Generalized magnetic susceptibilities in metals: Application of the analytic tetrahedron linear energy method to Sc

A detailed study of the generalized susceptibility $\chi \left(\overrightarrow{\mathrm{q}}\right)$ of Sc metal determined from an accurate augmented-plane-wave method calculation of its energy-band structure is presented. The calculations were done by means of a computational scheme for $\chi \left(\overrightarrow{\mathrm{q}}\right)$ derived as an extension of the work of Jepsen and Andersen and Lehmann and Taut on the density-of-states problem. The procedure yields simple analytic expressions for the $\chi \left(\overrightarrow{\mathrm{q}}\right)$ integral inside a tetrahedral microzone of the Brillouin zone which depends only on the volume of the tetrahedron and the differences of the energies at its corners. Constant-matrix-element results have been obtained for Sc which show very good agreement with the results of Liu, Gupta, and Sinha (but with one less peak) and exhibit a first maximum in $\chi \left(\overrightarrow{\mathrm{q}}\right)$ at $(0, 0, 0.31)\frac{2\pi }{c}$ [vs $(0, 0, 0.35)\frac{2\pi }{c}$ obtained by Liu et al.] which relates very well to dilute rare-earth alloy magnetic ordering at ${\overrightarrow{\mathrm{q}}}_m=(0, 0, 0.28)\frac{2\pi }{c}$ and to the kink in the LA-phonon dispersion curve at $(0, 0, 0.27)\frac{2\pi }{c}$.