  The quantum magnet NiCl$_2$-4SC(NH$_2$)$_2$ is a candidate for observing
Bose-Einstein Condensation of spin degrees of freedom in applied magnetic
fields. An XY antiferromagnetic ordered state occurs in a dome-shaped region of
the temperature-field phase diagram between H$_{c1}$ = 2.1 T and H$_{c2}$ =
12.6 T and below 1.2 K. BEC corresponds to the field-induced quantum phase
transition into the ordered state. We investigate magnetostriction in single
crystals of this compound at dilution refrigerator temperatures in magnetic
fields up to 18 T, and as a function of magnetic field angle. We show that
significant changes in the lattice parameters are induced by magnetic fields,
and argue that these result from antiferromagnetic couplings between the Ni
spins along the tetragonal c-axis. The magnetic phase diagram as a function of
temperature, field, and field angle can be extracted from these data. We
discuss the implications of these results to Bose-Einstein Condensation in this
system.
