Single-Particle Excitations in a Dense Electron Gas Containing a Positive Point Charge

We have investigated the perturbative solution of the equation for the one-particle Green's function for the ideal problem of a dense infinite electron gas with neutralizing uniform positive background and a static "source" consisting of a fixed positive point charge of atomic number $Z$. The densities appropriate to the perturbation expansion are so high as to limit the quantitative applicability of the model to very dense metals or dense degenerate astronomical systems such as white dwarfs. Lowest order expressions for the non-Hermitian effective Hamiltonian of the single-particle excitation spectrum are derived. According to an interpretation discussed in a previous paper, the eigenvalues and eigenfunctions of the effective Hamiltonian correspond also to single-electron energy levels and wave functions associated with ground-state properties of the system. Some general properties of the induced charge density and of the corresponding polarization potential are discussed. The theory predicts the existence of a discrete spectrum of bound holes and its disappearance beyond a certain limiting value of the density, $n$: $n^{\frac{1}{3}}\simeq Z^{2}a_0^{}{}_{}{}^{-1\mathrm{}}$, where $a_0$ is the Bohr radius. This is a consequence of the fact that the lowest order polarization potential is a shielded Coulomb potential (Yukawa potential) with a range inversely proportional to the classical plasma frequency. This potential, derived here by a formal limiting process, is well known from the electron theory of metals where its derivation has been based on a linearized Thomas-Fermi treatment. In order for the discrete spectrum of bound holes to have physical reality it is necessary that the level width of these holes be less than the spacing of bound levels or less than the distance to the continuum limit. This condition is verified, at high densities, by a lowest order calculation of the level width in the same formal high-density limit that yielded the Yukawa potential. Approximate numerical estimates for the level width are then given for a considerably wider range of densities and values of $Z$. It is shown that, to a fair approximation, the level width depends on only two parameters: the ratio of the interparticle spacing to the Bohr radius and the ratio of the binding energy to the Fermi energy, provided that these parameters are less than or comparable to unity. It turns out that away from the limit of very small binding energies, the plasmon-emission mode gives an important contribution to the level width. An interesting consequence of the present work is that for low binding energies the "orbits" of bound holes may be considerably larger than the interparticle spacing. Some physical applications of the results, particularly to the problems of electron capture by a nucleus in a dense medium, and the x-ray spectrum of atoms in metals are briefly discussed.