Basis of a lattice

On the basis of a lattice model, upper and lower bounds have been established for the entropy of dispersion of spherical globules of radius r and volume fraction in the continuous phase (ref. 20). Here r and are the "actual" radius of the globules (including the adsorbed layer of surfactants) and the corresponding... 

A/ is calculated in what follows by neglecting the interactions among globules. Expressions for the entropy of dispersion of the globules in the continuous phase were derived by Ruckenstein and Chi [12] on the basis of a lattice model, assuming (as is usually done in this kind of model) that the volume of a site is equal to the volume of a molecule... 

Most of the theories were based on the self-consistent mean field approximation.12-32 The other ones include the scaling analysis, - molecular dynamics simulations and Monte Carlo simulations.40,41 The self-consistent mean field theories12-32 were developed along the following three lines (1) on the basis of a lattice model,12-22 (2) on the basis of a diffusion type equation,23-28 and (3) analytical approaches.29-32... 

Many theories were developed to describe the distribution of segments between two polymer brushes. On the basis of a lattice... 

On the basis of a lattice model, the following expressions for LC were derived ... 

It is also difficult to establish a relation for C. As for Af, expressions have already been derived, either on the basis of a lattice model (2 6), or on the basis of the CarnahairStarling approximation for hard spheres (27). These expressions could be introduced in Equa-tions (8) and (16) to relate r and toY and C, or perhaps even more meaningfully, to relate Y and C to r and. ... 

The vectors b, b2, and 63 form the basis of a lattice in a 3-D space. This lattice will be called the inverse lattice. The inverse lattice vectors are, therefore. 

In the present review a description is given of the phase behavior of clathrates on the basis of a solution theory. The treatment is restricted to those cases where the empty host lattice ( solvent") is indeed unstable, although many of the present considerations also apply to the few cases known where the host lattice is stable. An example of the latter is the chroman complex first discovered by Dianin9 and recently examined by Baker and McOmie and Powell and Wett ers.34... 

Reciprocal spacings (1 jdm — qm = (2 sin 6)1 X) were calculated from the positions of the powder lines, and the lines were indexed on the basis of a face-centered cubic lattice. Analysis of a few lines in the back-reflection region gave a preliminary value of 0-081409 A-1 for 1 ja0. A refinement of l/a0 was carried out by... 

The T) and T2 dependence is described by Eqs. (3.4.3) and (3.4.4) [34] where Qi and q2 are spin-lattice and spin-spin surface relaxivity constants, and S/ Vis the surface-to-volume ratio of the pore. These equations provide the basis of a methodology for crack detection in cement paste specimens [13]. 

Results of the ideal solution approach were found to be identical with those arrived at on the basis of a simple quasichemical method. Each defect and the various species occupying normal lattice positions may be considered as a separate species to which is assigned a chemical potential , p, and at equilibrium these are related through a set of stoichiometric equations corresponding to the chemical reactions which form the defects. For example, for Frenkel disorder the equation will be... 

Therefore, uniaxially oriented samples should be prepared for this purpose, which give so-called fiber pattern in X-ray diffraction. The diffraction intensities from the PPX specimen of P-form, which had been elongated 6 times at 285°C, were measured by an ordinary photographic method. The reflections were indexed on the basis of the lattice constants a=ft=2.052nm, c(chain axis)=0.655nm, a=P=90°, and y=120°. Inseparable reflections were used in the lump in the computation by the least square method. 

There are, obviously, no compounds to illustrate lattice-induced strains with GII 3> 0.2 vu. Such structures are unstable and cannot exist, but if it is possible to model structures of any arbitrary composition using the methods described in Chapter 11, it is possible to determine which compositions give rise to stable structures and which ones do not. A systematic exploration of different compositions occurring between a group of elements would then lead to an understanding of the phase diagram. For example, on the basis of a few simple rules, Skowron and Brown (1994) were able to predict most of the structures in the Pb-Sb-S phase diagram and their relative stabilities (Section 11.2.2.2). 

Chemical diffusion has been treated phenomenologically in this section. Later, we shall discuss how chemical diffusion coefficients are related to the atomic mobilities of crystal components. However, by introducing the crystal lattice, we already abandon the strict thermodynamic basis of a formal treatment. This can be seen as follows. In the interdiffusion zone of a binary (A, B) crystal having a single sublattice, chemical diffusion proceeds via vacancies, V. The local site conservation condition requires that /a+/b+7v = 0- From the definition of the fluxes in the lattice (L), we have... 

On the basis of thermodynamic considerations, some of the lattice sites in the crystal are vacant, and the number of vacant lattice sites generally is a function of temperature. The movement of a lattice atom into an adjacent vacant site is called vacancy diffusion. In addition to occupying lattice sites, atoms can reside in interstitial sites, the spaces between the lattice sites. These interstitial atoms can readily move to adjacent interstitial sites without displacing the lattice atoms. This process is called interstitial diffusion. The interstitial atoms may be impurity atoms or atoms of the host lattice, but in either case, interstitial atoms are generally present only in very dilute amounts. However, these atoms can be highly mobile, and in certain cases, interstitial diffusion is the dominant diffusion mechanism. 

Waugh et al.131 discussed the selective oxidation of benzene to maleic anhydride on the basis of a detailed study of maleic anhydride and benzene adsorption on a V-Mo oxide catalyst supported on a-Al203. Hydroquinone is found to be an intermediate in this reaction and p-benzoquinone, formed from the hydroquinone, is the main intermediate in the non-selective pathway. The maleic anhydride is observed to be immobile adsorbed and the surface oxidation reaction has a relatively low activation energy. From this the authors conclude that it is not lattice oxygen but weakly bound molecular 02 which is responsible for the selective oxidation and a detailed mechanism, in which use is made of orbital symmetry arguments, is presented. 

Chapter 8 provides a unified view of the different kinetic problems in condensed phases on the basis of the lattice-gas model. This approach extends the famous Eyring s theory of absolute reaction rates to a wide range of elementary stages including adsorption, desorption, catalytic reactions, diffusion, surface and bulk reconstruction, etc., taking into consideration the non-ideal behavior of the medium. The Master equation is used to generate the kinetic equations for local concentrations and pair correlation functions. The many-particle problem and closing procedure for kinetic equations are discussed. Application to various surface and gas-solid interface processes is also considered. 

Fig. 11. Crystal structure of graphite. The unit cell is shaded in green, (a) Top view of the surface layer. The hexagonal surface lattice is defined by two unit vectors u and v in the xy-plane with a length of 246 pm and an angle of 120° forming a honeycomb web of hexagonal rings. The basis of the lattice consists of two carbon atoms a, (white) and /3 (red) with a distance of 142 pm. (b) Perspective view, showing the layered structure. The distance between layers is 2.36 times the next-neighbor distance of atoms within one layer, and the bond between layers is weak. The a-atoms (white) are directly above an a-atom in the layer directly underneath at a distance of 334.8 pm the /3-atoms (red) are over hollow sites (h). The unit vector w is parallel to the z-axis with a length of 669.6pm.

Since a lattice is just a set of points, we will need another entity to describe the real crystal. That is, it is required to locate a set of atoms named basis in the vicinity of the lattice sites. Therefore, a crystal will be a combination of a lattice and a basis of atoms. In Figure 1.3, a representation of the operation... 

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