Scattering ellipsoids, from

We have calculated the scattering from an object of known shape like a sphere for which measurements of the angular dependence of intensity permits a determination of its radius. Similar calculations are possible for other shapes like rods or ellipsoids leading to measures of their size. For a polymer coil, the shape and size is not fixed, so a statistical description is necessary. In addition, chain branching will affect the shape and size of the polymer coil. 

Fujime, S., Kubota, K., Dynamic Light Scattering from Dilute Suspensions of Thin Disks and Thin Rods as Limiting Forms of Cylinders, Ellipsoid, and Ellipsoidal Shell of Revolution,Piop/iys. Chem., 1985,23, 1-13. 

Beck and coworkers have measured energy loss spectra for K scattering from CO and N2, for scattering angles greater than 90 and energies 0.34-1.24 eV and have also observed an isotope effect in the structure. They have successfully modeled their results with classical scattering from a hard ellipsoid. ... 

The early Hartley model [2, 3] of a spherical micellar stmcture resulted, in later years, in some considerable debate. The self-consistency (inconsistency) of spherical symmetry witli molecular packing constraints was subsequently noted [4, 5 and 6]. There is now no serious question of tlie tenet tliat unswollen micelles may readily deviate from spherical geometry, and ellipsoidal geometries are now commonly reported. Many micelles are essentially spherical, however, as deduced from many light and neutron scattering studies. Even ellipsoidal objects will appear... 

In an attempt to separate the domains from the cores, we used limited degradation with several proteases. CBH I (65 kda) and CBH II (58 kda) under native conditions could only be cleaved successfully with papain (15). The cores (56 and 45 kda) and terminal peptides (11 and 13 kda) were isolated by affinity chromatography (15,16) and the scission points were determined unequivocally. The effect on the activity of these enzymes was quite remarkable (Fig. 7). The cores remained perfectly active towards soluble substrates such as those described above. They exhibited, however, a considerably decreased activity towards native (microcrystalline) cellulose. These effects could be attributed to the loss of the terminal peptides, which were recognized as binding domains, whose role is to raise the relative concentration of the intact enzymes on the cellulose surface. This aspect is discussed further below. The tertiary structures of the intact CBH I and its core in solution were examined by small angle X-ray scattering (SAXS) analysis (17,18). The molecular parameters derived for the core (Rj = 2.09 mm, Dmax = 6.5 nm) and for the intact CBH I (R = 4.27 nm, Dmax = 18 nm) indicated very different shapes for both enzymes. Models constructed on the basis of these SAXS measurements showed a tadpole structure for the intact enzyme and an isotropic ellipsoid for the core (Fig. 8). The extended, flexible tail part of the tadpole should thus be identified with the C-terminal peptide of CBH I. 

We do not restrict ourselves to material particles ellipsoidal voids in an otherwise homogeneous medium scatter light in no way significantly different. from that of real particles. Therefore, we shall avoid using the term... 

Noctilucent cloud particles are now generally believed to be ice, although more by default—no serious competitor is still in the running—than because of direct evidence. The degree of linear polarization of visible light scattered by Rayleigh ellipsoids of ice is nearly independent of shape. This follows from (5.52) and (5.54) if the refractive index is 1.305, then P(90°) is 1.0 for spheres, 0.97 for prolate spheroids, and 0.94 for oblate spheroids. 

As has been pointed out (63), this is a rather artificial model and, moreover, its application is quite unnecessary. In fact, (a> can be calculated from the refractive index increment (dnjdc), as has extensively been done in the field of light scattering. This procedure is applicable also to the form birefringence effect of coil molecules, as the mean excess polarizability of a coil molecule as a whole is not influenced by the form effect. It is still built up additively of the mean excess polarizabilities of the random links. This reasoning is justified by the low density of links within a coil. In fact, if the coil is replaced by an equivalent ellipsoid consisting of an isotropic material of a refractive index not very much different from that of the solvent, its mean excess polarizability is equal to that of a sphere of equal volume [cf. also Bullough (145)]. 

Fig. 15. (a) Intramolecular hydrogen bonds in urea crystal with displacement ellipsoids at 50% probability, (b) Static deformation density obtained from the multipolar analysis of the experimental data corrected for the thermal diffuse scattering. Theoretical deformation density obtained using (c) the Hartree-Fock method (d) the DFT method by generalized gradient approximation (contours at 0.0675 eA-3) (reproduced with permission from Zavodnik et al. [69]). 

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