Question

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?

• A. (200)
• B. (220)
• C. (311)
• D. (222)

Hint:

In FCC lattices, use the condition “all odd or all even” for allowed planes and compare $h^2+k^2+l^2$ values with normalized $\sin^2 \theta$ data to find missing peaks.

Answer 1

The Correct Option is B

Step 1: Recall the Bragg condition for cubic systems.
For a cubic system: \[ \sin^2 \theta \propto \frac{h^2 + k^2 + l^2}{a^2} \] Here, $(hkl)$ are Miller indices and $a$ is the lattice constant.
Step 2: Allowed reflections for FCC lattice.
In FCC structures, reflections are allowed only when $h, k, l$ are either all odd or all even. Thus the allowed planes are: (111), (200), (220), (311), (222), (400), etc.
Step 3: Calculate relative $h^2+k^2+l^2$ values.
\[ \begin{aligned} (111) & : 1^2+1^2+1^2 = 3
(200) & : 2^2+0+0 = 4
(220) & : 2^2+2^2+0 = 8
(311) & : 3^2+1^2+1^2 = 11
(222) & : 2^2+2^2+2^2 = 12
\end{aligned} \]
Step 4: Normalize the given $\sin^2 \theta$ values.
Take ratios with respect to the first value ($0.136$): \[ \frac{0.185}{0.136} \approx 1.36, \frac{0.504}{0.136} \approx 3.71, \frac{0.544}{0.136} \approx 4.00 \]

Step 5: Compare with theoretical ratios.
Theoretical ratios (relative to 3 for (111)): \[ \frac{4}{3} \approx 1.33, \frac{8}{3} \approx 2.67, \frac{11}{3} \approx 3.67, \frac{12}{3} = 4.00 \] Given experimental ratios: \[ 1,\; 1.36,\; 3.71,\; 4.00 \] This matches closely with theoretical ratios: \[ 1,\; \frac{4}{3},\; \frac{11}{3},\; 4 \]
Step 6: Identify missing reflection.
The ratio $\frac{8}{3} = 2.67$ (corresponding to (220)) is absent in the experimental data. Final Answer: \[ \boxed{(220)} \]