Transverse nuclear spin relaxation due to director fluctuations in liquid crystals. II. Second-order contributions of the fluctuating director

Recently, we have introduced a slow-motional theory for transverse nuclear spin relaxation due to director fluctuations [D. Frezzato, G. Kothe, and G. J. Moro, J. Phys. Chem. B 105, 1281 (2001)]. This method is now generalized to second-order contributions of the fluctuating director. We consider the specific case in which the director is aligned orthogonal to the magnetic field. By exploiting the Gaussian character of director fluctuations, the stochastic Liouville equation for the coupled spin and director dynamics is solved in terms of a characteristic function whose time dependence is determined by a nonlinear integral equation. A convenient solution of the integral equation is obtained by decomposing the characteristic function according to the relaxation rates of the director fluctuations. In a first application, we evaluate the free induction decay and the corresponding absorption spectrum for quadrupolar probe nuclei in nematic liquid crystals. It is shown that the transverse magnetization is well represented by a monoexponential decay, i.e., a Lorentzian lineshape in the frequency domain. Explicit relations are derived for the linewidths and frequency shifts under slow-motional conditions where the Redfield theory cannot be applied anymore