 Solution

Diffraction thermal-expansion thermometry needs a stable monochromatic X-ray beam, controlled collimation, an accurately calibrated angular scale, and an X-ray-sensitive detector with enough counts to fit reproducible line centroids. Crystal monochromators or appropriate energy selection are relevant here; an apparatus for visible-light interference does not directly measure these X-ray peaks. Sample displacement and angular-zero drift should be checked with a suitable reference standard.

The first-order angular sensitivities are about $6.93\times 10^{-5}$ and $1.17\times 10^{-4}$ degrees per kelvin, respectively. A one-kelvin inference therefore requires comparable centroid stability, with high count statistics and control of systematic errors; a centroid can be fitted more precisely than the line width. Fractional wavelength drift also mimics lattice expansion, so the desired temperature precision constrains $\Delta \lambda /\lambda$ relative to ${\alpha }_L\Delta T$.

The powder must be at a known uniform temperature and allowed to equilibrate. Mechanical stress, clamping, composition changes, oxidation and phase changes can also shift or broaden peaks. Accurate work requires a calibrated $a(T)$, including the actual temperature dependence of the thermal expansion coefficient, rather than assuming the supplied constant describes every temperature exactly. Using both lines provides a consistency check on a common cubic expansion.