On the energy dependence of the real part of the heavy-ion optical potential

The energy dependence of the real part of the heavy-ion optical potential is attributed to the J(J + 1) dependence of the-potential which is caused by the rotation induced during the collision. The magnitude and the form of the energy dependence are calculated for several light nucleus-nucleus systems with the use of the distortion potential which is defined by the diagonalization of the coupling interaction. It is shown that the heavy-ion scatterings with the inelastic channels of high and low excitation energies lead to weak and strong energy dependence of the potential, respectively.     

The real part of the nucleon-nucleus optical potential

The nucleus²⁴Na has been investigated by studying the gamma-rays emitted following thermal neutron capture in²³Na, with curved crystal and Ge(Li) spectrometers. Of the 277 transitions assigned to²⁴Na, 216 were placed in the²⁴Na level scheme containing 45 levels, of which six (1,961, 1,977, 3,866, 5,810, 5,918, and 6,222 keV) are reported for the first time. An average gamma-ray multiplicity of 3.3 gammas per neutron capture was observed. The neutron binding energy was determined to be 6,959.73 (14) keV. The resulting level scheme is compared to shell and rotational model predictions.     

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.     

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.     

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.     

A simple double-folding computation of the complex heavy-ion optical potential

We have derived a spin- and isospin-independent complex effective force to compute the ¹⁶O + ¹⁶O optical potential by the double-folding procedure. To avoid reference to a wave function, we have used superposition models to compute the values of the matter and intrinsic kinetic energy densities necessary to the local specification of the effective force. The present results compare favourably with those of a previously published, fully antisymmetrized calculation. The physcial significance and the quantitative importance of self-energy effects 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.     

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.     

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.     

Real part of the optical potential for composite particles

The optical potential for a composite particle is most simply approximated by the sum of the optical potentials of the constituent nucleons. Restricting ourselves to the real parts of the potentials we use this model as a first approximation in a calculation of the potentials for d, ³He, α and ¹²C. We add corrections for (i) the energy dependence of the nucleon potentials, (ii) three-body terms, (iii) the Pauli principle. All corrections can be important and that for the Pauli principle can be very large. We obtain a good explanation of the following phenomena: (a) the deuteron potential is nearly the sum of the neutron and proton potentials, (b) the potential for ³He is about 20 % less than the sum of the potentials of the nucleons in the ³He projectile, (c) the volume integral of the potential for ³He falls at both high and low energies in the energy range 20–100 MeV, (d) shallow potentials with large radii are found for low energy (30 MeV) scattering of α-particles, (e) deeper potentials are found for higher energy α-particle scattering. We predict shallow potentials for ¹²C scattering from light targets but deeper potentials for heavier targets.     

The reactive two-body part of the pion-nucleus optical potential

We calculate the reactive component of the two-body contribution to the pion-nucleus optical potential from a two-nucleon pion absorption mechanism that predicts the total cross section and angular distributions for π⁺d → pp very well. At threshold the calculated absorptive parts explain most of the values obtained from pionic atom level widths, whereas the dispersive parts, which are very sensitive to wave function correlations are considerably more attractive than what the conventional phenomenological parameters would suggest.     

On the form of the imaginary part of the optical potential

The imaginary part of the optical potential is derived fromMigdal's theory of nuclear structure. It is shown that the strong enhancement of ImV(r) at the nuclear surface is mainly due to a suppression of the effective nucleon-nucleon force by collective effects in the nuclear interior.     

Volume integrals of real part of optical potentials for light projectiles

A semi-empirical formula for the real volume integrals of the optical potentials for composite projectiles is proposed based on real volume integrals for the nucleonnucleus optical potential derived from the microscopic calculation of Jeukenne et al. which has been cast into a 5 parameter formula for energies upto 50 MeV by adding one additional term for the compositeness of the projectiles. The three parameters of this term have been obtained using the phenomenological analyses for light composite projectiles such as deuteron, triton, helium-3, alpha and lithium-6. The resultant formula describes successfully the physically meaningful volume integral per interacting nucleon-pair to an accuracy of 7% upto the projectile energies around 50 MeV per nucleon for targets over the whole periodic table.     

Density-dependent interactions and the folding model for heavy-ion potentials

It is pointed out that the failure of the single-folding model to predict the correct real part of heavy-ion optical potentials is partially due to the omission of effects arising from any density dependence present in the underlying nucleon-nucleon interaction. An estimate is made of the error introduced in typical calculations.     

Correlation between the optical model potential and matter distribution parameters in heavy-ion elastic scattering

Starting from the analytical expression for the folding model potential for the heavy-ion scattering, a simple relationship between the potential and the nuclear density parameters is obtained. This relationship is consistent with the analysis of the proximity forces and is confirmed around the strong absorption radius through the results of the folding model calculations for¹⁶O projectile incident on a variety of targets ranging from⁴⁰Ca to²⁰⁸Pb. These relations help to deduceR _x the distance where the folding potential has a specific valuex and the densities of the target nuclei. The predictions are found to be in good agreement with the experiment.     

The imaginary part of the local potential equivalent to the non-local α-nucleus optical potential

A simple expression of the α-nucleus optical potential has been derived from the Feshbach formula by using a closure approximation for summing over the excited states of the target nucleus. It has been shown that the correction to the real folding model potential is small. The imaginary local potential equivalent to the non-local Feshbach potential has been studied in detail for Ca nuclei and shown to reproduce quite well the gross properties of empirical potentials above 100 MeV with, however, a lack of absorption in the surface region. The A-dependence of the imaginary potential volume integral has also been investigated.     

The imaginary part of the nucleon-nucleus optical potential in semiclassical approximation

We calculate in a novel Thomas-Fermi theory for multiparticle-multihole states the contributions of the correlation and polarization graphs to the imaginary part of the nucleon-nucleus optical potential $\text{W(ω,}\text{R}\text{,}\text{P}\text{)}$ depending on energy, radius and momentum. The present theory generalizes the older Fermi-gas and local-density approximations to this problem in the sense that we can assess precisely the validity of our approach. We show that it yields results which coincide with the average part of a corresponding quantal calculation. We use a gaussian finite-range effective interaction derived from the Gogny force and phenomenological mean-field potentials of the Woods-Saxon or harmonic-oscillator type. With these ingredients and no further adjustable parameters our results for the depth and volume integrals are in good agreement with the average trend of the elastic scattering data. Further, the resulting momentum dependence of W is strong, especially for small P. The influence of the Pauli principle is studied in detail.