Crystal dielectric permittivity tensor

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. 

If a nematic liquid crystal has negligible conductivity the results of Sections 11.2.1-11.2.5 for the Frederiks transition induced by a magnetic field may be directly applied to the electric field case. To this effect, it suffices to substitute H by E and all components of magnetic susceptibility tensor Xij hy correspondent components of dielectric permittivity tensor s,y. From the practical point of view the electrooptical effects are much more important and further on we discuss the optical response of nematics to the electric field. 

Liquid-crystalline molecules possess anisotropy of the electric polarizability, and nearly always a significant permanent dipole moment resulting from contributions from different bond moments, see Table 4.1. Therefore, the dielectric permittivity of liquid crystals is also a tensor quantity. Because of the assumed uniaxiality of the system under consideration, there are again only two principal elements of the dielectric permittivity tensor Szz=H and Bxx= yy= l- Subscripts II and 1 denote respectively the principal geometries of dielectric measurements, i.e. the probing electric field parallel and perpendicular to the director. 

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 ... 

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... 

Next we will introduce the optical dielectric impermeability tensor of a crystal. The coefficients (17, ) of this tensor depend on the distribution of bond charges in the material [15,71]. The 17, are found by taking the reciprocal of the relative permittivity or dielectric constant [71]. The 17, have been defined in terms of the refractive index of the crystal as [71]... 

A well-known example of this is that cubic crystals are optically isotropic, which means that the dielectric permittivity has spherical symmetry in a cubic crystal. Another example is that the thermal expansion coefficient of a cubic crystal is independent of direction. In fact, if it were not, the crystal would lose its cubic symmetry if it were heated. Thus, as far as thermal expansion is concerned, a cubic crystal looks isotropic just as it does optically. Since, according to Neumann s principle, the physical properties of a crystal may be of higher symmetry than the crystal, we will generally find that they range from the symmetry of the crystal to the symmetry of an isotropic body. A more general example of higher symmetry in properties is that such physical properties characterized by polar second rank tensors must be centrosymmetric, whether the crystal has a center of symmetry or not, cf. Fig. 27. For, if a second rank tensor T connects the two vectors p and q according to... 

In a high static dectric fidd, isotropic dielectrics are endowed with properties similar to those of an axial crystal consequently, in the absence of absorption and dispersion, only the three diagonal dements of the permittivity variation tensor (245) are non-zero, and two of them are mutually independent. Thus, assuming the strong field Ep to be applied along the z-axis, we have by (245) ... 

The dielectric properties of a material are properly specified by a symmetric second-rank tensor relating the three components of the electrical displacement vector D to those of the field E. By choosing axes naturally related to the crystal structure the six independent components of this tensor can be reduced to three and, taking account of the hexagonal symmetry of the ice crystal, only two independent components remain. These are the relative permittivities parallel and perpendicular to the unique c-axis direction and we shall denote them by e, and e. We shall discuss the experimental determination of these quantities when we come to consider dielectric relaxation, since some difficulties are involved. For the present we simply note the results which are shown in fig. 9.2. The often-quoted careful measurements of Auty Cole (1952) were made with polycrystalline samples and removed many of the uncertainties in earlier work. They represent, however, a weighted mean of the values of e, and Humbel et al. (1953 [Pg.201]

We report on the determination of the complete tensor of relative electric permittivity in single crystals of pTS, as a function of temperature and polymer contents. Since the dielectric measurements prove to be a suitable method, enabling one to follow the second- order phase transition in pTS, we compare the dielectric behavior of monomer and polymer of pTS and pFBS, looking for a possible signature of a phase transition in the latter crystals. 

E/Eq, is called the relative permittivity or dielectric constant of the material, a quantity that is unitless. Both the permittivity and dielectric constant of liquid crystals are tensor quantities. 

If a dielectric is anisotropic, like liquid crystals and most solids, the scalar susceptibility in eqn (4.2) must be replaced by a tensor. Hence, the permittivity must also be a tensor quantity. 

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