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 defineThe complementary copper probability is $3(1-\rho )/4$, so the corner-site atomic scattering factor isThe 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 $p_c+3p_f=1$. ThereforeAt $\rho =1$ the corner and face sites carry pure gold and pure copper; at $\rho =0$ they have identical statistical compositions. These formulas describe the branch with $0\le \rho \le 1$ and equivalent face-site averages.