Alternating lattice model

This first chapter summarizes the main bulk characteristics of insulating oxides, as a prerequisite to the study of surfaces. The foundations of the classical models of cohesion are first recapitulated, and the distinction between charge-transfer oxides and correlated oxides is subsequently established. Restricting ourselves to the first family, which is the subject of this book, we analyse the mixed iono-covalent character of the anion-cation bonding and the peculiarities of the bulk electronic structure. This presentation will allow us to introduce various theoretical and experimental methods - for example, the most common techniques of band structure calculation - as well as some models - the partial charge model, the alternating lattice model - which will be used in the following chapters. 

These relationships exemplify how the band edges are sensitive to the anion-cation electron delocalization. Second-neighbour hopping broadens the bands, but in a way which cannot be predicted by the alternating lattice model, because it hinders the Hamiltonian decoupling, (1.4.7) and (1.4.8). 

We now derive the expressions for the two first moments Mi/Mo and My Mo of the local density of states (LDOS), normalized to one atom and one spin direction, for an oxide M Om, in a tight-binding approach. For this purpose, we will not need the two basic assumptions of the alternating lattice model. We will note AA) and Cp) the anion and cation atomic orbitals, and e x and ec/i their energies. The basis set is assumed to be orthonormal. The first moment Mi of an anion LDOS reads ... 

Result of the alternating lattice model In the classical models of insulators, the gap width A is equal to the energy difference between the cation and anion orbital energies, corrected by the Madelung potential ... 

To summarize, within the assumptions of the alternating lattice model, the gap width has two contributions related to the anion-cation difference in electronegativity and to covalent effects. The covalent contribution is an increasing function of the resonance integrals, but it also depends upon the vectors at which the gap opens and closes in reciprocal space, i.e. upon the symmetry of the orbitals and of the lattice. In specific cases, the orbital crystal field splitting and second-neighbour delocalization effects may slightly modify this simple picture. 

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. 

FIG. 13 Phase diagram of a vector lattice model for a balanced ternary amphiphilic system in the temperature vs surfactant concentration plane. W -I- O denotes a region of coexistence between oil- and water-rich phases, D a disordered phase, Lj an ordered phase which consists of alternating oil, amphiphile, water, and again amphi-phile sheets, and L/r an incommensurate lamellar phase (not present in mean field calculations). The data points are based on simulations at various system sizes on an fee lattice. (From Matsen and Sullivan [182]. Copyright 1994 APS.)... 

Regarding the parametrization of the Hamiltonian Eq. (7), the present approach relies on the parameters of the underlying lattice model Eq. (5). However, one could envisage an alternative approach, similar to the one described in Refs. [66-69] for small molecular systems, where a systematic diabatiza-tion is carried out based on supermolecular electronic structure calculations as described in Sec. 2.2. 

The same problem has been solved in an alternate way for all dimensions [42]. From this solution one can calculate the number of tracer-vacancy exchanges up to time t. In two dimensions the distribution is geometric, with mean (log t)/tt. The continuum version of this problem has been considered as well in the form of an infinite-order perturbation theory [43] the solution matches the asymptotic form of the lattice model. 

Since it became clear from various observations that the librational motions of the molecules, even in the ordered a and y phases of nitrogen at low temperature, have too large amplitudes to be described correctly by (quasi-) harmonic models, we have resorted to the alternative lattice dynamics theories that were described in Section IV. Most of these theories have been developed for large-amplitude rotational oscillations, hindered or even free rotations, and remain valid when the molecular orientations become more and more localized. 

One way of limiting the conformational space available to a protein is to confine a model polypeptide to a lattice. In doing so, unrealistic distortions are imposed on protein structure. However, lattice models offer the possibility to enumerate the entire conformational space available to a polymer chain. A detailed atomic picture is not typically employed with lattice models. However, a variety of lattices of increasing complexity facilitate more detailed chain representations, A trade-off exists between the detail of the models and the ability to evaluate conformational alternatives exhaustively. 

An alternative approach to accounting for the maxima in the temperature dependence of p is based on the Kondo-lattice model (Lavagna et al. 1982). The periodic array of independent Kondo impurities, described by the single-ion Kondo temperature TK, provides a proper description at elevated temperatures, while a coherent state yielding a drop of the resistivity is attained when the system is cooled to below another characteristic temperature coh- Although this approach is suitable particularly for Ce compounds where the Kondo regime was identified inequiv-ocally, the coherence effects are probably significant also in narrow-band actinide materials, as indicated by an extreme sensitivity of the lower-temperature decrease of the resistivity to the presence of impurities. 

Are all quantitative predictions of the thermodynamics of liquid crystals correct. If not stop here. The reason for this step is that die theory (Flory-Huggins lattice model) also predicts the occurrence of the isotropic to nematic phase transition in liquid crystals. If the theory had predicted correctly the properties of glasses but had failed for liquid crystals we would have had to abandon it, especially since in both cases the cause of the transition is ascribed to the vanishing of the configurational entropy. Alternatively the correctness of the prediction for liquid crystals argues for the correctness of the prediction for glasses. Since we have not been stopped by steps 3 and 4 we proceed to step 5. 

---