Dielectric susceptibility tensor

Thus the spectral function L(z) of an isotropic medium is represented as a linear combination of two spectral functions determined for an anisotropic medium pertinent to longitudinal ( ) ) and transverse (K ) orientations of the symmetry axis with respect to the a.c. field vector E. It is shown in GT, Section V, that these spectral functions are proportional to the main components of the dielectric-susceptibility tensor. 

Substitution of this expression into Equation (3.36) gives AE(t) in terms of the change in the field and the solution dielectric susceptibility tensor... 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

As a consequence of the uniaxial symmetry all material properties of nematics have to be represented by tensors. For instance, the dielectric displacement D and E are connected by the dielectric susceptibility tensor e as ) = eoeE = tole E + i —e ) n-E)n]. Thus e depends in general on the local director orientation and is specified by two dielectric constants, ey and e (for E parallel and perpendicular to n, respectively). An analogous representation applies to the electric conductivity tensor [Pg.102]

In general, however, tensor Qij is biaxial but the biaxiaUty is small, on the order of yPo where is the length corresponding to nematic correlations. This correlation length may be found, for example, from the light scattering in the isotropic phase close to the transition to the nematic phase. Then, at each point, that is locally, the anisotropic part of dielectric susceptibility tensor is biaxial and traceless 8ei + 5e2 + 8e3 = 0 with 8e2 8E3. 

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

Phase-matching in dispersive media can be fulfilled in birefringent crystals which have two different refractive indices n, and n for the ordinary and the extraordinary waves. While the ordinary index n, does not depend on the propagation direction, the extraordinary index n depends on both the directions of E and k. The refractive indices can be illustrated by the index ellipsoid defined by the three principal axes of the dielectric susceptibility tensor. If these axes are aligned with the x, y, z axes we obtain the index ellipsoid. 

The refractive indices can be illustrated by the index ellipsoid defined by the three principal axes of the dielectric susceptibility tensor. If these axis are aligned with the x, y, z axes we obtain... 

Interestingly, the contributions from the gradient of the electromagnetic field across the interface, Tfg xx and Tfg zzz, which scale with the mismatch in the optical dielectric constants of the media forming the interface [37], only appear in the susceptibility tensor components nd xl zzz- Therefore, these contributions may be rejected with a... 

The dielectric susceptibility, like magnetic suscephbUity, is a second-rank tensor. The dielectric constant (also known as the relative permittivity), k, is defined as ... 

From Eq. 8.52, it is seen that the applied electric field, E, and the polarization, P, are related through a second-rank tensor called the dielectric susceptibility, Xe- Three equations, each containing three terms on the right-hand side, are needed to describe... 

X j(w, k) being the dielectric polarizability (susceptibility) tensor, so that... 

A number of exploitable effects exist, due to the non-linear response of certain dielectric materials to applied electric and optical fields. An applied field, E, gives rise to a polarization field, P, within any dielectric medium. In a linear material, the relationship between P and E may be characterized by a single (first-order, second-rank) susceptibility tensor... 

The magnetization was only taken as an example. Many other properties (dielectric susceptibility, electric and thermal conductivity, molecular diffusion, etc.) are also described by second rank tensors of the same (quadrupolar) type Microscopically, such properties can be described by single-particle distribution functions, when intermolecular interaction is neglected. There are also properties described by tensors of rank 3 with 3 = 27 components (e.g., molecular hyperpolarizability Yijk) and even of rank 4 (e.g., elasticity in nematics, ATiju) with 3 = 81 components. Microscopically, such elastic properties must be described by many-particle distribution functions. 

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. 

Symmetry point group Dielectric tensor Elastic tensor ors , Piezoelectric tensor dimn Elasto-optic tensor Pmn Electrooptic tensor rmk Nonlinear susceptibility tensors x > 1 X< > Table number... 

Electrooptic tensor General optical properties Refractive index Nonlinear dielectric susceptibility Second order Third order... 

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