Lattice dispersivity

Dullien (1992) and Ferrand (1992) have applied numerical particle tracking methods to compute dispersion coefficients for such networks. In the study by Ferrand (1992) the conductivity of each bond was calculated using an expression that included entrance and exit effects as fluid moves between larger sites and narrower bonds, as well as the resistance of the bond itself. The lattice dispersivity was shown to increase linearly as the geometric standard deviation of the bond-size distribution was increased. Although not widely used at present, this modeling approach offers much promise for future research on the interplay between pore shape and size distribution in determining the relationship between K and Pn. 

Another class of polarizability formulations is based not on the integral equation in Eq. (2.5) but on the notion of a set of point dipoles. Draine and Goodman [100] found an optimal 0 ((W) ) correction to the CM polarizability in the sense that an infinite lattice of point dipoles with such polarizability would lead to the same propagation of a plane-wave as in a homogeneous medium with a given refractive index. This polarizability was called the lattice dispersion relation (LDR) ... 

In this chapter, absorption and scattering efficiencies spectra will be presented for silver nanoparticles (NPs) with different shapes and dimensions. All the spectra are calculated in the discrete dipole approximation framework (see Chapter 2), with the Palik complex dielectric function e(lattice dispersion relation (LDR) prescription for the polarizability (see Sec. 2.4.3.2). For dimensions of the NPs smaller than the mean free path of the conduction electrons, the surface damping correction A sd is added to the Palik dielectric function (see Sec. 2.3), as several works have shown that the dielectric constant is strongly dependent on the size and the shape of the nanoparticle [21-23]. 

A related approach carries out lattice sums using a suitable interatomic potential, much as has been done for rare gas crystals [82]. One may also obtain the dispersion component to E by estimating the Hamaker constant A by means of the Lifshitz theory (Eq. VI-30), but again using lattice sums [83]. Thus for a FCC crystal the dispersion contributions are... 

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

Quartz also has modest but important uses in optical appHcations, primarily as prisms. Its dispersion makes it useful in monochromators for spectrophotometers in the region of 0.16—3.5 m. Specially prepared optical-quality synthetic quartz is requited because ordinary synthetic quartz is usually not of good enough quality for such uses, mainly owing to scattering and absorption at 2.6 p.m associated with hydroxide in the lattice. 

Figure 4 Schematic vector diagrams illustrating the use of coherent inelastic neutron scattering to determine phonon dispersion relationships, (a) Scattering m real space (h) a scattering triangle illustrating the momentum transfer, Q, of the neutrons in relation to the reciprocal lattice vector of the sample t and the phonon wave vector, q. Heavy dots represent Bragg reflections.

In the perfect lattice the dominant feature of the electron distribution is the formation of the covalent, directional bond between Ti atoms produced by the electrons associated with d-orbitals. The concentration of charge between adjacent A1 atoms corresponds to p and py electrons, but these electrons are spatially more dispersed than the d-electrons between titanium atoms. Significantly, there is no indication of a localized charge build-up between adjacent Ti and A1 atoms (Fu and Yoo 1990 Woodward, et al. 1991 Song, et al. 1994). The charge densities in (110) planes are shown in Fig. 7a and b for the structures relaxed using the Finnis-Sinclair type potentials and the full-potential LMTO method, respectively. 

In this scheme, digital particles are still wandering localized clusters of informa-tionl but (conventional) variables such as space, time, velocity and so on become statistical quantities. Given that no experimental measurement to date has yet detected any statistical dispersion in the velocity of light, the sites of a hypothetical discrete underlying lattice can be no further apart than about 10 cm. 

Iadda89] Ladd, A.J.C. and D.Frenkel, Dynamics of colloidal dispersions via lattice-gas models of an incompressible fluid, pages 242-245 in [mann89]. 

This type of liquid is characterized by direction independent, relatively weak dispersion forces decreasing with r-6, when r is the distance between neighbouring molecules. A simple model for this type of liquid, which accounts for many properties, was given by Luck 1 2> it is represented by a slightly blurred lattice-like structure, containing hole defects which increase with temperature and a concentration equal to the vapor concentration. Solute molecules are trapped within the holes of the liquid thus reducing their vapor pressure when the latter is negligible. 

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