Lorentz field polarization

The result can be compared with the macroscopic expression as given by Bottcher. which is of a similar form but with the factor eB omitted everywhere. This is an unexpected result, where the macroscopic expression has not been recovered exactly despite the assumptions which were introduced and which gave Eq. (1) exactly when the polarizabilities of the ions were neglected. The appearance of the B-factor corresponds to.a Lorentz field polarizing the particle a field E = gives rise to a local field... 

From Eq, (1) it is clear that a model of crystal polarization that is adequate for the description of the piezoelectric and pyroelectric properties of the P-phase of PVDF must include an accurate description of both the dipole moment of the repeat unit and the unit cell volume as functions of temperature and applied mechanical stress or strain. The dipole moment of the repeat unit includes contributions from the intrinsic polarity of chemical bonds (primarily carbon-fluorine) owing to differences in electron affinity, induced dipole moments owing to atomic and electronic polarizability, and attenuation owing to the thermal oscillations of the dipole. Previous modeling efforts have emphasized the importance of one more of these effects electronic polarizability based on continuum dielectric theory" or Lorentz field sums of dipole lattices" static, atomic level modeling of the intrinsic bond polarity" atomic level modeling of bond polarity and electronic and atomic polarizability in the absence of thermal motion. " The unit cell volume is responsive to the effects of temperature and stress and therefore requires a model based on an expression of the free energy of the crystal. 

Early powder diffraction experiments relied mostly on the Debye-Scherrer experiment to record a diffractogram. A broad film strip set into a cylindrical chamber produced the first known two-dimensional powder diffraction data. In contrast to modern methods the thin equatorial strip was the only part of interest and intensities merely optically and qualitatively analysed. This changed drastically with the use of electronic scintillation counters. Intensities were no longer a matter of quality but quantity. Inevitably the introduction of intensity correction functions long known to the single-crystal metier, i.e. Lorentz and polarization corrections (see Section 14.3), made their way into the field of powder diffraction. 

Fig. 7.4 Lorentz model for the local field. Polarization of an ellipsoidal form dielectric sample and appearance of depolarizing field Ei (a), Lorentz cavity field E2 and the field of individual molecules within the cavity E3 (b)...

From Fig. 7.4a follows that the macroscopic field in the sample E = Eo+ El = Eq— P. When polarization P is very high, the macroscopic field is cmisider-ably reduced. The Lorentz field E2 is parallel to the external field and, for a spherical cavity, is equal exactly to +4jtP/3. Therefore, when both the sample and the cavity are spherical,... 

The use of the Lorentz field for polar molecules with permanent momentsyU and polarizabilities a leads to the familiar Langevin-Debye formula... 

To complete the discussion of the relation between dipole moment and dielectric constant, we consider a material medium containing N molecules per cubic centimeter, all of the same kind and in the same state a (the extension to mixtures is obvious if more than one state need be considered, it must be weighted with the appropriate Boltzmann factor). The polarization of the medium will be P = ATiR. For an isotropic medium, we may use for the effective electric field the Lorentz field... 

Continuum models have a long and honorable tradition in solvation modeling they ultimately have their roots in the classical formulas of Mossotti (1850), Clausius (1879), Lorentz (1880), and Lorenz (1881), based on the polarization fields in condensed media [32, 57], Chemical thermodynamics is based on free energies [58], and the modem theory of free energies in solution is traceable to Bom s derivation (1920) of the electrostatic free energy of insertion of a monatomic ion in a continuum dielectric [59], and Kirkwood and Onsager s... 

The return to equilibrium of a polarized region is quite different in the Debye and Lorentz models. Suppose that a material composed of Lorentz oscillators is electrically polarized and the static electric field is suddenly removed. The charges equilibrate by executing damped harmonic motion about their equilibrium positions. This can be seen by setting the right side of (9.3) equal to zero and solving the homogeneous differential equation with the initial conditions x = x0 and x = 0 at t = 0 the result is the damped harmonic oscillator equation ... 

With most lines, however, an anomalous Zeeman effect is observed and the number of components is greater, in some cases reaching twelve or fifteen. They arc symmetrically arranged and symmetrically polarized. The displacements, as in the simpler case, are proportional to the magnetic field intensity H, and are always expressible, in wave numbers, as rational multiples of the displacemenl in the normal effect, which is 4.67 x I0 57f (reciprocal centimeter), a quantity known as the Lorentz unit. The Zeeman effects observed in sun spots give valuable information as to the magnetic conditions in those areas. 

The relation between p and E is linear when E is small, but becomes nonlinear as E acquires values comparable with interatomic electric fields (typically, 105 to 108 V/m). This may be explained in terms of the simple Lorentz model in which the dipole moment is p = —ex, where x is the displacement of a mass with charge —e to which an electric force — eE is applied. If the restraining elastic force is proportional to the displacement (i.e., if Hooke s law is satisfied), the equilibrium displacement x is proportional to E P is then proportional to E, and the medium is linear. However, if the restraining force is a nonlinear function of the displacement, the equilibrium displacement x and the polarization density P are nonlinear functions of E and, consequently, the medium is nonlinear. 

In this subsection, the connection is made between the molecular polarizability, a, and the macroscopic dielectric constant, e, or refractive index, n. This relationship, referred to as the Lorentz-Lorenz equation, is derived by considering the immersion of a dielectric material within an electric field, and calculating the resulting polarization from both a macroscopic and molecular point of view. Figure 7.1 shows the two equivalent problems that are analyzed. 

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 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... 

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],... 

In addition to the field generated by the adjacent dipoles, there is a macroscopic fieldE due to the presence of charges and of the average polarization P of the medium. In the Lorentz treatment of polarization, for a constant macroscopic field in a linear and homogeneous medium of dielectric constant e (hence satisfying P = e0(e — 1)E), the local field E n(fl at a site of a selected dipole is related to the macroscopic field E via19... 

One of the models follows the Lorentz—Debye theory, as summarized by Jackson20 and Frenkel.21 In a constant macroscopic field Eo, the macroscopic polarization P is... 

In traditional Lorentz—Debye theory, the local field that acts on a water dipole is given by E + P/3so, where E is the macroscopic field and P is the polarization [5], In the present case, the local field /ilocal should also include the field Ep generated by the neighboring dipoles of a water molecule. 

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... 

It was shown by Lorentz [44] that, for a medium of constant polarization and with a sufficient symmetric distribution of the neighboring molecules around the central one, the field Em vanishes. Then, the average dipole moment of the central molecule, m(z)=v0P(z), where i 0 is the volume occupied by a water molecule, is provided by [32] ... 

In Eq. (13), the term proportional to m(z0) is negligible (which shows that the field of the neighbors vanishes for a uniform polarization of the medium, as suggested long ago by Lorentz [44]), and the terms proportional to the derivatives of odd order vanish because of symmetry. Once the field Em has been determined, Eq. (9) should be replaced by ... 

The idea is that only the field from molecules near to the solute has to be considered, so that the complete system can be separated into microscopic and macroscopic regions. The molecules in the latter can be described by the average macroscopic properties. Lorentz derived a simple relation between the local field and the macroscopic polarization, which is still in use [113,133,134,135]. 

Table G Definitions of the Electric Field E, the (Di)electric Polarization P, the Electric Displacement D, the Magnetic Field H, the Magnetization M, the Magnetic induction or flux density B, statement of the Maxwell equations, and of the Lorentz Force Equation in Various Systems of Units rat. = rationalized (no 477-), unrat. = the explicit factor 477- is used in the definition of dielectric polarization and magnetization c = speed of light) (using SI values for e, me, h, c) [J.D. Jackson, Classical Electrodynamics, 3rd edition, Wiley, New York, 1999.]. For Hartree atomic u nits of mag netism, two conventions exist (1) the "Gauss" or wave convention, which requires that E and H have the same magnitude for electromagnetic waves in vacuo (2) the Lorentz convention, which derives the magnetic field from the Lorentz force equation the ratio between these two sets of units is the Sommerfeld fine-structure constant a = 1/137.0359895...

Many complex fluids contain orientable molecules, particles, and microstmctures that rotate underflow, and under electric and magnetic fields. If these molecules or microstructures have anisotropic polarizabilities, then the index of refraction of the sample will be orientation-dependent, and thus the sample will be birefringent. In general, the anisotropic part of the index of refraction is a tensor n that is related to the polarizability a of the sample. The polarizability is the tendency of the sample to become polarized when an electric field is applied thus P = a E, where P is the polarization and E is the imposed electric field. When the anisotropic part of the index of refraction is much smaller than the isotropic part (the usual case), the index-of-refraction tensor n can be related to a by the Lorentz-Lorenz formula ... 

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