Point defects defined

Sinee there are six unknowns and three equations, there are three independent variables. We ean associate these with any three elementary independent modes of point defect formation which conserve the numbers of atoms. These are like basis vectors for representing arbitrary point defect concentrations. Let us define them as follows ... 

A useful way to approach these individual point defect energies is to define the energy per mole or cohesive energy of perfect material with respect to separated free atoms, Cmoi-We can then arbitrarily divide this between the atoms of type A and B so that ... 

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

Point defects were mentioned in a prior chapter. We now need to determine how they aiffect the structure auid chemical reactivity of the solid state. We will begin by identifying the various defects which can arise in solids and later will show how they can be manipulated to obtain desirable properties not found in naturally formed solids. Since we have already defined solids as either homogeneous and heterogeneous, let us look first at the homogeneous t5 e of solid. We will first restrict our discussion to solids which are stoichiometric, and later will examine solids which can be classified as "non-stoichiometric", or having an excess of one or another of one of the building blocks of the solid. These occur in semi-conductors as well as other types of electronically or optically active solids. 

A considerable body of scientific work has been accomplished in the past to define and characterize point defects. One major reason is that sometimes, the energy of a point defect can be calculated. In others, the charge-compensation within the solid becomes apparent. In many cases, if one deliberately adds an Impurity to a compound to modify its physical properties, the charge-compensation, intrinsic to the defect formed, can be predicted. We are now ready to describe these defects in terms of their energy and to present equations describing their equilibria. One way to do this is to use a "Plane-Net". This is simply a two-dimensional representation which uses symbols to replace the spherical images that we used above to represent the atoms (ions) in the structure. 

A unit, or perfect, dislocation is defined by a Burgers vector which regenerates the structure perfectly after passage along the slip plane. The dislocations defined above with respect to a simple cubic structure are perfect dislocations. Clearly, then, a unit dislocation is defined in terms of the crystal structure of the host crystal. Thus, there is no definition of a unit dislocation that applies across all structures, unlike the definitions of point defects, which generally can be given in terms of any structure. 

Even when the composition range of a nonstoichiometric phase remains small, complex defect structures can occur. Both atomistic simulations and quantum mechanical calculations suggest that point defects tend to cluster. In many systems isolated point defects have been replaced by aggregates of point defects with a well-defined structure. These materials therefore contain a population of volume defects. 

Once the cluster expansion of the partition function has been made the remaining thermodynamic functions can be obtained as cluster expansions by taking suitable derivatives. Of particular interest are the expressions for the equilibrium concentrations of intrinsic point defects for the various types of lattice disorder. Since the partition function is a function of Nx, N2, V, and T, it is convenient for the derivation of these expressions to introduce defect chemical potentials for each of the species in the set (Nj + N2) defined, by analogy with ordinary Gibbs chemical potentials (cf. Section I), by the relation... 

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). 

When an energetic electron scatters inelastically, an electron from the (filled) valence band can be promoted to the (empty) conduction band creating an electron/hole pair. On recombination, the excess energy is released as a photon, the wavelength of which is well defined by the band-gap transition. The technique is powerful in catalysis it is diagnostic of the electronic/chemical state and is sensitive to point defects. It can be used to probe the distribution of dopants in catalytic oxides. 

The second type of impurity, substitution of a lattice atom with an impurity atom, allows us to enter the world of alloys and intermetallics. Let us diverge slightly for a moment to discuss how control of substitutional impurities can lead to some useful materials, and then we will conclude our description of point defects. An alloy, by definition, is a metallic solid or liquid formed from an intimate combination of two or more elements. By intimate combination, we mean either a liquid or solid solution. In the instance where the solid is crystalline, some of the impurity atoms, usually defined as the minority constituent, occupy sites in the lattice that would normally be occupied by the majority constituent. Alloys need not be crystalline, however. If a liquid alloy is quenched rapidly enough, an amorphous metal can result. The solid material is still an alloy, since the elements are in intimate combination, but there is no crystalline order and hence no substitutional impurities. To aid in our description of substitutional impurities, we will limit the current description to crystalline alloys, but keep in mind that amorphous alloys exist as well. 

Solid-solid reactions are as a rule exothermic, and the driving force of the reaction is the difference between the free energies of formation of the products and the reactants. A quantitative understanding of the mechanism of solid-solid reactions is possible only if reactions are studied under well-defined conditions, keeping the number of variables to a minimum. This requires single-crystal reactants and careful control of the chemical potential of the components. In addition, a knowledge of point-defect equilibria in the product phase would be useful. 

We wish here to obtain the thermodynamic equations defining the liquidus surface of a solid solution, (At BB)2, ). It is assumed that the A and atoms occupy the sites of one sublattice of the structure and the C atoms the sites of a second sublattice. For the specific systems considered here Sb and play the role of C in the general formula above. It is also assumed that the composition variable is confined to values near unity so that the site fractions of atomic point defects is always small compared to unity. This apparently is the case for the solid solutions in the two systems considered. Then it can be shown theoretically (Brebrick, 1979), as well as experimentally for (Hgj CdJ2-yTe)l(s) (Schwartz et al, 1981 Tung et al., 1981b), that the sum of the chemical potentials of A and C and that of and C in the solid are independent of the composition variable y ... 

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. 

The resulting equilibrium concentrations of these point defects (vacancies and interstitials) are the consequence of a compromise between the ordering interaction energy and the entropy contribution of disorder (point defects, in this case). To be sure, the importance of Frenkel s basic work for the further development of solid state kinetics can hardly be overstated. From here on one knew that, in a crystal, the concentration of irregular structure elements (in thermal equilibrium) is a function of state. Therefore the conductivity of an ionic crystal, for example, which is caused by mobile, point defects, is a well defined physical property. However, contributions to the conductivity due to dislocations, grain boundaries, and other non-equilibrium defects can sometimes be quite significant. 

Since the state of a crystal in equilibrium is uniquely defined, the kind and number of its SE s are fully determined. It is therefore the aim of crystal thermodynamics, and particularly of point defect thermodynamics, to calculate the kind and number of all SE s as a function of the chosen independent thermodynamic variables. Several questions arise. Since SE s are not equivalent to the chemical components of a crystalline system, is it expedient to introduce virtual chemical potentials, and how are they related to the component potentials If immobile SE s exist (e.g., the oxygen ions in dense packed oxides), can their virtual chemical potentials be defined only on the basis of local equilibration of the other mobile SE s Since mobile SE s can move in a crystal, what are the internal forces that act upon them to make them drift if thermodynamic potential differences are applied externally Can one use the gradients of the virtual chemical potentials of the SE s for this purpose ... 

BA and Ebc (Fig. 5-5b) are functions of time while the point defects in the surroundings relax and the (Coulombic) potential well shifts from (A)->(B). This shift is represented by g(t) as defined above. Designating the initial activation energies at t = 0 as Eab(0) and iTBA(0), one sees from Figure 5-5 b that... 

In physical and chemical metallurgy, the Kirkendall effect, which is closely related to point defect relaxation during interdiffusion, has been studied extensively. It can be quantitatively defined as the internal rate of production or annihilation of vacan-... 

Let us refer to Figure 5-7 and start with a homogeneous sample of a transition-metal oxide, the state of which is defined by T,P, and the oxygen partial pressure p0. At time t = 0, one (or more) of these intensive state variables is changed instantaneously. We assume that the subsequent equilibration process is controlled by the transport of point defects (cation vacancies and compensating electron holes) and not by chemical reactions at the surface. Thus, the new equilibrium state corresponding to the changed variables is immediately established at the surface, where it remains constant in time. We therefore have to solve a fixed boundary diffusion problem. 

We start with a bimolecular diffusion controlled point defect reaction of the form A + B = C and assume that the immobile B s are (unsaturated) sinks for A. If we then project each B with its surrounding A s onto the coordinate origin at r = 0, and define the individual reaction volume as (4-7t/3)-/-ab the time dependence of the (projected) A-concentration in space is given by the solution to Fick s second law with the following boundary conditions (setting u = cA(r, t)/cA r,0)) u(r,0) = 1 for r>rABl m(°°,0 = 1 (normalized) u(rAB,t) = 0. The solution reads (Fig.5-8)... 

Point Defects. Point defects are defined as atomic defects. Atomic defects such as metal ions can diffuse through the lattice without involving themselves with lattice atoms or vacancies (Figure 9), in contrast to atomic defects such as self-interstitials. The silicon self-interstitial is a silicon atom that is bonded in a tetrahedral interstitial site. Examples of point defects are shown in Figure 9. 

The sensitivity of the embryo to the induction of morphological defects is increased during the period of organogenesis. This period is essentially the time of the origination and development of the organs. The critical period graph (Figure 13.7) demonstrates this point and defines the embryonic and fetal periods. 

The catalyst particle is usually a complex entity composed of a porous solid, serving as the support for one or more catalytically active phase(s). These may comprise clusters, thin surface mono- or multilayers, or small crystallites. The shape, size and orientation of clusters or crystallites, the extension and arrangement of different crystal faces together with macrodcfects such as steps, kinks, etc., are parameters describing the surface topography. The type of atoms and their mutual positions at the surface of the active phase or of the support, and the type, concentration and mutual positions of point defects (foreign atoms in lattice positions, interstitials, vacancies, dislocations, etc.) define the surface structure. 

The doped semiconductor materials can often be considered as well-characterized, diluted solid solutions. Here, the solutes are referred to as point defects, for instance, oxygen vacancies in TiC - phase, denoted as Vq, or boron atoms in silicon, substituting Si at Si sites, Bj etc. See also -> defects in solids, -+ Kroger-Vink notation of defects. The atoms present at interstitial positions are also point defects. Under stable (or metastable) thermodynamic equilibrium in a diluted state, - chemical potentials of point defects can be defined as follows ... 

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