H.2 Euler-Maclaur in Sum Formula
21.5.3 Vacancies and Interstitials in Ionic Crystals
Vacancies and interstitials can occur in crystals with ionic bonding but their formation is subject to additional constraints to insure charge neutrality. We shall illustrate these considerations by treating alkali halides, such as NaCl, and silver halides with formulae of the form AgX in which Ag has the oxidation state +1 andX is a halogen.14We consider the following types of point defects:
Positive ion vacancy Nv+ in number, each being a region of negative charge−eand capable of existing onNv+sites.
Negative ion vacancy Nv− in number, each being a region of positive charge e and capable of existing onNv−sites.
14We exclude AgF2in which Ag has the oxidation state +2. At this stage, we do not treat the possibility of color centers in which localized electrons or holes can exist.
Positive ion interstitial Ni+ in number, each being a region of positive chargee and capable of existing onNi+sites.
Negative ion interstitial Ni−in number, each being a region of negative charge−eand capable of existing onNi−sites.
For the sake of simplicity, we will first treat the case in which only pairs of defects are needed to balance charge because the other two types of defects have values of Gibbs free energy per defect that are much larger. For example, we will only need to consider positive ion vacancies balancing the charge of negative ion vacancies ifgi+andgi−exceedgv+and gv−by amounts that are large compared tokBT. This will give rise to two types of vacancies, also known as Schottky defects. On the other hand, we will only need to consider positive ion vacancies balancing the charge of positive ion interstitials ifgv−andgi−exceedgv+and gi+by amounts that are large compared tokBT. Such vacancy-interstitial pairs are known asFrenkel defects. In these cases, the constraints on charge neutrality could be applied by immediately settingNv+ = Nv−in the Schottky case andNv+ =Ni+in the Frenkel case, but a general methodology that can be used if one needs to consider more than two defect types is to use a Lagrange multiplierλto apply the constraints.
Thus, for the Schottky case we can minimize the function
gv+Nv++gv+Nv+−kBTlnWvv−λ(Nv+−Nv−), (21.209) where
Wvv= Nv+! (Nv+−Nv+)!Nv+!
Nv−!
(Nv−−Nv−)!Nv−!. (21.210) This results in
Nv+eq=Nv+exp(−βgv++λ); Nv−eq=Nv−exp(−βgv−−λ). (21.211) These can be multiplied to eliminateλwhich yields
Nv+eqNv−eq=Nv+Nv−exp
−β(gv++gv−)
. (21.212)
But since the constraint requiresNveq+=Nveq−, we obtain15 Nv+eq=Nv−eq=(Nv+Nv−)1/2exp
−β(gv++gv−)/2
. (21.213)
Equation (21.213) depends only on the average ofgv+andgv−, so the smaller of the two compensates for the larger in establishing the effective activation energy. This case is typical for alkali halides.
For the Frenkel case, we can proceed in a similar manner to obtain Nv+eq=Ni+eq=(Nv+Ni+)1/2exp
−β(gv++gi+)/2
. (21.214)
This case typically occurs for silver halides. By replacing+with−in Eq. (21.214), we could get a case in which negative ion vacancies and negative ion interstitials are the dominant point defects. By replacing “v” with “i” in Eq. (21.213), we could get a case in which positive
15Alternatively we could have setNv+eq=Nv−eqin Eq. (21.211) and then solved for expλ.
396 THERMAL PHYSICS
ion interstitials and negative ion interstitials are the dominant point defects, but this case is not expected to occur because interstitials typically have higher activation energies than vacancies.
Example Problem 21.7. Investigate the case in whichgv−andgi+differ from one another by orderkBTbutgv+,gv−,gi+gi−. Thus, negative ion interstitials can be ignored, so there must be charge balance among the remaining three types of defects.
Solution 21.7. In this case, we apply the charge balance constraint by addingλ(Nv+−Nv−− Ni+)toGand minimizing to obtain
Nv+eq=Nv+exp(−βgv++λ); Nv−eq=Nv−exp(−βgv−−λ); Ni+eq=Ni+exp(−βgi+−λ). (21.215) By eliminatingλ, we obtain
Nv+eqNv−eq=Nv+Nv−exp
−β(gv++gv−)
; Nv+eqNi+eq=Nv+Ni+exp
−β(gv++gi+)
. (21.216) Adding the two equations in Eq. (21.216) and applying the constraintNveq+=Nveq−+Nieq+ allows us to solve for
Nv+eq=(Nv+)1/2exp(−βgv+/2)
Nv−exp(−βgv−)+Ni+exp(−βgi+)1/2
. (21.217)
Then combining this result with Eq. (21.216) gives Nv−eq= (Nv+)1/2exp(−βgv+/2)
Nv−exp(−βgv−)+Ni+exp(−βgi+)1/2Nv−exp(−βgv−) (21.218) and
Ni+eq= (Nv+)1/2exp(−βgv+/2)
Nv−exp(−βgv−)+Ni+exp(−βgi+)1/2Ni+exp(−βgi+). (21.219)
For ionic crystals, there are many other types of point defects, such as those that arise when a small number of Ca++ions are substituted for Na+ions in NaCl, thus stimulating the production of an equal number of Na+ vacancies. Such defects can strongly affect electrical conductivity because of vacancy-assisted diffusion of ions. There is also the possibility of color centers that involve localized electrons and holes that have a large influence on optical adsorption. The reader is referred to the book by Ashcroft and Mermin [58, p. 621] for a discussion of these and other defects.
22
Entropy for Any Ensemble
Until now we have introduced four ensembles that are used in statistical mechanics:
the microcanonical ensemble in Chapter 16, the canonical ensemble in Chapter 19, the grand canonical ensemble in Chapter 21, and the pressure ensemble in Section 21.5 of Chapter 21. The canonical ensemble and the grand canonical ensemble were derived from the microcanonical ensemble, although an alternative derivation of the canonical ensemble was presented. Moreover, in Chapter 15, we introduced the dis- order function D{pi} that gives a precise measure of information based on a set of probabilities {pi} that can be used to characterize a system. In the present chapter, we give a definition of the entropy of a system represented by any ensemble used to define its thermodynamic state statistically. This definition will be based on the methodology of the most probable distribution used in Section 19.1.3 to derive the canonical ensemble. Our definition of entropy will enable us to relate systematically a specific thermodynamic function with the logarithm of the partition function for that ensemble.