title: Relation of system dimensionality and order parameters.
abstract: Orientational order parameters are useful metrics for characterizing the probability distribution for vector-valued quantities such as the dipole moment or optical axis of molecules in materials such as liquid crystals and organic glasses. These parameters are the moments of the underlying orientational probability distribution. Many molecular systems can be characterized using a single centrosymmetric (even) moment. For dipolar systems, an applied electric or magnetic field can break the symmetry of the system, leading to nonzero acentric (odd) moments. For complex systems, it is difficult to characterize the nature of the bulk structures and to quantitatively understand the relationship between acentric and centrosymmetric moments. We have found that it is useful to relate the moments of the distribution in terms of an apparent dimensionality of the ordering process. Here we show that the idea of noninteger dimensionality, originally introduced by Stillinger, provides a useful method to characterize the relation between centrosymmetric and acentric orientational order parameters. Applying dimensional constraints is equivalent to removing rotational degrees of freedom or constraining rotation within a restricted volume. Simulations based on simple examples—using restoring potentials on arrays of independent dipoles—and on complex many-body Monte Carlo simulations of dipolar spheroids are described. An analysis of the results illustrates the utility of fractional dimensionality to describe ordering in materials.
authors: Department of Chemistry, University of Washington , Seattle, Washington 98195-1700, United States.
authors: Bruce H
authors: BH
authors: Robinson
authors: Lewis E
authors: LE
authors: Johnson
authors: Bruce E
authors: BE
authors: Eichinger