for single-pole dominance in heavy-ion scattering

Strongly absorbed particles like α-particles and heavy ions often exhibit large, oscillatory back-scattering, which at certain energies can be described by a single Regge pole in the elastic scattering amplitude. A simple procedure is presented which can be used to test any partial-wave decomposition or optical-model amplitude for the presence of such a pole. It determines the angular range over which the pole dominates the amplitude and provides an estimate of the position and width (in l) of the pole. Sample applications of the test are made to several amplitudes of current interest, including that of the Florida State l-dependent optical model for heavy-ion scattering.