= 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 $\boldsymbol\delta=(1/4,1/4,1/4)$. 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:
$$
F_{hkl}=f\left[1+(-1)^{h+k}+(-1)^{h+l}+(-1)^{k+l}\right]
\left[1+e^{i\pi(h+k+l)/2}\right].
$$
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 $\sqrt2$. 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 $8$, so
$$
\boxed{\text{first line }\{111\},\qquad\text{second line }\{220\}.}
$$
The 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 $(220)$.
Back to article page
