Semiclassical approach to relativistic nuclear mean field theory

Abstract

The nuclear many-body problem is nowadays being increasingly approached on the basis of a relativistic formalism. Microscopic Dirac-Brueckner-Hartree-Fock (DBHF) calculations starting from a realistic nucleon-nucleon interaction seem to be very promising. In the theory of quantum hadrodynamics, nucleons interact through the exchange of virtual mesons and the dynamics is specified by Lorentz covariant Lagrangian densities. In the simplest version, only a vector field (associated with the w meson) accounting for short-range repulsion and a scalar field ("sygma" meson) responsible for attraction are needed to describe saturation in nuclear matter. Calculations in mean field Hartree approximation, neglecting exchange terms and without including any contribution from antiparticles, give an accurate description of many of the features of nuclear systems.<br/><br/>The success of semi-classical models in non-relativistic nuclear physics provides a very strong motivation for investigating similar methods in the relativistic context, where only the pure Thomas-Fermi approximation had been studied. In this thesis we set up the semi-classical expansion in relativistic nuclear mean field theory, including gradient corrections of order h(2) to the Thomas-Fermi model, and investigate several applications to nuclear systems.<br/><br/>On the basis of Wigner transform techniques, a. recursive scheme to obtain the semi-classical h(2) expansion of the propagator associated with a time-independent single-particle Hamiltonian with matrix structure is presented. We focus our attention on the application of the method to a Dirac Hamiltonian related to relativistic nuclear mean field theory, i.e., including a position-dependent effective mass and the time-like component of a. four-vector field. Compared with the non-relativistic case, the procedure is considerably more complicated owing to the matrix structure of the Hamiltonian. For this reason the "h", expansion is pushed to order h(2) only. A detailed derivation is given of the h(2)-order Wigner-Kirkwood expansion of the relativistic density matrix, in terms of the gradients of the vector and the scalar field, as well as of the expansion of the particle and energy densities. The idempotency of the semi-classical density matrix to second order in "h" is proven. The Wigner-Kirkwood expressions, as they stand, are not suitable to be employed in a self-consistent problem. Therefore, we obtain the corresponding density functional results. In this case the energy densities are expressed as a functional of the local density, the effective mass and their gradients.<br/><br/>The accuracy of the Wigner-Kirkwood series is tested on a. relativistic harmonic oscillator and perfect agreement with the Strutinsky averaged observables is found even in the highly relativistic regime. The density functional version is shown to be slightly less accurate, a feature already known in the non-relativistic case. It turns out that the semi-classical expressions represent the different quantities on average, that is, quantum fluctuations are averaged out. This model study shows that, for positive energy states, the derived semi-classical expansions contain all the relativistic ingredients, the difference with quantal results being due mainly to shell effects.<br/><br/>Extended Thomas-Fermi calculations, which· include h(2)-order gradient corrections, are performed for relativistic non-linear "sigma"- "omega" models using two kinds of Lagrangians which differ in the form of the scalar coupling for the isoscalar sigma meson. Comparing the semi-classical results of order h(2) (TFh(2)) with the Hartree results, we find that the TFh(2) approximation yields some underbinding when the effective mass (mº) of the model is small, and some overbinding when mº is large. For a value around mº/m = 0.65, both TFh(2) and Hartree would roughly yield the same binding energy. However, since semi-classical and quantal results must differ in the so-called shell energy, this indicates that it is not properly estimated by the TFh(2) approximation.<br/><br/>When the h(2)-order gradient corrections are taken into account (TFh(2), we have found a numerical instability in the solution of the semi-classical Klein-Gordon equation obeyed by the scalar field in the case of parameterizations which have mº/m </= 0.60, which can be eliminated if the q-meson mass mº is reduced (with the ratio g(2-0)/m(2-0) unaltered).<br/><br/>Second-order corrections in "h" to the TFh(0) approximation improve the agreement with Hartree solutions in a sensitive way, always yielding more bound nuclei than within the Hartree approach. The sign of the h(2) corrections depends on mº, and they are found to vanish around mº/m = 0.75 for the models of the type considered here. In several respects, the semi-classical relativistic phenomenology quite resembles the one met in the non-relativistic regime using Skyrme forces, in spite of the different origin of mº in both situations. Extending the so-called expectation value method to the relativistic problem, and using the TFh(2) semi-classical mean field as a starting point, perturbative quantal solutions are found which are in good agreement with the Hartree results.<br/><br/>The semi-classical TFh(0) and TFh(2) density distributions do not present oscillations due to the absence of shell effects, but they average the Hartree results. In the interior of the nucleus the TFh(0) and TF1i2 densities are very similar. However, in the surface and the outer region the TF1i2 densities come appreciably closer than TFh(0) to the Hartree results, due to the gradient corrections incorporated by the TF1i2 functionals, and show an exponential drop off.<br/><br/>Liquid drop model coefficients are calculated for some parameter sets of the "sygma-omega" model. We have found reasonable results for the surface thickness and for the surface and curvature energies, which are within the range of the values obtained in non-relativistic calculations using density-dependent Skyrme forces. Therefore, the relativistic effects do not seem to avoid the disagreement of the calculated value of the curvature energy with the empirical value.<br/><br/>In this work we also study the effects of the density-dependent Dirac spinor for the nucleons, as is determined microscopically in the DBHF approach, on various properties of the structure and scattering of finite nuclei. To enable this, we construct a relativistic energy density functional that contains the semi-classical kinetic energy density of order h(2). The effective mass and the volume term in the potential energy arise from a DBHF calculation of nuclear matter. This volume term is supplemented by some conventional correction terms and the few free parameters are suitably adjusted. It turns out that the radii of nuclei calculated with the present approach agree better with the experimental value than those obtained in similar studies using a potential energy derived from a non-relativistic G-matrix. This demonstrates that the Dirac effects improve the calculation of ground-state properties of finite nuclei also in our relativistic extended Thomas-Fermi (RETF) approximation.<br/><br/>However, this study of ground-state properties is not the main goal of our investigation.<br/>The capabilities of our RETF functional are actually appraised in situations in which a full microscopic relativistic calculation, or even a phenomenological one, cannot be easily made, such as nuclear fission of rotating nuclei and heavy ion scattering. In these situations, the method constitutes a reliable tool. For the nuclear fission barriers, the present calculations are the first ones carried out with a relativistic model. We have shown that the model yields results comparable to the non-relativistic ones, with the conceptual-advantage of being relativistic and thus automatically incorporating the spin-orbit force. For the calculations of heavy ion elastic scattering cross sections, we have been able to improve previous results due to achieving a better description of the nuclear, densities.<br/><br/>Let us summarize the two apparent merits which the TFh(2) approximation has over the simple TFh(0) one. On the one hand, it provides fully variational densities that go exponential to zero. On the other hand, it takes into account non-local spin-orbit and effective mass contributions up to order h(2), yielding a more reliable average value.

<i>Se establece el desarrollo semi-clásico hasta orden h(2) en la teoría nuclear relativista de campo medio. Así, se obtienen las densidades semi-clásicas relativistas de partículas y de energía para un conjunto de fermiones sometidos a un campo escalar y a un campo vector, en las teorías de campo medio de Wigner-Kirkwood y de Thomas-Fermi, incluyendo correcciones en gradientes hasta orden h(2). El método semi-clásico se aplica a un oscilador armónico relativista. Después se utiliza en modelos T-W no lineales, para los cuales se resuelven las ecuaciones variacionales en núcleos finitos y en materia nuclear semi-infinita. Los resultados semi-clásicos son comparados con los correspondientes resultados cuánticos Hartree.<br/><br/>Para estudiar los efectos de los espinores de Dirac para los nucleones sobre diversas propiedades de la estructura y de la dispersión de núcleos finitos, se construye un funcional de la densidad de energía relativista. El funcional contiene la densidad de energía cinética relativista de orden h(2). La masa efectiva y la parte potencial se obtienen a partir de cálculos Dirac-Brueckner de materia nuclear. Se presta especial atención al cálculo de barreras de fisión de núcleos en rotación y del potencial óptico para la dispersión de iones pesados a energías intermedias. </i>