Schottky formation energy

Figure 2.2 Variation of Schottky formation energy of the sodium chloride structure halides MX M = Li, Na, K X = F, Cl, Br, I.

AGs is the Schottky formation energy. The diffusion coefficient can then be written as... 

The formation energy of Schottky defects is described further in Chapter 2. 

Some values for the enthalpy of formation of Schottky defects in alkali halides of formula MX that adopt the sodium chloride structure are given in Table 2.1. The experimental determination of these values (obtained mostly from diffusion or ionic conductivity data (Chapters 5 and 6) is not easy, and there is a large scatter of values in the literature. The most reliable data are for the easily purified alkali halides. Currently, values for defect formation energies are more often obtained from calculations (Section 2.10). 

Some experimental values for the formation enthalpy of Frenkel defects are given in Table 2.2. As with Schottky defects, it is not easy to determine these values experimentally and there is a large scatter in the values found in the literature. (Calculated values of the defect formation energies for AgCl and AgBr, which differ a little from those in Table 2.2, can be found in Fig. 2.5.)... 

Figure 2.5 Calculated variation of the formation energy of Schottky and Frenkel defects in the halides AgCl and AgBr as a function of temperature. [Redrawn from data in C. R. A. Catlow, Mat. Res. Soc. Bull., XIV, 23 (1989).]...

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 formation energy of Schottky defects in NiO has been estimated at 198 kJ mol-1. The lattice parameter of the sodium chloride structure unit cell is 0.417 nm. (a) Calculate the number of Schottky defects per cubic meter in NiO at 1000°C. (b) How many vacancies are there at this temperature (c) Estimate the density of NiO and hence the number of Schottky defects per gram of NiO. 

Here, c+ and c are the site fractions of the cation and anion vacancies, respectively, Ks is the equilibrium constant for the formation of the Schottky pair, and gs (= g+ + g, the sum of the individual defect formation energies) is the Gibbs free energy to form the pair. In the bulk of a pure crystal, the condition of electrical neutrality demands that the concentrations of each defect in the pair are equal that is ... 

This is an important result because it predicts that the defect concentrations in stoichiometric oxides or compounds for which the Schottky or Frenkel defect formation energies are much greater than 3 eV will most likely be dominated by impurities. 

Compare the concentration of positive ion vacancies in an NaCl crystal due to the presence of lO " mol fraction of CaCF impurity with the intrinsic concentration present in equilibrium in a pure NaCl crystal at 400 C. The formation energy Ah of a Schottky defect is 2.12 eV, and the mole fraction of Schottky defects near the melting point of 800°C is 2.8 x lO ". ... 

Calculations of defect formation energies [49] suggest that Schottky disorder... 

If the surface energy for (001) MgO is 1 J/m what is that in electron volts per oxygen ion on the surface. How does this number compare to the formation energy of the Schottky defect How would this number be different if the material were NaCl instead of MgO ... 

The mean number ElEj of electron - hole pairs generated in semiconductors— with a mean formation energy of , = 3.6eV in silicon, for example—normally recombine. The electric field inside depletion layers separates the charge carriers, and minority carriers can diffuse to the depletion layer and contribute to the charge collection /eg or electron-beam induced current. Depletion layers can be formed by p-n junctions parallel or perpendicular to the surface or by Schottky barriers formed by a nonohmic evaporated metal contact. Therefore, a scanning electron probe becomes a useful tool for qualitative and quantitative analysis of junctions and semiconductor parameters [227], which is demonstrated by the following examples ... 

Concentrations of Schottky defect can be measured with experiment of thermal expansion of metals, namely the determination of thermal expansion coefficient of both the whole crystal and lattice parameters, respectively. The thermal expansion coefficient of the whole crystal includes not only the thermal expansion of crystal lattice itself, but the formation of Schottky defect. Therefore, the difference of two results can reflect both the existence and concentration of Schottky defect. For instance, at conditions near to the melting point, the concentration of Schottky vacant for alumina is about 1 x 10, and formation energy of its vacant is about 0.6eV (leV = 1.60 x 10 J) while that of NaCl is 10 -10 and formation energy is 2 eV, respectively. 

On the other hand, the Schottky reaction of AI2O3 (equation 9.12) has a relatively low formation energy ... 

Dislocation theory as a portion of the subject of solid-state physics is somewhat beyond the scope of this book, but it is desirable to examine the subject briefly in terms of its implications in surface chemistry. Perhaps the most elementary type of defect is that of an extra or interstitial atom—Frenkel defect [110]—or a missing atom or vacancy—Schottky defect [111]. Such point defects play an important role in the treatment of diffusion and electrical conductivities in solids and the solubility of a salt in the host lattice of another or different valence type [112]. Point defects have a thermodynamic basis for their existence in terms of the energy and entropy of their formation, the situation is similar to the formation of isolated holes and erratic atoms on a surface. Dislocations, on the other hand, may be viewed as an organized concentration of point defects they are lattice defects and play an important role in the mechanism of the plastic deformation of solids. Lattice defects or dislocations are not thermodynamic in the sense of the point defects their formation is intimately connected with the mechanism of nucleation and crystal growth (see Section IX-4), and they constitute an important source of surface imperfection. 

In pure and stoichiometric compounds, intrinsic defects are formed for energetic reasons. Intrinsic ionic conduction, or creation of thermal vacancies by Frenkel, ie, vacancy plus interstitial lattice defects, or by Schottky, cation and anion vacancies, mechanisms can be expressed in terms of an equilibrium constant and, therefore, as a free energy for the formation of defects, If the ion is to jump into a normally occupied lattice site, a term for... 

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