Electron-spin polarization in the Thomas-Fermi and Thomas-Fermi-Dirac atoms.

Abstract

The Thomas-Fermi (TF) and Thomas-Fermi-Dirac (TFD) statistical models are extended by the electron-spin interaction with an external uniform magnetic field B. The statistical atom, which is spherical, is then described with the aid of two distributions: the electron density n(r) and the relative magnetization \ensuremath{\zeta}(r). In the TF atom the magnetic field polarizes the electron spins totally (\ensuremath{\zeta}=1) in a region near the atomic boundary, while the rest of the TF atom is only partially polarized; however, \ensuremath{\zeta}(r) and n(r) are continuous. A convenient description of this atom is done with the aid of a suitably defined reduced potential f(x) to which n(r) and \ensuremath{\zeta}(r) are related. At B>0 the neutral TF atom remains infinite. Its magnetic moment shows unphysical ${\mathit{B}}^{3/4}$ proportionality for small B. The inclusion of the exchange in the TFD model for B>0 results in two possible types of atom: the atom of the type I---which exists only for B\ensuremath{\le}1.3\ifmmode\times\else\texttimes\fi{}${10}^{7}$ G---has continuous n(r) and \ensuremath{\zeta}(r); in the type-II atom there is a discontinuity in both n(r) and \ensuremath{\zeta}(r) that makes this atom unusable in physical applications. The type-I atoms have finite radii and are assumed to represent approximately the crystal atomic cells. The appropriate differential equation and boundary conditions for \ensuremath{\zeta}(r), together with the relationship between \ensuremath{\zeta}(r) and n(r), have been derived for this atom. The solutions of the equations and the atomic radii have been obtained for a wide range of atomic numbers and magnetic fields. On the basis of these solutions, the coefficients for the volume magnetostriction, the spin susceptibilities, and the ionization energies have been calculated for elements of the whole periodic table. Qualitatively, the experimental tendencies are found to be represented well by the type-I TFD atoms. A modified virial theorem for the spin-polarized TF and TFD atoms has been proved.