Schottky Defects

N atoms from a perfect crystal of total number of atoms  are displaced to the surface of the crystal. Let ε₀ be the energy needed to displace one atom from the bulk of the crystal to the surface. Find the equilibrium number of defects N at low temperatures τ << ε₀, assuming

SOLUTION

When N atoms are displaced to the surface, they leave the same number of vacancies. Now there are N vacancies and atoms in  lattice points. The entropy as a function of N is

(1)

where we have used Stirling’s formula

The free energy may be written

(2)

The minimum of the free energy can be found to be

(3)

or

(4)

Since  we have

(5)

which is what one would expect.