Dielectric polarization vectors

The response of a medium to a macroscopic field E(t) generated by the superposition of a static and an optical component (E(f) = E° + E cos(< h)) is represented by the dielectric polarization vector (dipole moment per unit of volume) P(t) ... 

P is the dielectric polarization vector (dipole moment per volume, C m m = C m ) induced by the electric field and M the magnetization vector (magnetic dipole moment per volume, A m ) induced by the magnetic field. The constants in (5) and (6) are the vacuum permittivity o = 8.85419 x 10 C m and the vacuum permeability ixq = AttX 10 V s m It is implied by (5) and (6) that the response of the medium is purely local (dipole approximation). 

In the electrodynamics of dielectrics, and electric induction vector D(r) and a dielectric polarization vector P(r) are introduced for describing the fields acting in the medium. Correspondences between these objects and those coming from the GSCRF theory are established below. 

There are several methods in use. Their examination is deferred to the next section. Here we remark that during the charging process there is a progressive increase of the dielectric polarization vector P which reaches the correct, or equilibrium, value at the end of the process. 

These are the Fourier components of the polarization vector which are connected with the oscillations of the ions that are present in eqns. (44)-(46). In addition to this polarization which results from the motion of the nuclei, purely electronic polarization (i.e. the polarization of electrons at equilibrium positions of the nuclei) is also of importance. In the frequency region below the optical range, the purely electronic polarization can be expressed through the optical dielectric permeability (i.e. the dielectric permeability corresponding to the frequencies which are less than those in the optical absorption region, but exceed those of the nuclei vibrations). Optical frequencies considerably exceed those of the nuclear vibrations therefore, in the optical frequency region the nuclei do not, in practice, contribute to polarization. The connection of the Fourier component of purely electronic polarization with that of the induction of the electric field has the usual form... 

Marchi and co-workers [27,28] have applied Equation (1.79) in the context of classical MD by using a Fourier pseudo-spectral approximation of the polarization vector field. This approach provides a convenient way to evaluate the required integrals over all volume at the price of introducing in the extended Lagrangian a set of polarization field variables all with the same fictitious mass. They also recognized the cmcial requirement that both the atomic charge distribution and the position-dependent dielectric constant be continuous functions of the atomic positions and they devised suitable expressions for both. 

A connection to vector fields (1.119) is established by the notion that a is equal to the normal component of the polarization vector P(r) located on the external side of S. Polarization vanishes in the bulk of the medium provided the dielectric constant does not change there. The apparent charge cr(r) found in terms of numerical algorithms [12] is, in turn, a linear functional of p(r). Its computation is equivalent to a solution of the Poisson equation with proper matching conditions for [Pg.98]

Examples of the order parameter are the density of the liquid, p(r), polarization vector of a dielectric medium, P(r), magnetic moment, p(r), spin of the atom in the crystal lattice of a solid, S, and charge density in the electrolyte solution, q(r). 

It should be mentioned here that the term is a second-rank tensor and has nine components because it is related to all the components of the polarization vectors and electric field vectors. Therefore, the dielectric constant is also a tensor of rank 2. The optical response of a medium at an optical frequency m can be represented equivalently by the complex refractive index as ... 

The origin of the terms transverse and longitudinal dielectric relaxation times lies in the molecular theory of dielectric relaxation, where one finds that the decay of correlation functions involving transverse and longitudinal components of the induced polarization vector are characterized by different time constants. In a Debye fluid the relaxation times that characterize the transverse and longitudinal components of the polarization are T ) and rp = (ee/eslfD respectively. See, for example, P. Madden and D. Kivelson, J. Phys. Chem. 86, 4244 (1982). 

A key point in the elementary theory of dielectric materials is that the polarization vector can be replaced by an appropriate charge distribution, which consists of both a surface charge distribution at the dielectric boundaries [Eq. (11.6)] and a volume charge distribution in the dielectric material itself [7, 85]. The latter was ignored in the early development of PCMs [13, 59], but was finally treated carefully in the late 1990s by Chipman [13-15, 89]. Generalizing Chipman s treatment to an arbitrary value of we note that in the absence of the medium, the solute s electrostatic potential would satisfy the Poisson equation = P/ inside... 

It is clear that the two polar vectors respect the apolar nature of n. There is also an obvious analogy of the above mechanism with the orientation polarization of a liquid dielectric, which was used by Helfrich to relate the two flexocoefficients with molecular properties. The intrinsic splay or bend can be related to an appropriate angle and molecular dimensions. The relevant component of the electric dipole moment and the curvature elastic constant, viz., and the splay constant Ki or /ux and the bend constant It s figure in the estimation of the flexocoefficients. Nematic liquid crystals made of banana-shaped molecules have been studied only recently, and a comparison of the experimental measurements with the Helfrich formula leads to interesting inferences, as will be mentioned later in this chapter, and covered more thoroughly in the companion Chapter 3 by Jakli et al. ... 

In thick films, mode 3 is localized near the interface with the substrate, and its amplitude decays exponentially with increasing depth into the film. In ultrathin films, the polarization vector P of this mode is almost parallel to the film plane and approximately constant across the film. For a film on a metal substrate with the dielectric function e, the mode 3 frequency is given by ... 

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]. 

In Section 7.2.1 we discussed polarization of molecular isotropic liquids. We introduced the equations for dielectric permittivity s and dielectric susceptibility X and wrote the microscopic definition of the polarization vector P as a sum of dipole moments in the unit volume = pNJM (p is density, Np is Avogadro number, M is molecular mass) ... 

The polarization of a medium and the optieal eleetrie field applied to it are linked by the material s suseeptibility X, a tensor quantity. In the previous seetion we eonsidered the limit of small optical fields, where the suseeptibility is a function of the dielectric constants only and is independent of the field. In this case the polarization vector P is related to the optical electric field E by the expression... 

As the polarization vector reflects the polarization of a unit space of the dielectric, then... 

The dielectric polarization, or simply polarization, within dielectric materials is a vector physical quantity, denoted by P, and its module is expressed in C.m Electric polarization arises due to the existence of atomic and molecular forces and appears whenever electric charges in a material are displaced with respect to one another under the influence of an apphed external electric field strength, E. On the other hand, the electric polarization represents the total electric dipole moment contained per unit volume of the material averaged over the volume of a crystal cell lattice, V, expressed in cubic meters (m ) ... 

If the vacuum is replaced by a dielectric medium, the capacity of the sphere increases because of polarization of the dielectric. Electric fields polarize matter in two ways by orienting molecules with permanent dipoles and by deforming electron clouds of each molecule. The polarization vector P is related to the characteristics of individual dipoles by the relation ... 

In the spatially modulated case, (8.5) and (8.6) have to be modified using the polarization vector P = Px, Py) which is perpendicular to the tilt vector The constants C and Q. describe the strength of the bilinear and biquadratic coupling the P and P terms are necessary to stabilize the system, e has the physical meaning of an electric susceptibility, it can be written as = /o o with o being the vacuum permittivity and Xo (electric susceptibilities and dielectric constants are used in this chapter as dimensionless or relative quantities) corresponds to the electric susceptibility of the uncoupled system (C = = 0). Writing the Landau free energy g as g = ge + gpt results in a... 

Polymers can have dipoles in the monomeric unit that can be decomposed in two different components parallel or perpendicular to the chain backbone. The dipole moment parallel to the chain backbone giving rise to an "end-to-end" net polarization vector will induce the so-called dielectric normal mode dielectric relaxation that can be studied using theoretical models. The dipole moment perpendicular to the chain backbone will lead to the segmental a-relaxation that can only be described using empirical models, since no definitive theoritical framework exists for this universal process. 

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