  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.
