Investigation of an Extended Magnetization Vector Constraint

In continuum micromagnetics, the classical “fixed” length constraint for a magnetization vector
M
is well known, while at the same time, some experimental conditions suggest the condition clearly breaks down, e.g., finite temperature. Assuming a fixed length for an elementary moment and using the definition of a magnetization vector as its volume average, along with existing continuum exchange theory, it can be shown that this constraint does not explicitly follow, but rather an extended form of it which includes exchange. Using this extended constraint, consequent equilibrium (and dynamic) continuum micromagnetics formulations are numerically solved to examine differences in current and this “extended” formulation. The classical problem of the transition of nanoparticles from flower to vortex states is considered for illustration. The use of the extended constraint suggests greater potential for nonuniformity at smaller particle sizes. Considering dynamics, it is also observed that new contributions from exchange are present in the direction of “damping,” which have the effect of speeding up magnetization “motion,” thus minimally damped systems may still demonstrate fast reversal, contrary to current intuition based on the LLG.