Past exam of the materials science course of the University of Cambridge / 2017 / Paper 1 / 3 / b / ii

The $(111)$ reflection is allowed, with $d_{111}=0.3233\mathrm{nm}$ and $2{\theta }_B=27.56^{\circ }$. Its odd-parity amplitude is proportional to $f_{\mathrm{Cl}}-f_{\mathrm{Na}}$, generally weaker than the corresponding sum. Its normal makes ${\cos }^{-1}(1/\sqrt{3})=54.74^{\circ }$ with the $[100]$ surface normal.

It is therefore not obtained in the ordinary symmetric scan of that face, despite its modest Bragg angle. In a usual edge-aligned cubic mounting its normal also has an out-of-horizontal-plane component. A suitable multi-axis reorientation, or a crystal cut with a $(111)$ face, makes the reflection accessible; a thin specimen in transmission or a powder experiment is another route. Tilting the scattering vector does not by itself ensure front-face reflection: because $54.74^{\circ }$ exceeds ${\theta }_B=13.78^{\circ }$, incident and outgoing rays for that highly asymmetric orientation cannot both lie outside the same flat $(100)$ face. A suitable specimen shape/transmission path or recut face is needed.