## Irreversible Magnetic Relaxation

The magnetic moment of a ferromagnetic system magnetized in an external field and then left on its own often changes with time. This is one of the manifestations of the so-called ''irreversible magnetic relaxation phenomena'', which inevitably occur at finite temperatures in any magnetic system which is not in a thermal equilibrium state.

A typical experiment to observe irreversible magnetic relaxation is as follows: a system is magnetized in an external field so that it acquires some total magnetic moment in the direction of this field. The magnitude and (or) the direction of the applied field is then suddenly changed and the time dependence of the system magnetic moment is measured. For a wide class of magnetic systems (magnetic powders, some alloys, thin films, etc.) such measurements provide a nontrivial result: magnetization relaxation is not exponential (mz @ exp(—t/tc), which one would expect for the thermal relaxation of a system over an energy barrier) but rather can be described by a linear-logarithmic dependence mz = mo - Sln(t0) or d(nT) = -S(= Const) (L67)

where the coefficient S is called magnetic viscosity. This linear-logarithmic dependence fits in many cases experimental data measured over many time decades from seconds to years (!) quite well. Such relaxation is called ''anomalous'' in order to distinguish it from the simple exponential relaxation.

The first phenomenological explanation of this phenomenon for magnetic systems was provided by Street and Wooley (1949). These authors suggested that such an unusual (at that time) relaxation behavior was due to the wide distributions of the energy barrier heights in the system under study. To understand why such a distribution leads to the linear-logarithmic behavior, let us consider the simplest model, namely a system of noninteracting magnetic particles each of which has two equilibrium magnetization states separated by the energy barrier E. We assume that the height of this barrier changes from particle to particle, and that the fraction of particles dN with the energy barriers in the small interval from E to E + dE is dN = p(E) dE (in this case p(E) is called the distribution density of the energy barriers).

The irreversible magnetization relaxation for particles with the given energy barrier E is given by the simple exponential Arrhenius law mentioned above: mz @ exp(—t/tc). This means that for each such particle the probability p(t) to jump over the barrier during the time t is given by p(t) = 1 — exp(—t/tc). The relaxation time tc also exhibits an exponential dependence on the barrier height: tc = t0 exp(E/kT) where the prefactor is about t0 @ 10—9 s (see the discussion of the superparamagnetic phenomena given above).

For the observation time t, all particles with the relaxation time tc « t, had already relaxed almost surely (t/tc >> 1, exp(—t/tc) a 0 probability to jump p(t) a 1), so that their relaxation can no longer be observed. The particles with much larger relaxation times t/tc « 1 are still not yet relaxed almost surely (t/tc « 1, exp(—t/tc) a 1, probability to jump p(t) a 0), hence, their relaxation could no longer be observed either. Thus, the only particles whose relaxation we measure at the observation time t are those with the relaxation time tc (= t0 exp(E/kT)) @ t - that is, with the energy barriers E a Ec = kT ln(t/t0). Due to the very strong (exponential!) dependence of the relaxation time on the energy barrier height E, only those particles with barriers in a narrow interval DE @ kT around the so-called critical energy Ec(t) = kT ln(t/t0) make any substantial contribution to the magnetic relaxation, observed at time t (see Fig. 1.23a), where the probability to jump - that is, the probability P(E) to overcome the energy barrier - is shown as a function of the barrier height E. In other words, it is a very good approximation to treat the critical energy Ec, as the boundary between the already relaxed and not yet relaxed particles (Fig. 1.23b).

The width of the energy barrier distribution p(E) is normally much larger than the thermal energy kT. For this reason, inside the mentioned small interval DE @ kT around Ec this distribution can be treated as constant. Hence, the number of relaxed particles dn (and the magnetic moment decrease — dmz) in the time interval from t to t + dt can be calculated as the product of the corresponding value p(Ec) and the shift of the critical energy dEc during this interval (see Fig. 1.23c; we recall that the critical energy separates relaxed and nonrelaxed particles): dmz a —r(Ec) dEc(t). Substituting in this relationship the time dependence of the critical energy Ec(t) = kT ln(t/t0), so that dEc(t) = kT dt/1, we obtain dmz a —kTp(Ec) dt/t or dmz dmz dt d ln t

which coincides with Eq. (1.67) if we set S = kTp(Ec). This means that the magnetic viscosity is simply proportional to the value of the energy barrier distribution density for the critical energy Ec. Generally speaking, this depends on the observation time due to the time-dependence of Ec @ ln t. However, since this dependence is very weak (logarithmic), it can be neglected for nonpathological barrier densities p(E), and this leads to the almost constant magnetic viscosity (Eq. 1.67) observed experimentally.

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