Macroscopic polarization, definition

P is the macroscopic polarization. It consists of a lattice polarization b21 w originating from the electric dipole moment arising from the mutual displacement of the two sublattices, and of a second term b22 P originating from the pure electron polarization. According to definition, P and E are connected by... 

These definitions for P, Pi, and aN may now be substituted into Equation 31 to give Equation 22. The roles of the phenomenological relaxation times Tj and T2 are now clear. The macroscopic polarization P, and hence the components P - and Pi, relax to their equilibrium values of zero with a relaxation time T2. The population difference N relaxes to its equilibrium value aNq with a relaxation time Tj. 

The surface polarization can be measured by different means. The most straightforward one is based on the pyroelectric technique [15]. To measure P one has to deal only with one surface of a cell with uniform director alignment, either planar or homeotropic at both interfaces. The main idea is to use a spatially dependent temperature increment in order to separate the contributions to the pyroelectric response coming only from the surface under study and not from the opposite one. By definition, the pyroelectric coefficient is y = dPIdT where P is macroscopic polarization of a liquid crystal and T is temperature. If we are interested only in the polarization originated from the orientational order we can subtract the isotropic contribution to y and calculate P in the nematic or SmA phases by integrating the pyroelectric coefficient, starting from a certain temperature T, in the isotropic phase ... 

The new polar symmetry allows for the existence of macroscopic polarization, large or small, depending on the magnitude of the strain and molecular dipole moments shown by small arrows. Due to the distortions, the densest packing of our pears and bananas results in some preferable ahgimient of molecular skeletons in such a way that molecular dipoles look more up than down. By definition, the dipole moment of the unit volume is electric polarization. These simple arguments brought R. Meyer to the brilliant idea of piezoelectric polarization [25] ... 

Before going on to discuss recent developments in theory of static or equilibrium permittivity for specific models in the next section mention should be made of different kinds of formalism developed by Fulton (18) and by Felderhof and Titulacr (19) The reader may have noticed that in the theories described so far the macroscopic polarization is evaluated as a statistical mechanical average consistent with the basic definition for the Maxwell equations discussed inl.lp while the macroscopic and thence Cg is not so computed but is instead introduced by electrostatic or cavity arguments. Both Fulton and Felderhof-Tltulaer have dispensed with cavities in their treatments relating the permittivity to polarization fluctuations (as expressed by <(Zi ) > for... 

In spectroscopy we may distinguish two types of process, adiabatic and vertical. Adiabatic excitation energies are by definition thermodynamic ones, and they are usually further defined to refer to at 0° K. In practice, at least for electronic spectroscopy, one is more likely to observe vertical processes, because of the Franck-Condon principle. The simplest principle for understandings solvation effects on vertical electronic transitions is the two-response-time model in which the solvent is assumed to have a fast response time associated with electronic polarization and a slow response time associated with translational, librational, and vibrational motions of the nuclei.92 One assumes that electronic excitation is slow compared with electronic response but fast compared with nuclear response. The latter assumption is quite reasonable, but the former is questionable since the time scale of electronic excitation is quite comparable to solvent electronic polarization (consider, e.g., the excitation of a 4.5 eV n — n carbonyl transition in a solvent whose frequency response is centered at 10 eV the corresponding time scales are 10 15 s and 2 x 10 15 s respectively). A theory that takes account of the similarity of these time scales would be very difficult, involving explicit electron correlation between the solute and the macroscopic solvent. One can, however, treat the limit where the solvent electronic response is fast compared to solute electronic transitions this is called the direct reaction field (DRF). 49,93 The accurate answer must lie somewhere between the SCRF and DRF limits 94 nevertheless one can obtain very useful results with a two-time-scale version of the more manageable SCRF limit, as illustrated by a very successful recent treatment... 

It is obvious that such a definition of solvent polarity cannot be measured by an individual physical quantity such as the relative permittivity. Indeed, very often it has been found that there is no correlation between the relative permittivity (or its different functions such as l/sr, (sr — l)/(2er + 1), etc.) and the logarithms of rate or equilibrium constants of solvent-dependent chemical reactions. No single macroscopic physical parameter could possibly account for the multitude of solute/solvent interactions on the molecular-microscopic level. Until now the complexity of solute/solvent interactions has also prevented the derivation of generally applicable mathematical expressions that would allow the calculation of reaction rates or equilibrium constants of reactions carried out in solvents of different polarity. 

Most modem discussions of solvent effects rely on the concept of solvent polarity. Qualitative ideas of polarity are based on observations such as like dissolves like and are well accepted and understood. However, quantifying polarity has proven to be extraordinarily difiicult. Since the macroscopic property polarity is the sum of all possible (non-specific and specific) intermolecular interactions between a solute and the solvent, excluding such interactions leading to definite chemical changes (reactions) ofthe solute, this is perhaps unsurprising. Hence, it is important that care is taken when discussing the polarity of any liquid to ensure that it is clearly understood what is actually being referred to. Attempts to compare polarities measured by different techniques that are sensitive to different properties of the solvent often only lead to confusion. 

Any set of constitutive relationships such as (10) and (11) or (13) and (14) is a reflection on the macroscopic scale of the microscopic behavior of the individual molecules which comprise the substance. The connection between the macroscopic and the microscopic is provided through the statistical interpretation one can give to the polarization vector P and the magnetization vector M which relate D with E and B with H through the auxiliary definitions... 

The polarization P, which has both a macroscopic and a molecular definition, is the link between experiment and molecular interpretation of the experimental results. 

The description of the behavior of particulate materials relies on constitutive equations, functions of stress, strain, and other physical quantities describing the system. It is rather difficult to extract macroscopic observables like the stress from experiments, e.g. in a two-dimensional (2D) geometry with photo-elastic material, where stress is visualized via crossed polarizers [6, 7]. The alternative is, to perform discrete element simulations [2, 4] and to average over the microscopic quantities in the simulation, in order to obtain some averaged macroscopic quantity. The averages over scalar quantities like density, velocity and particle-spin are strai tforward, but for the stress and the deformation gradient, one finds slightly different definitions in the literature [3, 8-11]. 

Ferroelectric domains are those macroscopic (greater than 20 A) regions in a solid having natural crystal polarity which is constant, but whose direction of polarity can be altered irreversibly by a practically imposable electric field. Implied in this definition is the existence within the material of a spontaneous electrical dipole moment hence the existence of piezoelectricity, and the reversing effects of applied fields hence the existence of hysteresis, and of the probability that at some temperature before complete crystal disruption the spontaneous polarization will be destroyed or directionally randomized on an atomic scale. At this temperature the dipolar instability between the decision of order or disorder will usually produce a high peak value of incremental permittivity. 

For Sb and A1 on HOPG, and for Si and Ge on crystalline SiNx/Si(lll), most grown-up crystallites have definite polar (also azimuthal for some) orientation alignment with the substrate. In contrast, for Ge on graphite, and Si and Ge on amorphous SiNx/Si(001), the orientation of the nanocrystals seems completely random, and high-index facets are observed quite often. Surface reconstructions not formed on bulk crystals are observed on some facets. These observations reflect the unique capacity of nanoscale facets to accommodate certain surface superstructures that are not observable on a macroscopic scale. 

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