The alternating lattice model

the Hamiltonian H can be split into two parts a diagonal term Hd, which involves the site energies a and ec and a non-diagonal term Hnd associated with the resonance integrals. The eigenfunctions ip )  

The resolution of (1.4.6) is thus equivalent to finding the eigenfunctions and and eigenvalues Fj = ( - 6a)(% - c) of on one sub-lattice. Due to the positive definite character of the operator the F values are larger than or equal to zero. 

For a given Bloch vector k, the degeneracy of F is ndc on the cation sub-lattice and md on the anion one. When ndc is different from md, there are two types of solutions for xpf there are no non-trivial solutions (no = min. ndc,mdA)) and ni solutions (m = max. ndc,mdA) — no) for which either (if mdA ndc) or (if mdA ndc) identically vanishes. The eigenenergies for the latter are equal to ec or a, respectively. In order to be specific, we will assume in the following that mdA ndc and that the diagonalization of is performed on the anion sub-lattice. 

Once the solutions of are obtained, i.e. after the determination of the band dispersion Fjj, of the range of existence of Fj [F in, Fmax] and of the density of states M(F), it is possible to deduce the total density of states JV( ) and the local densities of states Na(E) and Nc( ) on the anion and cation sub-lattices, thanks to the following relationships ( is the Kronecker symbol)  

The alternating lattice method is easy to use on all simple or more complex lattices, when the atomic orbitals have an s character, because the resonance integrals p are isotropic. For example, in an alternating linear chain, with cells of size a, indexed by the integer n, the application of Hnd to an anion orbital gives //nd A ) = P( C ) + C +i)), i.e.  

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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. 

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. 

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. 

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. 

Applying a Monte Carlo simulation to the 3D lattice model (Figure 12.19), the authors were able to predict a percolation occurring at a volume fraction of attapulgite of 0.02, or, alternatively, a weight fraction of 3-4 wt%. The slight divergence of this value from the one found for sepiolite nanoclays in PP can be explained by the lower aspect ratio of the latter. In fact, the percolation threshold is inversely proportional to the aspect ratio. 

The Onsager theory, or second virial expansion, is very successful in predicting the qualitative behavior of the isotropic-nematic phase transition of hard rods. H owever, it is an exact theory in the limit ofI/D—>oo. Straley has estimated that, for L/D < 20, the contribution of the third virial coefficient to the free energy in the nematic phase should be at least comparable to that of the second virial coefficient [22]. For more concentrated solutions, an alternative approach such as the Flory lattice model [3, 5] is required. The full phase diagram of rod-like colloidal systems has been obtained in computer simulations [23, 24]. The effects of polydispersity of rods [19, 25] and charged rods [10] are also important in the phase transitions. The comparison between Onsager theory and experimental results has been summarized by Vroege and Lekkerkerker [26]. 

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. 

The relationship between tlie lattice gas and the Ising model is also transparent in the alternative fomuilation of the problem, in temis of the number of down spins [i] and pairs of nearest-neighbour down spins [ii]. For a given degree of site occupation [i]. 

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.)... 

An alternative type of tip-induced nanostructuring has recently been proposed. In this method, a single-crystal surface covered by an underpotential-deposited mono-layer is scanned at a close tip-substrate distance in a certain surface area. This appears to lead to the incorporation of UPD atoms into the substrate lattice, yielding a localized alloy. This procedure works for Cu clusters on Pt(l 11), Pt(lOO), Au(l 11), and for some other systems, but a model for this type of nanostructuring has not been available until now. (Xiao et al., 2003). 

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