17-Phys-B7 Structure of Materials · December 2016
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. 98-Phys-B7 Structure of Materials, National Examination December 2016 — a closed-book examination (Casio or Sharp approved calculators only; all necessary equations, constants, the error-function table and the Cu–Ag phase diagram are supplied in the paper's own appendix). Candidates attempt any five of the seven questions, each worth 20 marks; every question is nonetheless answered in full below so the paper remains a complete study resource. This sitting numbers its questions with Roman numerals (Question I–VII) while sub-items inside each question use Arabic numerals (1., 2., 3.).
Reference texts. W. D. Callister Jr. & D. G. Rethwisch, Materials Science and Engineering: An Introduction, 10th ed. (atomic bonding, crystal structure and packing, point defects, diffusion, dislocations and slip, mechanical properties, phase diagrams and the lever rule, X-ray diffraction); D. J. Griffiths, Introduction to Quantum Mechanics, 3rd ed. (Bohr model, de Broglie wavelength, Heisenberg uncertainty).
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.
Part 1 — Given. BCC iron, $2\theta_{110}=44.704^\circ$, $\lambda=0.1541$ nm ($n=1$).
Find. Lattice constant $a$, and the incidence angle $\theta_{211}$ for $\{211\}$.
Part 2 — Given. Two electron-beam imaging techniques: SEM and TEM.
Find. Their differences in principle, beam energy, resolution/magnification, and 3-D imaging ability; which reveals sub-surface dislocations.
| Aspect | SEM | TEM |
|---|---|---|
| (a) Physical principle | A finely focused beam is raster-scanned across the sample surface; secondary and backscattered electrons ejected from a shallow surface layer are collected to build the image | A broad, high-energy beam is transmitted THROUGH an electron-transparent thin foil; the transmitted/diffracted electrons form the image (bright-field/dark-field diffraction contrast) |
| (b) Typical beam energy | $\sim1$–30 keV | $\sim100$–300 keV (much higher, needed to transmit through the sample) |
| (c) Resolution / magnification | $\sim1$–20 nm; up to $\sim10^5\times$ | Sub-nanometre (near-atomic); up to $\sim10^6\times$ or beyond — substantially finer than SEM |
| (d) 3-D imaging | Large depth of field gives a strongly topographic, pseudo-3-D appearance from a single image | Produces a 2-D projection through the foil thickness; no inherent depth information from one image (3-D reconstruction needs tilt-series tomography) |
Because dislocations are internal, bulk crystal defects, only a technique that images through the material can reveal them: $$\boxed{\text{TEM}}$$ (via diffraction contrast in a thin foil) is the standard technique for imaging sub-surface dislocation activity; SEM only probes the near-surface region and cannot directly resolve internal dislocations.
| Quantity | Value |
|---|---|
| $d_{110}$ | $0.2026$ nm |
| Lattice constant $a$ | $0.2865$ nm |
| $d_{211}$ | $0.1170$ nm |
| $\theta_{211}$ ($2\theta_{211}$) | $41.2^\circ$ ($82.4^\circ$) |
| Reveals sub-surface dislocations | TEM |
