Discussion
A student performed X-ray diffraction experiment on a FCC polycrystalline pure metal. The following \(\sin^{2}\theta\) values were calculated from the diffraction peaks.
\[ \sin^{2}\theta = 0.136,\ 0.185,\ 0.504,\ 0.544 \]
However, the student was negligent and missed noting one of the peaks. Which one of the following Miller indices corresponds to the missing peak?
Answer: B
The missing peak is (220).
For a cubic crystal, \(\sin^{2}\theta = \frac{\lambda^{2}}{4a^{2}}(h^{2} + k^{2} + l^{2})\). In FCC, \(h, k, l\) must be all odd or all even, so the first reflections are (111), (200), (220), (311), (222), with \(h^{2} + k^{2} + l^{2}\) = 3, 4, 8, 11, 12.
Divide each measured value by these numbers to find the constant \(\lambda^{2}/4a^{2}\):
\[ \begin{aligned} 0.136/3 &= 0.0453 \\ 0.185/4 &= 0.0463 \\ 0.504/11 &= 0.0458 \\ 0.544/12 &= 0.0453 \end{aligned} \]
All four give the same constant, about 0.0455, when matched to (111), (200), (311) and (222). The reflection with sum 8 is the one not recorded: (220), expected at \(\sin^{2}\theta \approx 8 \times 0.0455 = 0.364\). (200), (311) and (222) are already in the list.
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