Transverse nuclear spin relaxation due to director fluctuations in liquid crystals - A slow-motional theory

Transverse nuclear spin relaxation measurements, employing the Carr-Purcell-Meiboom-Gill sequence, can provide detailed information on the dynamics of director fluctuations in liquid crystals. In principle, a full characterization of the rate dispersion of the director modes can be derived from such measurements. However, the rigorous analysis of these experiments is generally hampered by the lack of a time-scale separation between the slow director fluctuations and the transverse magnetization decay. Under these conditions, the use of a fast-motion theory (i.e., Redfield theory) is no longer justified and one should resort to a slow-motional approach based on the stochastic Liouville equation for the simultaneous evolution of the stochastic variables (i.e., the director field) and the spin degrees of freedom. In this paper, explicit expressions for transverse deuteron spin relaxation times are derived without invoking any time-scale separation, on condition that (i) the stochastic variables are described as a multidimensional Gaussian process and that (ii) the spin Hamiltonian linearly depends on the stochastic variables. These are precisely the conditions usually adopted for the modeling of director fluctuations and their relaxation effects in the harmonic approximation. Thus, the present theory allows for a rather general analysis of the transverse magnetization decay in different kinds of experiments. In particular, analytical expressions are derived for the transverse relaxation times in Carr-Purcell-Meiboom-Gill sequences and evaluated as a function of the pulse spacing and the number of cycles in the sequence. It is shown that in the limit of an infinite number of pulses, one can interpret the measured asymptotic relaxation time as a superposition of independent contributions evaluated according to the Luz-Meiboom equation (Luz, Z.; Meiboom, S. J. Chern. Phys. 1963, 39, 366), originally derived within the Redfield limit.