Particle scattering disks

In fact, with the help of Krein s trace formula, the quantum field theory calculation is mapped onto a quantum mechanical billiard problem of a point-particle scattered off a finite number of non-overlapping spheres or disks i.e. classically hyperbolic (or even chaotic) scattering systems. 

In a dilute suspension of thin circular disks of radius R, the contributions by individual particles scattering independently can be derived in a manner similar to that for thin rods, but here we will simply present the result (see Kratky and Porod11 for the derivation) ... 

The particle scattering factor P(0, or form factor, has already been introduced in Chapter 9 (F(0) Section 9.7.1). It describes the average conformation of an individual polymer chain and model functions exist for a variety of particle shapes such as spheres, disks, or rodlike particles. For a random coil, it is expressed in terms of the Debye equation (Chapter 9, Equation 9.20). 

Fig. 3—Measurement of surface by HDI surface reflectance analyzer. In electromagnetic radiation (light), the polarization direction is defined as the direction of the electric field vector. The incident polarization of the light can be controlled. The instrument uses a variety of detectors to analyze the reflected polarization state of the light. (U.S. Patent 6,134,011). (a) Plane of the disk The SRA uses a fixed 60 degree (from the surface normal) angle of incidence. The plane of incidence is the same as the paper plane (b) Pit on a surface detected by reflected light channels of HDI instrument (c) Scratches on disk surface measured by HDI surface reflectance analyzer (d) Particles on the surface of disk detected by reflected light (black spot) and by scattered light (white spot) [8].

Transmission IR (TIR) spectroscopy if the solid in question is IR transparent over an appreciable range of wavelength. This is often used on supported metal catalysts, where the large metallic surface area permits a high concentration of adsorbed species to be sampled. The sample consist typically of 10-100 mg of catalyst, pressed into a self-supporting disk of approximately 1 cm2 and a few tenths of a mm in thickness. The support particles should be smaller than the wavelength of the IR radiation, otherwise scattering losses become important. 

Other classically chaotic scattering systems have been shown to have repellers described by a symbolic dynamics similar to (4.10). One of them is the three-disk scatterer in which a point particle undergoes elastic collisions on three hard disks located at the vertices of an equilateral triangle. In this case, the symbolic dynamics is dyadic (M = 2) after reduction according to C)V symmetry. Another example is the four-disk scatterer in which the four disks form a square. The C4 symmetry can be used to reduce the symbolic dynamics to a triadic one based on the symbols 0,1,2), which correspond to the three fundamental periodic orbits described above [14]. 

The pressed-salt method has attained wide application in studies of the infrared spectra of solids. In this method the solid sample is mixed with a powdered halide salt such as KI or KBr and the mixture is pressed into a disk at high pressures 53-55). This method reduces scattering because solid-gas interfaces are replaced by solid-salt interfaces. When this method is used,-the particle size of the solid is not of critical importance and most ordinary silica or alumina catalysts can be used without the necessity of any particle-size separation. Although it is simple experimentally, the pressed-salt method will probably never attain a major importance in catalytic work, because once the sample is embedded in the salt, it cannot be subjected to further treatment. 

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])...

Figure 2. Scattering factor of disk-like particles where K = 4it sin(0/2)/l, and L, the particle diameter. P represents the remaining higher order terms.

The output of the disk centrifuge, a curve of optical density versus time, can be converted to a particle size distribution by using Stokes law to convert the time axis to a size axis and by the application of light-scattering theory to calculate the particle frequencies from the optical densities. As described herein, one obtains mass distributions with linear size increments or the equivalent area distributions with logarithmic increments. 

Light scattered off the disk surface is perhaps the best independent tracer of particle size distributions in disks. This is because particle size and structure not only affect the scattering efficiency, but also change the phase function and polarization of the scattered light. These extra sets of observables can often be exploited to... 

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