Additive refraction approximation

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). 

The treatment of anisotropic crystalline solutions can also be carried out very simply. For such solutions consisting of two types of isotopic molecules and having two molecules per unit cell the dielectric constant tensor has been derived above in the additive refraction approximation (see eqn 5.42). This equation shows... 

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 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 ... 

Sodium antimonate contains less antimony than either antimony trioxide or pentoxide and is thus less effective. However, its unique pH and low refractive index makes the antimonate the most desirable synergist for polymers that hydrolyze when processed with acidic additives or in polymers for which deep color tones are specified. Sodium antimonate costs approximately 3.30—4.40/kg and can be obtained from either Elf Atochem NA under the Thermoguard name or from Anzon Inc. as a Timinox product. 

Materials. For holographic information storage, materials are required which alter their index of refraction locally by spotwise illumination with light. Suitable are photorefractive inorganic crystals, eg, LiNbO, BaTiO, LiTaO, and Bq2 i02Q. Also suitable are photorefractive ferroelectric polymers like poly(vinyhdene fluoride-i o-trifluorethylene) (PVDF/TFE). Preferably transparent polymers are used which contain approximately 10% of monomeric material (so-called photopolymers, photothermoplasts). These polymers additionally contain different initiators, photoinitiators, and photosensitizers. 

The factor f reduces the oscillation amplitude symmetrically about R - R0, facilitating straightforward calculation of polymer refractive index from quantities measured directly from the waveform (3,). When r12 is not small, as in the plasma etching of thin polymer films, the first order power series approximation is inadequate. For example, for a plasma/poly(methyl-methacrylate)/silicon system, r12 = -0.196 and r23 = -0.442. The waveform for a uniformly etching film is no longer purely sinusoidal in time but contains other harmonic components. In addition, amplitude reduction through the f factor does not preserve the vertical median R0 making the film refractive index calculation non-trivial. 

To obtain a measure of the dielectric constant and anisotropy of thin films, the refractive index of thin film samples was measured. It has been shown that the measured dielectric constant is approximately the square of the refractive index at 633 nm wavelength [the actual relationship is roughly e (refractive index) -i- 02.] and the anisotropy is obtained from the difference between the in-plane and out-of-plane refractive index [97]. The measured anisotropy of foamed polyimides is lower than that observed for non-foamed polyimides. In addition, a drop in refractive index of the samples was observed upon foaming. The polyimide PMDa/3FDA has a measured dielectric constant of ca. 2.9 at 70 °C. A foamed sample of PMDA/3FDA derived from copolymer 6f showed a drop in dielectric constant of 2.3 [97]. 

The spectrophotometer measures the transmission and, if an absorption measurement is carried out, converts the transmission into absorbance using these equations. This conversion works fine for samples where there is no reflection, either specular or diffuse, as is the case for nonturbid solutions. However, for films there is invariably some reflection, which is often quite large, particularly for films of high dielectric constant (or refractive index) materials, such as PbS and PbSe. Additionally, if the films are not completely transparent, then scattering introduces an extra element of reflection. Therefore, to measure the real absorption of a film, a reflection measurement must also be carried out and correction for this reflection made. The correction will be approximate and depends on the nature of the film itself. However, that most commonly used is... 

Coleman et al. 2471 reported the spectra of different proportions of poly(vinylidene fluoride) PVDF and atactic poly(methyl methacrylate) PMMA. At a level of 75/25 PVDF/PMMA the blend is incompatible and the spectra of the blend can be synthesized by addition of the spectra of the pure components in the appropriate amounts. On the other hand, a blend composition of 39 61 had an infrared spectrum which could not be approximated by absorbance addition of the two pure spectra. A carbonyl band at 1718cm-1 was observed and indicates a distinct interaction involving the carbonyl groups. The spectra of the PVDF shows that a conformational change has been induced in the compatible blend but only a fraction of the PVDF is involved in the conformational change. Allara M9 250 251) cautioned that some of these spectroscopic effects in polymer blends may arise from dispersion effects in the difference spectra rather than chemical effects. Refractive index differences between the pure component and the blend can alter the band shapes and lead to frequency shifts to lower frequencies and in general the frequency shifts are to lower frequencies. 

In order to assess the orientational stability of the poled state, the temperature dependence of the dipole mobility of the side groups was examined through dielectric relaxation measurements. (13) No low temperature relaxation below Tg was observed in the frequency range studied (100 Hz-100 kHz). In addition, the dielectric constant was approximately equal to the square of the refractive index, indicating that below T only electronic and no significant orientational contributions to the dielectric displacement are present. Thus, it was expected that a given orientational state of the ensemble would be stable at temperatures significantly below Tg. 

III. Electric Dipolar Polarizability and the Approximately Additive Molar Refract ivities... 

Thus, the additional approximations underlying the NEE are paraxiality both in the free propagator and in the nonlinear coupling, and a small error in the chromatic dispersion introduced when the background index of refraction is replaced by a constant, frequency independent value in both the spatio-temporal correction term and in the nonlinear coupling term. Note that the latter approximations are usually not serious at all. 

Optical properties are usually related to the interaction of a material with electromagnetic radiation in the frequency range from IR to UV. As far as the linear optical response is concerned, the electronic and vibrational structure is included in the real and imaginary parts of the dielectric function i(uj) or refractive index n(oj). However, these only provide information about states that can be reached from the ground state via one-photon transitions. Two-photon states, dark and spin forbidden states (e.g., triplet) do not contribute to n(u>). In addition little knowledge is obtained about relaxation processes in the material. A full characterization requires us to go beyond the linear approximation, considering higher terms in the expansion of h us) as a function of the electric field, since these terms contain the excited state contribution. 

By approximation other quantities are additive as well, such as the molar volume, molar heat capacity, molar heat of combustion and formation, molar refraction, etc. 

When the refractivities of gaseous compounds are calculated to standard conditions, the values of n -1 or (n -1)106 are often found to be nearly additively composed of those of their components, that is (n -1)106 is nearly equal to S(nA -1)106. The following table enables this comparison to be made between observed and calculated refractivities.2 The property is only approximately additive, the deviations from this relation being great in some cases.3,4... 

The refractive index of a compound is a property of some significance in regard to molecular constitution. The molar refraction, defined by Eq. (29-14), is a constitutive and additive property for a given compound it may be approximated by the sum of contributions of individual atoms, double bonds, aromatic rings, and other structural features. 

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