How many vacancies are there?

The equilibrium number of vacancies formed as a result of thermal vibrations may be calculated from thermodynamics:

where Ns is the number of regular lattice sites, kB is the Boltzmann constant, Qv is the energy needed to form a vacant lattice site in a perfect crystal, and T the temperature in Kelvin (note, not in oC or oF).

Using this equation we can estimate that at room temperature in copper there is one vacancy per 1015 lattice atoms, whereas at high temperature, just below the melting point there is one vacancy for every 10,000 atoms. Note, that the above equation gives the lower end estimation of the number of vacancies, a large numbers of additional (nonequilibrium) vacancies can be introduced in a growth process or as a result of further treatment (plastic deformation, quenching from high temperature to the ambient one, etc.)

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