Permittivity tensor

For many problems it is convenient to separate the piezoelectric (i.e., strain induced) polarization P from electric-field-induced polarizations by defining D = P + fi , where s is the permittivity tensor. For uniaxial strain and electric field along the 1 axis, when the material is described by Eq. (4.1) with the E term omitted. 

Let us consider a sphere composed of a material described by the constitutive relation (5.46). We assume that the principal axes of the real and imaginary parts of the permittivity tensor coincide this condition is not necessarily satisfied except for crystals with at least orthorhombic symmetry (Born and Wolf, 1965, p. 708). If we take as coordinate axes the principal axes of the permittivity tensor, the constitutive relation (5.46) in the sphere is... 

It is not difficult to generalize the results of this section to an anisotropic ellipsoid the axes of which coincide with the principal axes of its permittivity tensor. The principal values of the polarizability tensor of such a particle are... 

More general ellipsoidal particles in an anisotropic medium, where there is no restriction on the principal axes of either the real or imaginary parts of the permittivity tensors, have been treated by Jones (1945). 

Anisotropic fluids, of which nematic liquid crystals are the most representative and simplest example, are characterized by an anisotropic dielectric permittivity. The nematic phase has D,yuh symmetry, and in a laboratory frame with the Z axis parallel to the C , symmetry axis (the director) the permittivity tensor has the form ... 

The components of the permittivity tensor are related to Fourier-Laplace transforms, d>MA(k, to), of these TCFs by... 

The relative permittivity tensor for the system ik is defined (see, for example, Born and Wolf 1970 Landau et al. 1987) by the relation... 

The written set of equations has a simple solution for the components of the polarisation vector. We use them to write, in accordance to equation (10.1), the relative permittivity tensor... 

The written relations define the relative permittivity tensor for the system, which is formulated below to within second-order terms in the orientation tensor... 

In conformity with the significance of the terms employed by investigators of anisotropy (Tsvetkov et al. 1964), the effects associated with the first-order terms in equation (10.6) may be called the effects of intrinsic anisotropy, while the second-order effects may be referred to as the effects of mutual interaction. In the second approximation, the principal axes of the relative permittivity tensor do not coincide, generally speaking, with the principal axes of the orientation tensor. It is readily seen that interesting situations may arise when Aa < 0 in this case, the coefficients of the first- and second-order terms have different signs. 

Equation (10.6), formulated in the previous section, defines the relative permittivity tensor in terms of the mean orientation of certain uniformly distributed anisotropic elements, which we shall interpret here as the Kuhn segments of the model of the macromolecule described in Section 1.1. We shall now discuss the characteristic features of a polymer systems, in which the segments of the macromolecule are not independently distributed but are concentrated in macromolecular coils. 

As before, we shall consider each macromolecule to be divided into N subchains and assume that every subchain of the macromolecule is in the equilibrium. So, using the above formula relating the tensor of the mean orientation of the segments of the macromolecules (ejek) to the distance between the ends of the subchains, we arrive from relation (10.6), taken in the first approximation, at Zimm s (1956) expression for the relative permittivity tensor... 

Expression (10.8) for the relative permittivity tensor in terms of the normal co-ordinates introduced by means of equations (1.13), assumes the form... 

The last equation can be compared with equations (9.1) and (9.3) for the stresses in dilute solutions. On can see that, when internal viscosity is neglected ipv = 0), there is a relation between the permittivity tensor and stress tensor in the form... 

In this situation, which is also discussed in Section 7.5, we refer to experimental evidence according to which components of the relative permittivity tensor are strongly related to components of the stress tensor. It is usually stated (Doi and Edwards 1986) that the stress-optical law, that is proportionality between the tensor of relative permittivity and the stress tensor, is valid for an entangled polymer system, though one can see (for example, in some plots of the paper by Kannon and Kornfield (1994)) deviations from the stress-optical law in the region of very low frequencies for some samples. In linear approximation for the region of low frequencies, one can write the following relation... 

One admits that the relative permittivity tensor of the system is determined by the mean orientation of the segments, so that we consider expression (10.13) to be equivalent to the first-order terms of relation (10.6) and, at comparison, obtain the expression for the mean orientation of segments of macromolecules in an entangled system... 

The value of the refractive index n of light in the anisotropic medium depends on the direction of propagation s and on the direction of the polarisation of the light. For the given relative permittivity tensor ji, the refractive index can be determined from the relation (Born and Wolf 1970 Landau et al. 1987)... 

In the simplest cases, the optical anisotropy of polymer systems is studied under the conditions of simple elongation, when the elongation velocity gradient i/ii is given. The system investigated then becomes, generally speaking, a triaxial dielectric crystal with components of the relative permittivity tensor... 

For a system undergoing simple shear, when the velocity gradient v 2 7 0, the relative permittivity tensor is non-diagonal... 

Note that a frequency-dependent stress-optical coefficient C w) can be introduced by comparing the stress tensor and the relative permittivity tensor... 

Let us consider the anisotropy of polymer system undergoing simple steady-state shear. This situation can be realised experimentally in a simple way (Tsvetkov et al. 1964). The quantity measured in experiment are the birefringence An and the extinction angle x which are defined by formulae (10.19) and (10.20), correspondingly, through components of the relative permittivity tensor. 

One can turn to equation (10.10) to find the components of the relative permittivity tensor. Using expressions for the moments (2.42), one determines the gradient dependence of the quantities for dilute polymer solutions to within second-order terms... 

Now we refer to formula (10.13) for the relative permittivity tensor to determine the characteristic quantities in this case of strongly entangled linear polymers. We use expansions (7.32) and (7.43) for the internal variables to obtain the expression for the components of the tensor through velocity gradients... 

One can turn to discussion of the dynamo-optical coefficient, defined by equation (10.22). The expression for the relative permittivity tensor (10.10) and equation (2.41) for the moments allow one to write... 

X. Gonze and C. Lee, "Dynamical matrices, Bom effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory," Phys. Rev. B 55 (1997), 10355-10368. 

To conclude this discussion on short- and long-range interactions, let us calculate the dielectric permittivity tensor of the dipoles of the crystal. The wave vector K being fixed by the external field, we may write with the notation (1.43)... 

The optical response of a monomolecular layer consists of scattered waves at the frequency of the incident wave. Since the surface model is a perfect infinite layer, the scattered waves are reflected and transmitted plane waves. In the case of a 3D crystal, we have defined (Section I.B.2) a dielectric permittivity tensor providing a complete description of the optical response of the 3D crystal. This approach, which embodies the concept of propagation of dressed photons in the 3D matter space, cannot be applied in the 2D matter system, since the photons continue propagating in the 3D space. Therefore, the problem of the 2D exciton must be tackled directly from the general theory of the matter-radiation interaction presented in Section I. 

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