## Abstract

The stability of magnetic states and the mechanism for magnetic
transitions can be analyzed in terms of the shape of the energy
surface, which gives the energy as a function of the angles
determining the orientation of the magnetic moments. Minima on the
energy surface correspond to stable or metastable magnetic states and
can represent parallel, antiparallel or, more generally, non-collinear
arrangements. A rate theory has been developed for systems with
arbitrary number, *N*, of magnetic moments, to estimate the thermal
stability of magnetic states and the mechanism for magnetic
transitions based on a transition state theory approach. The minimum
energy path on the 2*N*-dimensional energy surface is determined to
identify the transition mechanism and estimate the activation energy
barrier. A pre-exponential factor in the rate expression is obtained
from the Landau–Lifshitz–Gilbert equation for spin dynamics. The
velocity is zero at saddle points so it is particularly important in
this context to realize that the transition state is a dividing
surface with 2*N* − 1 degrees of freedom, not just a saddle point. An
application of this rate theory to nanoscale Fe islands on W(110) has
revealed how the transition mechanism and rate depend on island shape
and size. Qualitative agreement is obtained with experimental
measurements both for the activation energy and the pre-exponential
factor. In particular, a distinct maximum is observed in the
pre-exponential factor for islands where two possible transition
mechanisms are competing: Uniform rotation and the formation of
a temporary domain wall. The entropy of the transition state is
enhanced for those islands making the pre-exponential factor more than
an order of magnitude larger than for islands were only the uniform
rotation is viable.

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