Quantum relaxation of magnetisation in magnetic particles

At temperatures below the magnetic anisotropy energy, monodomain magnetic systems (small particles, nanomagnetic devices, etc.) must relax quantum mechanically-thermal activation is ineffective. The discrete nature of the spectrum is important. This quantum relaxation must be mediated by the coupling to both nuclear spins and photons (and electrons if either particle or substrate is conducting). We analyze the effect of each of these couplings, and then combine them to get results for the physical relaxation of magnetic particles at low temperature and bias. This done for both conducting and insulating systems. The effect of electrons and phonons call be handled using ''oscillator bath'' representations; but the effect of the environment spins must be described using a ''spin bath'' representation of the environment, the theory of which was developed in precious papers. Conducting systems can be modelled by a ''giant Kondo'' Hamiltonian, with nuclear spins added in as well. At low temperatures, even microscopic particles oil a conducting substrate 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 substrate blocking of the spin dynamics at a well defined temperature. We analyze in turn the 3 different cases of (a) conducting substrate, conducting particle, (b) conducting substrate, insulating particle, and (c) conducting particle, insulating substrate. Insulating systems are quite different. The relaxation is strongly enhanced by the coupling to nuclear spins. At short times the magnetization 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. Depending on the parameters of the particles and the temperature, the crossover from nuclear spin-mediated to phonon-mediated relaxation can take place after a time ranging between fractions of a second up to months. The most surprising feature of the results is the pivotal role played by the nuclear spins. The results apply to any experiments on magnetic particles in which interparticle interactions are unimportant (we do not deal with the effect of interparticle interactions in this paper). They are also relevant to future magnetic device technology.