Dipole hardness electric field

Let us consider a hard sphere in a continuous matrix under an electric field. When the relative dielectric constant of particle s2 is larger than that of matrix s1, a point dipole in the particle is formed by application of an electric field. According to the classical theory [51], the point dipole moment is given by... 

General properties and definitions of polarizabilities can be introduced without invoking the complete DFT formalism by considering first an elementary model the dipole of an isolated, spherical atom induced by a uniform electric field. The variation of the electronic density is represented by a simple scalar the induced atomic dipole moment. This coarse-grained (CG) model of the electronic density permits to derive a useful explicit energy functional where the functional derivatives are formulated in terms of polarizabilities and dipole hardnesses. 

The concept of dipole hardness permit to explore the relation between polarizability and reactivity from first principles. The physical idea is that an atom is more reactive if it is less stable relative to a perturbation (here the external electric field). The atomic stability is measured by the amount of energy we need to induce a dipole. For very small dipoles, this energy is quadratic (first term in Equation 24.19). There is no linear term in Equation 24.19 because the energy is minimum relative to the dipole in the ground state (variational principle). The curvature hi of E(p) is a first measure of the stability and is equal exactly to the inverse of the polarizability. Within the quadratic approximation of E(p), one deduces that a low polarizable atom is expected to be more stable or less reactive as it does in practice. But if the dipole is larger, it might be useful to consider the next perturbation order ... 

Equation 24.110 permits to formulate the nonlinear dipole hardness /z3 of an atom in an electric field (Equation 24.18) in terms of the responses n- We assume a field applied along the x Cartesian direction. The nonlinear polarizability a3 (Equation 24.10) is... 

One of the models for the hydration force, the polarization model,5 assumes that the hydration force is generated by the local correlations between neighboring dipoles present on the surface and in water. The macroscopic continuum theory, in which water is assumed to be a homogeneous dielectric, predicts that there is no electric field above or below a neutral surface carrying a uniform dipolar density. However, at microscopic level the water is hardly homogeneous, and the electric interactions... 

We also compare in both figures the dielectric constant values of the dipolar fluid when adsorbed in the ionic and in the hard sphere matrix. A clear influence of the presence of charges in the matrix is then observed. The ionic matrix lowers the response of the dipoles to an external field, i.e. lowers the value of the dielectric constant of the fluid in the given state. This can easily be understood, since the local electric field that the matrix charges generate... 

The polarizability of a species is a measure of the degree to which it may be distorted, e.g. by the electric field due to an adjacent atom or ion. In the hard sphere model of ions in lattices, we assume that there is no polarization of the ions. This is a gross approximation. The polarizability increases rapidly with an increase in atomic size, and large ions (or atoms or molecules) give rise to relatively large induced dipoles and, thus, significant dispersion forces. Values of a can be obtained from measurements of the relative permittivity (dielectric constant, see Section 8.2) or the refractive index of the substance in question. 

The motion of ions in a buffer gas is governed by diffusive forces, the external electric field and the electrostatic interactions between the ions and neutral gas molecules. Ion-dipole or ion-quadrupole interactions, as well as ion-induced dipole interactions, can lead to attractive forces that will slow the ion movement, mainly due to clustering effects. The interaction potential can be calculated according to different theories, and three such approaches—the hard-sphere model, the polarization limit model, and the 12,4 hard-core potential model— were introduced here. Under... 

All materials are polarizable, but vacuum is not (although it can hardly be called a material). With only bound charges, an electric field can only displace charges so that dipoles are formed and the material is polarized. If the hiomaterial is dry, double layers (Section 7.5) will not be formed. A dry hiomaterial in contact with dry electrode metal forms an interface. With free charges, there are important additional effects from the migration of these in an electric field. 

The same integrals also appear in Problem 3.61, where they are needed to find spectroscopic transition intensities. In that case, a photon provides the applied electric field. This similarity allows polarizabilities to be determined in some applications (particularly for molecular ions and other molecules where dipole moments are hard to measure directly) by adding together transition intensities determined by spectroscopy. 

Figure 5. Response of polar dielectrics (containing local permanent dipoles) to an applied electric field from top to bottom paraelectric, ferroelectric, ferrielectric, antiferroelectric, and helielectric (helical anti-ferroelectric). A pyroelectric in the strict sense hardly responds to a field at all. A paraelectric, antiferro-electric, or helieletric phase shows normal, i.e., linear dielectric behavior and has only one stable, i.e., equilibrium, state for E=0. A ferroelectric as well as a ferrielectric (a subclass of ferroelectric) phase shows the peculiarity of two stable states. These states are polarized in opposite directions ( P) in the absence of an applied field ( =0). The property in a material of having two stable states is called bistability. A single substance may exhibit several of these phases, and temperature changes will provoke observable phase transitions between phases with different polar characteristics.

Correspondingly, the specific sensitivity parameters can be introduced diamagnetic susceptibility as the response under applied magnetic field, static dipole polarizability that accounts for the electronic cloud deformation under applied electric perturbation and the chemical hardness associated with the compactness of the electronic cloud by the nuclear influence, and possible applied electric perturbation. 

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