Particle scattering factor equation

C, the fourth parameter, represents the relationship between the first cumulant and the particlescattering factor. For values of 1/F( ) < 10, the double logarithmic plot of the first cumulant against the reciprocal particle-scattering factor yields a straight line, and the exponent v is related to the initial slope C oiF/q D, against by the equation... 

Therefore the derived equation for the particle scattering factor simultaneously gave an equation for the radius of gyration which is... 

In the last chapter, equations were derived for the particle-scattering factor, the mean-square radius of gyration, the diffusion coefficient and the first cumulant of the dynamic structure factor. All these have the common feature that, for homopolymers at least, they can be written in the following form ... 

For long rays, one has 0 as 1,1 - tp = b2q2/6, and Tb and Pi can be neglected compared to P2. This leads to the following equation for the particle-scattering factor of regular stars... 

Equation (D.2) shows Fourier transform of the pair distance distribution W (rn) for a path with its one end at r = 0 (the root) and the other at r (n-th generation). The particle scattering factor... 

Equation (E.8) is Stockmayer s well known result7 and the derived later by Whitney and Burchard117. Combination of (E.9) allows the particle-scattering factor of the cross-linked as117 other relationships were Eqs. (E.14), (E.10) and products to be rewritten... 

The modeling of a polymerization process is usually understood as formulation of a set of mathematical equations or computer code which are able to produce information on the composition of a reacting mixture. The input parameters are reaction paths and reactivities of functional groups (or sites) at monomeric substrates. The information to be modeled may be the averages of molecular weight, mean square radius of gyration, particle scattering factor, moduli of elasticity, etc. Certain features of polymerizations can also be predicted by the models. 

A measurement of physical parameters in solution for isolated macromolecules provides a manner by which the shape of a macromolecule can be determined. The approximate dimensions and axial ratio or radius can be calculated by applying Equations (4.3) through (4.17). As shown in Figure 4.10, the particle scattering factor for collagen molecules depicted in Figure 4.9 is more sensitive to bends than is the translational diffusion coefficient. 

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). 

Thus for a polydisperse solute, A/ <9 depends upon the weight-average molar mass and the z-average particle scattering factor. The latter can be examined further by applying Equation (3.137) and leads to... 

The atomic scattering factor for electrons is somewhat more complicated. It is again a Fourier transfonn of a density of scattering matter, but, because the electron is a charged particle, it interacts with the nucleus as well as with the electron cloud. Thus p(r) in equation (B1.8.2h) is replaced by (p(r), the electrostatic potential of an electron situated at radius r from the nucleus. Under a range of conditions the electron scattering factor, y (0, can be represented in temis... 

By that procedure, an additional factor V l appears in the equation of motion of pep [Eq. (4.29)]. This factor leads to the fact that the four-particle processes accounted for in this manner are not real and may vanish in the thermodynamic limit. At least this is true for four-particle scattering states. However, in the limiting case that we have only two-particle bound states, that is, the neutral gas, we can obtain a kinetic equation for the atoms if we use the special definition of the distribution function of the atoms (4.17) and (4.24). Using the ideas just outlined, the kinetic equation (4.62) was obtained. 

The implication of equation (7) as it relates to PCS, is that larger particles scatter considerably more light per particle than smaller particles. Thus the intensity distributions measured by PCS heavily emphasize the presence of larger particles. If the intensity distributions measured with PCS were extremely accurate, the fact that PCS measures intensity rather than volume or number distributions would be of little consequence, provided that the proper conversion factor,... 

In this relation, N is the number density of the scattering microemulsion droplets and S(q) is the static structure factor. Equation (2.12) is only strictly valid for the case of monodisperse spheres. However, for the case of low polydispersities the occurring error is small [63, 64]. S(q) describes the interactions between and the spatial correlations of the droplets. These are in general well approximated by hard sphere interactions in microemulsion systems [65], The influence of inter-particle interactions as described by S(q) canbe estimated at least for S(0) using the Carnahan-Starling expression [52,64,66]... 

To obtain the absolute sound attenuation in the coal slurry, the diffraction loss, the acoustic mismatch loss, the attenuation due to the Teflon window, and the oil coupling must be calculated. Thus, it is difficult to accurately determine the absolute attenuation. In practice, one measures the relative attenuation with respect to a standard. The attenuation of ultrasonic waves in a solid suspension is attributed to three major factors, namely, scattering, viscosity, and thermal effects. Although the presence of particles affects the fluid viscosity and thermal conductivity, the primary source of attenuation may be due to particle scattering. Hence, one may define the relative attenuation of the HYGAS coal slurry by comparing the slurry attenuation with that of the carrier fluid, i.e., the toluene/benzene mixture. This can be expressed by the equation... 

In this equation, the parameter 5agg( ) is the static structure factor which was introduced in Sect. 4.2.2.4. By assuming that the particle scattering obeys the RDG approximation, the scattering intensity can be written as ... 

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