The importance of the presence of a small fraction of vacancies in a crystal structure was demonstrated by considering thermodynamic stability. Effects due to distortion of the lattice structure surrounding the vacancy were included in the density functional theory, and it was shown that the free energy was lower when vacancies were present. Near to freezing point, this theoretical model predicted the equilibrium vacancy fraction in a hard-sphere crystal to be about 10–5. It decreased with increasing density.

Optimum Vacancy Concentration in a Crystal. S.P.Singh, C.Kaur, S.P.Das: Physical Review E, 2005, 72[2], 021603 (5pp)