Calculation of the imaginary part of the heavy ion potential

The paper contains a numerical evaluation of the expressions for the absorptive potential in heavy ion reactions given earlier. With a standard folding expression for the real part of the ion-ion potential general good agreement is found with experimental data for the angular distributions of elastic and inelastic scattering. Special interest is attached to the case of ¹⁶O + ²⁸Si where the calculated imaginary potential is very small at low bombarding energies.     

Folding computation of the 16O + 16O optical potential with a complex effective force

The ¹⁶O + ¹⁶O optical potential is obtained by using the folding method together with a previously defined complex effective nucleon-nucleon force closely related with the heavy-ion collision dynamics. This type of force allows the computation of both the real and the imaginary parts of the optical potential. Through the use of the folding method, finite-range effects are correctly incorporated. In that respect, the present results improve upon those obtained with the local density approximation. They also compare favourably with the phenomenologieal optical potentials.     

Surface and volume contributions to real and imaginary parts of the heavy-ion optical potential

Starting from the Feshbach expression for the optical potential, explicit formulae for the real and the imaginary parts of the optical potential between two heavy ions (HI's) are obtained. They are each composed of a volume and a surface term. The contributions to the volume term are calculated in two nuclear Fermi liquids which flow through each other starting from the realistic Reid soft core nucleon-nucleon (NN) interaction. Since the Fermi surface is formed by two spheres one obtains a complex Brueckner reaction matrix which is approximated by a complex, effective local interaction. It is used in a fully antisymmetrized double folding procedure to obtain the volume terms of the optical potential between the two HI's. The surface contributions are directly calculated in the collision of the two finite HI's. The collective surface vibrations (3^? octupole state and 2⁺, 4⁺ (T = 0) giant resonances for the ¹⁶O?¹⁶O collision) are included as intermediate states. This yields especially an imaginary contribution at the surface which reduces the transparency found with the volume terms alone. The method is applied to ¹⁶O?¹⁶O scattering at 83 and 332 MeV laboratory energy. The local approximations to the real and imaginary parts obtained in this way agree well with phenomenological fits. The surface terms improve the agreement of the differential cross section at 80 MeV where experimental data are available.     

Generator-coordinate study of elastic 16O + 40Ca scattering

Elastic ¹⁶O + ⁴⁰Ca phase shifts are calculated using a generator-coordinate method. A rotational band of molecular resonances is found with a rotational constant of about 30 keV. The width of these resonances should make them observable in the energy range 30 to 50 MeV. These features can be approximately reproduced by a shallow local potential. Properties of the imaginary part of the optical potential are discussed qualitatively. The optical potential should be transparent in a window of J-values comprised between about 16 and 25. Odd-even effects are shown to be negligible in the real part of the potential but might be important in its imaginary part. A possible confirmation of these predictions is found in an experimental excitation function.     

Coupled-channels study of projectile and target excitation in the scattering of 6Li from 12C and 16O

Coupled-channels calculations are presented tor elastic and inelastic ⁶Li + ¹²C scattering at E_c.m. = 16 MeV and 20 MeV, and for ⁶Li + ¹⁶O at 18.7 MeV. Excitation of states within ⁶Li, ¹²C and ¹⁶O are treated with rotational, rotation-vibration and vibrational models only. The 3⁺⁶Li and 2⁺¹²C states are strongly coupled to the elastic scattering and reduce the strengths of both the real and imaginary potentials. The 3^?16O state reduces only the strength of the imaginary potential. All other states are weakly coupled and have little effect on each other or the potential. The data are reasonably well described, with there being some preference for the 3^? state in ¹²C to be K = 0. Excitation of the 0₂⁺ state in ¹²C requires a combination of β-vibration and monopole breathing-mode form factors. The deformation lengths found are in poor agreement with those deduced from electron or proton scattering.     

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.     

Nucleus-nucleus scattering in the high-energy approximation and optical folding potential

A microscopic complex folding-model potential that reproduces the scattering amplitude of Glauber-Sitenko theory in its optical limit is obtained. The real and imaginary parts of this potential are dependent on energy and are determined by known data on the nuclear-density distributions and on the nucleon-nucleon scattering amplitude. For the real part, use is also made of a folding potential involing effective nucleon-nucleon forces and allowing for the nucleon-exchange term. Three forms of semimicroscopic optical potentials where the contributions of the template potentials—that is, the real and the imaginary folding-model potential—are controlled by adjusting two parameters are constructed on this basis. The efficiency of these microscopic and semimicroscopic potentials is tested by means of a comparison with the experimental differential cross sections for the elastic scattering of heavy ions ¹⁶O on nuclei at an energy of E ～ 100 MeV per nucleon.     

IWB analysis of scattering and fusion cross sections for the 12C+12C, 13C+16O and 16O+16O reactions for energies near and below the Coulomb barrier

Calculations are presented of the elastic scattering and fusion cross sections for the astrophysically interesting reactions ¹²C+¹²C, ¹²C+¹⁶O and ¹⁶O+¹⁶O. The calculations are performed using the incoming wave boundary condition (IWB) and a real ion-ion interaction potential. The results are compared with the available experimental data for the energy region near and below the Coulomb barrier. With values of two adjustable potential parameters (the radial position of the l = 0 barrier and the diffuseness) determined by fitting elastic scattering data, good agreement is obtained for the average energy dependence of the ¹²C+¹²C and ¹²C+¹⁶O fusion cross sections. In the case of ¹⁶O+¹⁶O, both the calculated absolute magnitude and the energy dependence of the fusion cross section are inconsistent with the data and this discrepancy is discussed.     

Pauli Correlation Effect for the Elastic Scattering of Protons by Different Nuclei at 800 MeV

The angular distributions of the elastic scattering of protons at an energy of 800 MeV by ¹⁶O and ²⁰Ne nuclei are described in terms of the optical model scattering theory. Single folding model is applied to calculate the optical potential taking the effective nucleon-nucleon interaction to be in two forms. One form includes the zero-range pseudo-potential term and the other includes a two-body Pauli correlation function. Analytical expressions for the real part of the optical potential are obtained for both forms. The imaginary part of the optical potential is taken to be of the Woods-Saxon's shape. It is found that introducing the Pauli correlation function improves the agreement with the experimental data for the elastic scattering differential cross-sections of protons with the target nuclei ¹⁶O and ²⁰Ne.     

Single Folding Optical Potential for Elastic Scattering of Protons from <Superscript>14</Superscript>N and <Superscript>16</Superscript>O in a Wide Range of Energies

Available experimental data for protons elastically scattered from ¹⁴N and ¹⁶O target nuclei are reanalyzed within the framework of single folding optical potential (SFOP) model. In this model, the real part of the potential is derived on the basis of single folding potential. The renormalization factor N _r is extracted for the two aforementioned nuclear systems. Theoretical calculations fairly reproduce the experimental data in the whole angular range. Energy dependence of real and imaginary volume integrals as well as reaction cross sections are discussed.     

An optical potential description of 16O + 28Si elastic scattering

The differential cross sections for ¹⁶O + ²⁸Si elastic scattering at seven energies between 21 and 35 MeV in the centre of mass are described well over the whole angular range from 20° to 180° by an optical potential whose real part consists of a double-folded potential supplemented by a phenomenological model-independent correction term. This surface correction is predominantly attractive and has structure which depends only weakly on the energy. The associated imaginary potentials imply surface transparency and have detailed structure which varies rapidly with energy. However there is a systematic trend for the absorptive region to extend to larger radii as the energy increases. A simple parameterization of this trend allows the main features of the excitation function for 180° scattering to be reproduced.     

Antisymmetry and channel coupling contributions to the absorption for p+αd+3He

To understand recently established empirical p+α potentials, RGM calculations followed by inversion are made to study contributions of the d+³He reaction channels and deuteron distortion effects to the p+α potential. An equivalent study of the d+³He potential is also presented. The contributions of exchange non-locality to the absorption are simulated by including an phenomenological imaginary potential in the RGM. These effects alone strongly influence the shape of the imaginary potentials for both p+α and d+³He. The potentials local-equivalent to the fully antisymmetrised-coupled channels calculations have a significant parity-dependence in both real and imaginary components, which for p+α is qualitatively similar to that found empirically. The effects on the potentials of the further inclusion of deuteron distortion are also presented. The inclusion of a spin-orbit term in the RGM, adds additional terms to the phase-equivalent potential, most notably the comparatively large imaginary spin-orbit term found empirically.     

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.     

Systematic analysis of 17,19F and 16,17O elastic scattering on 208Pb just below the coulomb barrier

The ¹⁷F, ¹⁷O + ²⁰⁸Pb elastic scattering at 90.4 and 78 MeV (just below coulomb barrier), respectively, are analyzed within the framework of the double folding approach. The folded potentials are constructed by folding the density independent M3Y effective nucleon-nucleon interaction over the nucleon density distribution of ¹⁷F and its mirror ¹⁷O nuclei. Gaussian oscillator (GO) density distribution is considered to represent the (core¹⁶O+nucleon) structure for ¹⁷F and ¹⁷O. The knock-on exchange potentials are introduced to construct the real semi-microscopic potentials. The imaginary potentials are supplemented to derive potentials in two forms: either phenomenological Woods-Saxon (WS) or the same form as real folded potentials and different strengths. Moreover, the analysis of ¹⁹F, ¹⁶O + ²⁰⁸Pb elastic scattering at 91, 78 MeV energies, in that order, is applied in comparison to the ¹⁷F, ¹⁷O + ²⁰⁸Pb systems in an endeavor to investigate the behavior and properties of ¹⁷F and ¹⁷O projectiles.     

Measurement and folding-potential analysis of the elastic α-scattering on light nuclei

Differential cross sections ofα-elastic scattering have been measured for the target nuclei¹¹B,¹²C,¹³C,¹⁴N,¹⁵N, and¹⁶O atE=48.7 and 54.1 MeV and for the nuclei¹⁷O,¹⁸O, and²⁰Ne atE=54.1 MeV. The experimental results were analysed in terms of the optical model using different complex potentials. Special emphasis is given to the application of the double-folding approach for the real part of the potential. The imaginary part is expressed in terms of Fourier-Bessel functions. Differential cross sections for theα-¹⁶O scattering over a wide energy range and for the elasticα-scattering for nuclei in the mass rangeA=11 up toA=24 atE=54.1 MeV are analysed by this method. A close correlation between the absorptive part of the potential and nuclear deformation is observed.