Theory of motional narrowing of EPR spectral density of magnetic ions

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Abstract
A theory of the EPR line shape of a magnetic impurity coupled to a lattice is presented that is capable of treating both the "slow" and "fast" regimes. A self-consistent equation of motion for the propagator of the ion motion is obtained under the assumption that the fluctuations of the lattice are not affected by the presence of the impurity. The equation is shown to correspond to the choice of the lowest-order term in a renormalized perturbation theory for the "self-energy" operator of the system. A diagrammatic representation of the perturbation theory is given which one may readily use to include higher-order corrections. The lowest-order approximation is analogous to the mode-coupling, or independent-mode approximation in extended systems, and is equivalent to the Kubo-Tomita approximation in the "fast" regime. A solution of a model problem that exhibits the motional-narrowing phenomenon is given. The theory may be generalized to arbitrary finite-dimensional systems coupled to a bath.