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21-Mat-A4 Deformation Behaviour and Properties of Materials · December 2016

Question 5 of 7: Microstructural Characterization (20 marks)

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. National Exams, December 2016 — 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 (dislocation theory) is genuinely deformation/mechanical-properties content; the paper as a whole is a broad introductory materials-science survey — electron structure, bonding, crystal structure, crystallographic directions/planes, solid solubility, XRD, microscopy 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 V — Microstructural Characterization (20 marks)

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.

V.1 — XRD lattice constant and the {211} angle

Given. BCC iron; $\{110\}$ peak at $2\theta=44.704^{\circ}$; wavelength $\lambda=0.1541$ nm.

Find. Lattice constant $a$; the diffraction (incidence) angle $\theta$ for $\{211\}$.

Approach. Bragg's law $\lambda=2d\sin\theta$ gives the $\{110\}$ interplanar spacing $d_{110}$; the cubic spacing relation $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$ (appendix) then gives $a$. The same $a$ substituted back into the spacing relation for $\{211\}$, then into Bragg's law, gives the new angle. (Both reflections satisfy the BCC selection rule $h+k+l=$ even: $1{+}1{+}0=2$ and $2{+}1{+}1=4$, so both are genuinely allowed BCC peaks.)

  1. {110} interplanar spacing. $$\theta_{110}=\frac{44.704^{\circ}}{2}=22.352^{\circ} \quad\Rightarrow\quad d_{110}=\frac{\lambda}{2\sin\theta_{110}}=\frac{0.1541}{2\sin(22.352^{\circ})}=0.2026\ \text{nm}$$
  2. Lattice constant. $$a=d_{110}\sqrt{1^2+1^2+0^2}=0.2026\sqrt2=\boxed{0.2865\ \text{nm}}$$ (matches the 0.2866 nm quoted directly for BCC iron in Question III.1, confirming the method.)
  3. {211} spacing and angle. $$d_{211}=\frac{a}{\sqrt{2^2+1^2+1^2}}=\frac{0.2865}{\sqrt6}=0.1170\ \text{nm}$$ $$\sin\theta_{211}=\frac{\lambda}{2d_{211}}=\frac{0.1541}{2(0.1170)}=0.6587 \quad\Rightarrow\quad \theta_{211}=\boxed{41.2^{\circ}}\ \ (2\theta_{211}=82.4^{\circ})$$
Question V.1 — summary
QuantityResult
$d_{110}$0.2026 nm
Lattice constant $a$0.2865 nm
$d_{211}$0.1170 nm
Incidence angle $\theta_{211}$41.2°

V.2 — SEM vs. TEM

Scanning vs. transmission electron microscopy
SEMTEM
(a) Physical principleA finely focused electron beam is rastered across the sample surface; secondary and backscattered electrons ejected from near the surface are collected to build a point-by-point image.A broad, high-energy electron beam is transmitted through an electron-transparent thin foil sample; the transmitted/diffracted beam intensity, magnified by projector lenses, forms the image (analogous to an optical slide projector).
(b) Typical electron energy∼1–30 keV∼100–300 keV (needs enough energy to punch through a thin foil)
(c) Resolution / magnification∼1–10 nm resolution; up to about 100,000×∼0.1–0.2 nm (near-atomic) resolution; up to about 1,000,000×
(d) 3-D imagingYes — large depth of field gives images with a strongly three-dimensional, topographic appearance.No — the image is a 2-D projection through the full foil thickness; depth information is lost (though can be partly recovered by tomography/tilt series).

Sub-surface dislocation activity. TEM is the technique that reveals it. Because the electron beam is transmitted through a (typically 50–200 nm thick) electron-transparent foil, diffraction contrast from the strain field around each dislocation is imaged throughout the foil's full thickness — not just its outer surface — making TEM the standard technique for directly imaging dislocation lines, networks and their evolution under load. SEM, imaging only secondary/backscattered electrons from a shallow near-surface layer, cannot see dislocation structure buried beneath the surface.