Correlation Functions in Dipolar Absorption-Dispersion

BY C. BROT

1 Introduction

The absorption and dispersion due to permanent dipole moments is one of the oldest techniques for the study of the orientational dynamics of molecules. The first paper by Debye on the subject was published more than sixty years ago. However, two important kinds of progress have been made in recent years: the first one is the extension of the experimentally accessible frequency range into the far infrared; the second one, which is theoretical, is the development of the correlation function formalism, which allows the theoretical models to be worked out in equilibrium.

Dielectric absorption is perhaps the case where the fluctuation-dissipation theorem finds its most direct case of applicability, as long as one stays on the macroscopic level. The difficult part of the problem is rather to link macroscopic fluctuations with molecular fluctuations. This problem involves dynamical intercorrelations of the electric moments of the molecules arising both from the long-range forces (reaction field) and from the short-range forces. The motions which are revealed by the dielectric measurements are approximately of a unimolecular character when the second type of inter-correlation is negligible (there are reasonably good methods for taking into account the first type). It is thought at present that such a simplification is justified in the case of simple, non-associated, molecular liquids; this is inferred from the near absence of static intercorrelations in these liquids; however, such an absence does not necessarily imply the non-existence of dynamical intercorrelations. In the Reporter's opinion it should be one of the important tasks of the dielectricians in the next few years to explore the possible existence of dynamical intercorrelations, e.g. by comparison of their results with one-molecule correlation functions obtained from vibrational spectroscopy.

As another preliminary remark, we note the growing interest in the relatively short time region of the dipolar motion, which is mainly reflected in the far-infrared part of the spectrum. Indeed, this is the time region where the dipolar motion is the most sensitive to the detail of the molecular environment. By contrast, on the one hand at very short times the orientational motion is the same as for a (classical) molecular gas, whereas on the other hand at long times it becomes stochastic, and then the single figure which in many cases is sufficient to describe the rapidity of the orientational randomization conveys very little information and not too much when combined with other deductions.

As implied in the title, and in conformity with a remark made above, the stress in this Report will be put on the time correlation functions, both macroscopic and molecular. This approach has already been used by Wyllie in an excellent Report written for this series. It is hoped that the present contribution will be complementary rather than redundant with respect to it. Indeed, we will dwell in much more detail on the relations between macroscopic and microscopic correlation functions, i.e. dynamical dielectric theory. Concerning the choice of illustrative models, the ones quoted by Wyllie are of great pedagogic value; we will rather make a Report of the models specialized to the description of orientational motion in different physical situations.

The next section of this Report will concern essentially the macroscopic correlation functions and their relations with the complex electric permittivity. The third section will give the proposed relations between macroscopic and molecular correlation functions, both individual and collective. The fourth section will deal with all that can be said rigorously about the behaviour of the molecular correlation functions, and this concerns the short-time behaviour only. Section 5 will be devoted to establishing relations between the correlation-function language and other descriptive formalisms. For the long-time behaviour of the orientational motion, one has to resort to models; these will be described in the last section; some of these models, probably familiar to the reader under an older formulation, will be cast into the correlation-function language; in passing, experimental evidence of the approximate validity of each model in specific cases will be mentioned.

Some of the approaches used to derive known results are thought to be original: for example, the extensive use of the notion of 'rigid' dipole first introduced by Frohlich, the demonstration of the Kramers-Kronig relations employing the fluctuation-dissipation theorem applied to a thin rod, and the extension of Gordon's sum rule to interacting systems using the equipartition of the rotational kinetic energy and its diagonality with respect to the molecules.

2 Macroscopic Applications of the Fluctuation-Dissipation Theorem

Response of a Dielectric Sample and 'External Field Susceptibility'. — It is demonstrated in statistical mechanics that if a relatively small external force F(u), which depends on time u but not on co-ordinates and momenta, is applied to a material system, the response of any quantity B which depends only on the co-ordinates and momenta of the particles and not directly on the time is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where one has assumed F(—[infinity) = 0 and where the real function φBA(t), called the (impulsive) response function, depends only on the correlated time-fluctuations of B and A at equilibrium. Here A is the quantity which, when multiplied by the applied force, yields the (small) increment in the Hamiltonian H of the system, i.e. ΔH = – AF(u).

Since in the case of dielectric phenomena the applied force is an external field Ee = F(u), the quantity A must be the component M along Ee of the electric moment of the system. Since it is also customary to study the response of a dielectric by measurement of its polarization (or moment per unit volume), the observed quantity B is also the electric moment, so that (phi)BA(t) reduces to φAA(t) and the correlations of fluctuations to be considered are autocorrelations of the total electric moment of the system along Ee.

Dropping now the subscript AA, and excluding the case of ferroelectrics where the equilibrium value of M does not vanish, equation (1) becomes:

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