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Physical Review C

Approach to the relativistic extended Thomas-Fermi expansion for Green’s functions, phase-space densities, and densities

D. Von-Eiff, S. Haddad, M. K. Weigel

Published Jul 1, 1992• 11 citations
Publisher

Abstract

We present an extension of the semiclassical Thomas-Fermi model to relativistic systems. These are obtained by application of the gradient expansion scheme on the Wigner transformed Dyson equation. Explicitly we give the expansion of the Green's functions, phase-space densities, and densities for a system of nucleons in a vector and scalar potential to second order.

DOI
10.1103/physrevc.46.230

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We present an extension of the semiclassical Thomas-Fermi model to relativistic systems. These are obtained by application of the gradient expansion scheme on the Wigner transformed Dyson equation. Explicitly we give the expansion of the Green's functions, phase-space densities, and densities for a system of nucleons in a vector and scalar potential to second order.
Semiclassical expansions for Green functions, self-energy, phase-space density and density are given and discussed. The many-body problem was treated in the relativistic Hartree-Fock approximation with a Lagrangian with a standard OBE potential structure including the possibility of space-dependent couplings. The expansions are obtained by formulating the many-body problem in the mixed position-momentum (Wigner) representation and application of the h(cross)-Wigner-Kirkwood expansion scheme. The resulting self-consistency problems for the zeroth and second order are formulated in three versions.
We present a new technique for evaluating the gradient expansion of the one-particle Green's function of a system of noninteracting particles. The method is based on the fundamental differential equation for the Green's function and is applicable to relativistic systems as well as to nonrelativistic ones. Explicitly we derive the gradient expansion of the relativistic Green's function of a system of electrons in an arbitrary time-independent external four potential to second order.
Using the $\ensuremath{\hbar}$ expansion of the Green's function of the Hartree-Fock-Bogoliubov equation, we extend the second-order Thomas-Fermi approximation to generalized superfluid Fermi systems by including the density-dependent effective mass and the spin-orbit potential. We first implement and examine the full correction terms over different energy intervals of the quasiparticle spectra in calculations of finite nuclei. Final applications of this generalized Thomas-Fermi method are intended for various inhomogeneous superfluid Fermi systems.
In the application of the extended Thomas-Fermi (ETF) method to a many-fermion system, only those terms in the ft expansion will be included that give finite contribution to the energy. The ETF model yields the smooth part of the level density and other quantities. It will be derived from the E. Wigner-J. G. Kirkwood. It discusses the ETF density variational method and its use for obtaining in a self-consistent way the average parts of the energy and other observables of a many-fermion system in its ground state. Finally, a numerical smoothing of the level density may also be done, by replacing each delta function spike by an appropriate smoothing function, like a Gaussian or a Lorentzian. Since the TF approximation is obtained by replacing the sum over the quantum states by the integral over classical phase space, it is not surprising that it amounts to taking the inverse Laplace transform of the classical canonical partition function.
A systematic method is presented for deriving the Thomas-Fermi equation for an atom and the quantum corrections from the many-body description. The novel feature of the method is that it does not require any a priori assumptions about the assignment of electrons to fully occupied single-particle states or about the distribution of electrons in phase space, but shows instead that the distribution which is usually assumed, or derived from the assumption of fully occupied single particle states, is a direct consequence of specifying that the many particle system is in its ground state. The procedure used in the derivation is the expansion of the mixed position-momentum representation of the Green's function in a series of powers of $\ensuremath{\hbar}$. The lowest order term is found to correspond with the Thomas-Fermi density. The form of the higher order terms, which are to be considered as corrections to zeroth order term, depends on the approximations made in the many-body equations for obtaining the Green's function. This paper deals only with the Hartree-Fock approximation, but the methods presented here allow generalization to other approximations which can include correlation effects.