Lattice substitutional disorder

Independently, Burger [231] develops analytical equations for lattice models without substitutional disorder. His results are special cases of the models presented by Ruland. 

So far, the solids that we have studied have been ordered, in the sense that they possess perfect translational symmetry. However, this perfection is really an idealization and, in reality, an actual crystal can be expected to have some sort of disorder, which breaks the long-range periodicity of the lattice. There are a number of ways in which disorder can arise. For instance, interstitial disorder occurs when an impurity atom is placed in the vacant space between two substrate atoms, which remain at their original locations in the lattice. Another situation is that of structural disorder, where the substrate atoms move away from their positions on the perfect lattice. However, the situation of interest in this chapter is that of substitutional disorder. Here, a perfect lattice of one type of atoms (say, A) has some of its members randomly replaced by another type (B). The result is a structurally periodic lattice, but with the constituent atoms A and B randomly placed on the lattice sites. The relative numbers of A and B atoms can be represented by the concentrations ca and cB, with ca + cB = l. The randomness of this type of solid introduces a level of difficulty into the theory, that we have not yet encountered. 

In substitutional disorder, two or more types of atoms randomly occupy one set of lattice nodes. Substitutional disorder of different atoms, in size and/or charge, would cause displacements not only in their lattice node, but also in the neighbor sites. The most striking effect of substitutional disorder is normally a thermal motion of the framework atoms which is apparently anomalously high. 

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] ... 

Microscopic disorder We consider a lattice the sites of which have disordered resonance energies, with a distribution of width Ae, but have the same intersite interactions (same dipole orientation and oscillator strength) as the perfect lattice. This is the so-called substitutional disorder model.122 We assume the disorder width to be smaller than the excitonic bandwidth (4< Be) and examine the bottom of the excitonic band, where the emitting and the absorbing K 0 states lie. In a renormalization-group scheme, we split the lattice into isometric domains of n sites, on which the excitation is assumed to be localized, and write the substitutional-disorder hamiltonian in this basis we thereby obtain a new disorder width An Aen-1/2 and a... 

Ab initio density functional methods using pseudopotentials and plane wave basis set are naturally well-adapted to treat periodic lattice models of the zeolite catalyst. Although, in the light of the above discussion, it does not correspond entirely to the real situation of a zeolite framework with substitutional disorder, such a model still offers the most of the guarantee that none of the physico-chemically important interactions are missed by our calculations. 

For a system where the donors are distributed at random it can be shown mathematically that there exists a critical concentration C (also called "percolation concentration") below which the percofation (edge-to-edge connectivity) has a probability of zero and above which the percolation probability (P ) rises sharply with donor concentration (C). A mathematical relation ( ), for a substitutionally disordered binary lattice, is ... 

In contrast to interstitial defects, the replacement of lattice atoms with dopant species is referred to as substitutional disorder. Since this type of displacement involves extensive diffusion of solvent and solute atoms, a number of requirements must be satisfied. These are known as the William Hume-Rothery Rulest ... 

Partial occupancy of atom sites is a relatively common special case of substitutional disorder, and non-coordinating solvent molecules are frequently found to occupy only about half of the voids in the crystal lattice. The presence of half waters in protein stractures is a typical example. Unusually high displacement parameters are a sign for partially occupied solvent molecules however, one should take into account that, due to their mobility, even fully occupied non-coordinating solvent molecules tend to show relatively high displacement parameters. Therefore, the ADPs should be drastically larger to justify a reduction of the occupancy factors. The residual electron density map, which shows negative electron density at or around the nuclear positions if the true occupancy is lower than one, is a better criterion. 

If a crystal is constructed by the stepwise addition of single particles, and if now and then a regular lattice site is left unoccupied, then we say that the crystal contains vacancies. If atoms or ions are introduced into the normally unoccupied spaces between regular lattice sites, then we speak of interstitial atoms or interstitial ions. Finally, regular lattice sites can be occupied by foreign particles. In this case we have substitutional disorder. In principle, all possible point defects have now been listed. 

Cain, J. E. Lattice and Substitution Disorder in Oriented Membrane Systems, Structure of the Chromatophore Membrane by X-Ray Diffraction. Federation Proceedings 33, 1241 (1974)... 

The anion sublattice is occupied only by As atoms, while the cation sublattice is occupied by Ga and Mn atoms, and also by As antisite defects. We consider only substitutional disorder on the cation sublattice which in turn is described within the coherent potential approximation (CPA) . We thus neglect local environment effects and lattice relaxations. 

In this approach [15] to the study of alloy electronic structure and energetics, one simulates a substitutionally disordered alloy by ensuring that the distribution of atoms of different species over the sites of a lattice preserve the first select few spatial correlation functions of the real disordered system. For example, the distribution may preserve the overall concentration and the nearest-neighbor pair correlation function. 

In any situation where there is substitutional disorder of any kind in a crystal the diffraction pattern consists of two parts the diffraction pattern of the averaged structure, which is confined to the Bragg reflections and whose analysis in terms of occupancy I have discussed and superimposed on this the diffraction pattern of the difference structure i.e. a distribution of electron density everywhere equal to the difference between the average and the actual structure. Even in the case of complete disorder this will be non-zero, but its diffraction pattern will be very weak everywhere because it will be diffused over all diffraction directions. Only if there is some kind of ordering on subsets of the crystallographic sites, which subsets define a superlattice of the crystal lattice, will the intensity of the difference pattern concentrate into definite directions and be easily detectable and interpretable. However if such a superlattice regularity is confined to 1 or 2 dimensions or to very small 3-dimensional domains then the difference diffraction will be smeared out. It may then be difficult to detect, and even when detected it can be difficult to interpret unambiguously. 

Chemical or substitutional disorder results from fractional occupancy of certain lattice sites by different elements of similar metallic radius. This causes a variable amount of these elements in the structure and extends the stability range of the corresponding phase in the phase diagram. In Mg32(Al,Zn)49, for example, three atom sites can be occupied by either A1 or Zn atoms, leading to an extension of the stability range in the Al-Mg-Zn system over a wide spread of values at almost constant Mg content. 

The greatest broadening effects are actually observed in substitutionally disordered crystals. Sub stitutional disorder in crystals occurs when two or more constituent ions have interchangeable lattice positions. Whilst the stoichiometry and periodicity of the crystal is maintained, the local environment varies throughout the structure. Optically active impurity ions experience different crystal fields as a... 

The ideal chain-structures employed in theoretical models bear little resemblance to real materials. Disorder has been modelled by either a distribution of conjugation lengths or electron-lattice interactions. Bond length and substitutional disorder has been shown to lead to the disappearance of bond alternation and closure of the band gap, though with a low density of states at the Fermi energy. A simple model has been developed for the influence of defects on the electrical conductivity of doped samples. Pinning of solitons by defects has been discussed, but it is clear that the role of disorder requires further study. ... 

Materials that contain defects and impurities can exhibit some of the most scientifically interesting and economically important phenomena known. The nature of disorder in solids is a vast subject and so our discussion will necessarily be limited. The smallest degree of disorder that can be introduced into a perfect crystal is a point defect. Three common types of point defect are vacancies, interstitials and substitutionals. Vacancies form when an atom is missing from its expected lattice site. A common example is the Schottky defect, which is typically formed when one cation and one anion are removed from fhe bulk and placed on the surface. Schottky defects are common in the alkali halides. Interstitials are due to the presence of an atom in a location that is usually unoccupied. A... 

There are three different substituted benzene compounds with the formula C6H4F2. Assume that the benzene rings pack similarly into their crystal lattices. If the positions of the H and F atoms are statistically disordered in the solid state, which isomer will have the least residual molar entropy ... 

---