Fundamentals of Materials Science and Engineering: An Integrated Approach, 3e

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132 • Chapter 5 / Imperfections in Solids

Fe2+

Vacancy

O2–

Fe2+

Fe3+

Figure 5.4 Schematic

representation of an Fe^2 +

vacancy in FeO that results

from the formation of two

Fe^3 +ions.

oxide and, in fact, its chemical formula is often written as Fe 1 −xO (wherexis some

small and variable fraction substantially less than unity) to indicate a condition of

nonstoichiometry with a deficiency of Fe.

Concept Check 5.1

Can Schottky defects exist in K 2 O? If so, briefly describe this type of defect. If they

cannot exist, then explain why.

[The answer may be found at http://www.wiley.com/college/callister (Student Companion Site).]

The equilibrium numbers of both Frenkel and Schottky defects increase with and

depend on temperature in a manner similar to the number of vacancies in metals

(Equation 5.1). For Frenkel defects, the number of cation-vacancy/cation-interstitial

defect pairs (Nfr) depends on temperature according to the following expression:

Nfr=Nexp

(

−

Qfr

2 kT

)

(5.3)

HereQfris the energy required for the formation of each Frenkel defect, andNis the

total number of lattice sites. (Also, as in the previous discussion,kandTrepresent

Boltzmann’s constant and the absolute temperature, respectively.) The factor 2 is

present in the denominator of the exponential because two defects (a missing cation

and an interstitial cation) are associated with each Frenkel defect.

Similarly, for Schottky defects, in an AX-type compound, the equilibrium num-

ber (Ns) is a function of temperature as

Ns=Nexp

(

−

Qs

2 kT

)

(5.4)

whereQsrepresents the Schottky defect energy of formation.

EXAMPLE PROBLEM 5.2

Computation of the Number of Schottky Defects in KCl

Calculate the number of Schottky defects per cubic meter in potassium chloride

at 500◦C. The energy required to form each Schottky defect is 2.6 eV, while the

density for KCl (at 500◦C) is 1.955 g/cm^3.