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.     

The reaction mechanism of heavy ion scattering at intermediate energies

The contribution to the real and imaginary nucleus-nucleus (N-N) optical potential from nucleon-nucleon scattering in the medium is calculated in a local density approximation from a two Fermi sphere nuclear matter picture for the N-N collision. This reaction mechanism is shown to be dominant for ¹²C + ¹²C scattering at all considered energies (160 MeV < E_lab < 2250 MeV) giving a weakly energy dependent reaction cross section of about 900 mb. Inclusion of the collective 2⁺, 3^? excitations in a coupled channel calculations gives good agreement for both the measured elastic and inelastic 2⁺ cross section at E_lab = 1016 MeV. This fully microscopic parameter free calculation indicates that the energy dependence of the reaction cross section for this system is mostly due to the decrease of the collective contribution with increasing energy contrary to current theoretical models.     

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.     

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.     

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.     

Theoretical analysis of <Superscript>6</Superscript>He + <Superscript>12</Superscript>C scattering by means of the model of microscopic optical potential

The ⁶He + ¹²C elastic scattering at E = 3.0, 38.3, and 41.6 MeV/nucleon is analyzed using the microscopic model of optical potential. According to this approach, two or three parameters are fitted that renormalize the depth of real, imaginary, and surface parts of the calculated optical potential. In this case, the ambiguity of the obtained sets of fitting parameters remains, but can be reduced by introducing an additional criterion of selection: the dependence of the volume integrals of the optical potential on the energy. The structure of the obtained optical potential, the role of the nuclear medium, the formation of the imaginary part of the optical potential, and the interconnection between the surface potential and the ⁶He breakup channels are discussed.     

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.     

Large-angle enhancements in the 16O(6Li, α)18F reaction at 48 MeV

The elastic scattering of ⁶Li + ¹⁶O at 48 MeV has been measured and fitted with an optical model calculation. Measurements have been made of the ¹⁶O(⁶Li, α)F reaction at 48 MeV populating the 1⁺ g.s., 3⁺ 0.927 MeV and 5⁺ 1.122 MeV states in ¹⁸F. The data exhibit cross sections at large angles comparable to those at forward angles, and have been compared with exact finite-range DWBA calculations. Exchange contributions were included for the 1⁺ g.s. and were unable to account for the large-angle data. Calculated statistical compound nucleus cross sections were approximately a factor of 100 below the data. The same conclusions are reached for previously published data at 34 MeV.     

An imaginary potential with universal normalization for dissipative processes in heavy-ion reactions

In this work we present new coupled channel calculations with the São Paulo potential (SPP) as the bare interaction, and an imaginary potential with system and energy independent normalization that has been developed to take into account dissipative processes in heavy-ion reactions. This imaginary potential is based on high-energy nucleon interaction in nuclear medium. Our theoretical predictions for energies up to ≈100 MeV/nucleon agree very well with the experimental data for the p,n+nucleus, ¹⁶O + ²⁷Al, ¹⁶O + ⁶⁰Ni, ⁵⁸Ni + ¹²⁴Sn, and weakly bound projectile ⁷Li + ¹²⁰Sn systems.     

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.     

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.     

The optical potential for vector-polarized deuterons of 52 MeV

The vector analysing power for elastic scattering of 52 MeV polarized deuterons was measured for ¹²C, ¹⁶O, ²⁰Ne, ²⁸Si, ³²S, ⁴⁰Ar, ⁵⁸Ni, ⁹⁰Zr and ¹⁹⁷Au. The data were analysed together with previously measured differential cross sections in the framework of the optical model. Best-fit and average optical-model parameters were obtained both for surface and volume absorption. Fits with surface absorption are superior to those with volume absorption, especially for heavier nuclei. Typical parameters of the spin-orbit part of the best-fit optical potentials are found to be V_s.o. ～- 5.5 MeV, r_s.o. ～- 1.15 fm and a_s.o. ～- 0.4 fm, and there is no evidence for an imaginary component. The volume integrals and rms radii of real, imaginary and l · s potentials show a smooth mass dependence and differ insignificantly for different sets of potentials.     

The 12C+24Mg elastic scattering: An example of anomalous transparency at coulomb barrier energies

Fifteen complete angular distributions of the elastic scattering of ¹²C+²⁴Mg were measured at energies around the Coulomb barrier (E_cm = 10.67–16 MeV). The angular distributions are strongly oscillating and could be well described by an optical potential family, whose real part was determined without continuous ambiguity. The imaginary part of this optical potential is very shallow. At four energies the inelastic scattering angular distributions leading to the 2⁺ state of the ²⁴Mg were also measured and analysed with coupled-channels calculations. The volume integrals of the optical potentials used in the coupled-channels calculations present the threshold anomaly in their energy dependence, with a clear Q-value dependence.     

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.     

Imaginary spin-orbit potential in the inelastic scattering of polarized protons from12C and16O

Analyzing powers and cross sections for elastic and inelastic scattering of 20, 30 and 40 MeV polarized protons from¹²C and¹⁶O have been analyzed in the coupled-channels collective model using the full Thomas deformed spin-orbit term. Evidence is given that there is an imaginary spin-orbit potential in the proton optical potential for these nuclei.     

The energy dependent nucleus-nucleus optical potential in a nuclear matter approach

We calculated the energy dependent real and imaginary part of the nucleus-nucleus optical potential in a lab. energy domain between 50MeV and 225MeV per nucleon for the systems¹²C-¹²C and⁵⁸Ni-⁵⁸Ni. In our model we assume that the energetical behaviour of the colliding nucleons inside of the two overlapping nuclei is locally that of two colliding nuclear matter systems. Therefore the effective nucleon-nucleon interaction is calculated using a Bethe-Goldstone equation that includes the Pauli blocking caused by the two Fermi spheres and takes account of the presence of the other nucleons by their mean field. Neglecting finite range effects one can calculate the real and imaginary part by a folding procedure.     

Large-angle 6Li + 28Si elastic and inelastic scattering at 27 and 34 MeV

Elastic and inelastic scattering data extending to θ_c.m ≈ 175° are reported for ⁶Li + ²⁸Si at 27 and 34 MeV. Optical model analyses of the elastic data were made using a variety of real potential forms. The large-angle data cannot be fitted with a Woods-Saxon real potential, but are well described by Woods-Saxon squared, double-folded or Fourier-Bessel potentials. The real potential is the same at both energies, but the imaginary potential is weaker at 27 MeV. The inelastic data were analyzed using the DWBA and coupled channels techniques with folded real form factors and deformed Woods-Saxon imaginary potentials, with the deformations taken from electron scattering. The 2⁺ state was fitted well at both energies with the DWBA, while the prediction decreased too rapidly at large angles for the 4⁺ state. The large-angle 4⁺ data were better described when two-step excitations were included in the coupled-channels calculations. The forward-angle 2⁺ data are sensitive to the interference between Coulomb and nuclear scattering and show that the nuclear and Coulomb deformation parameters β₂ are equal for this transition.