Hubbard Approximation Research Articles

Screening of a Fixed Charge in the Electron Liquid

The static dielectric function given by Kleinman and Langreth has been used to calculate the screening charge density around a fixed foreign charge in an electron liquid for values of ${r}_{s}$ which correspond to metallic densities. The results are compared with the earlier results based on the Hubbard approximation, and with those of Singwi and Tosi obtained by a self-consistent procedure.

Local Stability, Dielectric Constant, and Spin Susceptibility of Ferromagnetic Itinerant Electrons

By the perturbation calculation within the Hartree-Fock approximation, the free energy of electrons in metals is expanded as a power series of deviations of electron and spin densities from these in the ferromagnetic state up to the fourth powers. From the expansion, the wavevector-dependent dielectric constant and spin susceptibilities are calculated. It is shown that in ferromagnetic itinerant electrons the external electric (magnetic) field induces not only an electron (spin) density but also a spin (electron) density. The discontinuity of the longitudinal spin susceptibility at zero wavevector in the case of a δ-function type interaction has been explained by the nature of interaction. Coefficients in the expansion are calculated at 0°K in the cases of sinusoidal distributions of electron and spin densities and of a helical distribution of spin density in a paramagnetic gas, where the effective mass approximation and the Hubbard approximation for the exchange terms are used. It is found that the coefficients of the fourth powers of these density waves diverge at a wavenumber 2kf, where kf is the Fermi momentum divided by ℏ.

Approximate Screening Functions in Metals

A variational principle is used to solve an approximate integral equation for the screening functions of the electron gas. It is shown that the simplest possible trial function gives a solution to the equation for the dielectric constant which is exact in both the limits of small and large momentum transfers. The results are compared with other calculations. It is shown that the approximation developed recently by Kleinman is quite good in the static large-$k$ limit, but otherwise incorrect. The dielectric constant derived from the variational calculation is used to derive an expression for the ground-state energy; this expression is similar in its essential features to the interpolation schemes of Hubbard and of Nozi\`eres and Pines, even though the Hubbard approximation considerably underestimates the exchange enhancement of the vertex function at large $k$. Finally, it is suggested that similar variational principles may have other uses, as in the paramagnon problem.

Approximate Screening Functions in Metals

A variational principle is used to "solve" an approximate integral equation for the screening functions of the electron gas. It is shown that the simplest possible trial function gives a solution to the equation for the dielectric constant which is exact in both the limits of small and large momentum transfers. The results are compared with other calculations. It is shown that the approximation developed recently by Kleinman is quite good in the static large-$k$ limit, but otherwise incorrect. The dielectric constant derived from the variational calculation is used to derive an expression for the ground-state energy; this expression is similar in its essential features to the interpolation schemes of Hubbard and of Nozi\`eres and Pines, even though the Hubbard approximation considerably underestimates the exchange enhancement of the vertex function at large $k$. Finally, it is suggested that similar variational principles may have other uses, as in the paramagnon problem.

Screening of a Fixed Charge in the Electron Liquid

Using a modified dielectric function which includes short-range correlations, numerical calculations of the screening charge density around a fixed foreign charge in an electron liquid are presented for values of ${r}_{s}$ of interest in the metallic density range. These results are compared with the earlier calculations based on the random-phase and the Hubbard approximations. Sizeable differences are found at distances corresponding to the first two shells of ions around the impurity.

THERMALIZATION TIME OF POSITRONS IN METALS

The thermalization rate of positrons in metals is computed as a function of the electron density parameter r, for the entire range of metallic density 2 < rg < 5.7. The calculation is based on the propagator technique of many-body perturbation theory. In this formalism the rate of energy loss is essentially determined by the imaginary part of the positron self-energy operator which we treat both in the random phase approximation and the Hubbard approximation. In general, we find that the time required for the positron to drop to an energy of 0.025 eV is not as short as is commonly believed, although at room temperature there can be little doubt that complete thermalization has occurred. For aluminium at 100 °K, however, the thermalization time is longer than the annihilation time and it is suggested that this effect should be detectable in an experiment similar to Stewart and Shand's recent positron effective mass experiment in sodium.

The Pair Distribution Function in the R.P.A. and in the Hubbard Approximation

AbstractThe pair distribution function of the metallic electron gas is calculated exactly in the R.P.A. and in the Hubbard approximation. The results obtained are discussed and compared with those of previous approximate calculations and a slight modification to Hubbard's interpolation method is proposed.