External macroscopic electric field

Physically the simplest, though certainly not the easiest, of the electrostatic problems is the response of a system to constant external electric field. In general, there is no principal objection to incorporating one more potential external to the SYstM of electrons into the HKS equations the ionic potential V ° (r) can readily be replaced by some V ° (r) + V (r), where V is the potential of he field external to the crystal. The difficutly arises when V is to represent a macroscopic field constant in space it comes from the impossibility for plane-wave expansions to handle any quantity which does not have the periodicity of the lattice or superlattice. The problem was met already in Sections 6 and 4.3.10, and it is not new in solid state physics. The standard solution used in other contexts, viz. evaluating the q 0 limit, is not practical in the direct approach, because the smallest q s which can be handled are likely to be still too large for evaluating the q 0 limit numerically. 

The solution proposed in Ref. 65 consists of a kind of reconciliation of periodic with aperiodic. Instead of a constant electric field we choose as V (r) the saw-like potential given in Fig. 8.1.1. This field is periodic on a supercell which repeats the elementary unit cell m-times and yet there will be a few elementary cells for which the external potential is a straight line and which will feel a constant electric field it is in this part of supercell that the response to constant macroscopic field is to be studied. It is understood that the size of this small part of the supercell has to be equal to at least one elementary unit cell. 

In order to better visualize the screened electric field and to determine the slope of the underlying saw in Fig. 8.1.2, we have subtracted from Fig. 8.1.2 the self-consistent potential corresponding to the unperturbed situation, l.e. to the one without external field the difference AV(x) shown in Fig. 8.1.3 represents the screened saw-like external potential of the Fig. 8.1. (As.mentioned already in Section 5.1, the screened potential 

V + V + V + V, has to be considered in dealing with all dielectric quantities.) Comparison of the slopes in Figs. 8.1.1 and 8.1.3 yielded the ratio e = 19.08 for Ge and 13.60 in GaAs, which in both cases is 20 % above the experimental values. 

All local variations have their origin in the microscopic inhomogeneity of solid they are traditionally studied by the linear response theory, which relates them to the off-diagonal elements of the inverse dielectric matrix this is explained in the articles by A. Baldereschi and R. Resta in the present volume. A quantitative compai sc of the local variations in charge density with the predictions using the RPA dielectric matrix showed excellent agreement between both approaches, both in magnitude and detailed shape. 

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Let us first review the basics of the Lorentz theory for polarization. If one assumes that a constant macroscopic field is applied to a homogeneous medium of dielectric constant s, the polarization through the medium will be uniform. However, the polarization of a molecule is not proportional to the macroscopic electric field (created by sources external to the dielectric), but to the local electric field, which contains also the field generated by all the other molecules of the dielectric. To account for the latter, one can separate the medium in a spherical cavity (in which the central molecule and its molecular neighbors reside, see Fig. 1 A) and the rest of the medium, which... 

Similarly as the trace, the anisotropy of the polarizability tensor of diatomic colli-sional systems can also be related to some macroscopic properties, namely to the refractive properties of atomic gases. The so-called Kerr constant, the anisotropy of the refractive index in the parallel and perpendicular directions to the external static electric field is given by,... 

In his semi-macroscopic theory Kirkwood considers a dielectric sphere of macroscopic size, of volume V containing N molecules. The spherical sample of the isotropic, homogeneous dielectric of electric permittivity e is immersed in a uniform external field 0 applied in vacuum (of permittivity The mean macroscopic electric field E existing within the sphere is given by ... 

The long-wavelength field can be easily found if we take into account that in a medium without external charges the longitudinal component of the induction vector T> vanishes, and the macroscopic electric field is longitudinal, if the retardation, as assumed in the theory of Coulomb excitons, is not taken into account. From this considerations we obtain... 

First, let us consider the measurement of CVR When the density of the particles Pp differs from that of the medium Pjjj, the particles move relative to the medium under the influence of an acoustic wave. This motion causes a displacement of the internal and external parts of the double layer (DL). The phenomenon is usually referred to as a polarization of the DL (6). This displacement of opposite charges gives rise to a dipole moment. The superposition of the electric fields of these induced dipole moments over the collection of particles gives rise to a macroscopical electric field which is referred to as the colloid vibration potential (CVP). Thus, the fourth mechanism of particles interaction with sound leads to the transformation of part of the acoustic energy to electrical energy. This electrical energy may then be dissipated if die opportunity for electric current flow exists. 

Electrowetting concerns the use of an externally applied electric field to actuate or manipulate small volumes of liquid by altering its interfacial tension and hence the macroscopic contact angle or by inducing bulk liquid motion through an interfacial electric stress. 

The external static electric field applied to a dielectric material induces fire polarisation P, that is the dipole moment per unit volume. For low fields P is prt rtional to the electric field E [1>3], P = 8o(8s - 1) E, where 8s is the relative dielectric permittivity or dielectric constant and 8o is the dielectric permittivity of free space. All these quantities concom the macroscopic volume of the dielectric medium. In order to relate them to the relevant microscopic param ers (for example dipole moment and polarisability) the local electric field Eioc acting on a molecule must be known. The relation between Eioc and E is the crucial problem of the physics of dielectrics and has not been solved in general. For isotropic fluids the Onsager theory is commonly used [4]. 

The EMD studies are performed without any external electric field. The applicability of the EMD results to useful situations is based on the validity of the Nemst-Planck equation, Eq. (10). From Eq. (10), the current can be computed from the diffusion coefficient obtained from EMD simulations. It is well known that Eq. (10) is valid only for a dilute concentration of ions, in the absence of significant ion-ion interactions, and a macroscopic theory can apply. Intuitively, the Nemst-Planck theory can be expected to fail when there is a significant confinement effect or ion-wall interaction and at high electric... 

A paraelectric substance is not polarized macroscopically because the dipoles are oriented randomly. However, they can be oriented by an external electric field (orientation polarization). The orientation is counteracted by thermal motion, i.e. the degree of polarization decreases with increasing temperature. 

To relate the complex permittivity s of a polar medium with the complex susceptibility % provided by motions of the dipoles, we suggest that a polar medium under study is influenced by the external macroscopic time-varying electric field Ee(f) = Re[Em exp(imf)], where Em is the complex amplitude. This field induces some local field EM(f) = Re[ ) exp(icof)] in a cavity surrounding each polar molecule. A given molecule directly experiences the latter field. 

This longitudinal relaxation time differs from the usual Debye relaxation time by a factor which depends on the static and optical dielectric constants of the solvent this is based on the fact that the first solvent shell is subjected to the unscreened electric field of the ionic or dipolar solute molecule, whereas in a macroscopic measurement the external field is reduced by the screening effect of the dielectric [73]. 

In order to formulate a theory for the evaluation of vibrational intensities within the framework of continuum solvation models, it is necessary to consider that formally the radiation electric field (static, Eloc and optical E[jc) acting on the molecule in the cavity differ from the corresponding Maxwell fields in the medium, E and Em. However, the response of the molecule to the external perturbation depends on the field locally acting on it. This problem, usually referred to as the local field effect, is normally solved by resorting to the Onsager-Lorentz theory of dielectric polarization [21,44], In such an approach the macroscopic quantities are related to the microscopic electric response of... 

In the absence of external fields the suspension under consideration is macroscopically isotropic (W = const). The applied field h (we denote it in the same way as above but imply the electric field and dipoles as well as the magnetic ones), orienting, statically or dynamically, the particles, thus induces a uniaxial anisotropy, which is conventionally characterized by the orientational order parameter tensor (Piin h)) defined by Eq. (4.358). (We remind the reader that for rigid dipolar particles there is no difference between the unit vectors e and .) As in the case of the internal order parameter S2, [see Eq. (4.81)], one may define the set of quantities (Pi(n h)) for an arbitrary l. Of those, the first statistical moment (Pi) is proportional to the polarization (magnetization) of the medium, and the moments with / > 2, although not having meanings of directly observable quantities, determine those via the chain-linked set [see Eq. (4.369)]. 

In principle, the electric fields to be inserted in Eq.(7) are the electric fields at the location of the molecule. Instead of the local electric fields oc the external fields E are usually used. Therefore, local field correction factors have to account for the electric field screening of the surrounding material when going from the macroscopic susceptibilities to the molecular hyperpolarizabilities as shown below. 

In the relations between the macroscopic susceptibilities y , y and the microscopic or molecular properties a, ft, y, local field corrections have to be considered as explained above. The molecule experiences the external electric field E altered by the polarization of the surrounding material leading to a local electric field E[oc. In the most widely used approach to approximate the local electric field the molecule sits in a spherical cavity of a homogenous media. According to Lorentz the local electric field [9] is... 

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