21-Mat-A3 Structure and Characterization of Materials, Dec-12-Mtl-A3 2018, Q6

Question 6 (10 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.

Given.

Quantity	Value
$2\theta_1$	44.4°
$2\theta_2$	64.6°
$2\theta_3$	81.7°
Radiation	CuK$\alpha$, $\lambda\approx0.1541$ nm

Find. Whether the powder is BCC or FCC.

Approach. Compute $\sin^2\theta$ for each peak from Bragg's law, form the ratio to the first peak, and compare the integer ratio pattern to the cubic selection-rule sequences: BCC ($h+k+l$ even) gives $h^2+k^2+l^2 = 2,4,6,8,\ldots$ (ratio 1:2:3:4); FCC ($h,k,l$ all odd or all even) gives $3,4,8,11,\ldots$ (ratio 1:1.33:2.67:3.67).

Compute $\sin^2\theta$ for each peak. $\theta_1=22.2^\circ,\ \theta_2=32.3^\circ,\ \theta_3=40.85^\circ$: $$\sin^2\theta_1=0.1427,\quad\sin^2\theta_2=0.2856,\quad\sin^2\theta_3=0.4269$$
Normalize to the first peak. $$\frac{\sin^2\theta_2}{\sin^2\theta_1}=2.00,\qquad\frac{\sin^2\theta_3}{\sin^2\theta_1}=2.99$$ giving the ratio sequence $1:2:3$.
Match against the cubic selection rules. The BCC sequence $2:4:6\ (\div2\Rightarrow1:2:3)$ matches exactly (peaks are the (110), (200), (211) reflections). The FCC sequence $3:4:8\ (\div3\Rightarrow1:1.33:2.67)$ does not match the observed $1:2:3$ at all. $$\boxed{\text{BCC}}$$
Bonus — lattice parameter (not requested, included as a cross-check). From the (110) peak, $d_{110}=\lambda/(2\sin\theta_1)=0.1541/(2\times0.3778)=0.2039$ nm, and $a=d_{110}\sqrt{2}=0.288$ nm — close to $\alpha$-iron's $a=0.2866$ nm, a physically reasonable BCC metal.

Quantity	Value
$\sin^2\theta$ ratio pattern	1 : 2 : 3
Matches	BCC $(110),(200),(211)$
Crystal structure	BCC
Lattice parameter (cross-check)	$a\approx0.288$ nm

Concept & Theory

Topic: X-ray diffraction structure-factor selection rules for cubic lattices.

Key relations: Bragg's law $\lambda=2d\sin\theta$; cubic interplanar spacing $d=a/\sqrt{h^2+k^2+l^2}$; BCC allowed reflections require $h+k+l$ even; FCC allowed reflections require $h,k,l$ all odd or all even.

Why this works: The structure factor $F_{hkl}$ vanishes for forbidden $(hkl)$ combinations because atoms at the body-center (BCC) or face-centers (FCC) scatter exactly out of phase with the corner atoms for those reflections; only the allowed combinations survive, producing the characteristic $\sin^2\theta$ ratio "fingerprint" used here.

Common pitfall: Comparing raw $2\theta$ or $\theta$ ratios directly instead of $\sin^2\theta$ ratios — Bragg's law is linear in $\sin\theta$, not $\theta$, so only the $\sin^2\theta$ ratios reduce to the small-integer $h^2+k^2+l^2$ pattern.

Source: Cullity & Stock, Elements of X-Ray Diffraction, Ch. 4 (cubic structure-factor rules); Callister & Rethwisch, Ch. 3.19–3.20.

Question 6 of 8

Question 6 (10 marks)