Statistical Physics, Second Revised and Enlarged Edition

116 Entropyin other situations

Asweshouldexpect, thisisagaintheBoltzmanndistribution. However, thereis
animportant difference from (2.12), in that now there is noβto be discussed.The
− 1 /kkkBTcomesinfrom the outset,from the macrostate. Themultiplierαhas the same
m∑eaningas earlier, andisdeterminedbysubstitutingbackinto the number condition
nnj=N.

10 3 Vacancies in solids

Wehavejust seen that thefree energymethodgives the correct answer to an already
familiar problem. That is comforting, but not very exciting. Of greater importance is
that we can now tackle new problems, one ofwhichconcerns the number ofvacancies
which exist in equilibrium in a solid.
Avacancy occurs when an atom in the solid leaves its normal lattice position. The
atom mustgo somewhere else, asindicatedinFig. 10.1. Either thesolidwillexpand
alittle, the extra atom beingaccommodated on a normal lattice site (case 1). Or it
will sit, albeit rather uncomfortably, on an interstitial site (case 2). In the first case
the vacancywillusually beformedinitiallyat the surface andisthen abletomove
through the solid byatomic diffusion. In the second case, both the vacancyand the
interstitial defect, once formed, can diffuse through the solid. In fact the existence
ofvacanciesisthe mechanismbywhichatomicdiffusionisabletotakeplace, since
only one atom has to move in order for a vacancy and a neighbouring atom to change
places.
The questionis: why do vacanciesform, when clearlytheyrequire energytodo so?
The answer concerns entropy, and the minimization ofF.CertainlyUmustincrease
when a vacancyisformed. But at ahighenoughT,itis possiblefor thisincrease
tobe more than matchedbyadecreasein(−TS),givingan overall loweringofF(=
U−TS). The vacancy (and its interstitial in case 2) are free to roam, and so give
considerabledisorder to thesolid. Note that thewholediscussionhere couldnotbe
startedwithout thefree energyapproach,sinceitisTandnotUwhichisfixed.
Tobespecificletusdevelopasimplifiedmodelofcase1vacancies(case2vacancies
reappear as Exercise 10.5). Consider a simple solid ofNatoms, whichcontainsn
vacancies at temperatureT.TheproblemistofindhownvarieswithT.Suppose that

×××××× × ××

×××××× × ××

×××××× × ××

×××××× × ××

×× ××

×

×× × ×

× ××××××

1

2

Fig. 10.1Two types of vacancies which can occur in solids. Case 1: surface type. Case 2: interstitial type.