There have been several papers about Fermi Surface nesting, but it's hard to rebuild. What is the connection between Fermi surface nesting and the elastic anomaly?
Article Ab initio calculations of elastic constants of the bcc V–Nb ...
At low energies, an electron in a metal can only really interact with other electrons when both are close (within k_B T) to the Fermi surface, which means that contributions to the q-dependent static (=low frequency or low energy) susceptibility will come from q vectors that connect one state on the fermi surface with a state somewhere else on the fermi surface. Such a connection of different bits of the Fermi surface is called "Fermi surface nesting". In particular, if you connect two parallel sheets of Fermi surface, this will give you a peak in the susceptibility, which in term means that the electrons have a strong tendency to respond to external perturbations with that q vector.
If the nesting is strong enough (=big peak in susceptibility), the system might be so susceptible that it spontaneously breaks the symmetry by that q-vector even though no external pertubation is around. This is what happens in these papers is that, under pressure, the Fermi surface ends up having so strong nesting features that the system spontaneously wants to change its symmetry, and this gives rise to the lowering (softening) of the elastic constants. The fact that a cubic elastic constant goes negative means that it is energetically favourable to distort the lattice.
In this case, you can also apply a kind of Jahn-Teller like argument, since what happens when the lattice distorts is that portions of the fermi surface merge over the zone boundary, so what happens is that you remove some of the degeneracy of the Fermi level by distorting the lattice (this is what they show in Fig. 6a of the JPCS paper, the DOS drastically goes down under pressure when the lattice is distorted).
If you find the generalized susceptibility tricky to evaluate, you can always try the method we used in the attached paper here, where we just studied the "nesting intensity" as function of q along some axis. This contains basically the same information as the generalized susceptibiliy as far as the Fermi surface is concerned.
Article Theoretical studies of the incommensurate magnetic structure...
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