Frenkel Defects

N atoms are arranged regularly to form a perfect crystal. If one replaces n atoms among them from lattice sites to interstices of the lattice, this becomes an imperfect crystal with n defects (of the Frenkel type). The number  of interstitial sites into which an atom can enter is of the same order as N. Let ε₀ be the energy necessary to remove an atom from a lattice site to an interstitial site. Show that, in the equilibrium state at temperature τ such that τ << ε₀, the following relation is valid:

SOLUTION

We assume that the number of defects created around one lattice site does not affect the process of creating new defects. In other words, all configurations of the system are independent (not a very realistic assumption in general, but since τ << ε₀ and the number of defects it can be used as an approximation). The vacancies and interstices then can be distributed in ω_v and ω_i ways, respectively:

(1)

The total number of possible configurations of the system, ω, is given by

(2)

The entropy, S, may be written

(3)

Using Stirling’s formula

(4)

we obtain, from (3),

(5)

Using (5) and the fact that the total energy of the system E = nε₀, we have

(6)

or

(7)

and

(8)

The condition ε << τ implies that  and therefore

(9)