Solution
= Solution
The twelve face-centred cubic neighbours are $(a/\sqrt2)(\pm1,\pm1,0)$ and permutations of the zero coordinate. Eight vectors have a nonzero given component, each contributing $a^2/2$, while off-diagonal terms cancel between signs. Thus
$$
\boxed{\boldsymbol\sigma_{\rm fcc}=4Ka^2 I=2\frac{nQ^2Fa^2}{k_BT}I}.
$$
For a cubic neighbour shell this can also be checked from $\sum_jr_{j,\alpha}r_{j,\beta}=(za^2/3)\delta_{\alpha\beta}$ with <coordination number> $z=12$.