 Solution

ID: past-exam-of-the-condensed-matter-physics-course-of-the-university-of-oxford/2022/1/d/i/solution

Specify the normalization of the alloy sublattice order parameter rather than assuming that its endpoint values uniquely define it. Let

p_{c}

be the probability of gold occupation at a corner site and define

ρ = \frac{p _{c} - 1/4}{1 - 1/4}, p_{c} = \frac{1 + 3 ρ}{4} .

The complementary copper probability is

3 (1 - ρ) /4

⟨ f_{c} ⟩ = \frac{1}{4} [(1 + 3 ρ) f_{Au} + 3 (1 - ρ) f_{Cu}] .

The stoichiometry provides one gold chemical atom per four-site unit cell. If all three face sublattices are equivalent, their gold occupation probability

p_{f}

obeys

p_{c} + 3 p_{f} = 1

. Therefore

p_{f} = \frac{1 - ρ}{4}, 1 - p_{f} = \frac{3 + ρ}{4}, ⟨ f_{f} ⟩ = \frac{1}{4} [(1 - ρ) f_{Au} + (3 + ρ) f_{Cu}] .

ρ = 1

the corner and face sites carry pure gold and pure copper; at

ρ = 0

they have identical statistical compositions. These formulas describe the branch with

0 \leq ρ \leq 1

and equivalent face-site averages.

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