Enthalpy of Frenkel defects

The Gibbs energy, AGcf, is often replaced by the enthalpy of Frenkel defect formation, AHcP, as described above, to give... 

TABLE 2.2 Formation Enthalpy of Frenkel Defects in Some Compounds of Formula MX and MX2... 

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.)... 

Table 3.4 The formation enthalpy of Frenkel defects, AHf, in some compounds of formula MX and MX2...

The enthalpy of formation of Frenkel defects (kJ mol-1) is AgCl, 140, AgBr, 109, Agl, 58. The compound with the greatest number of Ag+ interstitials is ... 

Estimate the number of Frenkel defects in AgBr (NaCl structure) at 500°C. The enthalpy of formation of the defect is 110 kJ/mol, and the entropy of formation is 6.6/ . The density and molecular weights are 6.5g/cm and 187.8 g mol. respectively. State all necessary assumptions. 

Calculate the enthalpy of formation of Frenkel defects in sodium bromide, NaBr, using the data on the number of defects, n, present given in Table 3.5. 

Calculate the number of Frenkel defects present in a crystal of silver chloride, AgCl, at 300 K, given that the material has a cubic unit cell of edge 0.555 nm that contains four silver atoms. Assume that the interstitial atoms occupy any of eight tetrahedral sites in the unit cell. The enthalpy of formation of a Frenkel defect in AgCl is 2.69 x 10 J. 

The favored defect type in strontium fluoride, which adopts the fluorite structure, are Frenkel defects on the anion sublattice. The enthalpy of formation of an anion Frenkel defect is estimated to be 167.88 kJ mol-1. Calculate the number of F- interstitials and vacancies due to anion Frenkel defects per cubic meter in SrF2 at 1000°C. The unit cell is cubic, with a cell edge of 0.57996 nm and contains four formula units of SrF2. It is reasonable to assume that the number of suitable interstitial sites is half that of the number of anion sites. 

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. 

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

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

The activation energy for self-diffusivity of the Ag cations by the interstitialcy mechanisms is the sum of one-half the Frenkel defect formation enthalpy and the activation enthalpy for migration,... 

Frenkel defects and impurity ions can diffuse through the silver halide lattice by a number of mechanisms. Silver ions can diffuse by a vacancy mechanism or by replacement processes such as collinear or noncollinear interstitialcy jump mechanisms [18]. The collinear interstitial mechanism is one in which an interstitial silver ion moves in a [111] direction, forcing an adjacent lattice silver ion into an interstitial position and replacing it The enthalpies and entropies derived from temperature-dependent ionic conductivity measurements for these processes are included in Table 4. The collinear interstitial mechanism is the most facile process at room temperature, but the other mechanisms are thought to contribute at higher temperatures. 

Since the data were obtained in the transition region where intrinsic and extrinsic defects are contributing to the total defect concentration, the calculation of an enthalpy of motion cannot be made in a simple way because the temperature dependence of the Frenkel constant is not known. However, Ail. probably increases with temperature while the extrinsic defect concentration decreases with temperature. If, to a first approximation, these two trends cancel, then the enthalpy of motion is just the experimentally determined activation energy. Using this value from (16) and the defect concentration shown in Table VII, the preexponential constant Dq and hence the diffusion coefficient can be determined. 

An example of the kinds of result obtainable can be seen in the work of Harding (1985) on calcium fluoride. The calculated defect parameters for the formation enthalpy and entropy of the anion Frenkel defect are 2.81 eV and 5.4k, respectively. This compares with the experimental values of Jacobs and... 

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. 

Ef corresponds to the enthalpy of formation of a defect pair Frenkel or Schottky, cf. Chapter 3). In an alkali halide, E( is typically about 3 eV implying a very low defect concentration, of the order of 10 at room temperature. E is the enthalpy of defect migration, usually several eV in dense structures, and / is the correlation or Haven factor. Its value varies between 0 and 1 and takes into account unfruitful attempts at transport in a given direction, caused by random jumps of mobile species and the particular geometry of each site. In other words, this factor takes into account the correlation effects (/ = 1 when correlations are absent). Usually / is of the order of 0.5-0.8 and plays a role in determining the transport mechanism. 

From Eq. (8) it can be seen that the intrinsic defect concentrations will increase with increasing temperature and they will be low for high enthalpies of defect formation. The application of these equations to some specific systems would be illustrative. From thermodynamic measurements on cerium hydride CeH2, it was deduced that the intrinsic defects were anti-Frenkel defects -f Vj) and a value of 3.0 X 10 was computed for K p at 600°C. This compound has the fluorite structure which contains one octahedral interstice per Ce atom. Therefore a = 1. Since the compound is a dihydride, s = 2. Equation (25) then can be written... 

Let us consider a crystal similar to that discussed in Sections 1,3.3 and 1.3.4, which, in this case, shows a larger deviation from stoichiometry. It is appropriate to assume that there are no interstitial atoms in this case, because the Frenkel type defect has a tendency to decrease deviation. Consider a crystal in which M occupies sites in N lattice points and X occupies sites in N lattice points. It is necessary to take the vacancy-vacancy interaction energy into consideration, because the concentration of vacancies is higher. The method of calculation of free energy (enthalpy) related to is shown in Fig. 1.12. The total free energy of the crystal may be written... 

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