Past exam of the condensed matter physics course of the University of Oxford / 2013 / 1 / a / Solution

The sites form a face-centered cubic lattice decorated by an identical two-atom atomic basis of a crystal, at fractional displacements $0$ and $\mathbf{\delta }=(1/4,1/4,1/4)$. This is the diamond cubic structure; the eight atoms in its conventional cube are not eight independent Bravais-lattice points.

For identical silicon atomic scattering factors $f$, the conventional-cell structure factor separates into the FCC factor and the two-site motif factor:The first factor vanishes unless $h,k,l$ are all even or all odd. For all-even indices, the motif also vanishes if $h+k+l=4r+2$; it survives if the sum is divisible by four. All-odd indices survive, with motif modulus $\sqrt{2}$. Thus $(100)$ and $(110)$ are absent by centering, $(200)$ is absent by the diamond basis, and $(111)$ and $(220)$ are allowed.

For a cubic cell, $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$. At fixed X-ray wavelength, Bragg's law makes the smallest scattering angles correspond to the smallest allowed positive values of $h^2+k^2+l^2$. These start with $3$ and $8$, soThe braces denote whole powder families, including their symmetry-equivalent plane normals. The next family is $\{311\}$, with squared-index sum $11$; using the FCC rule alone would incorrectly insert a $(200)$ line before $(220)$.