Noninteracting defects

First approaches to the quantitative understanding of defects in stoichiometric crystals were published in the early years of the last century by Frenkel in Russia and Schottky in Germany. These workers described the statistical thermodynamics of solids in terms of the atomic occupancies of the various crystallographic sites available in the stmcture. Two noninteracting defect types were envisaged. Interstitial defects consisted of atoms that had been displaced from their correct positions into normally unoccupied positions, namely, interstitial sites. Vacancies were positions that should have been occupied but were not. 

The inherent problems associated with the computation of the properties of solids have been reduced by a computational technique called Density Functional Theory. This approach to the calculation of the properties of solids again stems from solid-state physics. In Hartree-Fock equations the N electrons need to be specified by 3/V variables, indicating the position of each electron in space. The density functional theory replaces these with just the electron density at a point, specified by just three variables. In the commonest formalism of the theory, due to Kohn and Sham, called the local density approximation (LDA), noninteracting electrons move in an effective potential that is described in terms of a uniform electron gas. Density functional theory is now widely used for many chemical calculations, including the stabilities and bulk properties of solids, as well as defect formation energies and configurations in materials such as silicon, GaN, and Agl. At present, the excited states of solids are not well treated in this way. 

The assumption that the defects are noninteracting allows the law of mass action in its simplest form, with concentrations instead of activities, to be used for this purpose. In this case, the equilibrium constant K for this reaction is... 

To construct such a diagram, a set of defect reaction equations is formulated and expressions for the equilibrium constants of each are obtained. The assumption that the defects are noninteracting allows the law of mass action in its simplest form, with concentrations instead of activities, to be used for this purpose. To simplify matters, only one defect reaction is considered to be dominant in any particular composition region, this being chosen from knowledge of the chemical attributes of the system under consideration. The simplified equilibrium expressions are then used to construct plots of the logarithm of defect concentration against an experimental variable such as the log (partial pressure) of the components. The procedure is best illustrated by an example. 

N is here the number of lattice defects (vacancies or interstitials) which are responsible for non-stoichiometry. AHfon is the variation of lattice enthalpy when one noninteracting lattice defect is introduced in the perfect lattice. Since two types of point-defects are always present (lattice defect and altervalent cations (electronic disorder)), the AHform takes into account not only the enthalpy change due to the process of introduction of the lattice defect in the lattice, but also that occurring in the Redox reaction creating the electronic disorder. 

Van der Waals sought to address two basic defects of the KMT noninteracting point mass picture (i) neglect of the finite molecular volume that distinguishes molecules from mathematical points and (ii) neglect of the intermolecular attraction that leads to condensation (liquid formation) at sufficiently low temperature. Whereas the ideal gas equation (2.2) exhibits no vestige of condensation phenomena, the Van der Waals equation (2.13) is intended to provide a unified description of gas-liquid ( fluid ) behavior, exhibiting the essential commonality that must be shared by these disparate forms of matter at the molecular level. 

If macroscopic thermodynamics are applied to materials containing a popnlation of defects, particnlarly nonstoichio-metric compounds, the defects themselves do not enter into the thermodynamic expressions in an exphcit way. However, it is possible to construct a statistical thermodynamic formahsm that will predict the shape of the free energy-temperatnre-composition curve for any phase containing defects. The simplest approach is to assnme that the point defects are noninteracting species, distributed at random in the crystal, and that the defect energies are constant and not a ftmction either of concentration or of temperatnre. In this case, reaction eqnations similar to those described above, eqnations (6) and (7), can be used within a normal thermodynamic framework to deduce the way in which defect populations respond to changes in external variables. 

This describes the situation where oxygen is lost from the crystal, and the surplus electrons, two per lost oxygen, reside on separate metal atoms in the crystal. In formal ionic terms, two M + cations are transformed to M+ cations. In semiconductor terms, two electrons are trapped on separate M + cations. Chemically this process may be considered as equivalent to doping of MO with M2O. The law of mass action, applied in its simplest form with concentrations instead of activities, in view of the supposed noninteraction of the defects, then gives an expression for the equilibrium constant, (9) ... 

In the absence of chemical and physical defects, the transport mechanism in both conducting polymers and molecular single crystals results from a delicate interplay between electronic and electron-vibration (phonon) interactions [3]. The origin and physical consequences of such interactions can be understood by simply considering the tight-binding Hamiltonian for noninteracting electrons and phonons ... 

Fig. 4.11. Probability distribution functions of the soliton defects in the noninteracting limit on a 102-site chain for the 1B state. Left defect, or soUton (filled symbols), right defect, or antisoliton (open symbols) extrinsic dimerization, = 0 (circles), Je = 0.1 (squares) and A = 0.1.

Birefringence of Phantom Networks. This theoiy is the basis for all theories that deal with birefringence of elastomeric pol3uner networks. It is based on the phantom network model of rubber-like elasticity. This model considers the network consisting of phantom (ie, noninteracting) chains. Consider the instantaneous end-to-end distance r for the ith network chain at equilibrium and at fixed strain. For a perfect (ie, no-defects) phantom network, the birefringence induced because of a deformation is defined as (50)... 

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