Causality and the energy dependence of microscopic nucleus-nucleus potential

Microscopic nucleus-nucleus potentials have been calculated by the Tübingen Group, using the energy-density formalism, for several systems. If these potentials are to be interpreted as optical potentials which generate the elastic-scattering wave function and phase shifts, they would be expected to satisfy causality. It is shown by investigations of the energy dependence of the real and imaginary parts of the microscopic potentials that they are consistent with causality requirements.     

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.     

Energy dependence of the optical potential

It is important to take into account the intrinsic energy dependence of the optical potential arising from the energy dependence of the separation distance of the two-body effective interactions, used for calculating the optical potential. The consequence of such an energy dependence both for the direct and the exchange term is discussed.     

The temperature dependence of the Hi optical potential

Within a realistic effective nucleon-nucleon interaction, derived from the Reid soft-core potential at zero temperature, the dependence on temperature of the heavy ion optical potential is investigated for the systems ⁴⁰Ca + ⁴⁰Ca and ²⁰⁸Pb + ²⁰⁸Pb. The hot nuclear densities are generated using the Thomas-Fermi model at a finite temperature. It is found that both the real and the imaginary parts of the optical potential become more attractive and extend to large distances when the temperature increases. The barrier heights are lowered and shifted outwards.     

Giant resonance polarization terms in the nucleus-nucleus potential

Effective polarization potentials for adiabatic Coulomb coupling to giant resonances are derived for heavy-ion collisions. The isovector giant dipole resonance and isosealar giant quadrupole resonance are found to contribute attractive terms varying as Z₁²Z₂²/r⁴ and Z₁²Z₂²/r⁶ respectively to the effective nucleus-nucleus potential. Because of the strong Z-dependence these terms become significant in the collision of very heavy ions, dominating the far tail of the ion-ion potential. Analysis of a recently determined surface potential for ⁴⁰Ar + ¹⁶⁰Gd suggests the presence of such polarization terms. We calculate that for a ²⁰⁸Pb + ²⁰⁸Pb collision the adiabatic polarization potential exceeds the real nuclear ion-ion potential estimated by the proximity theorem for center-of-mass separation distances $\text{r ?1.4(A}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}{}_1\text{+ A}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}{}_2\text{)}$. Furthermore, at the point where the polarization and nuclear potentials become equal ? V_pol may be 2–3 MeV for very heavy systems. Therefore the adiabatic polarization represents a measureable effect which may be expected to play a significant role for heavy-ion reactions in which discrete final states are observed.     

The effect of deformation on the nucleus-nucleus optical model potential and how it produces pockets

A recently derived nucleon-nucleon interaction has been used to calculate both the real and imaginary parts of the optical-model potential for the two nuclear pairs Pb+U and U+U. Three different bombarding energies per projectile nucléon (6 MeV, 11.8 MeV and 20.9 MeV) have been considered. We found a dramatic dependence of both the real and imaginary parts of the optical-model potential on relative orientation of the colliding nuclei. For the two considered pairs we predicted pockets whose depths depend strongly on orientation.     

Mass dependence of hadron distributions in ultrarelativistic nucleus-nucleus collisions

(Anti-) baryon and kaon distributions in nucleus-nucleus reactions at 200 AGeV are studied in the framework of the transport model RQMD. Production mechanisms for strangeness and baryon pairs are tested by comparing their projectile and target mass dependence to available experimental data. RQMD contains two collective production processes, fusion of overlapping strings into highly charged chromoelectric ropes and hadronic rescattering. It turns out that both rope formation and hadronic rescattering are of importance for creation-and annihilation-of strangeness and antibaryons.     

Diabatic interaction potential for nucleus-nucleus collisions

Within a refined method for the construction of diabatic states allowing for the treatment of the full spin-orbit coupling, characteristic features of the diabatic potential for nucleus-nucleus collisions are investigated. Approximately 90% of the strong repulsion results from diabatic particle-hole excitations, while only 10% is due to compression. The diabatic interaction potential describes a physical situation intermediate between adiabatic and sudden approximations.     

Causality and the threshold anomaly of the nucleus-nucleus potential

According to the causality principle, a scattered wave cannot be emitted before the arrival of the incident wave. This principle implies the existence of a dispersion relation between the real and the imaginary parts of the optical potential. We discuss the difference between the dispersion relations which hold for nucleus-nucleus scattering on the one hand and for nucleon-nucleus scattering on the other hand. In the case of nucleus-nucleus scattering, the dispersion relation predicts that the modulus of the real part of the optical potential has a bell-shaped maximum, as a function of energy, when the imaginary part approaches zero, i.e. for energies near the top of the Coulomb barrier. The shape of this apparent anomaly is investigated in the framework of several models. It is shown that there exists an algebraic model which is at the same time simple and sufficiently accurate in the sense that the difference between its outcome and that of more realistic models is smaller than the uncertainties introduced by the assumptions which have to be made. Various systems are discussed, in particular ¹⁶O + ²⁸⁰Pb and α + ⁴⁰Ca. Several implications of the anomaly are pointed out, including its effect on the sub-barrier fusion of two heavy ions.     

Energy dependent nucleus-nucleus potential in heavy ion collisions

The energy dependence of the real part of the nucleus-nucleus potential is evaluated in the proximity picture of Blocki et al. using Seyler-Blanchard two-body effective interaction. This energy dependent potential is used to study the fusion excitation functions and deep inelastic collisions. These results are compared with those calculated with the energy independent proximity potential and it is observed that the energy dependence of the potential does not affect the results significantly except for the reduction of the high energy fusion cross sections.     

One-nucleon exchange and the effective local potential in nucleus-nucleus scattering

The influence of one-nucleon exchange on the local nucleus-nucleus potential is investigated using the resonating group method. Neutron scattering on ⁴He and ¹⁶O nuclei is examined in detail. The results are compared to microscopic calculations.     

The absorptive part of the nucleus-nucleus potential in a semiclassical approach

The imaginary part of the optical potential for nuclear ion-ion scattering in the energy range 20 MeV ?E/A ? 200 MeV is derived using Feshbach's projection formalism. It is defined as the effective absorptive potential in the projected one-body Schroedinger equation for the relative motion of the colliding nuclei. Calculations are done in the Thomas-Fermi approximation, which accounts in a simple way for all phase-space effects as well as for the finite size of the ions. Intrinsic excitations are considered to be of one particle — one hole type in either of the ions, the other remaining in its ground state. The effective two-body interaction is taken to be of finite range. Further simplifications of the model consist in neglecting antisymmetrization between the mutual wave functions of the two ions and in the omission of the Coulomb energy.     

Target mass dependence and intranuclear cascade in high energy nucleus-nucleus collisions

Intranuclear cascade, one of the most important non-linear effects in high energy nucleus-nucleus collisions, is investigated, its influence on the target mass dependence of the distributions of final state particles is discussed. Using hydrodynamical model to treat non-linear effects, the main features of the rapidity distributions and their dependence on the target mass are naturally explained. The results of numerical calculation for¹⁶O-A central collisions at 60 and 200A GeV agree with the existing data.     

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.     

On the various derivations of the energy dependence of the empirical optical potential

The predictions of the energy dependence of the real part of the empirical optical potential implied by the formulations based on nuclear reaction theories and on the multiple scattering approach are compared. The recent data on the nucleon-nucleon forward scattering amplitude obtained from phase-shift analyses, direct measurements and dispersion relations are used. The meaning of the local expression obtained in the multiple scattering formalism is discussed and it is shown that its implications agree with those derived by the nuclear-theories approach.     

The nuclear response and the imaginary potential for nucleus-nucleus collisions

The Fermi-gas model is used in this paper to study the nucleus-nucleus collision. The field produced by one of the nuclei is considered to act on nucleons in the other nucleus, which is treated as a Fermi gas of radius R. The imaginary part of the (non-local) nucleus-nucleus potential is then computed by evaluating the energy-conserving second-order term in which the intermediate states are particle-hole excitations produced in the Fermi gas. The equivalent local potential, obtained by using the Perey-Saxon method, is compared with phenomenological imaginary potentials. Later it is shown that, in the limit of small range of non-locality, the imaginary potential can be related to the nuclear response function. With this, one can write the nuclear friction coefficient that is used in phenomenological analyses of heavy-ion collisions in terms of the imaginary potential.