Lorentz 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. 

The molecular field acting at the location of a water molecule is composed of a macroscopic field, E = — (dip/ dr), a local field (the Lorentz field El) that occurs because each molecule is surrounded by a dielectric medium (constituted of all the other water molecules), El = P/3eo, plus an additional field, Ep, due to the neighboring dipoles.18,27 Assumingthat the additional fieldis generated only by the neighboring molecules from the same and adjacent icelike water layers (see Figure 2), at each of the sites (I or II) of layer j the field is provided by13... 

The Ewald series for the three-dimensional crystal can also be differentiated. The first derivative yields expressions for the Madelung electric field Fm (due to local charges). The second derivative yields the Madelung field gradient, or, equivalently, the internal or dipolar or Lorentz field FD (due to local dipoles) [68-71],This second derivative can also generates the dimensionless 3x3 Lorentz factor tensor L with its nine components Lv/t ... 

With regard to dipolar liquids and thdr solutions in non-dipolar solvents, Piekara developed a complete theory of dipolar couplings calculating all three correlation factors Rcn, Pk, and Rs- These studies revealed that nitrobenzene molecules coalesce momentarily into antiparalld aggregates, which then tend to couple mutually into almost parallel pairs. Feterlin and Stuart performed a detailed study of the influence on Ae not orjly of Debye dipole couplings but moreover of anisotropy of the local Lorentz field. 

The microscopic Lorentz field equations can be written in the following form (in rationalized gaussian units) ... 

Ho is the value of the magnetic field at the nucleus generated by an external magnet and is effectively zero away from a large magnet. The next term, —DM, is the demagnetising field and fttM is the Lorentz field (the coefficient being strictly applicable for cubic symmetry only) but both are small. arises as a result of the interaction of the nucleus with an imbalance in the s-electron spin density at the nucleus, and can be written as... 

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... 

A linear relation between the field at the muon site, corrected for the (calculated) Lorentz field Rl nd the magnitude of the c-component of the R magnetic moment could be established for the full series (fig. 90) of R2pei4B compounds. Large corrections are quite apparent for the cases (Er, Tm) where the moments are oriented perpendicular to the c-axis. 

Fig. 90. Local magnetic field at the muon site (collected for the Lorentz field) as a fimction of die magnetic moment of the R ion RjFej B compounds. Open circles represent the c-axis component of those moments. From Niedermayer et aL (1990).

The classical dipole field and the Lorentz field also have to be included. Hirosawa and Nakamura (1982a) performed a careful and detailed analysis of the Co... 

In the literature, several EMTs have been reported. These EMTs originate from different assumptions of the shape, construction, number, and mutual orientations of the basic unit cells and various types of interactions between them (in addition to or instead of the Lorentz field — the long-range dipole-dipole field involved in the CMLL model). Rather than attempt to review this large body of theoretical work (see, e.g., Refs. [183, 195-197]), the results of only a few of them will be discussed. 

E2 is the Lorentz field coming from the charges at the inner surfaces of the cavity and... 

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 lattice sums in parameters Bpq absolutely converge and may be calculated by direct summation. Special care is needed only when calculating conditionally convergent sums in parameters of quadrupole components of the point charge field, which are proportional to the parameters Qap( ) of the Lorentz field ... 

If the applied field is directed along the c-axis of a sphere-shaped axial ferromagnetic crystal, and its value does not exceed the Lorentz field Hi, the resonance frequent Vj Hoz differ from eq. (255). At Hoz>Ht, NMR frequencies are calculated according to the formula... 

Interchanging the dot and the cross and introducing the magnetic moment we obtain (11-13). We have been assuming in the previous equation that the molecules are in an infinitely dilute gas. If the molecules are in solution we approximate the nonspecific effects of the solvent by the use of an effective electric field strength in the solvent. This is usually done by using the Lorentz field thus we write... 

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