21-Mat-A4 Deformation Behaviour and Properties of Materials · May 2016
Question 6 of 8: X-ray Diffraction (20 marks)
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. National Exams, May 2016 — 10-Met-A4, Structure of Materials. Three hours, closed book, one approved calculator (Casio or Sharp). Eight questions of 20 marks each (Roman numerals I–VIII); the rubric asks for any five, with only the first five in the answer book marked. All eight are solved here, because this set is a study resource rather than an exam script. All necessary equations, constants and an error-function table are provided in the exam's own appendix (reproduced where used below).
Note on the exam title. The printed exam header reads 10-Met-A4, Structure of Materials. Only Question VIII (mechanical deformation) is genuinely deformation/mechanical-properties content; the paper as a whole is a broad introductory materials-science survey — bonding, crystal structure, crystallographic directions/planes, imperfections, phase diagrams, XRD and diffusion — and is answered as such below.
Reference texts. The answers below are keyed to the works normally recommended for this syllabus code:
W. D. Callister and D. G. Rethwisch, Materials Science and Engineering: An Introduction, 10th ed. — atomic bonding, crystal structure, crystallographic directions/planes, imperfections, diffusion, XRD, mechanical properties, phase diagrams.
G. E. Dieter, Mechanical Metallurgy, 3rd ed. — stress–strain behaviour, elastic–plastic response.
Approach. Bragg's law gives $\sin^2\theta \propto (h^2+k^2+l^2)$ for a cubic system. The ratio of $\sin^2\theta$ for the first two peaks distinguishes the structures: BCC's first two allowed reflections are $\{110\}$ ($h^2+k^2+l^2=2$) and $\{200\}$ (=4), ratio 2.00; FCC's are $\{111\}$ (=3) and $\{200\}$ (=4), ratio 1.333.
Compute $\sin^2\theta$ for the first two peaks. $\theta_1=20^{\circ}$, $\theta_2=29^{\circ}$:
$$\sin^2\theta_1=0.1170,\qquad \sin^2\theta_2=0.2350$$
Ratio.
$$\frac{\sin^2\theta_2}{\sin^2\theta_1}=\frac{0.2350}{0.1170}=2.01\approx\boxed{2.0\ \Rightarrow\ \text{BCC}}$$
(a ratio near 1.33 would indicate FCC; 2.01 matches BCC's $\{110\}/\{200\}$ ratio of 2 to within rounding of the printed angles).
VI.2 — Lattice constant
Approach. Use Bragg's law $n\lambda=2d\sin\theta$ with $n=1$ and the cubic interplanar spacing $d=a/\sqrt{h^2+k^2+l^2}$, applied to the first peak $\{110\}$.
Interplanar spacing from the $\{110\}$ peak.
$$d_{110}=\frac{\lambda}{2\sin\theta_1}=\frac{0.154}{2\sin20^{\circ}}=\frac{0.154}{0.6840}=0.2251\ \text{nm}$$
Lattice constant.
$$a=d_{110}\sqrt{1^2+1^2+0^2}=0.2251\sqrt2=\boxed{0.318\ \text{nm}}$$
Cross-check from the $\{200\}$ peak: $d_{200}=\lambda/(2\sin29^{\circ})=0.1588$ nm, $a=d_{200}\sqrt4=0.318$ nm — consistent.
VI.3 — First two angles for the other structure (FCC)
Approach. Keeping the same lattice constant $a=0.318$ nm (same real element, hypothetically re-indexed under the other structure's first two reflections), compute $2\theta$ for FCC's $\{111\}$ and $\{200\}$.
The four diffracting-plane families: BCC {110} (a face-diagonal plane through the cell) and {200} (a cube-face plane, the (100) plane doubled by the {200} reflection order); FCC {111} (an octahedral body-diagonal plane) and {200} (a cube-face plane).
