= Solution
{c}
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
$$
\rho=\frac{p_c-1/4}{1-1/4},\qquad p_c=\frac{1+3\rho}{4}.
$$
The complementary <copper> <probability> is $3(1-\rho)/4$, so the corner-site <atomic scattering factor> is
$$
\boxed{\langle f_c\rangle=\frac14\left[(1+3\rho)f_{\rm Au}+3(1-\rho)f_{\rm Cu}\right].}
$$
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 $p_c+3p_f=1$. Therefore
$$
p_f=\frac{1-\rho}{4},\qquad 1-p_f=\frac{3+\rho}{4},\qquad
\boxed{\langle f_f\rangle=\frac14\left[(1-\rho)f_{\rm Au}+(3+\rho)f_{\rm Cu}\right].}
$$
At $\rho=1$ 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 $0\leq\rho\leq1$ and equivalent face-site averages.
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