Disordered lattice

Atoms are not rigidly bound to the lattice, but vibrate around their equilibrium positions. If we were able to look at the crystal with a very short observation time, we would see a slightly disordered lattice. Incident electrons see these deviations, and this, for example, is the reason that in LEED the spot intensities of diffracted beams depend on temperature at high temperatures the atoms deviate more from their equilibrium position than at low temperatures, and a considerable number of atoms are not at the equilibrium position necessary for diffraction. Thus, spot intensities are low and the diffuse background high. Similar considerations apply in other scattering techniques, as well as in EXAFS and in Mossbauer spectroscopy. 

In disordered lattices, every ion or molecule is attracted to the rest of the particle less firmly than a corresponding unit belonging to a perfect lattice. Hence, the former escapes into the solution more easily than the latter. Careful experiments to test this explanation would be welcome. 

Fig. 4 Possible adatom (xmfigurations for the coadsorption of two atomic species (e.g. C,0) on the square lattices of preferred adsorption sites on (100) surfaces of b.c.c. transition metals. The two atomic species are denoted by small open or filled circles, respectively, (a) shows the top layer of the substrate and possible adsorption sites the solid lines connect centers of the substrate atoms in this layer, (b) shows the c(2 x 2) structure with random (xxupation of the sites by the two species (c) ordered structure I (the (2x1) structure) (d) ordered structure II [ordered c(2 x 2) structure] (e) and (f) show the disordered lattice gas and lattice liquid states, respectively. (From Lee and Landau .)...

Yh fyT = X)( z) > as cubic and completely disordered lattices (incidentally, the type of arrangements for which in the classical Lorentz cavity-field calculation the contribution of the dipoles inside the small sphere vanishes) these lattice sums vanish for large spherical samples. The sums and... 

However, in all the papers mentioned above the authors analyzed only three-dimensional (3D) systems, while a two-dimensional (2D) case is also experimentally observed surfaces of various absorbers, heterogeneous catalysts, photocatalysts, etc. In [137], Fel dman and Lacelle examined the quenched disorder average of nonequilibrium magnetization, i.e., a free induction decay G(t) and its relative fluctuations for dipolar coupled homonuclear spins in dilute substitutionally disordered lattices. The studies of NMR free induction decays and their relative fluctuations revealed that the functional form of the disorder average (G(t))c depends on the space-filling dimentionality D of the lattice. Explicit evaluations of these averages for dilute spin networks with D = 1, 2, 3 were presented in [137] ... 

Optical Response and Emission from a 2D Disordered Lattice... 

Section IV is devoted to excitons in a disordered lattice. In the first subsection, restricted to the 2D radiant exciton, we study how the coherent emission is hampered by such disorder as thermal fluctuation, static disorder, or surface annihilation by surface-molecule photodimerization. A sharp transition is shown to take place between coherent emission at low temperature (or weak extended disorder) and incoherent emission of small excitonic coherence domains at high temperature (strong extended disorder). Whereas a mean-field theory correctly deals with the long-range forces involved in emission, these approximations are reviewed and tested on a simple model case the nondipolar triplet naphthalene exciton. The very strong disorder then makes the inclusion of aggregates in the theory compulsory. From all this study, our conclusion is that an effective-medium theory needs an effective interaction as well as an effective potential, as shown by the comparison of our theoretical results with exact numerical calculations, with very satisfactory agreement at all concentrations. Lastly, the 3D case of a dipolar exciton with disorder is discussed qualitatively. 

This approximation consists in replacing the real disordered lattice by a crystal, translationally invariant, with molecules of average polarizability < > ... 

Alloys are also usually poorer conductors of electricity. The diminished electrical conductivity of an alloy compared to the parent metal is best understood by considering the wave character of a conducting electron. Electron waves, which move easily through a pure crystal, are scattered by the disordered lattice of an alloy. 

SO that the alkali metals are intercalated in between the tubes within the bundles. Disordered lattices are observed after intercalation of Rb and Cs ions. Simulations of the X-ray powder data for potassium intercalation compounds indicate that three K ions occnpy the triangular cavities between the tubes. Various physical measurements including in situ measurements of the resistivity show a large drop in the resistance on intercalation and the temperature dependence snggests metallic behavior. 

In recent years there has been an explosion of interest in the electron properties of disordered lattices. The more common line of approach to this kind of problem is to study the mean resolvent of the random medium, and the memory function methods can be of remarkable help for this purpose. Otherwise one can investigate by the memory function methods (basically the recursion method) a number of judiciously selected configurations this line of approach is particularly promising because it allows one to overcome some of the limitations inherent in the mean field theories. In this section we de-... 

Lattice methods are well suited to treatment of the intermolecular, or steric , part of the configuration partition function for a system comprising particles or molecules in which volume exclusion plays its usual, dominant role. In principle, these well established methods are no less applicable to systems consisting of species of asymmetric shape, although certain modifications of the conventional procedures are required in order to accommodate highly asymmetric molecules on a lattice when they maintain a degree of orientational disorder. Lattice theory has been adapted to this purpose and the practicability of this approach has been demonstrated... 

FIG. 3 Models for the structure of carbonaceous adsorbents, (a) "Disordered lattice. (From Ref. 71.) (b) Crumpled paper sheets. (From Refs. 72 and 73.)... 

Solids may be either crystalline or non-crystalline. The crystalline state is characterized by a perfectly ordered lattice and the non-crystalline (amorphous) state is characterized by a disordered lattice. These represent two extremes of lattice order and intermediate states are possible. The term degree of crystallinity is useful in attempts to quantify these intermediate states of lattice order. 

An active form of the compound, that is, a very fine crystalline precipitate with a disordered lattice, is generally formed incipiently from strongly over-saturated solutions. Such an active precipitate may persist in metastable equilibrium with the solution and may convert ( age ) only slowly into a more stable inactive form. Measurements of the solubility of active forms give solubility products that are higher than those of the inactive forms. Inactive solid phases with ordered crystals are also formed from solutions that are only slightly oversaturated. 

Figure 7.2. Solubility and saturation. A schematic solubility diagram showing concentration ranges versus pH for supersaturated, metastable, saturated, and undersaturated solutions. A supersaturated solution in the labile concentration range forms a precipitate spontaneously a metastable solution may form no precipitate over a relatively long period. Often an active form of the precipitate, usually a very fine crystalline solid phase with a disordered lattice, is formed from oversaturated solutions. Such an active precipitate may persist in metastable equilibrium with the solution it is more soluble than the stable solid phase and may slowly convert into the stable phase.

Another problem which obscures the analogy between different phase transitions is the fact that one does not always wish to work with the corresponding statistical ensembles. Consider, for example, a first-order transition where from a disordered lattice gas islands of ordered c(2x2) structure form. If we consider a physisorbed layer in full thermal equilibrium with the surrounding gas, then the chemical potential of the gas and the temperature would be the independent control variables. In equilibrium, of course, the chemical potential jx of subsystems is the same, and so the chemical potential of the lattice gas and that of the ordered islands would be the same, while the surface density (or coverage 9) in the islands will differ from that of the lattice gas. The three-dimensional gas acts as a reservoir which supplies adsorbate atoms to maintain the equilibrium value of the coverage in the ordered islands when one cools the adsorbed layer through the order-disorder transition. However, one often considers such a transition at... 

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