Polarizability approximation

The mean-polarizability approximation, discussed in detail by Agranovitch,16 presents the same advantages (simplicity, arbitrary concentrations, etc.), and the same limitations as the average-locator approximation in particular, this theory provides two bands of persistence behavior for all values of the parameters. This may be checked on the example of a cubic crystal, where the local field has a very simple form The modes of the mixed crystal are given by... 

Figure 4.22. Polariton solutions for a 3D mixed crystal in the mean-polarizability approximation (4.117). In strong local field (A3), one obtains a resonance of the virtual crystal cAwA + cBwB another solution, strongly shifted, exists at low frequencies. On the contrary, in weak local fields (A,), the frequencies of the pure A and B crystals, slightly shifted, are solutions. We note that for cB - 0, one of the solutions tends, for any strength of the local field A, to ojb, which is the frequency of B unshifted by the interaction with the lattice A.

A. D, Buckingham and K. L. Clarke-Hunt. The pair polarizability anisotropy of SF in the point-atom-polarizability approximation. Molec. Phys., 40 643-648 (1980). 

T. Keyes and B. M. Ladanyi. The role of local fields and interparticle pair correlation in light scattering by dense fluids IV. Removal of the point-polarizability approximation. Molec. Phys., 55 1271-1285 (1977). 

P, J. Berkhout and I. F. Silvera. Mixing of rotational states, breakdown of the independent polarizability approximation and renormalized interactions in the solid hydrogens under pressure. Communic. Phys., 2 109-114 (1977). 

If the impurity distribution is assumed to be uniform the local field, on the average, can be approximated (in the mean polarizability approximation) by... 

Before we proceed to the case of crystals with two molecules per unit cell we should note that the so-called mean polarizability approximation which makes it possible to idealize the crystal as an array of molecules having mean polarizability given by eqn (5.35) is, actually, a very old approach. In former times (see, for instance, Section 6 of (15)), this approximation when applied to molecular systems with the van der Waals interaction was referred to as the additive refraction approximation . It should be stressed that, although eqn (5.38) provides just a very convenient extrapolation procedure, its accuracy increases with decreasing c so that the term linear in c in eqn (5.39) proves to be exact (but it does not take into account the effect of excitation delocalization discussed below). 

In this approximation (the additive refraction or mean polarizability approximation) eqn (5.42) with the addition of eqn (5.37) fully determines the dependence of the dielectric constant tensor on the impurity concentration c. The optical properties of mixed crystals with large impurity concentrations are discussed in Section 5.6. As we did above for crystals with one molecule per unit cell, we shall discuss here the case of small values of c when we can ignore terms of the order of c2, c3, etc. in the expansion of the tensor (5.42) in powers of c. Then we obtain ... 

Now we shall discuss some features of the electromagnetic wave spectrum in mixed crystalline solutions using the equations for the dielectric constant tensor of the solution derived in the mean polarizability approximation. We know that the effects of concentration broadening of absorption spectra are lost in this approximation. However, Onodera and Toyozawa (16), Dubovsky and Konobeev (17), Hoschen and Jortner (18), and Hong and Robinson (19), who have actually studied the corrections to the mean polarizability approximation, have shown that... 

Moreover, the mean polarizability approximation can yield highly accurate results for dispersion and optical anisotropy of crystalline solutions outside the absorption band. This is due to the fact that the concentration broadening in crystals of this type (with only van der Waals interactions between the molecules) does not affect the integral oscillator strength of a transition. The mean polarizability approximation served as the basis for the procedure developed by Obreimov for the analysis of the composition of multicomponent systems as applied to a wide variety of isotopic mixtures, both liquid and crystalline (for the details, see (20)). 

Let us illustrate the mean polarizability approximation by applying it to the spectrum of a binary mixture of isotopically related molecules in a cubic crystal. In this case eqn (5.12) gives... 

It should be stressed that the results obtained here for the positions of the lines and the intensities of absorption in such solutions coincide with the results of Broude and Rashba (21) which were derived in the framework of the theory of small-radius excitons using an approximation identical with the mean polarizability approximation (the additive refraction approximation the entire tensor ci j(ui, k) was not found in this study). 

If we take into account only one of the excited states in the molecule this equation is replaced by eqn (5.21). In the region of transparency many of the excited molecular states make comparable contributions to the polarizability of the molecule so that the approximation (5.21) becomes insufficient. A similar situation occurs also in crystalline solutions. In this case, in the mean polarizability approximation (the additive refraction approximation) we should write, for instance, for isotopic mixtures,... 

In the mean polarizability approximation indicated above there exists a Lorenz-Lorentz relation (Lorenz-Lorentz formula)... 

Fluorine substitution generally substantially reduced the dielectric constant in cases with symmetric substitution. It was estimated that around 50% of the observed decreasing trend in the dielectric constant versus the percentage of fluorene can be attributed to a reduction in absorbed moisture as the result of an increase in hydrophobicity. The remaining 50% can be apportioned, on average, between the increased free volume (approximately 25%) and reduced total polarizability (approximately 25%). Little effect of fluorene substitution on dynamic thermal or thermo-oxidative stability was observed for a series of fluorinated polyimides [48]. Low dielectric constant polyimides and low water absorption polyimides were reported by several researchers [49-51]. The correlation between a high free volume and dielectric constant in polyimides has been previously reported in the literature [52]. 

If we substitute this expression into (IIIB-67), we find that the (Xii 1 term does not contribute to By assuming cylindrical S5Tnmetry for the polarizability tensor we have decreased the information needed for its use from six parameters (Equation IIIB-73) to two the direction of the S5mimetry axis 63 and the difference in polarizability parallel and perpendicular to this axis (< 33 — 11) We thus have a polarizability approximation to the rotational contribution from far ultra-violet transitions in group j... 

We therefore can use a polarizability approximation for the interaction between many transitions in one group with a particular transition in another. However, we cannot approximate the interactions between many transitions in each of two groups by use of the polarizabilities of both groups. 

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