Orientation distribution fourth moment

Broad-line NMR derivative spectra were obtained using a Brucker HFX-90 spectrometer to record the resonance at 84.67 MHz. The specimens, made by compacting granular PTFE into preforms, sintering at 380°C, and cooling slowly at a rate of 0.02 deg/min had a specific gravity of 2.205. The second moment of tire NMR line shape is of interest because the fourth moment of the orientation distribution function is proportional to it. 

Evidently, Raman scattered light contains information about both the second and the fourth moments of the orientation distribution function. This is in contrast to birefringence and dichroism measurements, which respond only to anisotropies in the second moments. 

The optical apparatus used in this work was described in section 8.6 and has the capability of providing both Raman scattering and birefringence measurements simultaneously. The Fourier expansion of the overall Raman scattering signal is given by equation (8.51), and the coefficients are given by equations (8.52) to (8.54). In these expression, a simple, uniaxial form for the Raman tensor was assumed. From these coefficients, the anisotropies in the second and fourth moments of the orientation distribution can be solved as... 

From slow-shear-rate solutions of the Smoluchowski equation, Eq. (11-3), with the Onsager potential, Semenov (1987) and Kuzuu and Doi (1983, 1984) computed the theoretical Leslie-Ericksen viscosities. They predicted that ai/a2 < 0 (i.e., tumbling behavior) for all concentrations in the nematic state. The ratio jai is directly related to the tumbling parameter X by X = (1 -h a3/a2)/(l — aj/aa). Note the tumbling parameter X is not to be confused with the persistence length Xp.) Thus, X < I whenever ai/a2 < 0. As discussed in Section 10.2.4.1, an approximate solution of Eq. (11-3) predicts that for long, thin, stiff molecules, X is related to the second and fourth moments Sa and S4 of the molecular orientational distribution function (Stepanov 1983 Kroger and Sellers 1995 Archer and Larson 1995) ... 

Hence we see that the stress depends on only the second and the fourth moments of the orientation distribution, (pp) and (pppp). However, in general, it is necessary to know the complete orientation distribution function Mp, t) to calculate exact results for these moments by means of (2 104). 

In another type of application a low molecular fluorescent probe is added to a system containing macromolecules. As would be expected, the rotation of a small species is insensitive to the molecular weight of high polymers, but depends on the "microscopic viscosity" which is a function of free volume. For instance, Nishijima has shown that the microscopic viscosity of liquid paraffin hydrocarbons levels off for molecular weights above 1000 and that the microscopic viscosity of polystyrene containing 10 volume"/ benzene is only 200 times as high as that of benzene (15). Nishijima also showed that the emission anisotropy is a useful index of molecular orientation. Since both the excitation and the emission are anisotropic, the method yields the fourth moment of the distribution function of orientations, while other optical properties (dichroism, birefringence) depend on the second moment (15). 

Polarized Raman spectroscopy can be used to probe the moleeular orientation in polymers. The advantages of using Raman polarization are that it can be used for thick samples if care is taken not to scramble the polarization in the sample. This generally means that the sample must not be turbid. Secondly, Raman can determine both the second and fourth moments of the orientation distribution function while IR and birefringence measurements only report the second moment. Thirdly, Raman spectroscopic measurements (like IR) can reveal the orientation of not only the crystalline phase but also the amorphous phase. Raman polarizations can be made on polymeric fibers that are difficult with IR. Polyethylene [83,84]. and poly(ethylene terephthalate) have been studied [85]. 

Langkilde, F.W, M. Glsin, E.W. Thulstrup, and J. Michl. 1983. Alignment of solutes in stretched polyethylene. Determination of the five second and fourth moments of the orientation distribution of 2-fluoropyrene from polarized fluorescence. Additional evidence for the twisting of weak transition moments by the solvent environment. J. Phys. Chem. 87 2901-2911. 

As expected from our experience with a particle in a box, three quantum numbers are necessary to describe the spatial distribution or electrons in atoms. To describe an electron in an atom completely, a fourth quantum number, m called the spin quantum number must be specified. This is because every electron has associated with it a magnetic moment which is quantized in one or two possible orientations parallel with or opposed to an applied magnetic field. The magnitude or the magnetic moment is given by the expression... 

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