Fermi Surface, Pseudopotential Coefficients, and Spin-Orbit Coupling in Lead

The de Haas-van Alphen periods in lead have been studied in 200 kG impulsive fields, employing techniques with sensitivity and selectivity which are far improved over those used in an earlier study of this metal, and several new sets of oscillations have been discovered. The results confirm in detail the correctness of a nearly-free-electron Fermi surface based on four conduction electrons per atom, and the experimentally determined Fermi surface has been described in terms of an interpolation scheme using four orthogonalized plane waves for each wave vector k. As would be expected for a heavy metal such as lead, it is necessary to allow for the large spin-orbit interaction in order to achieve an accurate description of all portions of the Fermi surface. The four adjustable parameters required in the description have been determined by a least-squares fit to eight observed extremal areas of cross section, and are found to be (in Ry): Fermi energy $E_f=0.718\mathrm{}\pm 0.001$; Fourier coefficients of the pseudopotential $V_{111}=-0.084\mathrm{}\pm 0.002$ and $V_{200}=-0.039\mathrm{}\pm 0.002$; and spin-orbit interaction $\lambda =0.096\mathrm{}\pm 0.002$. These values of the parameters refer to a specific interpolation scheme and assume that the mass in the kinetic-energy matrix elements is the free-electron mass (i.e., no attempt has been made to consider explicity many-body effects or the electron-phonon interaction). When the above parameters are used, the total occupied volume is calculated to correspond to 4.02±0.02 electrons per atom, and the model confirms the experimental finding that the fourth zone is empty. The calculated dispersion curves $E\left(\mathrm{k}\right)$ reflect the `inert-pair' behavior which is so well known in the chemistry of lead salts, in that there is a large energy gap between a filled $6s$-like band and the lowest branch of the $6p$-like bands which is never less than 0.13 Ry. The predictions of the model as regards the detailed orientation dependence of the de Haas-van Alphen periods are found to be in excellent agreement with experiment; comparison is also made with the Fermi-surface dimensions, cyclotron masses, magneto-resistance, etc., as determined by other experiments.