Dielectric permittivity tensor

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. 

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. 

To show this, it is necessary to insert the Fourier components E(q) of the dielectric permittivity tensor e( ) of the cholesteric into the general formula for the scattering cross section a oc (r s(q) f) as already discussed for nematics in Section 11.1.3. Here f and r are polarization vectors for the incident and reflected light, q is the wavevector of scattering coinciding in this simple geometry with the wavevector of the reflected wave [2]. 

The detailed analysis of light propagation in the cholesteric helix is quite complex. It consists in the search for eigenmodes of Maxwell equations in a medium with the position-dependent dielectric permittivity tensor. 

The helical structure which can develop in thick cells of chiral smectic C phases having planar surface alignment conditions can be used to obtain measurements of the components of the dielectric permittivity tensor [29], but the technique is restricted to chiral smectic phases. Measurements are made (see Fig. 9) of the homeotropic state, as above, and additionally the helical state (Fig. 12), and the uniformly-tilted state ob-... 

Figure 84. The principal values of the dielectric permittivity tensor at 100-kHz measured by the short pitch method. The material is the mixture SCE12 by BDH/Merck (from Buivydas [155]).

Maradudin and coworkers [18] have demonstrated that the macroscopic low-frequency static dielectric permittivity tensor y(w) which gives the infrared spectra main contribution, is a sum of both an ionic part and a limit value of a pure electronic contribution. According to Gonze and Lee [6], one has... 

Figure 10-2. A typical frequency dependence of real and imaginary parts of the components of the dielectric permittivity tensor for a nematic material...

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. 

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