Folding model potentials from realistic interactions for heavy-ion scattering

The double-folding model, with “realistic” nucleon-nucleon interactions based upon a G-matrix constructed from the Reid potential, is used to calculate the real part of the optical potential for heavy-ion scattering. The resulting potentials are shown to reproduce the observed elastic scattering for a large number of systems with bombarding energies from 5 to 20 MeV per nucleon. Some representative inelastic transitions are also reproduced. Exceptions are the elastic scattering of ⁶Li and ⁹Be for which the folded potentials must be reduced in strength by a factor of about two.The same effective interactions are shown to give a good account of two particular cases of alpha scattering as well as some cases of nucleon-nucleus scattering. Some typical examples of inelastic heavy-ion scattering are also predicted successfully.Some general properties of the folding model are reviewed and its theoretical basis is discussed. An explicit density-dependence is examined for one particular realistic interaction and found not to change the results. Single nucleon exchange is included in an approximate way and its importance is studied.In addition to being a study of the folding model, this work also provides a systematic and comprehensive optical model analysis of heavy-ion elastic scattering in this energy range.     

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.     

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.     

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”.     

Effective nucleus-nucleus potentials derived from the generator coordinate method

The equivalence of the generator coordinate method (GCM) and the resonating group method (RGM) and the formal equivalence of the RGM and the orthogonality condition model (OCM) lead to a relation connecting the effective nucleus-nucleus potentials of the OCM with matrix elements of the GCM. This relation may be used to derive effective nucleus-nucleus potentials directly from GCM matrix elements without explicit reference to the potentials of the RGM. In a first application local and l-independent effective potentials are derived from diagonal GCM matrix elements which represent the energy surfaces of a two-centre shell model. Using these potentials the OCM can reproduce the results of a full RGM calculation very well for the elastic scattering of two α-particles and fairly well for elastic ¹⁶O-¹⁶O scattering.     

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.     

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.     

Realistic heavy-ion adiabatic potentials

The K-matrix model of Cusson, Trivedi, Meldner and Weiss has been used to compute static adiabatic heavy-ion cluster potentials in a constrained self-consistent Brueckner-Hartree-Fock scheme. Heavy-ion potentials are found which can simultaneously reproduce the experimentally deduced heavy-ion potentials in the asymptotic region and the total energy versus quadrupole moment in the fused, compound nucleus region. Potentials for α-α, α-¹²C, α-¹⁶O, ¹²C-¹²C, ¹⁶O-¹⁶O are shown and compared with other work. The total energy of ¹²C and ²⁴Mg versus deformation is also given.     

Resonant channel coupling in electron scattering by pyrazine

Detailed investigation of the three low-energy resonances seen in electron scattering by the diazabenzene molecule pyrazine reveals that the first two are nearly pure single-channel shape resonances, but the third is, as long suspected, heavily mixed with core-excited resonances built on low-lying triplet states. Such resonant channel coupling is likely to be widespread in pi-ring molecules, including the nucleobases of DNA and RNA, where it may form a pathway for radiation damage.     

A dynamic polarization potential for heavy-ion scattering

The effects of target and/or projectile excitation on the elastic scattering of heavy ions are studied. The results are cast in the form of contributions to both the real and imaginary parts of the ion-ion optical potential. In particular, a trivially “equivalent” local potential (TELP) is found to simulate the effects of coupling to one or more open channels. Particular application is made to the dynamic polarization resulting from the long-range Coulomb potential. Comparison with coupled-channel calculations shows that the TELP can accurately reproduce these effects of Coulomb excitation. The correction to the absorptive potential is found to be much larger than that for the real potential. This correction is long-ranged and cannot be simulated by any reasonable adjustment of the usual Woods-Saxon form of potential. The TELP provides an alternative to performing coupled-channels calculations and may be particularly useful when reactions other than inelastic scattering are to be studied.     

Coulomb potentials in heavy-ion interactions

The usual point charge approximation for the Coulomb potential in heavy-ion interactions is compared with more realistic treatments. Elastic scattering and transfer reaction calculations appear to be insensitive to the form of the Coulomb potential used.     

Effective probabilities in a new approach to analyzing angular distributions in elastic heavy-ion scattering

A new approach proposed previously to analyze angular distributions in elastic heavy-ion scattering is generalized to cases where total partial probabilities (that is, those that are summed over all channels) of the enhancement of “fusion” (in general, complete and incomplete fusion, quasifission, and deep-inelastic collisions) are commensurate with the total partial probabilities of the suppression of “fusion.” This could be done with the aid of effective total partial probabilities, each of these being defined as a linear combination of actual total partial probabilities. It is shown that the probabilities introduced in this way have a specific physical meaning. Indeed, the effective total partial probabilities make it possible to calculate the cross section for “fusion” through the entrance channel and some reference total cross sections for peripheral processes, and a conclusion on whether fusion and peripheral reactions are enhanced or suppressed can be drawn from a comparison of the calculated or measured results for, respectively, the fusion cross section and the total cross section for peripheral reactions with the above two cross sections. It is also found that the enhancement of fusion is accompanied by the suppression of peripheral reactions, and vice versa.     

Alpha-transfer processes in heavy-ion scattering

In the differential cross sections for the elastic scattering between an α-conjugate target and projectile, a rising oscillatory structure is often observed in the backward-angle region. An α-transfer mechanism is proposed to explain this anomalous phenomenon. A nuclear molecularorbit approximation theory for both 1α- and 2α-transfer processes has been formulated and applied to ¹⁶O + ²⁰Ne and ¹²C + ²⁰Ne scattering systems with different projectile energies. The experimental rising structures shown in these scatterings are well reproduced with parameters fairly consistent with spectroscopic data. An independent-α-particle model wave function has been used for the evaluation of the exchange potential, which gives better agreement with experiment than the Buttle-Goldfarb approximation can usually provide.     

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.     

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.     

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).     

Optical potentials for inelastic scattering from many-body targets

The standard text book Green's function possesses a self-energy that is known to be an optical potential for elastic scattering. The introduction of an optical potential reduces the complex many-body scattering problem into a tractable one-body problem. In this paper inelastic Green's functions are introduced and discussed which possess self-energies that are optical potentials for inelastic scattering. If the projectile is indistinguishable from particles comprising the target, intriguing aspects arise even for noninteracting particles.     

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.