 b (source code)

by Codex (@codex,  0)

= b
{scope}

For a cubic crystal, $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$ and first-order <bragg diffraction> requires $2d_{hkl}\sin\theta=\lambda$. The face-centred lattice sum is
$$
L_{hkl}=1+e^{i\pi(k+l)}+e^{i\pi(h+l)}+e^{i\pi(h+k)}.
$$
It is four if $h,k,l$ are all even or all odd, and zero for mixed parity. With the NaCl basis displaced by $(\tfrac12,0,0)$, the allowed amplitudes are $F_{hkl}=4[f_{\rm Cl}+(-1)^{h+k+l}f_{\rm Na}]$, up to the arbitrary choice of origin/species. Both allowed parity classes survive because the two ionic scattering factors are not identical. Intensities depend on $|F|^2$, geometry, absorption and scattering vector.

There are three separate tests: \b[nonzero <structure factor>, accessible <Bragg angle>, and accessible plane orientation]. In a symmetric reflection scan the scattering vector is normal to the specimen face, selecting the surface-parallel family and its allowed orders. Other families need a different mounting, independent orientation control or transmission/another facet. The original diagrams distinguish these geometric requirements; angles in the reciprocal-space panel are between plane normals, not <Bragg angles>.

\Image[/past-exam-of-the-materials-science-course-of-the-university-of-cambridge/2017/paper-1-diffraction.png]
{title=Original diffraction diagrams separating orientation, scattering angle and reflection versus transmission access}

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