Macroscopic electric field

The dielectric constant is the electrostatic expression of the interaction of atoms and molecules with macroscopic electric fields rather than with the exceedingly strong fields of individual atoms and molecules. The interaction between the homogeneous outside field and electrically asymmetrical (polar) molecules results in a finite effect, since in these molecules the contributions of positive and negative charges do not cancel. 

Figure 11. Values of the macroscopic electric field as calculated from the simulation for one overpotential. From Ref. 36, by permission.

In this section, a simple description of the dielectric polarization process is provided, and later to describe dielectric relaxation processes, the polarization mechanisms of materials produced by macroscopic static electric fields are analyzed. The relation between the macroscopic electric response and microscopic properties such as electronic, ionic, orientational, and hopping charge polarizabilities is very complex and is out of the scope of this book. This problem was successfully treated by Lorentz. He established that a remarkable improvement of the obtained results can be obtained at all frequencies by proposing the existence of a local field, which diverges from the macroscopic electric field by a correction factor, the Lorentz local-field factor [27],... 

Hereby, the branches with E - and / -symmetry are twofold degenerated. Both A - and / d-modes are polar, and split into transverse optical (TO) and longitudinal optical (LO) phonons with different frequencies wto and wlo, respectively, because of the macroscopic electric fields associated with the LO phonons. The short-range interatomic forces cause anisotropy, and A - and / d-modcs possess, therefore, different frequencies. The electrostatic forces dominate the anisotropy in the short-range forces in ZnO, such that the TO-LO splitting is larger than the A -E splitting. For the lattice vibrations with Ai- and F -symmetry, the atoms move parallel and perpendicular to the c-axis, respectively (Fig. 3.2). 

Another limitation of the Poisson-Boltzmann approach is that the interaction between two surfaces immersed in water might not be exclusively due to the electrolyte ions. For instance, water has a different structure in the vicinity of the surface than in the bulk and the overlapping of such structures generates a repulsion even in the absence of electrolyte [20]. In this traditional picture, the hydration repulsion is not related to ion hydration actually it is not related at all to electrolyte ions. However, as recently suggested [21], this hydration interaction can still be accounted for within the Poisson-Boltzmann framework, assuming that the polarization is not proportional to the macroscopic electric field, but depends also on the field generated by the neighboring water dipoles and by the surface dipoles. 

Figure 4. (a) The electric potential (circles) as a function of the distance from the surface for the system described in the text. The continuous line represents a Spline interpolation, (b) The average polarization of a water molecule (squares) as a function of the distance from surface for the system described in text the macroscopic electric field (triangles) obtained through the numerical derivative of the potential is not proportional to the average polarization. 

In this paper a model was presented, which allowed one to calculate both the electric potential and the polarization between two surfaces, without assuming, as in the traditional theory, that the polarization and the macroscopic electric field are proportional. An additional local field, due to the interaction between neighboring dipoles, was introduced in the constitutive equation which relates the polarization to the local field. The basic equations were also derived using a variational approach. 

The average polarization of a water molecule, m(z), between two identical, charged parallel plates separated by a distance 2d is related to the macroscopic electric field, E(z), and to the local field produced by the neighboring dipoles via23... 

The traditional double-layer theory combines the Poisson equation with the assumption that the polarization is proportional to the macroscopic electric field, and uses Boltzmann distributions for the concentrations of the ions. The potential of mean force, which should be used in the Boltzmann distribution, is approximated by the mean value of the electrical potential. The macroscopic field E and the polarization P are related via the Poisson equation... 

In a typical macroscopic assumption of proportionality between polarization and applied electric field, P = e0(c — 1 )E, where e is the dielectric constant, and eq3 reduces to the traditional Poisson—Boltzmann equation (the concentrations cH and c0h being in general much smaller than ce). However, if the correlations between neighboring dipoles are taken into account, the following constitutive equation relating the polarization to the macroscopic electric field is obtained7... 

In a continuous and homogeneous liquid, the polarization is related to the macroscopic electric field via a local dielectric constant e(r),... 

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... 

The prime in Eq. (3-62) indicates that the sum is restricted to sites that do not belong to the same molecule. Depending on the specific implementation the tensors T(1) are multiplied with appropriate /e factors for the associated atoms. The last term in Eq. (3-59), efacM, is the macroscopic electric field. This completes the most usual form of vpo1, i.e., the potential of the dipoles due to the total field at the polarizable sites is made a part of the effective Hamiltonian and Eq.(3-24) is solved self-consistently. Since the induced dipoles M in the solvent (MM) part are self-consistent for any field E, i.e., also for intermediate fields during the iterative process for solving Eq. (3-24), in this way we obtain an overall self-consistent solution, similar to, e.g., the HF or Kohn-Sham procedure. Extension to post-HF methods are straightforward because the reaction potential (RP) is formally a one-particle... 

Consider a suspension of Vp identical spherical soft particles in a general electrolyte solution of volume V. We define the macroscopic electric field in the suspension (E), which differs from the applied electric field E. The field (E) may be regarded as the average of the gradient of the electric potential (= in the... 

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 ... 

Reiss has questioned whether the definitions of the macroscopic electric fields used in the experiments have always been correctly identified. A more general definition of the field would be,... 

It is important to note that these treatments, both at the DFPT and SOS level, pertain to hUed bands that is, for insulators or cold semiconductors the treatment of partially filled bands requires the characterization of scattering effects as well. Moreover, since incident rather than macroscopic electric fields (or vector potentials) generally enter into the perturbation Hamiltonian [Eqs. (4) and (6)], the various linear and nonlinear susceptibihties have to be corrected if there are no imphcit or explicit local field corrections. 

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