Optimum vacancy concentration in a crystal

The importance of the presence of a small fraction of vacancies in a crystal structure is demonstrated from considerations of thermodynamic stability. We include in the density functional theory the effects due to the distortion of the lattice structure surrounding the vacancy and show that the free energy is less when vacancies are present. Near freezing point, our theoretical model obtains the equilibrium vacancy fraction in the hard sphere crystal to be ~10-5 and it decreases with increase of the density