Quantum Oscillations of Elastic Moduli and Softening of Phonon Modes in   Metals
  In this paper we present a theoretical analysis of the effect of magnetostriction on quantum oscillations of elastic constants in metals under strong magnetic fields.   It is shown that at low temperatures a significant softening of some acoustic modes could occur near peaks of quantum oscillations of the electron density of states (DOS) at the Fermi surface (FS). This effect is caused by a magnetic instability of a special kind, and it can give rise to a lattice instability. We also show that the most favorable conditions for this instability to be revealed occur in metals whose Fermi surfaces include nearly cylindrical segments. 
Natalya A. Zimbovskaya