Particle scattering spheres

Figure A3.11.4. Trajectory associated with a particle scattering ofT a hard sphere potential.

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. 

A special anisotropic particle scattering problem has been treated by Roth and Dignam (1973), who considered an isotropic sphere coated with a uniform film with constitutive relations... 

There are both similarities and differences between scattering by spherical and by nonspherical particles. Near the forward direction, where scattering may be associated primarily with diffraction, external reflection, and twice-refracted transmission (Hodkinson and Greenleaves, 1963), nonspherical particles scatter similarly to area-equivalent spheres, in general. Forward scattering by large particles is dominated by diffraction, which depends on particle... 

Large nonspherical particles scatter similarly to area-equivalent spheres near the forward direction. 

Particle size and distributions can be determined by a number of different methods. The technique described here is light scattering. Different measurement methods produce different results which can be correlated experimentally. The absence of distribution standards for light scattering particle sizing instruments precludes any determination of size accuracy. This is further complicated by particles of non-spherical shape which makes the concept of size very difficult to define. However, for particulate materials encountered in most industrial processes, the assumption that particles are spheres produces quite useful results that are repeatable and relate to important parameters of many processes. 

Figure 2. Scattering from silica particles bound with strongly cationic polymers ( ), compared with that from free silica particles (+). The particles are spheres of precipitated silica with a radius of 19 nm In water at pH near 7 they bear 0.3 negative charge per nm of surface, most of which Is compensated by adsorbed counterions (15). The polymers are AM-CH copolymers with a ratio of cationic to total monomers equal to 0.3 the total amount of polymer In the floe approximately compensates the chemical charge borne by the silica particles (9).

Particle-scattering functions for random coils and spheres are indistinguishable at values of Z near unity (Figure 2). Likewise, the radii of random coils and spheres are also similar in the range of Z values less than 1.2 (Figure 3). This is reflected in the similarity of radii calculated for random coils and spheres (columns 7 and 8 of Table II). The largest difference between radii calculated for these two shapes is 8.5% for... 

Figure 2. Reciprocal of particle-scattering functions for random coils (-------) and spheres (------).

In Fig. 4.16 the particle scattering function P(0) is plotted for random coils, spheres, and rods as a function of respective parameters. 

Figure 5.1 Single particle scattering intensity I (q) for a solid sphere of radius R.

Prior to the 1970s, methods such as light scattering and surface area measurements were used to obtain average particle size of colloidal silica. Both of these methods assume that the particles are spheres. Light scattering... 

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). 

Apart from the solid regular particles dealt with in 2,3 and 4 above, other simple particle scattering laws include those for hollow spheres, ellipsoids and discs. These have been discussed and summarised in other publicationsand reference to these should be made for further details. 

What could be the cause of such a large difference in thermodynamic stability After all, the number of Ni -N coordinate-covalent bonds is six in both the products of these two reactions, so the enthalpy changes (Ai/) involved when these bonds are formed should be fairly similar. That seems to leave entropy as the major explanation for the effect. Indeed, the rationale for the chelate effect can be understood in two ways, both related to the relative probabilities that the two reactions will occur. First, consider the number of reactants and products in the two cases. As written more explicitly in Equations (6.11) and (6.12), it is apparent that the number of ions and molecules scattered throughout the water structure in the first reaction stays the same (seven in both the reactants and the products). In the second reaction, however, three ethylenediamine molecules replace six water molecules in the coordination sphere, and the number of particles scattered at random throughout the aqueous solution increases from four to seven. The larger number of particles distributed randomly in the solution represents a state of higher probability or higher entropy for the products of the second reaction. Therefore, the second reaction is favored over the first due to this entropy effect. 

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