## Abstract

It is well known that in a nematic liquid crystal the local director is liable to fluctuate in direction, and that the fluctuations contribute a frequency-dependent term to the nuclear magnetic relaxation rate, $1/T_{1}$ (Pincus 1969). Previous attempts to calculate this term are criticized, and an alternative calculation is presented, based on the continuum model developed in earlier papers in this series. The spectral density function $\tilde{J}_{2}(2\omega _{0})$ which occurs in the conventional formula for $1/T_{1}$ is found to contribute a term proportional to ln $(\omega _{0}\tau _{\text{c}})$, where $\tau _{\text{c}}$ is the relaxation time for fluctuation modes at the cut-off wave number $q_{\text{c}}$. This term, hitherto ignored, may be comparable with the leading term proportional to $(\omega _{0}\tau _{\text{c}})^{-\frac{1}{2}}$, which is contributed by $\tilde{J}_{1}(\omega _{0})$, especially in low-$T_{\text{c}}$ nematics at temperatures close to $T_{\text{c}}$, where $\tau _{\text{c}}$ should be large and the order parameter $S_{2}$ is small. In addition, the coefficient of the leading term is found to involve $S_{2}^{2}(S_{2}^{-\frac{1}{3}}-1)$/ln$S_{2}^{-\frac{1}{3}}$, rather than $S_{2}^{2}$ as has hitherto been supposed. A formula is also derived for the rotating-frame relaxation rate, $1/T_{1\rho}$, which includes an unconventional term in ln $(\omega _{1}\tau _{\text{c}})$. This theory, like the theories it is intended to replace, is strictly applicable only to the ring protons on the nematic molecules, and only to the intramolecular as opposed to intermolecular relaxation rates. An attempt is made to compare it with some of the data recorded in the literature, but further experiments on suitably deuterated materials would be needed to check it effectively.

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