Theory of Exchange-Narrowed Magnetic-Resonance Linewidth at Finite Temperature

General formulas are derived for the width of an exchange-narrowed magnetic-resonance absorption line at a finite temperature where the Zeeman and/or exchange energies can be comparable to the thermal energy $\mathrm{kT}$. The perturbation which gives rise to line broadening also is allowed to be comparable to $\mathrm{kT}$ in our treatment, and this marks a departure from previous theories of exchange narrowing. The relaxation-function approach of Kubo and Tomita is used, and it is assumed, as in the infinite-temperature limit, that the appropriate correlation times are sufficiently short. We show that the correlation function $〈M_x\mathrm{}\left(t\right)\mathrm{}{M}_x\mathrm{}\left(0\right)\mathrm{}〉$ of the $x$ component of the magnetization has an apparent relaxation rate which is given by an obvious extension of the Van Vleck moments formula to finite temperature. This relation holds for an arbitrary ratio of perturbation energy to $\mathrm{kT}$. However, it appears that only if the perturbation energy is much less than $\mathrm{kT}$ can this relaxation function be simply related to ${\overline{M}}_x\mathrm{}\left(t\right)\mathrm{}$, which describes relaxation of the macroscopic magnetization. If this is the case, then an unambiguous result for the linewidth is obtained which is clearly related to Van Vleck's infinite-temperature theory.