  The magnetic instability at the front of the spin avalanche in a crystal of
molecular magnets is considered. This phenomenon reveals similar features with
the Darrieus-Landau instability, inherent to classical combustion flame fronts.
The instability growth rate and the cut-off wavelength are investigated with
respect to the strength of the external magnetic field, both analytically in
the limit of an infinitely thin front and numerically for finite-width fronts.
The presence of quantum tunneling resonances is shown to increase the growth
rate significantly, which may lead to a possible transition from deflagration
to detonation regimes. Different orientations of the crystal easy axis are
shown to exhibit opposite stability properties. In addition, we suggest
experimental conditions that could evidence the instability and its influence
on the magnetic deflagration velocity.
