Correlations in Magnetic Exchange-Lattice Relaxation

Abstract

At sufficiently low temperatures, the bottleneck in magnetic relaxation can be between the exchange system and the lattice for substances with an exchange interaction of the same order of magnitude as the Zeeman interaction. We extend Griffiths' original theory by including correlations between different pairs of spins in a calculation of the exchange-lattice relaxation rate. A density-matrix approach is used in which the exchange and lattice systems are each assumed to possess a temperature. Results are restricted to temperatures for which Curie's law holds. Variation of exchange interaction with interatomic spacing is assumed to be the dominant source of relaxation, and single- and two-phonon processes are analyzed as well as atomic diffusion with a slow jumping rate. For linear chains with nearest-neighbor interactions, correlations have a marked effect on the phonon relaxation rate at temperatures well below the Debye $\theta$. Two-phonon processes give a $T^9$ temperature dependence instead of the $T^7$ dependence found by Griffiths, and the relaxation rate for the direct process is less than that calculated by Griffiths by a factor of the order of ${\left(\frac{\hslash J}{k\theta }\right)}^2$. This could be of more than academic interest, since a number of paramagnetic materials in which exchange-lattice relaxation might be observable appear to have linear chain structures as far as magnetic properties are concerned. In three-dimensional magnetic lattices the correction to Griffiths' formula is about 20% for the two-phonon processes. If the source of lattice motion is atomic diffusion with a jumping frequency $\frac{1}{{\tau }_c}$ much less than the exchange frequency, then correlations are shown to be negligible. In this instance Garwin and Landesman's result $T_{\mathrm{EL}}\approx {\tau }_c$ is reestablished.