Polarization density vector

The properties of a dielectric medium through which an electromagnetic (optical) wave propagates are completely described by the relation between the polarization density vector P(r, t) and the electric-field vector E r, t). It was suggested that P(r, t) could be regarded as the output of a system whose input was E(r, t). The mathematical relation between the vector functions P(r, t) and E(r, t) defines the system and is governed by the characteristics of the medium. The medium is said to be nonlinear if this relation is nonlinear. 

As a consequence of the translational invariance of the matrix e, K ) is an eigenstate, so that the induced dipoles build up a plane wave K>. The macroscopic polarization-density vector is given by... 

Now let us introduce passive ion channels in the membrane of a polarized cell (Figure 5.7). The channels are normally closed, but now suddenly opened. Due to the potential difference, cations will immediately start to migrate into the negative cell interior. A current density field is suddenly created both intra- and extracellularly. The extracellular current density vector field J and the potential field are related by Eq. 2.1 V = — J/a. The current is generated by the ionic flow, and it terminates on the membrane capacitor in a discharge/charge process. 

The variation in absorption due to the electric field modulation (Equation 19.16) is a nonlinear optical effect. We now consider the origin of nonlinear behavior in materials. In a classical description [89-91], the electric field interacts with the charges (q) in an atom through the force (qF). which displaces the centre of the electron density away from the nucleus. This results in charge separation and thus in a field-induced dipole pi. For an assembly of atoms, the average summation over all atoms ultimately gives rise to the bulk polarization P vector of the material. P opposes the externally applied field and is given by ... 

In Eq. 2, D is the electric displacement or electric flux density vector, E is the electric field vector, P is the electric polarization vector, and is the permittivity of vacuum. In many isotropic materials the induced polarization is directly proportional to the applied field strength, except for the case of very high fields. We can write [71]... 

Let us consider the case to divide into two regions. There are 8 horizontal operators considering the direction of the boundary. If we state column direction as north and south, and row direction as east and west, then the operators are classified to north, south, east, west, northeast, northwest, southeast, southwest operators which are illustrated in Figure 8.5. The north operator is described for example. The boundary of the north operator is along the row direction, namely east and west. Applied voltages toward northern part of the electrodes are -E [V] while applied voltages to southern part of the electrodes are E [V]. The current density vectors directed to north, because southern part propels the molecules while northern part repels the molecules. We can think of other operators in the same manner. The directions of boundary are row, column and diagonal (in two ways), and for each case, there are two kinds of polarities. 

The classical scheme for dichroism measurements implies measuring absorbances (optical densities) for light electric vector parallel and perpendicular to the orientation of director of a planarly oriented nematic or smectic sample. This approach requires high quality polarizers and planarly oriented samples. The alternative technique [50, 53] utilizes a comparison of the absorbance in the isotropic phase (Dj) with that of a homeotropically oriented smectic phase (Dh). In this case, the apparent order parameter for each vibrational oscillator of interest S (related to a certain molecular fragment) may be calculated as S = l-(Dh/Di) (l/f), where / is the thermal correction factor. The angles of orientation of vibrational oscillators (0) with respect to the normal to the smectic layers may be determined according to the equation... 

Figure 1 3. Contour plot of the electron density of CO, showing the magnitudes and directions of atomic and charge transfer dipoles (arrow length is proportional to magnitude). Arrow heads point to the negative end. The molecular dipole moment is given by the vector sum of charge transfer terms (p.c.t.) and the atomic polarizations ( ra p). Values were obtained at the DFT level using the B3LYP functional and the 6-31 1+G(3df) basis set. The SCF molecular dipole = 0.096 D the computed molecular dipole ( Jtc.t.[0] + Aa.p.[0] + Hc.JC] + Aa.p.[C]) = 0.038 au = 0.096 D, close to the experimental value of 0.1 10 D (15).

If the considered molecule cannot be assimilated to a sphere, one has to take into account a rotational diffusion tensor, the principal axes of which coincide, to a first approximation, with the principal axes of the molecular inertial tensor. In that case, three different rotational diffusion coefficients are needed.14 They will be denoted as Dx, Dy, Dz and describe the reorientation about the principal axes of the rotational diffusion tensor. They lead to unwieldy expressions even for auto-correlation spectral densities, which can be somewhat simplified if the considered interaction can be approximated by a tensor of axial symmetry, allowing us to define two polar angles 6 and

symmetry axis of the considered interaction) in the (X, Y, Z) molecular frame (see Figure 5). As the tensor associated with dipolar interactions is necessarily of axial symmetry (the relaxation vector being... 

The state of polarization is determined by the pair of complex numbers e and e2 the quantities ei 2 and e2 2 represent probability densities of a definite (linear or circular) polarization of the photon as determined by the unit vectors Xi and x2- Since ej and e2 are related by the normalization condition... 

From the response of each substrate it is thus possible to calculate the electric displacement density (D) due to the polarization of the material when subjected to E vector. 

Remember that e = D/E, where the displacement vector is given by D = E -F 4 rP (CGS units), E being the macroscopic electric held and P the macroscopic polarization. Considering a density of atoms N, the macroscopic polarizahon is P = N (p) = 7Vq (Eioc> (where the symbol ( ) indicates an average valne) and so D = E + dTriVa (Eioc>. Now assnming (Eioc) = E, we obtain ... 

Hirshfeld (1971) was among the first to introduce atom-centered deformation density functions into the least squares procedure. Hirshfeld s formalism is a deformation model, in which the leading term is the unperturbed IAM density, and the deformation functions are of the form cos" 0jk, where 9jk is the angle between the radius vector r7 and axis k of a set of (n + l)(n + 2)/2 polar axes on each atom /, as defined in Table 3.8 (Hirshfeld 1977). The atomic deformation on atom j is described as... 

For compactness, the subscript M for the electronic density parameters has been omitted in Eq. (8.49). The polar coordinate system has the z axis of the local Cartesian coordinate system as the polar axis, and the vector RMP is referred to this local coordinate system. 

Here I0 is the intensity of the x-ray beam, r0 = e2/mc2 is the classical electron radius (2.82 x 10 15 m)., P(9,) is the polarization of the x-rays it depends on the angle between the polarization and the scattering vector. For horizontally polarized x-rays, it takes the form P(0, < >) = 1 - sin220 sin2t)>, where 20 is the scattering angle and < > the azimuthal angle with respect to the vertical direction. The formfactor ) is the Fourier transform of the atomic electron density ... 

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