Past exam of the materials science course of the University of Cambridge / 2018 / Paper 1 / 3

For a cubic phase, $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$ and Bragg's law givesThe first six FCC-allowed squared-index sums are $3,4,8,11,12,16$, from indices all odd or all even. The labels1–6 are thereforeRemember that the plot reports $2\theta$, not $\theta$. The first peak givesThe last peak provides an independent estimate,Thus $a\simeq 0.3638\mathrm{nm}$ is appropriate to the supplied precision. The two estimates differ by about $0.08\%$; high-angle data are less sensitive to a fixed angular error, so the $400$ estimate is useful for the composition calculation. The printed angles do not justify many more significant figures or an exact fit of all six positions.

Sample A is the Cu–Au alloy. At the indicated copper-rich composition, the supplied Cu–Au diagram permits one copper-rich solid-solution phase. The Cu–Ag diagram instead gives two equilibrium solid phases at room temperature. A single consistent set of FCC peaks therefore identifies A under the stated equilibrium assumptions; similar pure-metal lattice parameters alone would not distinguish gold from silver.

Use Vegard's law as an approximate atomic-fraction interpolation:With the higher-angle estimate from part(a),This is about $5.1$ atomic percent, not weight percent. Using rounded molar masses $M_{\mathrm{Au}}\simeq 197$ and $M_{\mathrm{Cu}}\simeq 63.55\mathrm{g}\mathrm{mo}{\mathrm{l}}^{-1}$,Using the first peak alone instead gives roughly $12.6wt\%$. The justified conclusion is therefore about $13$–$14wt\%$ gold, with uncertainty from angle rounding, calibration and the approximate linear lattice-composition relation.

Sample B should show two FCC peak families, one from a copper-rich phase and one from a silver-rich phase. Their lattice parameters are set by the compositions of the coexisting phases, not by one lattice parameter at the overall alloy composition.

For corresponding Miller indices the silver-rich phase has a larger lattice parameter and hence a larger interplanar spacing, so its reflection occurs at smaller $2\theta$. Some reflections may overlap or be weak. Relative intensities depend on phase fractions, atomic scattering factors, absorption and texture; a raw peak-height ratio is not directly a mass fraction.

At room temperature the mutual solubilities in the supplied Cu–Ag diagram are small, so the two families would be near the elemental copper and silver positions. Slow cooling is important: a metastable supersaturated single phase would be a different interpretation. Quantitative phase or composition analysis would require calibrated diffraction and an appropriate refinement, not merely counting the visible peaks.

SEM with energy-dispersive X-ray spectroscopy can identify gold versus silver through their characteristic X-ray energies. Acquire spectra and maps from representative regions: two compositionally distinct regions would support the two-phase interpretation of B. With standards or validated matrix corrections it can estimate local compositions and, with representative sampling, an overall composition. Surface topography and the interaction volume limit spatial interpretation.

X-ray fluorescence provides a complementary more bulk-averaged elemental measurement. Gold and silver produce different characteristic lines; calibrated intensities can yield their mass fractions, with corrections for absorption, excitation and specimen geometry. It does not by itself determine whether an element occupies one or two crystallographic phases.

Both methods supply elemental information missing from a lattice-parameter-only identification. A purely morphological micrograph could reveal phase separation but would not alone give a reliable numerical chemical composition.