Orientation averages determination

The half-height of this peak is higher than the peak at 998cm The transition moment angle of this peak is estimated to be 23° with the chain axis [4]. As shown in Figure 2(c), the Herman s orientation averages determined from this peak is always lower than the peaks at 841 and 981 cm", respectively, and this suggests that this peak is even more sensitive to the disordered material. In order to improve the curve... 

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,... 

The viscosity of the gas has been determined 2 to be 0-0001470 at 0° C. 0-0001552 at 15° C. and 0-0001997 at 100° C. Rankine has used these values to calculate the mean collision area, that is the average area presented by the molecule in all possible orientations, which determines the frequency of molecular collisions the value obtained was 0-985 x 10 1S sq. cm. 

The density of pure copper is 8.93 g/cm, and the densities of the Claudian quadrantes average about 8.91 0.04 g/cm. None of the coins has any appreciable quantity of voids or internal pores. Die orientations were determined by rotating each coin about its obverse vertical axis. The reverse vertical axis was then found to point in a given direction, which was identified by the equivalent direction of the hour hand of a clock. All of the 84 Claudian quadrantes in the chemical compositional group. 

Experimentally-determined properties naturally relate to laboratory axes and the connection to the above properties, tacitly assumed to be defined in relation to molecular axes, is made by taking classical orientational averages (in quantum-mechanical terms we would be taking into account rotation) and this has been discussed in detail in Ref. [2], For example, for 7777 (commonly referred to as Y), if Z is a space fixed axis and ka is the cosine of the angle between Z and the molecular axis a, we have... 

The effects of orientation on the mechanical properties of polymers at both small and large deformations depend on the mode of orientation, which determines the preferred average chain alignment. For example, the mechanical properties obtained after uniaxial orientation (which biases the chain end-to-end vectors in one favored direction) differ from those obtained by biaxial orientation (which biases these vectors in two favored direction). Furthermore, the mechanical properties obtained after simultaneous equibiaxial orientation (where orientation in the two favored directions is imposed simultaneously, at equal rates, and to equal extents) often differ from those obtained after sequential orientation in the two favored directions, as well as after orientation by different amounts and/or at different rates in those two directions. See Seitz [35] for a review of the effects of uniaxial and biaxial orientation on the fracture of polystyrene, which fails by brittle fracture or crazing, under uniaxial tension and impact. 

Equation (3.21) implies that the determination of an optical parameter at a given time consists of three steps (1) orientation averaging for a cluster possessing a given particle number(2) statistical averaging over all... 

There are two possible sources for the discrepancies between measured and predicted elastic constants when the orientation averages used are determined from models for molecular orientation the models may be incorrect and neither Voigt nor Reuss averaging may be appropriate. In order to examine the second of these possibilities more directly it is necessary to determine the orientation averages experimentally. 

As would be determined 70 years later, orientationally averaged projections are exactly equal to ft (assuming hard-sphere interactions) for and only for contiguous bodies that lack concave surfaces and thus permit neither self-shadowing nor multiple scattering of gas molecules.For other bodies, ft always exceeds the projection. No polyatomic molecule is truly convex because of crevices between the atoms. However, the effect of such small locally concave areas on ft is only a few percent and PA is often passable for largely convex shapes. This is not true for objects... 

While photographic techniques may allow one to obtain some average orientation values for a deformed crystalline polymer there is need for a quantitative measure of this distribution. In general, the distribution in orientation is determined for a single (hkl) plane—usually a (/lOO), (OkO) or (OOO plane if sufficient diffraction exists. The data are then presented either in a pole figure or may be used to determine the Herman s orientation function defined as... 

Anisotropic g and Hyperfine interaction. The hyperfine tensor A for each nucleus is a real 3x3 matrix that can always be diagonalized. The components of the diagonalized hyperfine tensor consist of an isotropic part, Oo, and a purely anisotropic part, a, whose orientational average is zero. Thus, the a components are averaged out in fluid media and can only be determined in the solid state or in the case of highly restricted molecular motion. The diagonal elements... 

One source of such constraints could be the orientationally averaged cross section obtained from imi mobility. At the end of Sect. 3.2.2 we discussed the use of drift tube ion mobdity as a conformational filter for cryogenic ion spectroscopy but didn t mention the information content obtained from IMS. The cross section determined from the drift time in such an experiment could, in principle, be used as a filter in the conformatimial search procedure by comparing the cross sections of candidate structures to the measured value. While this would require a fast calculation of the cross section, for use as a conformational filter this could be done with reduced accuracy. 

Another fundamental class of molecular transformations features the complexation (association or dissociation) of two species. The potential of mean force needs to be determined as a function of the inter-species separation in this case. For two spherical atoms or ions a one-dimensional reaction coordinate is appropriate and W(r ) will reveal important information, e.g. on the existence of contact and solvent-separated ion pairs and on the intervening barriers. The treatment of non-spherical substrates can either involve some geometrical restrictions on the approach path or include appropriate orientational averaging with r as a characteristic distance such as the centers-of-mass separation. Since it will generally be desirable to cover substantial ranges of r, importance sampling or statistical perturbation theory can be used to obtain W(r ). 

The above relation shows that the orientation-averaged extinction cross-section for macroscopically isotropic and mirror-symmetric media is determined by the diagonal elements of the transition matrix in the particle coordinate system. The same result can be established if we consider an ensemble of randomly oriented particles (with 0 and 0) illuminated by a linearly polarized plane wave (with real polarization vector Cpoi). 

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