Group theoretical analysis of nuclear spin relaxation in liquid crystals and molecular solids

Abstract

A comprehensive and general group theoretical analysis is presented of the time correlation functions (TCFs) governing nuclear spin relaxation in anisotropic systems such as liquid crystals and molecular solids. Without invoking specific dynamic models, the consequences of time-reversal invariance (detailed balance) and rotational symmetry are explored systematically. By constructing TCFs from irreducible tensor components adapted to the crystal symmetry and making use of group theoretical orthogonality relations, the most general and concise expression of the orientation dependence of the labframe TCFs is derived. Depending on the crystal symmetry, there are between 2 and 15 coefficients in this expression: these are the irreducible crystal-frame TCFs associated with the irreducible representations of the crystal point group. The corresponding irreducible crystal-frame spectral densities constitute the complete model-independent information content derivable from spin relaxation experiments. Explicit expressions for the irreducible crystal-frame TCFs and the associated angular functions are given for all crystallographic point groups. The combined effect of crystal symmetry and local (microstructure) symmetry is also investigated, as is the effect of symmetry on TCF contributions from motions on multiple time scales. Furthermore, the relation of the TCFs to the independent order parameters is established. While many of the new results are applicable to a wide variety of spectroscopic techniques and systems, the explicit results have their greatest potential for spin relaxation studies of oriented lyotropic liquid crystals.