Three-dimensional ensemble averages for tensorial interactions in partially oriented, multi-particle systems

Abstract

In a variety of three-dimensional, multi-particle systems, interactions of tensorial form occur between individual components and an applied stimulus that operates uniformly throughout the ensemble. When each material component has an identical, fixed orientation, its own response is replicated in the observed form of behaviour by the system as a whole, with respect to the angular disposition of the stimulus. The same complete correlation between microscopic and macroscopic response does not, however, operate in other systems where there is a degree of orientational freedom amongst the microscopic components. One limiting case is where the extent of such freedom allows random orientations, such conditions delivering a system that is macroscopically isotropic. When each particle has incomplete orientational freedom, the formulation is in general less mathematically tractable; here, the theory required to describe ensemble response yields analytically solvable equations only in cases where the distribution function takes a relatively simple form, such as a scalar product, and few explicit results are available. This paper addresses general systems in which there is an orientation-dependent complex exponential factor, weighting the response of each component. By means of irreducible tensor decomposition, results are determined for tensor interactions up to and including rank 2, where the weighting exponent is also of any rank up to the same order. Illustrative applications are drawn from the theory of laser particle orientation and photon absorption.

Laser optical separation of chiral molecules

“…As a result of this anisotropic interaction with the laser beam, the mean potential energy of the trapping interaction is subject to a weighted orientational distribution. To analytically evaluate the potential energy with the appropriately weighted averages introduces some demanding mathematics: a complete theoretical representation has only recently become possible [18], finding its first application in connection with optical binding [19]. The initial step is represented by the following; …”

“…(4.2) in ref. [18] is omitted since B 1 0 and, additionally, n m since C 1 0.) On application of the relevant rank of rotational average [22,23] for each term in Eq.…”

Optically tailored access to metastable electronic states

“…The selection rules procedure we have employed follows a pattern established previously on nonlinear optical processes, for example Refs. [12][13][14][15][16][17][18], and considers the irreducible Cartesian tensor components of the polarization and molecular response tensors [11]. An irreducible Cartesian basis affords direct connectivity with the directional properties of the radiation and of the electronic transitions, while also affording a transparent linkage with the symmetry-based irreducible representations of the molecular electronic states, using the rules of angular momentum coupling.…”

Electromagnetic trapping of chiral molecules: orientational effects of the irradiating beam

“…The colons denote an inner product between two such tensors, and the chevrons signify an ensemble average [41]; A and C refer to the optical stimuli (in the laboratory frame); and B and D to the molecular response (molecule frame). This expression represents the dynamic analog to the much simpler physics associated with calculating a mean dipole in a static field, which includes a weighting distribution involving only rank-1 tensor (vector) contractions [42], and whose general form of weighting leads to the familiar Langevin function [43].…”