Enthalpy of Schottky defects

The Gibbs energy, AGs, is often replaced by the enthalpy of Schottky defect formation, AHs, as in previous sections, to give... 

TABLE 2.1 Formation Enthalpy of Schottky Defects in Some Alkali Halide Compounds of Formula MX"... 

Calculate the migration enthalpy for Na ion migration and the enthalpy of Schottky defect formation from the data shown in Fig. 7.3. Discuss all assumptions. 

Table 3.3 The formation enthalpy of Schottky defects, AHs, in some alkali halide compounds of formula MX...

For the alkah and lead halides, a simple empirical relation exists between the formation enthalpy of Schottky defects and the melting temperature Tnj, i.e.,... 

Comparison of this expression with Eq. (95) shows that the enthalpy of Schottky defect formation is decreased by interactions, and therefore the concentration of defects should increase. However, the effect is usually quite small. For example, in NaCl at high temperatures, the change in AHs(j, is only 0.8 kcal/mole from a total value of about 48 kcal/mole. 

Since the association of defects is usually an exothermic reaction (rather than endothermic as in point defect formation). A// is negative. In alkali halides the enthalpy of Schottky defect pairing is about —20 kcal/mole. 

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.)... 

The following table gives the values of the fraction of Schottky defects, S/N, in a crystal of NaBr, with the sodium chloride structure, as a function of temperature. Estimate the formation enthalpy of the defects. 

For comparative purposes, table 4.3 lists defect energies (enthalpies) of Schottky and Frenkel processes in halides, oxides, and sulfides. The constant Kq appearing in the table is the preexponential factor (see section 4.7) raised to a power of 1/2. 

Applying equation 4.75, we obtain an enthalpy of formation of Schottky defects of 2.64 eV, which is somewhat higher than the value derived by Ottonello et al. (1990) with static potential calculations (see table 4.2) ... 

TABLE 5.1 The formation enthalpy of Schottky and Frenkel defects in selected compounds... 

Estimate the molar fraction of Schottky defects in a crystal of metal oxide MO at 2000 °C given that the formation enthalpy of a single defect is 2eV. [Answer 6.08 x 10 3]... 

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... 

In this case, AHs is in J mol and represents the enthalpy required to form 1 mole of Schottky defects, and R is the gas constant (JK mol ). The fraction of vacant sites in a crystal as a result of Schottky disorder is given by ... 

Calculate the number of vacancies in a crystal of nickel oxide, NiO, at 1000 °C, given that the enthalpy of formation of Schottky defects is 160kJmol , and the density is 6670 kg m . ... 

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... 

Pure potassium bromide, KBr, which adopts the sodium chloride structure, has the fraction of empty cation sites due to Schottky defects, ncv/Nc, equal to 9.159xl0-21 at 20°C. (a) Estimate the enthalpy of formation of a Schottky defect, Ahs. (b) Calculate the number of anion vacancies per cubic meter of KBr at 730°C (just below the melting point of KBr). The unit cell of KBr is cubic with edge length a = 0.6600 nm and contains four formula units of KBr. 

If we assume, at first approximation, that AHs does not change with T, it is obvious that the increase in defect concentration with T is simply exponential. Table 4.4, for instance, hsts Schottky defect concentrations calculated in this way at various T by Lasaga (1981c) for NaCl and MgO, assuming defect enthalpies of 2.20 and 4.34 eV, respectively. 

Table 5.1 lists some enthalpy-of-formation values for Schottky and Frenkel defects in various crystals. 

A plot of InoT vs. T in this case will give a greater value for the activation energy, E, because it will actually depends on two terms, the activation energy for the cation jump, and the enthalpy of formation of a Schottky defect ... 

Table 5.9 gives the variation of defect concentration with temperature for Csl. Determine the enthalpy of formation for one Schottky defect in this crystal. 

For Na ion migration in NaCl, AHl, is 77kJ/mol, while the enthalpy and entropy associated with the formation of a Schottky defect are, respectively, 240kJ/mol and 10 K (see Table 6.2). 

Schottky defects are thus easier to produce. The energy required is composed of the sublimation enthalpy and an energy of relaxation of the lattice. With these energies, the concentration of lattice vacancies in thermal equiUbrium can be calculated. Estimates for the density of vacancies yield for anthracene at 300 K 1.5 cm and for naphthalene 7 10 cm T At the melting point, for anthracene one finds 2 10 cm and for naphthalene, 3 10 cm ([Ml], p. 223). Notably, the density at 300 K roughly corresponds to the minimum density of specific impurihes which can be achieved with the best purification processes and detected with the best analytic methods (cf Sect. 3.2). 

Note for Schottky defects, AH is AHg, the enthalpy of formation of a Schottky defect, and Ei and Ei are the energy barriers to he surmounted for vacancy diffusion by cations and anions, respectively for Frenkel defects, AH is AHf, the enthalpy of formation of a Frenkel defect, and Ei and E are the energy barriers to be surmounted for interstitial and vacancy diffusion, respectively. 

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