Equilibrium Population of Vacancies in a Monatomic Crystal

If vacancies exist in the crystal under equilibrium, the Gibbs energy must be less than it is in a perfect crystal. However, if the introduction of more and more defects causes the Gibbs energy to continually fall, then ultimately no crystal will remain. 

Defects in Solids, by Richard J. D. Tilley Copyright 2008 John Wiley Sons, Inc. 

This implies that the form of the Gibbs energy curve must pass through a minimum if defects are to be present at equilibrium (Fig. 2.1a). 

The change in the Gibbs energy of the crystal by an amount AGy, due to the introduction of n v vacancies distributed over N possible atom sites is given by 

As a first approximation, assume that the total entropy change, ASy, is made up solely of configurational entropy. This can be determined by using the Boltzmann equation for the entropy of a disordered system  

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