Question 5 of 7: Microstructural Characterization

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(a) — First Three XRD Peaks for Powder Nickel (FCC)

Given.

Find. The lattice parameter $a$, the first three allowed $\{hkl\}$ families, their $d$-spacings, and the corresponding $2\theta$ diffraction-peak positions.

Approach. Get $a$ from the FCC hard-sphere relation, apply the FCC structure-factor reflection rule ($h,k,l$ all odd or all even) to find the first three allowed planes in order of increasing $2\theta$ (decreasing $d$), compute $d_{hkl}$ from the cubic relation, and solve Bragg's law for $2\theta$.

1. Lattice parameter (FCC, appendix relation $a=2\sqrt2R$). $$a=2\sqrt2(0.125)=\boxed{0.3536\ \text{nm}}$$
2. Reflection rule and first three allowed planes. FCC reflects only when $h,k,l$ are all odd or all even (structure factor vanishes otherwise). Ranked by increasing $h^2+k^2+l^2$ (decreasing $d$, increasing $2\theta$), the first three allowed families are $\{111\}$, $\{200\}$, $\{220\}$ — note $\{100\}$ and $\{110\}$ are both forbidden (mixed parity).
3. Interplanar spacings, $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$ (appendix). $$d_{111}=\frac{0.3536}{\sqrt3}=0.2041\ \text{nm},\quad d_{200}=\frac{0.3536}{\sqrt4}=0.1768\ \text{nm},\quad d_{220}=\frac{0.3536}{\sqrt8}=0.1250\ \text{nm}$$
4. Bragg angles, $n\lambda=2d\sin\theta$ (appendix, $n=1$). $$\theta_{hkl}=\sin^{-1}\!\left(\frac{\lambda}{2d_{hkl}}\right)$$ $$\theta_{111}=\sin^{-1}\!\left(\frac{0.1542}{2(0.2041)}\right)=22.19^{\circ}\ \Rightarrow\ \boxed{2\theta_{111}=44.38^{\circ}}$$ $$\theta_{200}=\sin^{-1}\!\left(\frac{0.1542}{2(0.1768)}\right)=25.86^{\circ}\ \Rightarrow\ \boxed{2\theta_{200}=51.72^{\circ}}$$ $$\theta_{220}=\sin^{-1}\!\left(\frac{0.1542}{2(0.1250)}\right)=38.08^{\circ}\ \Rightarrow\ \boxed{2\theta_{220}=76.17^{\circ}}$$

V.1(b) — Single Crystal with (100) Parallel to the Surface

In the powder pattern above, every grain orientation is present, so all three allowed families (111), (200), (220) diffract into the detector simultaneously. A single crystal oriented with its (100) plane parallel to the surface, scanned in the usual symmetric $\theta$–$2\theta$ (Bragg–Brentano) reflection geometry, can only diffract from lattice planes that are themselves parallel to the surface — i.e. only the $(h00)$ family. Since $(100)$ itself is forbidden by the FCC reflection rule (mixed parity would not even apply here, but a single unit index is odd — not "all even"), the first allowed reflection from this family is $(200)$, at the same $2\theta_{200}=51.72^{\circ}$ computed above. The (111) and (220) peaks would disappear entirely (those planes are not parallel to the surface in this orientation and cannot satisfy the reflection geometry), leaving essentially a single peak in the scan range considered, at 51.72°, instead of the three-peak powder pattern.

V.1(c) — Distinguishing FCC from BCC by XRD

The two structures obey different reflection (structure-factor) rules: FCC reflects only when $h,k,l$ are all odd or all even, giving allowed $\sin^2\theta$ ratios of $3:4:8:11:12:16:19:20\,\ldots$ (from $h^2+k^2+l^2=3,4,8,11,12,\ldots$); BCC reflects only when $h+k+l$ is even, giving ratios $2:4:6:8:10:12:14:16\,\ldots$ (from $h^2+k^2+l^2=2,4,6,8,\ldots$). Practically, a candidate indexes the measured peaks, computes $\sin^2\theta$ for each peak, forms the ratio of each value to the first (smallest), and compares the resulting integer sequence to the two lists above — a $3:4:8$ pattern of successive ratios (as computed for Ni in V.1(a): $3:4:8$ for (111):(200):(220)) identifies FCC, while a $1:2:3$ (i.e. $2:4:6$ normalized) pattern identifies BCC. This structure-factor-driven "systematic absence" is the standard XRD phase-identification method, requiring no prior knowledge of the lattice parameter.