Limitations of the Method of Moments for Unlike-Spin Broadening

Abstract

The magnetic resonance line-shape analysis based on the second and fourth moments of the frequency distribution, as originally formulated by Van Vleck, is shown to give incorrect results in the presence of the combined conditions (a) rapidly modulated (i.e., narrowed) interactions with unlike-spin species and (b) like-spin broadening comparable to (or greater than) that from mechanism (a). A new line-shape analysis is given based on density-matrix perturbation theory which specifically treats the case of rapidly modulated unlike-spin coupling. The results are compared with the moment analysis and applied to several examples from the literature. The case of the ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$ NMR linewidth in antiferromagnetic Mn${\mathrm{F}}_2$ is discussed in considerable detail, including a detailed evaluation of the ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$-${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$ Suhl-Nakamura coupling. Experimental results are reported for the inhomogeneous ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$ line broadening in a Mn${\mathrm{F}}_2$ crystal of excellent quality, allowing a comparison of measured and calculated free-induction decay times. Contrary to the earlier "method-of-moments" analysis, in which the ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$ line was thought to be narrowed by the combined effect of ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$-${}_{}{}^{55}{}_{}{}^{}\mathrm{Mn}$ dipolar coupling and ${}_{}{}^{55}{}_{}{}^{}\mathrm{Mn}$-${}_{}{}^{55}{}_{}{}^{}\mathrm{Mn}$ Suhl-Nakamura interaction, we find the linewidth to be accounted for by ${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$-${}_{}{}^{19}{}_{}{}^{}\mathrm{F}$ spin-spin interactions alone. This conclusion is in agreement with the new line-shape theory.