Schottky defect equilibrium number

The estimation of the number of Frenkel defects in a crystal can proceed along lines parallel to those for Schottky defects by estimating the configurational entropy (Supplementary Material S4). This approach confirms that Frenkel defects are thermodynamically stable intrinsic defects that cannot be removed by thermal treatment. Because of this, the defect population can be treated as a chemical equilibrium. For a crystal of composition MX, the appropriate chemical equilibrium for Frenkel defects on the cation sublattice is... 

The papers of Wagner and Schottky contained the first statistical treatment of defect-containing crystals. The point defects were assumed to form an ideal solution in the sense that they are supposed not to interact with each other. The equilibrium number of intrinsic point defects was found by minimizing the Gibbs free energy with respect to the numbers of defects at constant pressure, temperature, and chemical composition. The equilibrium between the crystal of a binary compound and its components was recognized to be a statistical one instead of being uniquely fixed. 

Self-diffusion of Ag cations in the silver halides involves Frenkel defects (equal numbers of vacancies and interstitials as seen in Fig. 8.116). In a manner similar to the Schottky defects, their equilibrium population density appears in the diffusivity. Both types of sites in the Frenkel complex—vacancy and interstitial— may contribute to the diffusion. However, for AgBr, experimental data indicate that cation diffusion by the interstitialcy mechanism is dominant [4]. The cation Frenkel pair formation reaction is... 

A population of vacancies on one subset of atoms created by displacing some atoms into normally unoccupied interstitial sites constitute a second arrangement of paired point defects, termed Frenkel defects (Figure 2(b), (c)). Because one species of atom or ion is simply being redistributed in the crystal, charge balance is not an issue. A Frenkel defect in a crystal of formula MX consists of one interstitial cation plus one cation vacancy, or one interstitial anion plus one anion vacancy. Equally, a Frenkel defect in a crystal of formula MX2 can consist of one interstitial cation plus one cation vacancy, or one interstitial anion plus one anion vacancy. As with the other point defects, it is found that the free energy of a crystal is lowered by the presence of Frenkel defects and so a popnlation of these intrinsic defects is to be expected at temperatures above 0 K. The calculation of the number of Frenkel defects in a crystal can proceed along lines parallel to those for Schottky defects. The appropriate chemical equilibrium for cation defects is ... 

Calculate the equilibrium number of Schottky defects n in an MO oxide at 1000 K in a solid for which the enthalpy for defect formation is 2 eV. Assume that the vibrational contribution to the entropy can be neglected. Calculate AG as a function of the number of Schottky defects for three concentrations, namely, n, all assumptions. Plot the resulting data... 

Let us consider an MXs crystal with Schottky defects (this is one of the more easily defined types, mathematically). We define S=l. The types of intrinsic defects was given in 2.6.1., the equilibrium constants of intrinsic defects was given in Table 2-1, numbers of intrinsic defects was given in... 

Under equilibrium conditions, the Gibbs energy of a crystal, G, is lower if it contains a small population of Schottky defects, similar to the situation shown in Figure 3.12. This means that Schottky defects will always be present in crystals at temperatures above 0 K, and hence Schottky defects are intrinsic defects. The approximate number of Schottky defects, ns, in crystal with a formula MX, at equilibrium, is given by ... 

The presence of a small number of Frenkel defects reduces the Gibbs energy of a crystal and so Frenkel defects are intrinsic defects. The formula for the equilibrium concentration of Frenkel defects in a crystal is similar to that for Schottky defects. There is one small difference compared with the Schottky defect equations the number of interstitial positions that are available to a displaced ion, N, need not be the same as the number of normally occupied positions, N, from which the ion moves. The number of Frenkel defects, np. present in a crystal of formula MX at equilibrium is given by ... 

We give some experimental values for the enthalpy of formation of Schottky defects in Table 11.4. We can use these numbers to calculate equilibrium defect concentrations as we have for NaCl in Table 11.5. The population of point defects is very low, but it is clear from Eq. Box 11.1 that vacancies are stable in the crystal at any temperature above absolute zero. Because energies for point defect formation in stoichiometric oxides such as... 

In order to illustrate the problem, let us take a KCl crystal which has been doped with CaCl2. The concentration of dopant is much greater than the thermal Schottky disorder of the pure crystal. Thus, in the absence of interaction between the defects, the number of dissolved Ca ions is virtually equal to the number of cation vacancies V. However, as described in section 3.2.2, neutral associates will form by virtue of elastic and electrical interactions. Their energy of formation is of the order of 1 kcal/mole. Therefore, at constant P and T, there exists a dynamic equilibrium of the form... 

If the temperature of a potassium chloride crystal is suddenly decreased, then the number of Schottky defects present will be greater than the new equilibrium value. That is, the crystal is supersaturated with respect to defects. Equilibration is then achieved by the reaction ... 

Similarly, for Schottky defects, in an AX-type compound, the equilibrium number (Ns) is a fimction of temperature as... 

The formation of the combination of defects may be described as a chemical reaction and thermodynamic equilibrium conditions may be applied. The chemical notations of Kroger-Vink, Schottky, and defect structure elements (DSEs) are used [3, 11]. The chemical reactions have to balance the chemical species, lattice sites, and charges. An unoccupied lattice site is considered to be a chemical species (V) it is quite common that specific crystal structures are only found in the presence of a certain number of vacancies [12]. The Kroger-Vink notation makes use of the chemical element followed by the lattice site of this element as subscript and the charge relative to the ideal undisturbed lattice as superscript. An example is the formation of interstitial metal M ions and metal M ion vacancies, e.g., in silver halides ... 

An intrinsic defect is one that is in thermodynamic equilibrium in the crystal. This means that a population of these defects cannot be removed by any forms of physical or chemical processing. Schottky, Frenkel, and antisite defects are the best characterized intrinsic defects. A totally defect-free crystal, if warmed to a temperature that allows a certain degree of atom movement, will adjust to allow for the generation of intrinsic defects. The type of intrinsic defects that form will depend upon the relative formation energies of all of the possibilities. The defect with the lowest formation energy will be present in the greatest numbers. This can change with temperature. 

The compound will be stoichiometric, with an exact composition of MX10ooo when the number of metal vacancies is equal to the number of nonmetal vacancies. At the same time, the number of electrons and holes will be equal. In an inorganic compound, which is an insulator or poor semiconductor with a fairly large band-gap, the number of point defects is greater than the number of intrinsic electrons or holes. To illustrate the procedure, suppose that the values for the equilibrium constants describing Schottky disorder, Ks, and intrinsic electron and hole numbers, Kc, are... 

The number of interstitial atoms Np in the Frenkel type and the number of vacancies TYj in the Schottky type at thermal equilibrium can be obtained, following a similar calculation to that for the concentration of point defects of elements mentioned in Section 1.3.1, as... 

Particle irradiation effects in halides and especially in alkali halides have been intensively studied. One reason is that salt mines can be used to store radioactive waste. Alkali halides in thermal equilibrium are Schottky-type disordered materials. Defects in NaCl which form under electron bombardment at low temperature are neutral anion vacancies (Vx) and a corresponding number of anion interstitials (Xf). Even at liquid nitrogen temperature, these primary radiation defects are still somewhat mobile. Thus, they can either recombine (Xf+Vx = Xx) or form clusters. First, clusters will form according to /i-Xf = X j. Also, Xf and Xf j may be trapped at impurities. Later, vacancies will cluster as well. If X is trapped by a vacancy pair [VA Vx] (which is, in other words, an empty site of a lattice molecule, i.e., the smallest possible pore ) we have the smallest possible halogen molecule bubble . Further clustering of these defects may lead to dislocation loops. In contrast, aggregates of only anion vacancies are equivalent to small metal colloid particles. 

In spite of a great number of investigations aimed at the preparation of photocatalysts and photoelectrodes based on the semiconductors surface-modified with metal nanoparticles, many factors influencing the photoelectrochemical processes under consideration are not yet clearly understood. Among them are the role of electronic surface (interfacial) states and Schottky barriers at semiconductor / metal nanoparticle interface, the relationship between the efficiency of photoinduced processes and the size of metal particles, the mechanism of the modifying action of such nanoparticles, the influence of the concentration of electronic and other defects in a semiconductor matrix on the peculiarities of metal nanophase formation under different conditions of deposition process (in particular, under different shifts of the electrochemical surface potential from its equilibrium value), etc. 

In thermal equilibrium, some ionic crystals at a temperature above absolute zero enclose a certain number of Schottky pair defects, that is, anion and cation vacancies in the structure (see Section 5.7.1) [13]. Since the concentration of Schottky pair defects at equilibrium at an absolute temperature, T, obeys the mass action law, then [16]... 

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