Polarizability principal values

With the new coordinate system only the three diagonal components axx, ayy, and olzz referred to as principal values of a are nonzero. The halfaxes of the ellipsoid are a 2, ay]/2, and aj1/2. If the polarizability ellipsoid... 

It is interesting to compare and contrast an isotropic ellipsoid and an anisotropic sphere the polarizability of both particles is a tensor, the principal values of which are... 

It is not difficult to generalize the results of this section to an anisotropic ellipsoid the axes of which coincide with the principal axes of its permittivity tensor. The principal values of the polarizability tensor of such a particle are... 

It is important to note that the form effect is proportional to the square of the dielectric contrast, Ae, and will always be positive for prolate particles (Ll > L2), and negative for oblate shapes (L2 > Lx). The intrinsic contribution can change sign depending on the relative magnitudes of the principal values of the polarizability tensor of the particle. 

The Gas Phase Polarizability Anisotropy. Murphy50 has measured the depolarization ratio for Rayleigh scattering, pR, and analysed the intensity distribution in the rotational Raman spectrum of the vapour at 514.5 nm. The ratio R20 of the invariants of the a,-,aA/ tensor can be determined by fitting the rotational Raman distribution, and a is known (from the Zeiss-Meath formula). Knowledge of the three quantities, a, pR and R2o, allows the polarizability anisotropy, Aa, and the three principal values of the tensor to be calculated. The polarizability anisotropy invariant is numerically equal to the quantity,... 

Proceeding from the above discussion, let us now find the general form of the energy of the dipole-dipole resonance radiationless interaction in anisotropic crystals. To do this we must generalize the relations of Section 5.4. Assume that a (w) are the principal values of the polarizability tensor of the individual molecule (y = 1, 2,3) and t are the directions of its principal axes. Then we can write this tensor as... 

Here, a, is the principal value of the tensor of the generalized polarizability of one adsorbed molecule [Eq. (1.35)] and (W ), is the number of the adsorbed molecules per unit area that contribute to the /th component of the polarizability. The quantity on the right of Eq. (1.89) is called surface susceptibility. 

Such axes are called principal axes of polarizability and the associated at, i = h 2, 3, the principal values of the polarizability. For the isotropic case, it is apparent that 01 = 02 = 03 = a. These axes are clearly the most convenient ones to use in the discussion of induced dipoles, and will be used extensively throughout the remainder of this section. 

The determination of the principal values of the polarizability requires the solution of a third-order secular determinant, entirely analogous to the secular determinant described in Chap. 2. 

Expansion of the Polarizability in the Normal Coordinates. Thus far, the components of the polarizability have been assumed to be independent of the time. For a molecule executing small vibrations, however, the polarizability may be expanded in terms of the normal coordinates. Thus the principal values could be written as... 

Principal a, os, of inertia, 284 of polarizability ellipsoid, 44 Principal moments of inertia, 284 Principal values, of moments of inertia, 284... 

We can substitute for the sum in (IIIB-67), but it is more useful to first make a further simplification. Knowledge of the polarizability tensor a means knowledge of the three principal values and directions of the tensor. 

Equation (IIIB-83) is identical to (MFK-Equations 20, 21). It is also identical to Kirkwood s (1937) equation. Using the dipole approximation to ViQaiiot nd the definition of a polarizability tensor (Equation IIIB-68), we see that (IIIB-83c) can be written formally as a function of at (v) and a/ (v). These are tensors for groups t and j whose principal values and principal directions are frequency dependent. If we know the a s at each frequency we know the ji s and we can use (III B-83c) as it stands. 

Equation (5.6) reduces to the result for an ordinary isotropic liquid when S = 0. In the nematic, S 7 0 and the projection of the polarizability takes on its extreme values, a and a, when = 0° and 90°, respectively. If we consider a mesogen with cylindrical symmetry having a principal value of the polarizability along the long molecular axis, = an), and a unique value transverse to the 1 axis,... 

In Eq. (5.7) we have substituted S for the principal value of the order tensor Su. It should be clear from the results of Eq. (5.7) in conjunction with the relationship between the polarizability and the refractive index (Eq. (5.5), leaving aside complications associated with anisotropic internal-field corrections), that n w 7 In short, a nematic liquid may be readily distinguished from an ordinary liquid because... 

Maier and Meier [35] worked out these ideas more quantitatively by applying Onsager s theory of static polarization for nematic liquid crystals. The Onsager theory relates the dielectric constant to molecular properties, namely the molecular polarizability a and the permanent electrical dipole moment /i. In the case of nematic liquid crystals, however, the polarizability has to be treated as a tensor with principal values an and a and an angle ]8 has to be introduced to describe the... 

The induced dipole is parallel to the external field vector. The quantities o x, Oyy and a2 z are called principal values of the molecular polarizability, and the axes x, y and z ... 

The Raman measurements provide values directly for P)mn, the coefficients of the Legendre expansion related to coordinates axes chosen with respect to the principal axes of the differential polarizability tensor, hence the superscript r. The coefficients Pimn for the orientation of the units of structure must then be obtained by further calculation from the P)mn. 

Recently, Heiberg et al. [26] have studied polarizabilities of the intermolecular contacts in bis(ethylenedithiolo)tetrathiafulvalene (BEDT-TTF) and bis(ethylenedioxy)tetrathiafulvalene (BEDO-TTF) molecular crystals by polarizing microscope techniques. The principal refractive indices and the corresponding optical axes have been calculated by tensorial addition of the bond polarizabilities of all bonds in the molecules. Comparison of calculated and measured values of the relative polarizabilities showed that the polarizabilities of the molecules only cannot yield the measured indicatrix and axes angle. Thus polarizabilities with other orientations must be involved. From the crystal structure of the molecular crystals it is known that 10 and four different contacts exist between the molecules of BEDT-TTF and BEDO-TTF, respectively, with contact distances lower than van der Waals distances. Assigning of polarizabilities of these contacts can explain the measured behavior. 

As a result of the low ionization enthalpies for the outer electrons and the sphericity and low polarizability of the resulting M+ ions, the chemistry of these elements is principally that of their +1 ions. No other cations are known or, in view of the values of the second ionization enthalpies, expected. The ions M, where the s shell is filled, are discussed in Section 3-4. 

In the discussion of Raman and ROA theory an important concept is that of the invariants. The values of invariants, the simplest example of which is the mean polarizability a, do not alter if the principal axes of the molecule are rotated. 

In the general case, when fi acts at known finite angles with the 1,2, and 3 directions, the components /xx, /x2, and /x3 must be inserted into (26), and extraction of the b, s from (22), (26), and (27) becomes more tedious. Some typical polarizability ellipsoids, specified by their principal semiaxes, and drawn from values, are shown in Table 21. On p. 291... 

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