In magnetic fields applied parallel to the anisotropy axis, the magnetization of Mn$_{12}$ has been measured in response to a field that is swept back and forth across the resonances corresponding to steps $N=4,5,...9$. The fraction of molecules rema
ining in the metastable well after each sweep through the resonance is inconsistent with expectations for an ensemble of identical molecules. The data are consistent instead with the presence of a broad distribution of tunnel splittings. A very good fit is obtained for a Gaussian distribution of the second-order anisotropy tunneling parameter $X_E=-ln(mid Emid/2D)$. We show that dipolar shuffling is a negligible effect which cannot explain our data.
In magnetic fields applied parallel to the anisotropy axis, the relaxation of the magnetization of Mn$_{12}$ measured for different sweep rates is shown to collapse onto a single scaled curve. The form of the scaling implies that the dominant symmetr
y-breaking process that gives rise to tunneling is a locally varying second-order anisotropy, forbidden by tetragonal symmetry in the perfect crystal, which gives rise to a broad distribution of tunnel splittings in a real crystal of Mn$_{12}$-acetate. Different forms applied to even and odd-numbered steps provide a distinction between even step resonances (associated with crystal anisotropy) and odd resonances (which require a transverse component of magnetic field).
