Correlation Functions for the Director Fluctuations

When director fluctuations have small angular amplitudes, the correlation functions of Eq. (6.7) give 

Note that the damping rate [r q )] has been modified by the contribution of translational diflFusion D) from the form of Eq. (6.21) under the one-constant approximation. The reduced spectral density ji j (a ) due to director fiuctuations is obtained by Foirrier transformation of q (t) 

in fact, simpler first to perform the Fourier transformation to give 

For thermotropic nematics, it is often assumed that ujq a c, so that the cutoff function has a limiting value of unity. This produces the characteristic dependence in the spin relaxation rate [6.3, 6.20]. For lyotropic 

In uniaxial phases, terms involving q 0 in the above equation vanish and S may be approximated by decoupling the coordinate systems to give [6.35] 

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