Sampson-Seitz Procedures and the Static Responses of Normal Fermion Systems

It is shown that a differential characterization of Sampson-Seitz methods for calculating the static responses of a normal fermion system at zero temperature yields expressions for these quantities which are identical with those deduced by Landau from a semiphenomenological basis and by the author, in several instances, from spherical, time-independent, many-body perturbation theory. This connection is explicitly demonstrated for the Galilean invariance, magnetic susceptibility, and compressibility of a normal fermion system with translation-invariant interactions. The calculation of the magnetic susceptibility and compressibility of a dense electron gas is examined from both points of view. In the case of the magnetic susceptibility, it is shown that the contribution to ${\alpha }_c$ from graphs with two virtual excitations vanishes to $O\left(r_s\right)$ in agreement with the Sampson-Seitz calculation of Brueckner and Sawada. Although this demonstration is trivial by Sampson-Seitz methods, it is only a consequence of detailed calculation in Landau's formulation. It is found that $\left(\frac{1}{K}\right)\left(\frac{2}{\rho R}\right)={\alpha }_c-0.0676\mathrm{}\ln r_s+0.255\mathrm{}$, where $K$ is the compressibility, $\rho$ is the density, and $R$ is the rydberg.