MATTER IN A MAGNETIC-FIELD IN THE THOMAS-FERMI AND RELATED THEORIES

We present a general discussion of the Thomas-Fermi (TF) and Thomas-Fermi-Dirac (TFD) approximations for the ground state properties of matter in a magnetic field taking all Landau levels into account. In the course of doing this we review some facts that are common to all theories of the TF type. Such theories are defined by specifying the energy density w of the electron gas as a function of the electron density n subject to some mild general requirements. Convexity of w is not needed, but singularities in ∇n occur if d2w dn2 is not strictly positive. We also point out that the no binding theorem of TF theory holds irrespective of the shape of w. In TF theory with a magnetic field d2w dn2 vanishes when new Landau bands begin to be populated and singular features in density profiles show up at such densities. These singularities are a rigorous consequence of quantum mechanics in the sense that TF theory becomes exact in the limit when the nuclear charges and the number of electrons tend to infinity, provided the magnetic field is scaled by the same factor as the charges to the power 4 3. Apart from these features an atom with nuclear charge Z in a field of the order of 109 × Z 4 3 gauss exhibits a distinct shell structure associated with the Landau bands. The exchange energy of a homogenous electron gas in a magnetic field is computed in a Hartree-Fock approximation. In particular we obtain closed expressions for the exchange energy for abitrary Landau bands. The inclusion of the exchange energy leads in TFD theory to jumps in the electronic density at which the density of electrons in some Landau band changes discontinously from zero to a finite value. A gradient correction (von Weizsäcker term) in the energy functional smooths out the discontinuities, but divergences in the density gradient reappear if the nuclear charges and the magnetic field tend to infinity as above. © 1992.