Past exam of the physics A course of the University of Cambridge / 2019 / Paper 2 / Section C / C10 / e / Solution

Use temperature-dependent X-ray diffraction, with the crystal in a controlled heating stage and a detector measuring diffracted directions. A reflection occurs when the scattering wavevector matches a reciprocal lattice vector, or equivalently when Bragg's law $2d_{hkl}\sin \theta =\lambda$ is satisfied. Record peak positions and intensities while heating to infer changes in lattice spacings and structure.

A monatomic conventional BCC cell has nonzero crystal basis structure factor only when $h+k+l$ is even. For FCC the indices must be all even or all odd. The change in reflection selection and peak positions therefore distinguishes the phases. For instance, BCC starts with a $(110)$ family whereas FCC starts with $(111)$, when indexed in their own conventional cells.

The transformation can create multiple orientation variants, grains, strain and phase coexistence. A reciprocal point that formerly satisfied the diffraction geometry may disappear or move off a fixed detector; a loss of one spot alone does not identify the new phase. Thermal expansion shifts peaks, thermal motion weakens coherent intensities, and hot iron may oxidize without a protective atmosphere.

After transformation, scan crystal angle and scattering angle or use an area detector to locate the new reflections. If single-crystal order has been lost, powder X-ray diffraction or a sufficiently broad orientation survey collects rings or many grain reflections instead of relying on the original alignment. Index several new peaks using the FCC selection rule and fit their common lattice parameter; this probes the new structure even when the original crystal orientation is no longer usable.