21-Mat-A4 Deformation Behaviour and Properties of Materials · December 2018
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. National Exams, December 2018 — 10-Met-A4, Structure of Materials. Three hours, closed book, one approved calculator (Casio or Sharp). Seven questions of 20 marks each (Roman numerals I–VII); the rubric asks for any five, with only the first five in the answer book marked. All seven 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 VI.2(c) (grain-size strengthening) touches mechanical/deformation properties directly; the paper as a whole is a broad introductory materials-science survey — atomic structure, bonding, crystal structure/directions/planes, point defects, XRD, dislocations and phase diagrams — and is answered as such below.
Reference texts. The answers below are keyed to the works normally recommended for this syllabus code:
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
Given. $2\theta$ peaks at 40°, 58°, 73°, 86.8°, 100.4°, 114.7°; $\lambda=0.154$ nm.
Find. (a) BCC or FCC; (b) lattice constant $a$.
Approach. Compute $\sin^2\theta$ for each peak and form the ratio to the first peak's value. For a cubic structure, $\sin^2\theta \propto h^2+k^2+l^2$, and the two structures give characteristically different ratio sequences: BCC (only $h+k+l$ even reflect) gives ratios $1:2:3:4:5:6:7:8\dots$; FCC ($h,k,l$ all-even or all-odd) gives $1:1.33:2.67:3.67:4:5.33\dots$. Once the structure is identified, Bragg's law $n\lambda=2d\sin\theta$ with $d=a/\sqrt{h^2+k^2+l^2}$ gives $a$ from each peak.
| $2\theta$ (°) | $\theta$ (°) | $\sin^2\theta$ | Ratio to peak 1 | Nearest integer |
|---|---|---|---|---|
| 40.0 | 20.0 | 0.1170 | 1.000 | 1 |
| 58.0 | 29.0 | 0.2350 | 2.009 | 2 |
| 73.0 | 36.5 | 0.3538 | 3.025 | 3 |
| 86.8 | 43.4 | 0.4721 | 4.036 | 4 |
| 100.4 | 50.2 | 0.5903 | 5.046 | 5 |
| 114.7 | 57.35 | 0.7089 | 6.060 | 6 |
| $hkl$ | $h^2{+}k^2{+}l^2$ | $d$ (nm) | $a$ (nm) |
|---|---|---|---|
| (110) | 2 | 0.2252 | 0.3184 |
| (200) | 4 | 0.1590 | 0.3177 |
| (211) | 6 | 0.1296 | 0.3171 |
| (220) | 8 | 0.1123 | 0.3170 |
| (310) | 10 | 0.1004 | 0.3169 |
| (222) | 12 | 0.0916 | 0.3168 |
| Aspect | Scanning Electron Microscopy (SEM) | Transmission Electron Microscopy (TEM) |
|---|---|---|
| (a) Physical principle | A finely focused beam is rastered across the sample surface; secondary and backscattered electrons emitted from near the surface are collected to build a point-by-point image. | A beam is transmitted through an ultra-thin sample; the transmitted/diffracted electrons form a projected image of the internal structure, much like X-ray radiography. |
| (b) Typical accelerating energy | $\sim$1–30 keV | $\sim$100–300 keV (much higher, needed to penetrate the sample) |
| (c) Resolution / magnification | $\sim$1–20 nm resolution; up to $\sim$50,000× | $\sim$0.1–0.2 nm (near-atomic) resolution; up to $\sim$1,000,000× |
(d) Only TEM can reveal sub-surface dislocation activity in a metallic thin sample — SEM images only the surface (or near-surface, via secondary electrons), whereas TEM's transmitted-beam geometry images the full thickness of the electron-transparent foil, so dislocations, stacking faults and precipitates throughout the sample's volume produce diffraction contrast directly visible in the image (this is precisely how dislocation densities and slip activity are studied experimentally, tying back to Question VI's dislocation content).
