Exchange Narrowing: Magnetic Resonance Line Shapes and Spin Correlations in Paramagnetic KMn${\mathrm{F}}_3$, RbMn${\mathrm{F}}_3$, and Mn${\mathrm{F}}_2$

Analyses of line shapes and correlation functions are made to investigate the ${\mathrm{Mn}}^{++}$ EPR and ${\mathrm{F}}^{19}$ NMR in Mn${\mathrm{F}}_2$, KMn${\mathrm{F}}_3$, and RbMn${\mathrm{F}}_3$. Quantitative comparison of theory and experiment is now possible for these systems, and the standard theoretical approximations have proven to be inadequate. However, the form of the exchange interaction and broadening mechanisms is simple enough to allow us to extend earlier work by calculating higher-order moments and higher-order times in the short-time expansion of the correlation function than are usually feasible. We investigate several model line profiles consistent with the second and fourth moments for EPR in the infinite temperature limit. Agreement with experimental line-widths is obtained only for line shapes whose wing structure is non-Lorentzian, in contrast to the truncated Lorentzian forms usually adopted. The correlation functions characteristic of these profiles are obtained for the purpose of comparison with direct calculations later. We present short-time expansions (to fourth order) of the correlation functions, and we find that they decay more slowly than the Gaussian form usually assumed. The NMR correlation can be obtained for all times by a small interpolation between the short-time solutions and the predictions of a diffusion model for long times. The resulting function is consistent with that obtained from the line-profile analysis and both are in substantial agreement with previous spin-correlation studies. The linewidths predicted from this analysis and the moment method are both in agreement with experiment.