Anisotropic scatterers

The rotational phase shift 5, which cannot exceed a mean angle of a molecular rotation during collisional time (anc), is certainly small in the case of non-adiabatic collisions. This condition is exactly that needed for anisotropic scattering (or IR absorption) spectrum narrowing, just as vibrational dephasing must be weak for an isotropic spectrum to narrow. 

The Hubbard straight line corresponds to rotational diffusion, and quasistatic straight lines (6.65) to quasi-free rotation. One type of motion substitutes for the other in the vicinity of the minimum point of curve xe,2 (tj) and is accompanied by collapse of the anisotropic scattering spectrum. 

Temkin S. I., Burshtein A. I. On the shape of the Q-branch of Raman scattering spectra in dense media. Anisotropic scattering, Chem. Phys. Lett. 66, 62-4 (1979). 

Buckingham, 1974 Barron, 1982). Within this model, therefore, isotropic scattering makes zero contribution to the ROA intensity, which is generated exclusively by anisotropic scattering, and the CID Eqs. (2) reduce to... 

Peaks in Isotropic and Anisotropic Scattering Patterns 8.2.2.1 Isotropy and Anisotropy... 

Both for isotropic and anisotropic scattering patterns line profile analysis can be performed. If a curve from an anisotropic pattern is analyzed, the results are limited... 

If these concepts of curve analysis shall be applied to the anisotropic scattering of polymer fibers, one should choose to study either the longitudinal or the transversal density fluctuations. According to the decision made, the fiber scattering must be projected either on the fiber axis or on the cross-sectional plane. This results in scattering curves with a one- or a two-dimensional Porod s law. Because modern radiation sources always feature a point-focus, the required plots for the separation of fluctuation and transition zone are readily established (cf. Table 8.3). 

Frequent malpractice is the application of the azimuthal averaging algorithm to anisotropic scattering data for the purpose of isotropization. The result appears isotropic, but the chosen integration is incorrect. Only in the case of low anisotropy this procedure is permitted, because then the introduced error is kept small. 

We notice that anisotropic scattering patterns can be calibrated to absolute intensity, as well. 

Equation (8.59) defines the ID interference function of a layer stack material. G (s) is one-dimensional, because p has been chosen in such a way that it extinguishes the decay of the Porod law. Its application is restricted to a layer system, because misorientation has been extinguished by Lorentz correction. If the intensity were isotropic but the scattering entities were no layer stacks, one would first project the isotropic intensity on a line and then proceed with a Porod analysis based on p = 2. For the computation of multidimensional anisotropic interference functions one would choose p = 2 in any case, and misorientation would be kept in the state as it is found. If one did not intend to keep the state of misorientation, one would first desmear the anisotropic scattering data from the orientation distribution of the scattering entities (Sect. 9.7). 

For anisotropic scattering patterns and the multidimensional case Vonk ( [168] and [22], p. 302) has proposed to utilize a multidimensional correlation function. It is not frequently applied. 

Tchoubar. For the application to anisotropic scattering patterns Stribeck [26] has extended this principle to a space of deliberate dimensionality. Available technology constricts its practical use to the scattering of materials with fiber symmetry, and the fiber-symmetrical CDF... 

Whenever we are considering orientation, we are dealing with anisotropic scattering data. Orientation is most frequently analyzed in 2D scattering data with fiber symmetry or in pole-figure data recorded by means of a texture goniometer. 

Rigid Structural Entities. If the initial structure is described by rigid, anisotropic structural entities which are oriented at random, the evolution of anisotropic scattering is readily studied by means of the methods presented in Chap. 9. A practical example is the study of growing orientation in fiber-reinforced materials. 

Transferred to the observation that the reflections in moderately anisotropic scattering images are found on ellipses3, it appears reasonable to parameterize such images in elliptical coordinates ( , v). The transformation relations are [266]... 

These equations are approximately valid also for anisotropic scattering when a is substituted according to Eq. (8.9) by aet( = (1 -g)a. The validity, however, decreases with increasing absorptivity of the sample. 

It should be mentioned that Eqs. (IV-35) and (IV-36) neglect the anisotropic scattering of the coiling chains in the network. It is, however, very reasonable to suppose that these chain contributions are minor. They certainly will not cause an angular dependence in Hv, because they are not correlated over large distances. 

Fig. 33. Experimental curves of the angular dependence of anisotropic light-scattering of some chemically crosslinked polyurethane rubbers. The first number in the code refers to the average number of oxypropylene units between crosslinks and the second number to the average number of urethane links per chain. More such links seem to yield more anisotropic scattering and thus more "bundle like"...

These equations use Cartesian tensor notation in which a repeated Greek suffix denotes summation over the three components, and where ay7 is the third-rank antisymmetric unit tensor. For a molecule composed entirely of idealized axially symmetric bonds, for which [3 (G )2 = /3(A)2 and aG1 = 0 [13, 15], a simple bond polarizability theory shows that ROA is generated exclusively by anisotropic scattering, and the CID expressions then reduce to [13]... 

Pulverized-Coal-Fired Furnaces Application of the Monte Carlo Method to Anisotropic Scatter. Int. J. Heat Mass Transfer, 26, 1649. 

Trivic, D.N., O Brien, T.J., and Amon, C.H. Modeling the radiation of anisotropically scattering media by coupling Mie theory with Finite volume method. International Journal of Heat and Mass Transfer, 2004. 47, 5765-5780. 

To see why this is the case, we first consider the portion of the response that arises from llsm. According to Equation (10), we can express (nsm(t) nsm(0)> in terms of derivatives of llsm with respect to the molecular coordinates. Since in the absence of intermolecular interactions the polarizability tensor of an individual molecule is translationally invariant, FIsm is sensitive only to orientational motions. Since the trace is a linear function of the elements of n, the trace of the derivative of a tensor is equal to the derivative of the trace of a tensor. Note, however, that the trace of a tensor is rotationally invariant. Thus, the trace of any derivative of with respect to an orientational coordinate must be zero. As a result, nsm cannot contribute to isotropic scattering, either on its own or in combination with flDID. On the other hand, although the anisotropy is also rotationally invariant, it is not a linear function of the elements of 11. The anisotropy of the derivative of a tensor therefore need not be zero, and nsm can contribute to anisotropic scattering. 

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