Random orientation correlation

In many cases, however, azimuthally dependent patterns are obtained as, for example in annealed polytetrafluoroethylene film. Such scattering may be treated in terms of the previously described non-random orientation correlation theories. An alternate approximate approach is that proposed by Keijzers et in which a scattering pattern... 

The extension of the random orientation correlation theory to the description of the scattering from oriented polyethylene films was described by Stein and Hotta. This application is not strictly correct since it was made prior to the development of non-random orientation correlation theory and it is now realised that random theory is not adequate for the description of such films. In fact, it is now felt that random correlations in oriented systems are quite unlikely and that the degree of non-randomness increases with orientation. While the non-random correlation theory has been generalised to permit the description of oriented systems, it is still rather complex so that it has not been actually applied. In principle eqn. (118) could be used to describe oriented systems, by using the van Aartsen-Stein expression for / and the Stein-Hotta expression for /r in the oriented state. It is likely that 4>s and may also vary with orientation. So far, this has not been done. 

This was done by Stein and Wilson (2) and is called the random orientation correlation approach. The essential assumption of these authors is that all correlations are functions of the distance only and do not depend upon the direction of the vector j. Cross correlations are assumed to be absent. This model was used successfully in a number of cases (4), (5), but proved to be too simple. The assumption mentioned above implies that the scattering will be cylinder-symmetrical around the incident beam and so will depend on 9 only. Photographs of the scattering pattern have shown, however, that there are a number of cases where this evidently does... 

A first attempt to formulate such a theory was made by Stein et al (3), but this was a two-dimensional approach only. Moreover the resulting equations are rather cumbersome. A different approach was made by Keyzers et al (11), who interpreted their measurements on semi-crystalline polypropylene and polystyrene on the basis of a linear combination of the random orientation correlation approach and the spherulite model of Stein and Rhodes (6). Although this worked rather well in practice, the combination of two fundamentally different kinds of theory for one sample is not entirely satisfactory. The present author has therefore developed a new theory which will be published separately (12). 

It can be shown that for y=z=o the equation given (eq. 9) reduces to the equations for the random orientation correlation approach. So the more general theory presented here does indeed contain the Stein-Wilson theory (2) as a limiting case and at the same time shows that two other functions of the scattering angle, y(h) and z(h) respectively, are involved. 

A comparison between equations 18 and 15 reveals two differences. In the first place the term containing the density correlation function is replaced by a term cT Dq, which means that the model calculations automatically lead to the conclusion that in this case the correlation function for random orientation correlations and that for random density fluctuation correlations are identical. This seems to be a general result. The second difference is the occurrence of cross correlation terms in the equations for B and D. In the case of model calculations these extra terms are connected with the other functions of the system in the way given above. In the general case the cross correlation terms occur only in the B and D value too, but then there need not be a connection with the other functions of the system. 

The interpretation of scattering data can be done in terms of correlation functions. In these Proceedings J. J. van Aartsen discusses a very general theoretical approach to the determination of such correlation functions (10). If the correlation between volume elements with polarizabilities and parallel and perpendicular to their optic axes depends only on the scalar distance of separation between the elements (.random orientation correlation) there is no information in the R and R envelopes that is not already contained in the so called and components. These... 

In the limit of zero scattering angle the rod model and the random orientation correlation approach can formally be connected. At small angles can be expanded as a series in h (18,17, ), yielding... 

This chapter considers the distribution of spin Hamiltonian parameters and their relation to conformational distribution of biomolecular structure. Distribution of a g-value or g-strain leads to an inhomogeneous broadening of the resonance line. Just like the g-value, also the linewidth, W, in general, turns out to be anisotropic, and this has important consequences for powder patterns, that is, for the shape of EPR spectra from randomly oriented molecules. A statistical theory of g-strain is developed, and it is subsequently found that a special case of this theory (the case of full correlation between strain parameters) turns out to properly describe broadening in bioEPR. The possible cause and nature of strain in paramagnetic proteins is discussed. 

For the initial time t = 0, the above formula (A2.7) is identical with the result of averaging over random orientations in a surface plane. In the course of time, the "memory" of the initial orientation fades, the condition t w 1 (w 1 is the average period between reorientations) permitting an independent averaging over ea(t) and e (0), and the correlation function (A2.7) tends to zero. 

Fig. 7e. Distinct neutron structure functions, H (s), for amorphous solid (...) and for liquid D2O. Calculated curves are for randomly oriented water molecules with molecular center correlations derived from X-ray diffraction. (From Ref. 27>)...

The function So (s) describes the neutron scattering from randomly oriented D2O molecules with the positional correlation between molecular centers specified by the correlation function hoo R). We expect the orientational correlation between pairs of water molecules to be of much shorter range than the positional correlation between molecular centers, and hence the functions 3o (s) and 3 (s) should be nearly equal for values of s <2 A-1. 

For randomly oriented radiators, the substitution, Eq. 2.84, will in general be possible so that the correlation function may be written as... 

The orientation of Cu3Si relative to the Si(100) face did correlate to rate and selectivity a random orientation gave the best results. One study found that the reaction was inhibited by a Si02 layer65 and a second study found no such inhibition66. SEM analysis of several surfaces after reaction with MeCl showed square reaction pits. 

Effect of Multiple Layers and Packing All correlations for the collection efficiency discussed so far are based on the ideal case of a single cylindrical collector. Now, let s examine a filter unit consisting of randomly oriented multiple layers. Consider an area (A) of filter at a right angle to the gas flow and with a depth dh. If the packing density a is defined as the volume of fiber per unit volume of filter bed, the velocity within the filter void space is equal to... 

The LDr correlates with the orientation of the transition moment of the dye relative to the reference axis, as quantified by the angle a. LDr is also proportional to an orientation factor S (S = 1 denotes perfect alignment of the dye, S = 0 random orientation). For an isolated, non-overlapping transition, Eq. (7) establishes the correlation between LDr, a and S. These definitions lead to the qualitative rule that with an angle a > 55°, a negative LD signal is observed, whereas with a < 55°, a positive signal appears in the spectrum. Thus, with an appropriate set-up the orientation of a chromophore relative to a reference axis can be determined. 

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