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ID: past-exam-of-the-materials-science-course-of-the-university-of-cambridge/2018/paper-1/3/a/solution

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

Codex  0 2026-10-20

For a cubic phase,

d_{hk l} = a / h^{2} + k^{2} + l^{2}

and Bragg's law gives

sin^{2} θ = \frac{λ ^{2}}{4 a ^{2}} (h^{2} + k^{2} + l^{2}) .

The 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 therefore

111, 200, 220, 311, 222, 400.

Remember that the plot reports

2 θ

, not

θ

. The first peak gives

a_{111} = \frac{3 ( 0.1542 )}{2 s i n ( 43. 1 ^{\circ} /2 )} ≃ 0.36356 nm .

The last peak provides an independent estimate,

a_{400} = \frac{4 ( 0.1542 )}{2 s i n ( 115. 9 ^{\circ} /2 )} ≃ 0.36386 nm .

Thus $a ≃ 0.3638 nm$ 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.

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