Lattice analogy

Fig. 9. Diagrammatic representation of a bulky host constitution (A) and (a)—(c) of crystal lattice-analogous arrangements of A (two-dimensional versions shaded areas represent the lattice voids)...

Many compounds will crystallise out with hydrogen peroxide in the crystal lattice, analogous to crystalline hydrates. A few (some of which are listed below) sequester... 

If the adsorption results in an organized deposit layer and a certain arrangement similar to the crystalline lattice we speak of epitaxy. A prerequisite of epitaxy is not only a two-dimensional lattice analogy of foreign lattice layers but also the mode of nucleus formation, fusing the nuclei, purity of crystal surface, and binding energy between lattice... 

Attempts to crystallize cation-water host lattices analogous to the per-alkyl ammonium salt hydrates, e.g., tetraalkyl borate hydrates, were unsuccessful [432]. 

The Born model [74], for example, satisfies the first condition however, it does not satisfy the second one because in it the longitudinal and transverse elastic constants of the linear chain of bonds (the lattice analog of a rod) decrease /V 1 however, the rods must behave more pliably in relation to transverse shifts (the elastic constant decreases L-3). Therefore, the Born scalar model leads to enhanced rigidity in the vicinity of pc. 

The small effective mass of unpaired atoms allows us to cool them to temperatures higher than that corresponding to the bottom of the diatomic band. The price is, however, that most of the atoms are discarded and only a small fraction of cl/periodic potential is a sparse-lattice analogy of the transition from Mott-insulator to a superfluid state in the fully occupied lattice, recently observed in Ref. [Greiner 2002],... 

Thermodynamic equations are formulated for the isomorphic behavior of A-B type random copolymer systems, in which both A and B comonomer units are allowed to cocrystallize in the common lattices analogous to, or just the same as, those of the corresponding homopolymers poly(A) or poly(B). It is assumed that, in the lattice of poly (A), the B units require free energy relative to the A units and vice versa. On the basis of the derived thermodyn-amie equations, phase diagrams are proposed for the A-B random copolymers with cocrystallization. The melting point versus comonomer composition curve predicted by this diagram is very consistent with that experimentally observed for the P(3HB-co-3HV) copolymers, as shown in Fig. 21.1. It is suggested that the minor comonomer unit with a less bulky structure cocrystallize thermodynamically simpler than that with a more bulky structure. 

The nature of the hydrocarbon and sulfur content is still not clear. However, calculations based on density data would seem to support earlier suggestions that the sulfur must be present as SO, or H O within the silica lattice. The optical characteristics of the mineral show that the organic matter occurs in films between the faces of the crystals. On the other hand, calculations based on the difference in densities of the original mineral and the pyrolyzed silica crystals show that the sulfur compounds at least must be within the crystal lattice. Kamb offers evidence (69) that the silica structure is a clathrate with SO,. HjO, and CH in the lattice analogous to the known 12 A gas hydrates of water, 6X 46HjO, where X is CH , H S, CO SOi, Cl, etc., and in fact the structure is the complete analogue of 6CI - H 0. 

Note CdCl2 also exists as a hexagonal lattice, analogous to Cdl2. 

Various functional forms for / have been proposed either as a result of empirical observation or in terms of specific models. A particularly important example of the latter is that known as the Langmuir adsorption equation [2]. By analogy with the derivation for gas adsorption (see Section XVII-3), the Langmuir model assumes the surface to consist of adsorption sites, each having an area a. All adsorbed species interact only with a site and not with each other, and adsorption is thus limited to a monolayer. Related lattice models reduce to the Langmuir model under these assumptions [3,4]. In the case of adsorption from solution, however, it seems more plausible to consider an alternative phrasing of the model. Adsorption is still limited to a monolayer, but this layer is now regarded as an ideal two-dimensional solution of equal-size solute and solvent molecules of area a. Thus lateral interactions, absent in the site picture, cancel out in the ideal solution however, in the first version is a properly of the solid lattice, while in the second it is a properly of the adsorbed species. Both models attribute differences in adsorption behavior entirely to differences in adsorbate-solid interactions. Both present adsorption as a competition between solute and solvent. 

Our discussion shows that the Ising model, lattice gas and binary alloy are related and present one and the same statistical mechanical problem. The solution to one provides, by means of the transcription tables, the solution to the others. Flistorically, however, they were developed independently before the analogy between the models was recognized. 

In this section, we concentrate on the relationship between diffraction pattern and surface lattice [5], In direct analogy with the tln-ee-dimensional bulk case, the surface lattice is defined by two vectors a and b parallel to the surface (defined already above), subtended by an angle y a and b together specify one unit cell, as illustrated in figure B1.21.4. Withm that unit cell atoms are arranged according to a basis, which is the list of atomic coordinates within drat unit cell we need not know these positions for the purposes of this discussion. Note that this unit cell can be viewed as being infinitely deep in the third dimension (perpendicular to the surface), so as to include all atoms below the surface to arbitrary depth. 

The arrangement of lattice points in a 2D lattice can be visualized as sets of parallel rows. The orientation of these rows can be defined by 2D Miller indices (hksee Figure lb). Inter-row distances can be expressed in terms of 2D Miller indices, analogous to the notation for 3D crystals. 

The initial configuration is set up by building the field 0(r) for a unit cell first on a small cubic lattice, A = 3 or 5, analogously to a two-component, AB, molecular crystal. The value of the field 0(r) = at the point r = (f, 7, k)h on the lattice is set to 1 if, in the molecular crystal, an atom A is in this place if there is an atom B, 0, is set to —1 if there is an empty place, j is set to 0. Fig. 2 shows the initial configuration used to build the field 0(r) for the simple cubic-phase unit cell. Filled black circles represent atoms of type A and hollow circles represent atoms of type B. In this case all sites are occupied by atoms A or B. 

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