  At temperatures below the magnetic anisotropy energy, monodomain magnetic
systems (small particles, nanomagnetic devices, etc.) must relax quantum
mechanically. This quantum relaxation must be mediated by the coupling to both
nuclear spins and phonons (and electrons if either particle or substrate is
conducting. We analyze the effect of each of these couplings, and then combine
them. Conducting systems can be modelled by a "giant Kondo" Hamiltonian, with
nuclear spins added in as well. At low temperatures, even microscopic particles
on a conducting substrate (containing only $10-50$ spins) will have their
magnetisation frozen over millenia by a combination of electronic dissipation
and the "degeneracy blocking" caused by nuclear spins. Raising the temperature
leads to a sudden unblocking of the spin dynamics at a well defined
temperature. Insulating systems are quite different. The relaxation is strongly
enhanced by the coupling to nuclear spins. At short times the magnetisation of
an ensemble of particles relaxes logarithmically in time, after an initial very
fast decay; this relaxation proceeds entirely via the nuclear spins. At longer
times phonons take over, but the decay rate is still governed by the
temperature-dependent nuclear bias field acting on the particles - decay may be
exponential or power-law depending on the temperature. The most surprising
feature of the results is the pivotal role played by the nuclear spins. The
results are relevant to any experiments on magnetic particles in which
interparticle dipolar interactions are unimportant. They are also relevant to
future magnetic device technology.
