Differential polarizability

Ratio of principal components of differential polarizability tensor... 

For simplicity, consider a single unit of structure with a cylindrical differential polarizability tensor. The scattered intensity Is is then... 

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. 

For a cylindrical differential polarizability tensor we can write ax = a2 and r = v.3l[Pg.91]

It can readily be appreciated that in the absence of any knowledge regarding the differential polarizability tensor, it is a difficult exercise to obtain precise information from the Raman measurements. However, if r is known, Equations (19) are six linear simultaneous Equations in the six quantities (ot2I0N0) 1, P)0o> P220) PIoo> P420 and... 

Die-drawing 143 Die-drawn fibers 144 Differential polarizability tensor 91 Differential scanning calorimetry 63, 165 Diffusion 189... 

The existence of these differential polarizabilities means (as indeed it does in benzenoid compounds) that a single cr+ parameter will not accu-... 

In Figure 6, we have plotted the differential methylene polarizability (the isotropic polarizability per methylene unit) of the n-alkanes. The most notable feature of this plot is the increase in the isotropic polarizability with the number of carbon atoms. From the decomposition of the polarizability along the principal directions in Figure 7 (with the Cartesian directions defined as for Figure 3), we note that the differential polarizability increases only in the x direction—that is, in the component parallel to the carbon chain. In the other two directions, the polarizability converges smoothly except for weak oscillations in the in-plane ayy component, oppositely directed to those in the other in-plane component axx. These oscillations, which are absent in the isotropic polarizability, may be linked to the alter-... 

The behavior of the polarizability of the n-alkanes may be rationalized with reference to the simple particle-in-a-box model. In this model, the polarizability increases as the fourth power in the length of the box, which implies that the polarizability per unit length (i.e., the differential polarizability) increases as the third power in the length of the box. Obviously, in the n-alkanes, the electrons do not at all behave according to this simple model, but we may still attribute the increasing differential polarizability of the n-alkanes to a more pronounced delocalization of the electrons in the longer chains, as the electrons become more loosely attached to the system. As the chain grows, this effect becomes less important and the differential polarizability becomes constant. 

If, in the n-alkanes, there is a small but noticeable increase in the differential polarizability with the chain length, we would expect this effect to be more pronounced in conjugated systems, where the delocalized 7r-electrons... 

It has been stated that, when specific hydrogen-bonding effects are excluded, and differential polarizability effects are similar or minimized, the solvent polarity scales derived from UV/Vis absorption spectra Z,S,Ei 2Qi),n, Xk E- ), fluorescence speetra Py), infrared spectra (G), ESR spectra [a( " N)], NMR spectra (P), and NMR spectra AN) are linear with each other for a set of select solvents, i.e. non-HBD aliphatic solvents with a single dominant group dipole [263]. This result can be taken as confirmation that all these solvent scales do in fact describe intrinsic solvent properties and that they are to a great extent independent of the experimental methods and indicators used in their measurement [263], That these empirical solvent parameters correlate linearly with solvent dipole moments and functions of the relative permittivities (either alone or in combination with refractive index functions) indicates that they are a measure of the solvent dipolarity and polarizability, provided that specific solute/ solvent interactions are excluded. 

For D D , the reciprocals of these correlation times are raised by an amount [(D /D ) — 1] as indicated in Eq. (7.55). As noted previously, K rriL, ttim) values at a particular temperature are computed from (P2) and (P4). The fourth-rank order parameter P4) cannot be directly measmed from a NMR spectrmn, but may be derived from measurements of the mean square value of a second-rank quantity [7.19-7.22]. In the Raman scattering technique [7.21], the second-rank molecular quantity is the differential polarizability tensor of a localized Raman mode. In fluorescence depolarization [7.19], the average of the product of the absorption and emission tensors is used to determine (P4). Since there is a lack of experimental determination of (P4) in liquid crystals, this may be calculated based on the Maier-Saupe potential... 

Karamanis, P., Pouchan, C., 8c Maroulis, G. (2008). Structure, stability, dipole polarizability and differential polarizability in small gallium arsenide clusters from all-electron ab initio and density-functional-theory calculations. Physical Review A - Atomic, Molecular, and Optical Physics, 77(1), 013201/1-013201/6. 

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