H.S. Köhler

Effect of two-body collisions on heavy ion dynamics

Abstract Two-body collisions are included in TDHF by the time-relaxation method. Calculations are made in a one-dimensional slab-geometry at E c.m A = 20 MeV . The relaxation time is assumed to be temperature dependent. Results arc shown for two functions of the temperature as well as for ordinary TDHF (infinite relaxation time). Temperature and the two components of the pressuretensor (perpendicular and parallel to the beam direction) are shown. The approach towards the hydrostatic limit (isototropic pressure) as the relaxation time decreases is shown. Relaxation time is expected to decrease with temperature, i.e. beam energy. The energy at which the pressure becomes isotropic is important for understanding the dynamics of heavy ion collisions.

TDHF with two-body dissipation

Abstract A modification of the time-dependent Hartree-Fock (TDHF) formalism is proposed in order to incorporate two-body dissipation. For this purpose a relaxation time, τ, is defined for a statistical redistribution of nucleon momenta due to the nucleon-nucleon scatterings. This process drives the nuclear system at each point towards isotropic Fermi distributions. The method is illustrated by showing results for the collision of one-dimensional nuclear slabs and comparison with TDHF is made. Microscopic calculations show τ to be of the order of 10 −21 s. For this value of τ and collisions of a few MeV/nucleon the two-body dissipation is comparable with one-body dissipation.

Mean field, occupation numbers and spectral functions in nuclear matter

Abstract Using the experimental free-scattering phase shifts as input, properties of nuclear matter are calculated by Brueckner many-body techniques. Previously reported calculations are extended to include second-order “rearrangement” contributions to the mean field, occupation numbers and spectral functions. Results are in close agreement with potential model calculations.

Mean field and effective cross sections in hot nuclei

Abstract Collisions between heavy nuclei produce nuclear matter of high density and excitation. Brueckner methods are used to calculate the momentum- and temperature-dependent mean field for nucleons propagating through nuclear matter during these collisions. The use of a potential model for the NN interactions is bypassed by calculating the Brueckner reaction matrix directly from the NN phase shifts using a version of Brueckner theory previously published by the author. Arndt phase-shift solutions up to 1600 MeV in energy are used. The binding energy of nuclear matter is normalized to −15.5 MeV by adjusting one free parameter. This gives a saturation density of 0.16 fm −3 and the compressibility is 200–250 MeV. The temperature and density dependence of the effective interaction is shown. The mean field is complex and the real part relates to “one-body” collisions while the imaginary part is related to the “two-body” collision term in transport theory. The error in using free cross-sections when calculating the imaginary part is in general less than 20% but can be at least as large as 75%. The influence of the mean field (effective mass) on the collision term is emphasized.

Heavy ion collisions in three dimensions by the relaxation-time method

Abstract A quantum transport equation with two-body collisions included via a relaxation-time method, earlier applied to two-dimensional (slab) collisions, is now extended to three-dimensional calculations A density matrix is constructed from self-consistent field wave functions and is time-evolved in cartesian coordinates. This dynamical model of the nucleus is applicable at all nonrelativistic energies. The semiclassical limit is discussed. Results are shown for 16 O- 16 O collisions between 40 and 200 MeV/ A lab energies. Hot spots and conditions for fragmentation are discussed. The threshold for breakup of the compound system formed in a head-on collision lies between 40 and 60 MeV/ A lab energies. At these energies, the maximum density-averaged thermal excitation energy is 7 and 10 MeV/ A (average temperatures 8 and 11 MeV), respectively, compared with a binding energy of 8 MeV/ A . The system does not thermalize completely, and the distribution in momentum space is not quite isotropic when breaking up.

Calculation of relaxation times in nuclear matter

Abstract The Boltzmann collision term with the Uehling-Uhlenbeck-Nordheim modification for fermion statistics is computed in nuclear matter. The initial distribution function is in momentum space either two Fermi spheres separated by a relative momentum as in a collision between two heavy ions or some other specified deformation. Relaxation times for the equilibration is obtained as a function of density and final temperature of the equilibrated system. The mode dependence of the relaxation times is calculated by expanding the angular dependence of the distribution in spherical harmonics. Transport coefficients for viscosity and thermal conductivity are also calculated as well as their temperature and density dependences.

Approximations to Brueckner theory for nucleon-nucleus collisions

Abstract The concept of quasiparticles has been very fruitful for the development of theories of many-body systems of strongly interacting particles. A central problem for a quantitative treatment is the interaction between the quasiparticles (the effective interaction between particles). The Brueckner reaction matrix theory and the Jastrow variational approach have been used extensively, especially for nuclei. Our present concern is the collisions between nuclei. An exact calculation of the effective interaction between the nucleons is then extremely complicated. Only the simplest extreme, the nucleon-nucleus optical potential, can be treated fully. Approximate treatments are called for. The present paper investigates some approximations to the Brueckner theory. We are primarily interested in the imaginary part of the one-body (one-nucleon) mean field, as it relates to the damping of collective motion. Of special interest is the increase at low density. Comparisons with Skyrme forces show that they exaggerate this density dependence.

From TDHF to hydrodynamics

Abstract The Time-Dependent Hartree-Fock theory provides a microscopic approach to the scattering of heavy ions. Fundamental in this theory is a mean(one-body) field. The calculation of this field from a two-body effective interaction makes the theory microscopic. Many-body effects are included by the Brueckner definition of this interaction; the reaction-matrix. In excited media it is in general complex allowing for decays. The imaginary part relates directly to the collision-term in a transport equation. We treat this term by the time-relaxation-method. This implies an extension of the TDHF-equation to include two-body collisions. Hydrodynamic equations are derived from this new equation. The solution of the two equations agree quantitatively for short-relaxation-times. Relaxation-times are calculated as a function of temperature.

A comparison between velocity- and density-dependent interactions in finite nuclei calculations

Abstract A comparison is made between Hartree-Fock calculations utilizing a velocity-dependent interaction of quadratic form and an interaction in which the velocity dependence is converted to a density dependence. The conversion is made so that it is exact for nuclear matter. For finite nuclei considerable differences between the two interactions occur when computing density distributions and energy spectra of level states. Total binding energies also give errors of 1.5 MeV/ A for 16 O and 1.0 MeV/ A for 40 Ca, while for 208 Pb the error is negligible. It is suggested to approximate the general non-locality or velocity dependence of the Brueckner K -matrix by a quadratic velocity dependence. This should be preferred to a density dependence as it is less drastic, but no more difficult to compute with.

Comparison of Brueckner and Jastrow methods for nuclear matter

Abstract Brueckner-Goldstone diagrams are calculated (or estimated) in an expansion up to four-hole lines for two interactions to compare with Jastrow variational calculations. The central force OMY potential gives a result close to that obtained in the Fermi hypernetted chain expansion of the Jastrow method. The tensor force Reid soft core potential also gives a result in good agreement with Jastrow variational calculations at the saturation density of nuclear matter (kF ≈ 1.4 fm−1). The calculated saturations differ for the two methods however, giving 18 MeV at kF = 1.6 fm−1 for the Brueckner method while reported Jastrow calculations give a binding of about 22–25 MeV at kF = 1.75 fm−.