Longitudinal Relaxation of Ferromagnetic Grains

We study the activated longitudinal dynamics of a small single-domain magnet with uniaxial anisotropy, coupled to quantum noise. The smallest finite eigenvalue ${\lambda }_1={\Gamma }_0{e}^{{-E}_B/k_{B}T}$ of the relaxation matrix is evaluated in a controlled approximation. For white noise we find ${\Gamma }_0\propto T^{-1\mathrm{}}$ at moderate temperatures and ${\Gamma }_0=\mathrm{const}$ at very low $T$. Coupling to elastic waves leads to a prefactor that is linear in $T$ or constant, depending on temperature. At very low $T$, the discreteness of the energy spectrum is crucial.