Statistical Theory of Spin-Spin Interactions in Solids

Abstract

A statistical theory of spin interactions is presented which takes its starting point from pair transitions rather than single-particle transitions. The approach through straightforward perturbation theory applied to pairs, and the approach through the time development of the pair-transition operator $S_x\mathrm{}\left(t\right)\mathrm{}$ are shown to be equivalent. The averaging over possible pair configurations is performed by means of a weight function which allows inclusion of the details of the lattice structure and of departures from random spin distribution. Concentration dependence and temperature dependence appear naturally in the formalism. The formalism is not intrinsically restricted to a particular type of spin interaction. Magnetic dipole forces, exchange of any specified magnitude and any specified finite range, and cross-relaxation effects can be included. Several aspects of moment theory are clarified. The existence of asymmetry in certain line shapes is indicated. The theory is applied to magnetic dipole interaction and exchange. The Fourier transform of the magnetic resonance line shape is derived in an exact, explicit, and semiclosed form, in which the details of the system under consideration appear parametrically. A formula is given for all the moments of the line, but it is shown that moments bear a direct relationship to the observed half-width only in the limit of very dense spin concentration. The results for limiting cases agree with those derived by Anderson and by Kubo and Tomita. In particular it appears that the line is always Lorentzian in the center and Gaussian in the wings, and that it approaches a pure Lorentzian shape as either the spin concentration or the effective nearest-neighbor distance become vanishingly small.