Energy of point defects

In order to be able to calculate the concentrations of point defects at thermodynamic equilibrium, it is necessary to know the change in free energy of the crystal which accompanies the formation of point defects, since the equilibrium is determined by the minimization of the free energy when the pressure, the temperature, and the other independent thermodynamic variables are given. A theoretical calculation of the free energy of formation of defects is still one of the most difficult problems in solid state physics and chemistry. The methods of calculation for each group of materials - metals, covalent crystals, ionic crystals - are all very 

Moving this molecule back from infinity to the surface of the crystal yields precisely the lattice energy C/l mole. Thus, by putting this all together and using eq. (1-2), we obtain as a first approximation to the energy of a mole of Schottky defects  

For sufficiently large values of e, 1/s is negligibly small compared to one. Assuming that n = 9 and that the Madelung constant aM 1.748 (NaCl structure), we obtain Us 0.35 This is of the right order of magnitude. 

In the last few years the ways of calculating defect states, configurations, and energies in ionic crystals have become much more sophisticated, both in mathematical and computational method and in underlying physical model [33]. An immediate understanding of the nature of the point defects has thus been provided for MgO, UO2, and Cap2. 

The discussion in this section has only been concerned with the enthalpy term. In order to determine the free energy, which is necessary for a calculation of the equilibrium defect concentration, the standard entropy change for the formation of a mole of defects may be estimated as follows. In the simplest case of the Einstein approximation for the limiting case of Dulong-Petit behaviour, the crystal with Nq lattice atoms is considered to be a system of 

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A.N. Orlov, Yu.V. Trushin, Energy of Point Defects in Metals, Metallurgiya Publ., Moscow, 1983 (in Russian)... 

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 ... 

R.A. Johnson. Empirical potentials and their use in calculation of energies of point-defects in metals. J. Phys. F, 3(2) 295-321, 1973. 

Pacchioni, G. and lerano, G., Ab initio formation energies of point defects in pme and Ge-doped Si02, Phys. Rev. B 56, 7304 (1997). 

First-principles calculations of formation energies of point defects were made on BaTiOs [723,724] and NaNbOs [725] crystals. Among the various fundamentally and technologically important oxides, SrTiOs is a simple structural prototype for many perovskites, in which the detailed investigation of native and dopant defects can lay the theoretical groundwork that can be applied to structurally and chemically more complex perovskite materials [726]. 

These observations are not new. A similar electronic correlation mechanism has been proposed as explanation for the underestimation of charge state transition energies of point defects in semiconductors. Further support for the interaction with the solvent band structure is provided by the comparison to implicit solvent models, which omit such interactions. Errors in the calculation of reduction potentials are significantly smaller for the very same GGA functionals (see, for example, [14]). The unfavorable comparison to implicit solvent models raises the question why allatom methods are used if the huge increase in computational costs only leads to deterioration of accuracy. The answer must be that there are situations where an allatom approach is required. The reactivity and transport of excess electrons in water almost certainly involve conducting states as intermediates. Localization of holes in water is still subject to debate but could possibly play a role in transport and reaction kinetics. Electrochemical interfaces are another example of systems for which interactions between localized and extended states are important. The DFTMD/... 

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