Point defects statistical thermodynamics

At the beginning of the century, nobody knew that a small proportion of atoms in a crystal are routinely missing, even less that this was not a mailer of accident but of thermodynamic equilibrium. The recognition in the 1920s that such vacancies had to exist in equilibrium was due to a school of statistical thermodynamicians such as the Russian Frenkel and the Germans Jost, Wagner and Schollky. That, moreover, as we know now, is only one kind of point defect an atom removed for whatever reason from its lattice site can be inserted into a small gap in the crystal structure, and then it becomes an interstitial . Moreover, in insulating crystals a point defect is apt to be associated with a local excess or deficiency of electrons. 

This article is concerned with the statistical mechanics of interactions between point defects in solids at thermodynamic equilibrium. The review is made entirely from the point of view of the... 

The question raised by Anderson (1970,1971) and Anderson et al (1973) as to whether anion point defects are eliminated completely by the creation of extended CS plane defects, is a very important one. This is because anion point defects can be hardly eliminated totally because apart from statistical thermodynamics considerations they must be involved in diffusion process. Oxygen isotope exchange experiments indeed suggest that oxygen diffuses readily by vacancy mechanism. In many oxides it is difficult to compare small anion deficiency with the extent of extended defects and in doped complex oxides there is a very real discrepancy between the area of CS plane present which defines the number of oxygen sites eliminated and the oxygen deficit in the sample (Anderson 1970, Anderson et al 1973). We attempt to address these issues and elucidate the role of anion point defects in oxides in oxidation catalysis (chapter 3). 

In this chapter, we discuss classical non-stoichiometry derived from various kinds of point defects. To derive the phase rule, which is indispensable for the understanding of non-stoichiometry, the key points of thermodynamics are reviewed, and then the relationship between the phase rule, Gibbs free energy, and non-stoichiometry is discussed. The concentrations of point defects in thermal equilibrium for many types of defect structure are calculated by simple statistical thermodynamics. In Section 1.4 examples of non-stoichiometric compounds are shown referred to published papers. 

STATISTICAL THERMODYNAMICS OF POINT DEFECTS hydrogen sites, respectively. Hence we have... 

Thus, lattice defects such as point defects and carriers (electrons and holes) in semiconductors and insulators can be treated as chemical species, and the mass action law can be applied to the concentration equilibrium among these species. Without detailed calculations based on statistical thermodynamics, the mass action law gives us an important result about the equilibrium concentration of lattice defects, electrons, and holes (see Section 1.4.5). 

Chapter 1 deals with classical non-stoichiometric compounds. By classical, the author means that the basic concept of the phase stability has been well established from a thermodynamical point of view, and does not mean that research in this field has been fully completed. In these compounds the origin of non-stoichiometry is point defects . In the first half of the chapter, the fundamental relation between point defects and non-stoichiometry is described in detail, based on (statistical) thermodynamics, and in the second half various examples, referred to the original papers, are shown. 

Chemical solid state processes are dependent upon the mobility of the individual atomic structure elements. In a solid which is in thermal equilibrium, this mobility is normally attained by the exchange of atoms (ions) with vacant lattice sites (i.e., vacancies). Vacancies are point defects which exist in well defined concentrations in thermal equilibrium, as do other kinds of point defects such as interstitial atoms. We refer to them as irregular structure elements. Kinetic parameters such as rate constants and transport coefficients are thus directly related to the number and kind of irregular structure elements (point defects) or, in more general terms, to atomic disorder. A quantitative kinetic theory therefore requires a quantitative understanding of the behavior of point defects as a function of the (local) thermodynamic parameters of the system (such as T, P, and composition, i.e., the fraction of chemical components). This understanding is provided by statistical thermodynamics and has been cast in a useful form for application to solid state chemical kinetics as the so-called point defect thermodynamics. 

Statistical thermodynamics can provide explicit expressions for the phenomenological Gibbs energy functions discussed in the previous section. The statistical theory of point defects has been well covered in the literature [A. R. Allnatt, A. B. Lidiard (1993)]. Therefore, we introduce its basic framework essentially for completeness, for a better atomic understanding of the driving forces in kinetic theory, and also in order to point out the subtleties arising from the constraints due to the structural conditions of crystallography. 

Monovacancy. Statistical thermodynamics requires that if a vacancy is formed by removing an atom from the crystal and depositing it on the surface, then the free energy of the crystal must decrease as the number of created vacancies increases until a minimum in this free energy is reached. Because a minimum in the free energy exists for a certain vacancy concentration in the crystal, the vacancy is a stable point defect. The following facts about vacancies have been obtained experimentally (12). 

The Silicon Self-Interstitial Atom. A similar consistent statistical thermodynamic analysis of the existence of self-interstitials shows that silicon self-interstitials are stable point defects. The following arguments further support the silicon self-interstitial. 

Alan Allnatt s research interests at Western Ontario have been concerned with the statistical mechanics of the transport of matter through crystals. His earliest work centered on obtaining methods for calculating the equilibrium distributions and thermodynamic properties of the point defects (vacancies, interstitials, solutes) that make transport possible. He first studied dilute systems, so the methods could be largely analytical. The methods for ionic crystals,... 

The statistical thermodynamic approach, along lines already indicated, has been more tractable and suggestive. Models have been based on the Fowler-Guggenheim treatment of localized monolayers, in which account is taken of energy terms arising from interaction between point defects in nearest neighbor... 

Walter Haus Schottky (1886-1976) received his doctorate in physics under Max Planck from the Humboldt University in Berlin in 1912. Although his thesis was on the special theory of relativity, Schottky spent his life s work in the area of semiconductor physics. He alternated between industrial and academic positions in Germany for several years. He was with Siemens AG until 1919 and the University of Wurzburg from 1920 to 1923. From 1923 to 1927, Schottky was professor of theoretical physics at the University of Rostock. He rejoined Siemens in 1927, where he finished out his career. Schottky s inventions include the ribbon microphone, the superheterodyne radio receiver, and the tetrode vacuum tube. In 1929, he published Thermodynamik, a book on the thermodynamics of solids. Schottky and Wagner studied the statistical thermodynamics of point defect formation. The cation/anion vacancy pair in ionic solids is named the Schottky defect. In 1938, he produced a barrier layer theory to explain the rectifying behavior of metal-semiconductor contacts. Metal-semiconductor diodes are now called Schottky barrier diodes. 

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. 

Point defects are amenable to analysis by equilibrium statistical thermodynamics. The simplest formula for estimating the void concentration in a... 

The appropriate thermodynamic and statistical-mechanical formalism for the application of molecular simulation to the study of point defects has been given only recently, by Swope and Andersen [90]. These workers identified the number of lattice sites M as a key thermodynamic variable in the characterization of these systems. A real solid phase is free to adopt a value for M that minimizes the system free energy, because it can in principle create or destroy lattice sites through the migration of molecules to and from the surface of the crystal. The resulting bulk crystal can thus disconnect the molecule number N from the lattice-site number M, and thereby achieve an equilibrium of lattice defects in the form of vacancies and interstitials. 

The concentration of defects can be derived from statistical thermodynamics point of view, but it is more convenient treat the formation of defects as a chemical reaction, so that equilibrium constant of mass action can be applied. For a general reaction, in which the reactants A and B lead to products C and D, the equation is given by ... 

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