Electron-Ion Interaction and the Fermi Surfaces of the Alkali Metals

A method is proposed whereby the partial-wave phase shifts which characterize the scattering of plane waves by the ionic cores in a metallic lattice may be deduced from experimental Fermi-surface data. The method is applied to an analysis of currently available experimental data on the shapes of the Fermi surfaces of the alkali metals. Starting from the full many-body theory of conduction electrons in a metallic lattice, it is shown that the shape of the Fermi surface may be derived from a one-electron-like Schrödinger equation which involves a nonlocal effective potential. The augmented-plane-wave method is applied to solve the nonrelativistic Schrödinger equation for the shapes of the surfaces of constant energy in k space, in the approximation where the effective potential may be represented by an angular-momentum-dependent potential of muffin-tin form. The partial-wave phase shifts of the muffin-tin potential are adjusted to bring the area distortions of the computed surface into agreement with the experimental data. The phase shifts deduced in this way are shown to be largely independent of the radius of the muffin-tin sphere. It is found that the shapes of the Fermi surfaces of the alkali metals are systematically influenced by the position of each metal in the periodic table. Lithium shows a strong $p$ phase shift, which anticipates the onset of the $p$ resonance in the second row of the periodic table. The phase shifts in sodium are found to converge rapidly in angular momentum, presumably because the $p$-like component of the ionic potential is largely cancelled by the $p$-like core states, while the $d$-like component of the ionic potential is rather weak. Potassium, rubidium, and cesium show increasingly strong $d$ phase shifts, which are associated with the positions of these metals at the heads of the $3d$, $4d$, and $5d$ transition series. Only for potassium are the experimental data sufficiently accurate to show the influence of a small $f$ phase shift. The method of phase-shift analysis has several advantages as a technique by which the radial distortions of the Fermi surface of a metal may be deduced from experimental data on the anisotropy of its extremal cross-sectional area. These include rapid convergence of the series of phase shifts, the use of a model surface generated in a way closely related to a first-principles band-structure calculation, and the possibility of applying the technique both to nearly-free-electron metals and to metals of the transition series. The radial distortions of the Fermi surfaces of the alkali metals are computed, and the results are compared with those of earlier calculations. Finally, it is shown that our results impose criteria which must be satisfied by any effective potential of muffin-tin form that may be proposed to represent the effects of electron exchange and correlation in the alkali metals.