CHAPTER 1
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:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
or
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2')
One has in classical mechanics:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3a)
where φ(t)=<M(O) · M(t)> is the equilibrium autocorrelation function; it is real and even.
One has in quantum mechanics:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3b)
where M is now an operator and p is the density matrix. Although φ(t) is again real and even, Tr[ρM(O) · M(t')] is not.
Equation (2) can be viewed as axiomatic: it expresses a superposition principle. It is, however, better to remember that it is a theorem resulting from the linear-response approximation of irreversible statistical mechanics, with cp being identified with either of expressions (3). It is precisely this identification which constitutes the fluctuation-dissipation theorem proper.
Except for a few molecules with very low moments of inertia, classical mechanics can be applied to the study of orientational motion and consequently of dipolar absorption. Hence we will use equation (3a) rather than (3b). When necessary, a first-order quantum correction can be added quite simply.
The most usual forms adopted for the time variation of the applied force, i.e. the electric field, are step functions and alternating fields.
Decreasing Step Function, i.e. Removal of a Constant Field. Remembering that one must have Ee( —∞) = 0, we adopt the following field variation, which yields an almost constant field for small negative times:
[ILLUSTRATION OMITTED]
Using equation (2) for positive u, noting that with the above field the upper limit of the integral in equation (2) can be put equal to zero, and using equation (3a), we obtain:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The second term is negligible since γ has been chosen arbitrarily small with respect to the decay rate of φ. Since also φ(+∞)= 0:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
In words, after removal of a constant field the moment vanishes following the same law as the regression of its spontaneous fluctuations.
Alternating Fields. We now write the field:
Ee(u)= Ee0 R exp[(iω + γ)u] (γ very small)
Hence:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
One defines an 'external field susceptibility' x*e(omega) by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Hence the susceptibility is given by a Fourier-Laplace transform:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
Adopting classical mechanics, i.e. using equation (3a), and integrating by parts:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
With this relation, the evaluation of Xe from is established, but it is necessary to examine the relation of Ee(omega) to the macroscopic (Maxwell) field E and the experimental susceptibility defined in terms of E.
Before going into this relation, we make the following remark: it is amusing to note that it is under its most elaborated form [equation (6)] that the fluctuation-dissipation theorem has been more or less implicitly exploited for a long time to probe the molecular motions. Relation (4) which gives directly, in the time domain, the autocorrelation of the spontaneous motion, has been put in use only very recently, when sharp enough step fields have been made available by the progress of electronics. It is also very recently that the even more direct method which consists in recording (or Fourier analysing) the spontaneous electrical noise of a dielectric has been exploited.
Relations with the Electric Permittivity. — In the same manner as the time-averaged quadratic moment of a sample depends on its shape because of the importance of boundary conditions in long-range (dipolar) forces, in the dynamical equation (6) the autocorrelation function φ(t) has different initial values and behaviours for different specimen shapes, so that the above-defined 'external field susceptibility' is shape-dependent. This is very inconvenient: one rather wishes to work with the Maxwell susceptibility X defined as the ratio of the polarization (moment per unit volume) to the Maxwell field E in the sample. This is shape-independent, but will be related by a shape-dependent relation to the shape-dependent autocorrelation of the total moment. The choice of the assumed sample shape will then be a matter of convenience for the purpose aimed at. For example, one guesses that for relating the susceptibility to the molecular motion, which in fluids is statistic-ally isotropic, the choice of a spherical sample will be a good choice. On the other hand, for demonstrating some macroscopic relations, other choices might be more convenient. Also, sometimes, comparison of results with two choices can help one's understanding and serve to check one's analysis.
We have to deal with quasi-electrostatics, i.e. a situation in which the finiteness of the velocity of radiation can be neglected. This condition is met in the following discussion (but presents a problem in infrared spectroscopy which has not yet been solved satisfactorily). We consider fields which vary sinusoidally in time. In steady state the polarization (moment per unit volume) developed in matter by the field will have also a sine variation. We note E and P, respectively, these two vectors (which have a complex amplitude). The constitutive relation of quasi-electrostatics is
ε*[member of]E =[member of]E+P([member of] + X*)E (7)
Here ε* = ε' – iε" is the complex relative permittivity (hereafter called' dielectric constant'); x* is the absolute susceptibility. member of] is a number which must be taken equal to 8.85 x 10-12 F m-1 in the SI system and to (4π)-1 in the usual (unrationalized) e.s.u. system. One must join to equation (7) the shape-dependant relation between the external field Ee and the Maxwell field E inside the sample. This reads, in the case where the polarization is uniform:
E = Ee–λP/4π[member of] (8)
P=M/V
where V is the volume of the sample, M its electric moment, and λ a shape-dependant 'depolarization factor' whose values are:
λ = 4π/3 for a sphere
λ = 4π for a plane slab perpendicular to the field
λ = 0 for a thin rod parallel to the field.
Combination of equations (7) and (8) yields:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
Recalling that, by definition,
x*e = VP/Ee, X* = P/E
we have:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
Kramers-Kronig Relations and the Macroscopic Sum Rule. — These integral relations, which are very general, link the real (dispersive) part of the permittivity to its imaginary (dissipative) part. They can be demonstrated by using only causality and postulating the superposition principle. It seems more in the spirit of this Report to give a demonstration which is based upon a particular application of the fluctuation-dissipation theorem. To do this we choose the case of a thin rod parallel to the external field. Then , λ = 0, and using a combination of equations (9), (10), and (5):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
Here φ(t) depends on the particular behaviour of the longitudinal component of the fluctuating moment of a thin rod, but this is of no importance for our purpose because we will eliminate it.
Remembering that <φ(t) is always real one has:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12a)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12b)
Fourier inversion of (12b) yields:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
so that:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
We write this:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The two last terms yield zero because these rapidly oscillating functions are modulated by [epsilon"](s), which tends to zero when s [right arrow] ∞. That [epsilon"](∞) [right arrow] 0 results from equation (12b) where
π(t) [right arrow] 0 when t [right arrow] ∞.
Hence:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13a)
As a more rigorous demonstration would make clear, this integral is to be understood as a Cauchy principal value. In the same manner one can show that:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13b)
Usually, at least for sufficiently rigid molecules, the fluctuations of the dipole moment of the sample occur in two very different time regions: that where the motion of the permanent dipole moment occurs ( > 10-13 s), and that where distortion (electronic and atomic) moments fluctuate (usually < 10-13 s). Conversely, there are several absorption-dispersion regions for ε(w); the lowest-frequency one is due to the permanent dipoles; the corresponding dispersion in ε' extends between the static permittivity ε0 and a high-frequency limit ε∞, which is usually reached at, or not far beyond, 100 cm-1. Under these circumstances the Kramers-Kronig integrals can be split into independent parts for the different absorption-dispersion regions. Keeping only the low-frequency (dipolar) part of primary interest to us, the above relations are made valid by replacing unity by ε∞ in the l.h.s. of equation (13a) and in the r.h.s. of equation (13b) (remembering also that the upper limits of the integrals then mean 'up to about 100 cm-1').
