Silver chloride crystallises in the rock-salt structure and disorders by the Frenkel…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction

Solid State Physics, Ch Point defects, dislocations and alloys

Problem

Silver chloride crystallises in the rock-salt structure and disorders by the Frenkel mechanism on the cation sublattice: a silver ion leaves a regular cation site (there are

N

such sites) for one of the tetrahedral interstitial voids of the close-packed anion sublattice, of which there are two for every cation site, so the number of available interstitial sites is

N'=2N

. Let

n

be the number of Frenkel pairs present in equilibrium at temperature

T

(equal to the number of cation vacancies and to the number of occupied interstitials), and let

E_F

be the energy to form one pair. ((a)) By counting the number of ways of distributing

n

vacancies over the

N

cation sites and, independently,

n

interstitial ions over the

N'

interstitial sites, and minimising the free energy

F=nE_F-k_BT\ln\Omega(n)

with Stirling's approximation for

n\ll N,N'

, show that

n=\sqrt{NN'}\,e^{-E_F/2k_BT}

. ((b)) Explain in one sentence why no separate charge-neutrality condition has to be imposed here, unlike for a Schottky pair in a binary ionic crystal. ((c)) Take

E_F=1.4\,\mathrm{eV}

and evaluate the equilibrium fraction

n/N

of cation sites that are vacant at

T=600\,\mathrm{K}

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