The 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)... 

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 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]).

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. 

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... 

In this case the dielectric permittivity e is a matrix rather than a scalar. When the principal axes of each material are perpendicular, this tensor can be written as... 

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. 

It can be seen that, D and E, in the general case, are connected by a second-rank tensor. This tensor can be either the permittivity tensor or the dielectric constant tensor, depending on how the individual components are expressed. In the above equation, the tensor is a permittivity tensor. To convert it to a dielectric constant tensor, each element, Stj is divided by the permittivity of free space, sq. 

The temperature dependence of average dielectric permittivity enters the equations both explicitly (term ksT) and through S (the additional contribution from h and F is weak) while is directly proportional to S. The latter corresponds to the uniaxial symmetry of the dielectric permittivity with a tensor form ofEq. 3.16. 

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. 

Material properties are characterized by the permittivity tensor Sy or the dielectric susceptibility tensor Xij describing relations between the field quantities by... 

Meng ZY, Cross LE (1985) Determination of the electrostriction tensor components in singlecrystal Cap2 from the uniaxial stress dependence of the dielectric permittivity. J AppI Phys 57 488... 

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 electric flux density D and electric field strength E are vectors, i.e. tensors of first order and therefore may be related via a tensor of second order with nine constants for the three dimensions. Due to the potential property also observed for electrostatic fields, the tensor is symmetric and thus contains six independent entries. The electrostatic constitutive relation can be expressed with the aid of the dielectric permittivity matrix e (to be distinguished from the strains e) or its inverse p ... 

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... 

Phases having biaxial symmetry (tilted smectic phases) exhibit dielectric biaxiality in particular. At frequencies of 1 MHz and below, the biaxiality becomes important and critically influences the electrooptic switching behavior of the SmC phase. It is therefore important to be able to measure the biaxiality at these frequencies. Being a symmetrical second rank tensor, the dielectric permittivity can always be diagonalized in a proper frame and described by three components along the principal directions. The three principal values can then be expressed by a single subscript and can be determined by three independent measurements performed at three different orientations of the director relative to the measuring electric field. In practice, this may not be that... 

When a uniform director profile is assumed, then the smectic director field n can be described by the three angles (p, 6, and 5. These angles are uniform throughout the sample and the dielectric tensor component yy corresponds to the dielectric permittivity p measured in the planar orientation. Under certain conditions this may be written... 

As.discussed in ibe previous section, the response of the system of charges rqnescoting a molecule, group, or even a crystallite to the action of an external force field may be treated as a perturbation, and quantum mechanical methods may be used, la this way the response of the dielectric material to a time-depeadent (e.g., periodic) force field can be delennined in terms of general complex susceptibilities, which are related directly to the complex permittivity tensor (Ret 12, pp. 178-201). 

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... 

Due to the uniaxial or symmetry of a nematic phase, the dielectric permittivity of a nematic is represented by a second rank tensor with two principal elements, 8 and 8 The component 8 is parallel to the macroscopic symmetry axis, which is along the director, and is perpendicular to this. According to a molecular field theory, they are approximated by... 

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