Dielectric linear

Figure Bl.5.5 Schematic representation of the phenomenological model for second-order nonlinear optical effects at the interface between two centrosynnnetric media. Input waves at frequencies or and m2, witii corresponding wavevectors /Cj(co and k (o 2), are approaching the interface from medium 1. Nonlinear radiation at frequency co is emitted in directions described by the wavevectors /c Cco ) (reflected in medium 1) and /c2(k>3) (transmitted in medium 2). The linear dielectric constants of media 1, 2 and the interface are denoted by E2, and s, respectively. The figure shows the vz-plane (the plane of incidence) withz increasing from top to bottom and z = 0 defining the interface.

Polarization which can be induced in nonconducting materials by means of an externally appHed electric field is one of the most important parameters in the theory of insulators, which are called dielectrics when their polarizabiUty is under consideration (1). Experimental investigations have shown that these materials can be divided into linear and nonlinear dielectrics in accordance with their behavior in a realizable range of the electric field. The electric polarization PI of linear dielectrics depends linearly on the electric field E, whereas that of nonlinear dielectrics is a nonlinear function of the electric field (2). The polarization values which can be measured in linear (normal) dielectrics upon appHcation of experimentally attainable electric fields are usually small. However, a certain group of nonlinear dielectrics exhibit polarization values which are several orders of magnitude larger than those observed in normal dielectrics (3). Consequentiy, a number of useful physical properties related to the polarization of the materials, such as elastic, thermal, optical, electromechanical, etc, are observed in these groups of nonlinear dielectrics (4). 

Linear combination of atomic orbitals (LCAO) method, 16 736 Linear condensation, in silanol polycondensation, 22 557-558 Linear congruential generator (LCG), 26 1002-1003 Linear copolymers, 7 610t Linear density, 19 742 of fibers, 11 166, 182 Linear dielectrics, 11 91 Linear elastic fracture mechanics (LEFM), 1 509-510 16 184 20 350 Linear ethoxylates, 23 537 Linear ethylene copolymers, 20 179-180 Linear-flow reactor (LFR) polymerization process, 23 394, 395, 396 Linear free energy relationship (LFER) methods, 16 753, 754 Linear higher a-olefins, 20 429 Linear internal olefins (LIOs), 17 724 Linear ion traps, 15 662 Linear kinetics, 9 612 Linear low density polyethylene (LLDPE), 10 596 17 724-725 20 179-211 24 267, 268. See also LLDPE entries a-olefin content in, 20 185-186 analytical and test methods for,... 

Figure 3.1 shows a simplified picture of an interface. It consists of a multilayer geometry where the surface layer of thickness d lies between two centrosymmetric media (1 and 2) which have two different linear dielectric constants e, and e2, respectively. When a monochromatic plane wave at frequency co is incident from medium 1, it induces a nonlinear source polarization in the surface layer and in the bulk of medium 2. This source polarization then radiates, and harmonic waves at 2 to emanate from the boundary in both the reflected and transmitted directions. In this model, medium 1 is assumed to be linear. 

Inside a rectangular well a dipole rotates freely until it suffers instantaneous collision with a wall of the well and then is reflected, while in the field models a continuously acting static force tends to decrease the deflection of a dipole from the symmetry axis of the potential. Therefore, if a dipole has a sufficiently low energy, it would start backward motion at such a point inside the well, where its kinetic energy vanishes. Irrespective of the nature of forces governing the motion of a dipole in a liquid, we may formally regard the parabolic, cosine, or cosine squared potential wells as the simplest potential profiles useful for our studies. The linear dielectric response was found for this model, for example, in VIG (p. 359) and GT (p. 249). 

This section presents a fundamental development of Sections V and VI. Here a linear dielectric response of liquid H20 is investigated in terms of two processes characterized by two correlation times. One process involves reorientation of a single polar molecule, and the second one involves a cooperative process, namely, damped vibrations of H-bonded molecules. For the studies of the reorientation process the hat-curved model is employed, which was considered in detail in Section V. In this model a hat-like intermolecular potential comprises a flat bottom and parabolic walls followed by a constant potential. For the studies of vibration process two variants are employed. 

The relative permittivity of a solvent depends also on the electric field E, but ordinary fields employed in the laboratory are rarely strong enough to cause an appreciable change of s. The phenomenon is called the non-linear dielectric effect. A relevant expression (Grahame 1953) is ... 

A linear dielectric medium is characterized by a linear relation between the polarization density and the electric field, P = e0xE, where eo is the permittivity of free space and x is the electric susceptibility of the medium. A nonlinear dielectric medium, on the other hand, is characterized by a nonlinear relation between P and E, as illustrated in Figure 4.19. 

Figure 4.19 The P-E relation for (a) a linear dielectric medium, and (b) a nonlinear medium...

Figure 16.9 (a) Linear dielectric image of a-c domain in PZT, (b) cross-sectional image along the A-A line. 

Here, 33, 333, 3333, and 33333 correspond to linear and nonlinear dielectric constants and are tensors of rank 2nd, 3rd, 4th and 5th, respectively. Even-ranked tensors including linear dielectric constant 33 do not change with polarization inversion, whereas the sign of the odd-ranked tensors reverses. Therefore, information regarding polarization can be elucidated by measuring odd-ranked nonlinear dielectric constants such as 333 and 33333. 

The interaction of the charge system of the donor (or acceptor) with the linear dielectric medium is described in the form... 

Similar expressions and properties of the free energy functional (1.118) hold for all other levels of the QM molecular theory the factor is present in all cases of linear dielectric responses. More generally, the wavefunctions that make the free energy functional (1.117) stationary are also solutions of the effective Schrodinger Equation (1.107). 

Continuum solvation models consider the solvent as a homogeneous, isotropic, linear dielectric medium [104], The solute is considered to occupy a cavity in this medium. The ability of a bulk dielectric medium to be polarized and hence to exert an electric field back on the solute (this field is called the reaction field) is determined by the dielectric constant. The dielectric constant depends on the frequency of the applied field, and for equilibrium solvation we use the static dielectric constant that corresponds to a slowly changing field. In order to obtain accurate results, the solute charge distribution should be optimized in the presence of the field (the reaction field) exerted back on the solute by the dielectric medium. This is usually done by a quantum mechanical molecular orbital calculation called a self-consistent reaction field (SCRF) calculation, which is iterative since the reaction field depends on the distortion of the solute wave function and vice versa. While the assumption of linear homogeneous response is adequate for the solvent molecules at distant positions, it is a poor representation for the solute-solvent interaction in the first solvation shell. In this case, the solute sees the atomic-scale charge distribution of the solvent molecules and polarizes nonlinearly and system specifically on an atomic scale (see Figure 3.9). More generally, one could say that the breakdown of the linear response approximation is connected with the fact that the liquid medium is structured [105],... 

These results are now in terms of correlation functions of P2Ct) and, in the field on cases, of P (t), as in linear dielectric and other response theories. The asymmetry given by the last term of the field on expression reveals the interesting possibility of obtaining both P correlations (from the field off response) and P correlations (from the asymmetry). One easily verifies that Benoit s result is obtained for diffusional reorientations, as then = (l/5)exp(-6D t), -(2/15)exp(-6Dt), and PoP2PlPl(t)> = (2/15)exp(-2D t)7 but this form of dynamics is not assumed in the development End effects of joint correlations Eire included in eqs.13 and 14. 

The term x Eo sin(cot) represents the response expected from a linear dielectric. The second term contains the components /2 Ej and — /2)E cos(2cot) the first of these represents a constant polarization which would produce a voltage across the material, that is rectification the second corresponds to a variation in polarization at twice the frequency of the primary wave. Thus the interaction of laser light of a single frequency with a suitable non-linear material leads to both frequency doubling (SHG) and rectification. 

Any linear dielectric response can be described as the sum (or integral) of damped harmonic oscillators in the form of Eq. (L2.318),... 

Fig. 5. Standard linear equivalent circuit of an assumed linear dielectric cell membrane can be modelled with simple standard components. This assumption breaks down if the field is amplified across the membrane as in Fig. 1 to a degree sufficient to produce nonlinearity...

In this discussion we have dealt with a linear dielectric where the polarisation is proportional to the applied field. As with most physical phenomena, the dielectric response of materials is non-linear. Fortunately the higher order... 

As we see, the parameter 1 results from Langevin reorientation of the polarizability ellipsoid and is always positive. The second of the above parameters, 2, corresponds to Bom s term in the Kerr effect and can be positive or negative, depending on the electric structure of the molecule. The third, the Debye parameter 3, has no counterpart in other phenomena of molecular orientation, and is specific to the non-linear dielectric behaviour of dipolar substances. 

The first non-linear polarizability is a tensor of rank 3 and possesses non-vanishing elements only in the case of molecules without a centre of symmetry. This Born distortion tensor was resorted to by Piekara for explaining the non-linear dielectric behaviour of nitrobenzene. In the case of axially symmetric molecules, the tensor of the first non-linear distortion (hyperpolarizability ) can be written as follows ... 

To show the influence of various microscopic and structural factors on linear and non-linear effects in dense dielectrics, it is convenient to apply first a semi-macroscopic treatment of the theory, and then to proceed to its molecular-statistical interpretation, assuming appropriate microscopic models. The semi-macroscopic method was initially applied by Kirkwood and modified by Frohlich in the theory of linear dielectrics, and has beat successfully used in theories of non-linear tUelectrics. "... 

The system BaO — T1O2 comprises 5 compounds, three of which have an incongruent melting point. The lowest eutectic melts at 1317 °C. Only two of the compounds mentioned find practical applications BaTi03 and BaTi409. The former is especially significant and will be discussed in detail below. The other compound is one of the correcting phases used in rutile ceramics (see above). In addition, it constitutes abase for linear dielectrics with a low temperature dependence of permittivity. The properties of some of the materials dealt with above are listed in Table 27. 

Here, as above, y is the Sack inertial parameter. Noting the initial condition, Eq. (238), all the cn j (0) in Eq. (256) will vanish with the exception n = 0. On using the integration theorem of Laplace transformation as generalized to fractional calculus, we have from Eq. (256) the three-term recurrence relation [cf. Eq. (240)] for the only case of interest q — 1 (since the linear dielectric response is all that is considered) ... 

As several works devoted to the nonlinear optical properties of metal nanoparticles include a size dependence of the linear dielectric function, it seems to us relevant to introduce and briefly comment now the most widespread approach used to describe such a dependence. It consists in modifying the phenomenological collision factor F in the Drude contribution (Eq. 2) as ... 

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