Scattering between free particles

Because of the existence of two-particle bound states the three-particle scattering states split into channels that means, we have to take into account scattering processes between free particles and bound pairs. To study the effect of composite particle scattering processes, we pay atten-... 

Thus, a susceptibility that depends on frequency and wave vector implies that the relation between P(x, t) and E(x, t) is nonlocal in time and space. Such spatially dispersive media lie outside our considerations. However, spatial dispersion can be important when the wavelength is comparable to some characteristic length in the medium (e.g., mean free path), and it is well at least to be aware of its existence it can have an effect on absorption and scattering by small particles (Yildiz, 1963 Foley and Pattanayak, 1974 Ruppin, 1975, 1981). 

The radial distance distribution in simple atomic and molecular fluids is determined essentially by the exclusion volume of the particles. Zemike and Prins [12] have used this fact to construct a one-dimensional fluid model and calculated its radial distance correlation function and its scattering function. The only interaction between the particles is given by their exclusion volume (which is, of course, an exclusion length in the one-dimensional case) making the particles impenetrable. The statistical properties of these one-dimensional fluids are completely determined by their free volume fraction which facilitates the configurational fluctuations. 

In order to construct a collision integral for a bound-state kinetic equation (kinetic equation for atoms, consisting of elementary particles), which accounts for the scattering between atoms and between atoms and free particles, it is necessary to determine the three-particle density operator in four-particle approximation. Four-particle collision approximation means that in the formal solution, for example, (1.30), for F 234 the integral term is neglected. Then we obtain the expression... 

Again using Eqs. (4.65) and (4.67) we may write the collision integral for the first process of (4.80), corresponding to an inelastic scattering between an atom and free particles. We then have... 

Simulated extinction spectra for Ag nanoparticles embedded in a polymer matrix to compare with experimental data shown in Figure 8.4. In theoretical calculations, we used the complex value of the optical constant CAg in the visible range [48] that was obtained by measurements on a set of fine silver particles. Such an approach [48] takes into account limitations imposed on the electron free path in particles of different size and electron scattering at the particle-insulator interface [49] and thus yields a more exact value of eAg than does the procedure of correcting optical constants for bulk silver [50], The complex value of Epmma for the polymer matrix was found elsewhere [42]. The extinction was calculated for particles of size between 1 and 10 nm (according to the MNP sizes in Figure 8.2). 

At the intermediate wavelengths no useful analytic forms of solution are known. On the other hand, (138) yields readily to numerical solutions. Such results show a smooth interpolation between characteristic hydrodynamic and free-particle behavior. Notice, however, that in this region it is essential to treat the collisions and molecular flow on equal footing for this reason it would be inappropriate to apply either (154) or (155). Since the intermediate k range is particularly relevant to neutron inelastic scattering studies of liquids and related computer molecular dynamics simulations, the validity of our kinetic model solutions is of interest. 

Comparing this with the research history of polyacetylene, both cases are analogously situated since both polyacetylene and polythiophene were intractable, their chemical and stmctural characterization was quite difficult in the conventional powder form [6,9], However, the polymer films of high quality prepared via either the Shirakawa method [7] or the electrochemical routes [10-12] have enabled easier characterization and the systematic research of polyacetylene and polythiophene. That is, the materials both in a thick free-standing form and in a thin on-substrate form are free from uneven particles which cause irregular scattering of light and yield contact resistance between those particles. This allows us to record sharply resolved spectra and to measure the intrinsic conductivity of the materials. 

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