The Fermi Surface and Phase Stability

Assuming a fixed band structure (the rigid band model), a decrease in the density of states is predicted for an increase in the electron/atom ratio for a Fermi surface that contacts the zone boundary. It will be recalled that electrons are diffracted at a zone boundary into the next zone. This means that A vectors cannot terminate on a zone boundary because the associated energy value is forbidden, that is, the first BZ is a polyhedron whose faces satisfy the Laue condition for diffraction in reciprocal space. Actually, when a k vector terminates very near a BZ boundary the Fermi surface topology is perturbed by NFE effects. For k values just below a face on a zone boundary, the electron energy is lowered so that the Fermi sphere necks outwards towards the face. This happens in monovalent FCC copper, where the Fermi surface necks towards the L-point on the first BZ boundary (Fig. 4.3f ). For k values just above the zone boundary, the electron energy is increased and the Fermi surface necks down towards the face. 

An interesting area still under debate in the field of metallurgy is the consequences of Fermi surface topology on the phase equilibria in alloy systems. Elucidation of the connection between these two, seemingly unrelated, features started with the work of William Hume-Rothery, who reported that the critical-valence electron to atom ratios. 

The connection follows the line of reasoning just presented. As a polyvalent metal is dissolved in a monovalent metal, the electron density increases, as does the Fermi energy and Fermi-wave vector. Eventually, the Fermi sphere touches the BZ boundary and the crystal stmcture becomes unstable with respect to alternative structures (Ra5mor, 1947 Pettifor, 2000). Subsequent work has been carried out confimiing that the structures of Hume-Rothery s alloys (alloys comprised of the noble metals with elements to the right on the periodic table) do indeed depend only on their electron per atom ratio (Stroud and Ashcroft, 1971 Pettifor and Ward, 1984 Pettifor, 2000). Unfortunately, the importance of the e/a ratio on phase equilibria is much less clear when it does not correspond precisely to BZ touching. 

---