Particle scattering cylinders

Another proj rty of powders that could affect the results obtained with diffraction based instruments is the shape of the particles. As many particles are approximately spherical or at least circular in projection this might not be a big problem. Ne le shaped particles or cylinders pose a completely different problem. Swithenbank et al have considered the effect of cylinders. Obviously, a cylinder, like a slit has a linear not a circular diffraction pattern. However, for randomly orientated cylinders an equivalent circular diffraction pattern was produced which gave a dimension 12% smaller than the cylinder diameter. The scattered light is insensitive to the cylinder length (assuming L/D>3). 

Synopsis of Experiment and Results. The material is irradiated during straining and relaxation. The example shows that a nanostructure which is hard to interpret from a series of scattering patterns may clearly reveal its complex domain structure after transformation to the CDF. Different structural entities are identified which respond each in a different way on mechanical load. The shape of the basic particles is identified (cylinders). The arrangement of the cylinders is determined. Thus the semi-quantitative analysis of the CDF provides the information necessary for the selection and definition of a suitable complex model which is required for a... 

For an ensemble of uncorrelated ID particles (cylinders, layers) with a Gaussian96 particle thickness distribution the ID scattering intensity is [197]... 

Recently, a low-resolution model of the chromatin core particle has been derived from a combination of single-crystal X-ray diffraction and electron microscopic data (Finch et al., 1977). The particle is described as a flat cylinder 110 A in diameter and 57 A in height. A similar shape and similar dimensions were found to be consistent with the low-angle neutron scattering from core particles in solution (Pardon et al., 1977 Suau et al., 1977). Some conclusions may be drawn concerning the conformation of the DNA. Presumably, the strong 28 A periodicity apparent in the crystal data (Finch et al., 1977) corresponds to the pitch of the DNA superhelix wound about the histone core. X-Ray and spectroscopic data suggest that the DNA super-... 

In this chapter we consider theories of scattering by particles that are either inhomogeneous, anisotropic, or nonspherical. No attempt will be made to be comprehensive our choice of examples is guided solely by personal taste. First we consider a special example of inhomogeneity, a layered sphere. Then we briefly discuss anisotropic spheres, including an exactly soluble problem. Isotropic optically active particles, ones with mirror asymmetry, are then considered. Cylindrical particles are not uncommon in nature—spider webs, viruses, various fibers—and we therefore devote considerable space to scattering by a right circular cylinder. 

There are many naturally occurring particles, such as some viruses and asbestos fibers, which are best represented as cylinders long compared with their diameter. Therefore, in this section we shall construct the exact solution to the problem of absorption and scattering by an infinitely long right circular cylinder and examine some of the properties of this solution. 

In Chapter 3 we derived a general expression for the amplitude scattering matrix for an arbitrary particle. An unstated assumption underlying that derivation is that the particle is confined within a bounded region, a condition that is not satisfied by an infinite cylinder. Nevertheless, we can express the field scattered by such a cylinder in a concise form by resolving the incident and scattered fields into components parallel and perpendicular to planes determined by the cylinder axis (ez) and the appropriate wave normals (see Fig. 8.3). That is, we write the incident field... 

Figure 8.7 Scattering cross section per unit particle volume for normally incident light polarized parallel (---) and perpendicular (...) to the axis of an infinite cylinder in air.

Scattering problems in which the particle is composed of an anisotropic material are generally intractable. One of the few exceptions to this generalization is a normally illuminated cylinder composed of a uniaxial material, where the cylinder axis coincides with the optic axis. That is, if the constitutive relation connecting D and E is... 

The classical method of solving scattering problems, separation of variables, has been applied previously in this book to a homogeneous sphere, a coated sphere (a simple example of an inhomogeneous particle), and an infinite right circular cylinder. It is applicable to particles with boundaries coinciding with coordinate surfaces of coordinate systems in which the wave equation is separable. By this method Asano and Yamamoto (1975) obtained an exact solution to the problem of scattering by an arbitrary spheroid (prolate or oblate) and numerical results have been obtained for spheroids of various shape, orientation, and refractive index (Asano, 1979 Asano and Sato, 1980). 

An infinite right circular cylinder is another particle shape for which the scattering problem is exactly soluble (Section 8.4), although it might be thought that such cylinders are so unphysical as to be totally irrelevant to real... 

Figure 2 shows the mean cascading velocity versus distance down the granular cascade for experiments run at the same tangential velocity. Despite a nearly fourfold difference in diameter, the velocity data all fall on nearly the same curve over the first 3 cm down the flowing layer. This agreement indicates that initial particle accelerations may be nearly equivalent, regardless of vessel size. Scatter in the experimental data shown in Figure 2 precludes direct calculation of accelerations, so least-square polynomials were fit to the experimental data. By differentiating the polynomial fit, we obtain an estimate of the downstream acceleration, shown in Figure 3. Over the initial upper third [0 to ( )F of the flowing layer, the acceleration profiles for all cylinders are nearly identical, with only mi-...

Fig. 8 Pair distribution functions of complexes of a cylindrical symmetry (57% styryl-methyl(trimethyl)ammonium, 16% methacrylic acid, 27% methyl methacrylate) and b disklike symmetry (79% styrylmethyl(trimethyl)ammonium, 13% methacrylic acid, 8% methyl methacrylate). The curves which were calculated from the scattering data are represented by triangles and squares. Solid lines represent the distribution functions of a an idealized cylinder with a diameter of 3.0 nm and of b a disk with a height of 2.2 nm. The insets depict idealized symmetries of the particles. (Adapted from Ref. [31])...

---