  Precise magnetic hysteresis measurements of small single crystals of
Mn$_{12}$ acetate of spin 10 have been conducted down to 0.4 K using a high
sensitivity Hall magnetometer. At higher temperature (>1.6K) step-like changes
in magnetization are observed at regularly spaced magnetic field intervals, as
previously reported. However, on lowering the temperature the steps in
magnetization shift to higher magnetic fields, initially gradually. These
results are consistent with the presence of a second order uniaxial magnetic
anisotropy, first observed by EPR spectroscopy, and thermally assisted
tunnelling with tunnelling relaxation occurring from levels of progressively
lower energy as the temperature is reduced. At lower temperature an abrupt
shift in step positions is found. We suggest that this shift may be the first
evidence of an abrupt, or first-order, transition between thermally assisted
and pure quantum tunnelling, suggested by recent theory.
