Calculation of NMR line shapes in calcium fluoride from modified moment expansions

Abstract

Theoretical second, fourth, sixth, and eighth moments of nuclear-magnetic-resonance absorption lines in calcium fluoride are used to examine the convergence of two different modified moment expansion for free-induction-decay (fid) curves. These expansions provide a systematic method of obtaining corrections to two initial approximations to a line shape which are obtained from either the local-field model, which gives a Gaussian fid curve, or the Abragam function. In the former case one obtains the Fourier transform of the Gram-Charlier expansion and in the latter case a Neumann expansion. These expansions may also be applied to the memory function, a local-field correlation function, rather than the fid function since, in general, the functional form of a memory function is insensitive to the form of a line shape and, in particular, these two curves are similar in shape for dipolar-broadened resonance lines. In analyzing the results of these expansions we are led to formulate a condition for oscillations in an fid curve. This condition is that local-field correlations persist for a time ${T}_{2}^{**}$ which is at least of the order of the mean beat period ${M}_{2}^{\ensuremath{-}\frac{1}{2}}$. Here ${M}_{2}$ is the second moment of the resonance line and ${T}_{2}^{**}$ is the relaxation time of the memory function. Also, a new trial function is proposed for Ca${\mathrm{F}}_{2}$ fid curves which gives the proper behavior at both long and short times.