Orientation averages

B1.3.2.5 THE MICROSCOPIC HYPERPOLARIZABILITY TENSOR, ORIENTATIONAL AVERAGING, THE KRAMERS-HEISENBERG EXPRESSION AND DEPOLARIZATION RATIOS... 

For an isotropic material, all orientations are equally probable and all such products that have an odd number of Tike direction cosines will vanish upon averaging-. This restricts the nonvanishing tensor elements to those such as xVaaa abba - Similarly for the elements Such orientational averaging is crucial in... 

Fig. 1. Orientationally averaged magnetic susceptibility of various forms of carbon [21].

Information on molecular orientation can be useful in two primary ways. First, it is possible to use the orientation functions or averages to gain an understanding of the mechanisms of plastic deformation. Secondly the orientation averages can provide a basis for understanding the influence of molecular orientation on physical properties, especially mechanical properties. 

Amorphous orientation average Crystalline orientation average Nuclear spin number Scattered intensity Scattered intensity Transmitted intensity... 

In this review the definition of orientation and orientation functions or orientation averages will be considered in detail. This will be followed by a comprehensive account of the information which can be obtained by three spectroscopic techniques, infra-red and Raman spectroscopy and broad line nuclear magnetic resonance. The use of polarized fluorescence will not be discussed here, but is the subject of a contemporary review article by the author and J. H. Nobbs 1. The present review will be completed by consideration of the information which has been obtained on the development of molecular orientation in polyethylene terephthalate and poly(tetramethylene terephthalate) where there are also clearly defined changes in the conformation of the molecule. In this paper, particular attention will be given to the characterization of biaxially oriented films. Previous reviews of this subject have been given by the author and his colleagues, but have been concerned with discussion of results for uniaxially oriented systems only2,3). 

It is appropriate to conclude this section of the discussion with a formal summary of the orientation averages Pimn which can be obtained for four combinations of sample symmetry and symmetry of the structural units which fall within the orthorhombic symmetry of sample and unit which can be defined by Equation (1). 

Another way to divide the overall orientation is to recognise the distinction between crystalline material which is all in the trans conformation, and amorphous material which is both trans and gauche. By combining infra-red and X-ray orientation data, together with the determination of the proportion of crystalline material, the amorphous orientation average fa = gauche = 0. which is seen from the above results to be very reasonable,... 

Fig. 9. Plot of amorphous orientation average fa, obtained from birefringence, etc., versus f, obtained from im etc- Triangles, draw temperature 80 °C circles draw temperature 85 °C open symbols, single-stage draw full symbols, two-stage draw. Reproduced from Polymer by permission of the publishers, Butterworth Co (Publishers) Ltd. (C)...

A comparison of the amorphous orientation averages fa obtained by these two different routes is shown in Fig. 9, and provides some support for the validity of these procedures. 

The equations of motion (75) can also be solved for polymers in good solvents. Averaging the Oseen tensor over the equilibrium segment distribution then gives = l/ n — m Y t 1 = p3v/rz and Dz kBT/r sNY are obtained for the relaxation times and the diffusion constant. The same relations as (80) and (82) follow as a function of the end-to-end distance with slightly altered numerical factors. In the same way, a solution of equations of motion (75), without any orientational averaging of the hydrodynamic field, merely leads to slightly modified numerical factors [35], In conclusion, Table 4 summarizes the essential assertions for the Zimm and Rouse model and compares them. 

A CDF is interpreted in the same way as a CLD or an IDF. All these functions exhibit the probability distributions of domain size and arrangement. Clearer than a CLD is the IDF, because it does not contain an orientation average but exhibits the topology in a selected direction. Clearer than an IDF is the CDF, because it visualizes the nanodomain topology in space, i.e., in more than one direction. 

The Boltzmann statistical weight for each rotamer is multiplied by (1 + aPt(cos 8)), with a being an adjustable parameter, before a particular rotamer is selected. Values of a, between zero and unity, will give rise to an order parameter for segment orientations. Average of Pt(cos 8) of 0.0 to 0.5. 

The values of these autocorrelation functions at times t = 0 and t = 00 are related to the two order parameters Legendre polynomial P2(cos/ ) and P4 (cos p). respectively, relative to the orientation p of the probe axis with respect to the normal to the local bilayer surface or with respect to the liquid crystal direction. The order parameters are defined as... 

Fig. 3. Orientationally averaged total stopping cross section St for protons impinging on H2O as a function of proton velocity along with the experimental results of Reynolds et ah (filled circles) [28]. 5 e, and represent the electronic, nuclear, and rovibrational contributions to the stopping, respectively.

The dry nematic solution exhibits a smooth texture after being annealed, with a field of disclinations at any glass-solution interface. A nematic phase produced by cooling the isotropic phase will exhibit a complex, mottled texture that slowly anneals to the smooth texture. The ratio Rvv(q)/ Hv(q) (for e = 20 deg.) for either morphology indicates appreciable orientational averaging of the orientation fluctuation. Photon correlation scattering on the... 

Selected entries from Methods in Enzymology [vol, page(s)] Additive properties of polarization, 246, 286 angle-resolved, assessment of peroxidation effects on membranes, 233, 274-275, 281-283, 287-288 binding isotherm construction, 246, 287-288 effect of inner filter effects, 246, 288 incoherent systems, 246, 263-264 orientational averaging, 246, 265, 269-270 Perrin equation, 246, 284-285 polarization of emission, 246, 284 rotational diffusion, 246, 9, 260 time-resolved, assessment of peroxidation effects on membranes, 233, 274, 283-285, 285-287. 

The orientational average of dipoles oriented at an angle 9 with respect to the direction of the linear light polarization depends on the factor ... 

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