Point formation energy

In terms of the point defect energies so defined, our stoichiometry-conserving defects have formation energies given by ... 

Several points are worth noting about these formulae. Firstly, the concentrations follow an Arrhenius law except for the constitutional def t, however in no case is the activation energy a single point defect formation energy. Secondly, in a quantitative calculation the activation energy should include a temperature dependence of the formation energies and their formation entropies. The latter will appear as a preexponential factor, for example, the first equation becomes... 

Formula for the chemical potentials have been derived in terms of the formation energy of the four point defects. In the process the conceptual basis for calculating point defect energies in ordered alloys and the dependence of point defect concentrations on them has been clarified. The statistical physics of point defects in ordered alloys has been well described before [13], but the present work represents a generalisation in the sense that it is not dependent on any particular model, such as the Bragg-Williams approach with nearest neighbour bond energies. It is hoped that the results will be of use to theoreticians as well as... 

What is the basis of atomistic simulation calculations of point defect formation energies ... 

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 formation energies of point defects in a pure metal are 1.0 eV (vacancies) and 1.1 eV (interstitials). The number of vacancies is ... 

In terms of formal point defect terminology, it is possible to think of each silver or copper ion creating an instantaneous interstitial defect and a vacancy, Ag and VAg, or Cu and Vcu as it jumps between two tetrahedral sites. This is equivalent to a high and dynamic concentration of cation Frenkel defects that continuously form and are eliminated. For this to occur, the formation energy of these notional defects must be close to zero. 

Scheme 11.2 Products of chlorine atom addition to propadiene (la) and heats of formation [kj mol-1] of addition products 2a and 3a [16], a> QCISD(T)/6—311+G(d,p)//QCISD(T)/6—31+G(d,p) MP2/6-31+G(d,p) for calculation of the zero point vibrational energy.

Figure 8. Formation energy versus Li concentration for three structures of Mn oxide (top) and Co oxide (bottom) ( ) 5-LiJM204-labeled spinel, (0) 7-LiJMn02-labeled layered, and (+) partially inverse spinel structure with 1/4M tetrahedral (ps-(LiJM)tet(LijM3)oct08 0 < x< 1 and 0 < y < 2) labeled 1/4 Mn tet. As the Li content is increased, the Li is added to the tetrahedral site first of p5-(LiJM)ter (LijM3)octOs, and then to the octahedral sites. For Mn, there also is the energy of (a) a structure with one-sixth of the Mn in tetrahedral sites at Xu =1/3 labeled 1/6 Mn tet with a triangle data point and (x) a structure with one-eighth of the Mn in tetrahedral sites at Xu =1/4 labeled 1/8 Mn tet.

The constant a is proportional to the step formation energy, f and b and c account for step-step interactions. Entropic, dipole and elastic interactions between steps give rise to the pfterm [24,25] whereas the p term may be due to electronic effects [26], The experimental data of fig. 1 can be fitted by y(0) = jo cosq f(p) over the whole range of orientations when all terms of eq. (1) are allowed [27]. More about this point in section 6. In principle, it is a matter of great interest to test the validity of eq. (1) and to determine the step as well as step interaction energies. 

To put it on a more quantitative basis, Johansson uses the expression for the critical point of the Mott transition as reformulated by Hubbard in terms of the bandwidth W[ and the polar state formation energy Uh (or effective intra-atomic correlation) (Eq. (36)). 

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