The crystal structure of a cubic metal can be determined by analyzing the positions of diffraction peaks in an X-ray powder diffraction pattern using Bragg's Law and the relationship between interplanar spacing ($d_{hkl}$) and the lattice parameter ($a$) for cubic systems.

Analyzing Diffraction Peaks

Bragg's Law is given by:

$n\lambda = 2d\sin\theta$

For cubic crystals, the interplanar spacing is:

$d_{hkl} = \frac{a}{\sqrt{h^2+k^2+l^2}}$

Combining these, we get:

$\sin^2\theta = \frac{\lambda^2}{4a^2}(h^2+k^2+l^2)$

Let $S = h^2+k^2+l^2$. Then:

$\sin^2\theta = \left(\frac{\lambda^2}{4a^2}\right) S$

The term $\left(\frac{\lambda^2}{4a^2}\right)$ is a constant for a given experiment. Therefore, $\sin^2\theta$ is directly proportional to $S$. We can calculate the ratio $S$ for each peak.

Calculating S Values

Given:

• Lattice parameter, $a = 3.92 \text{ Å}$
• Wavelength, $\lambda = 1.5405 \text{ Å}$
• Diffraction angles ($2\theta$): $39.7^\circ, 46.2^\circ, 67.5^\circ, 81.3^\circ$

First, calculate the constant factor:

$K = \frac{\lambda^2}{4a^2} = \frac{(1.5405 \text{ Å})^2}{4 \times (3.92 \text{ Å})^2} \approx \frac{2.3731}{61.4656} \approx 0.03860$

Now, calculate $\theta$ and $\sin^2\theta$ for each peak and then find $S = \frac{\sin^2\theta}{K}$.

The sequence of calculated $S = h^2+k^2+l^2$ values is approximately 3, 4, 8, 11.

Identifying Crystal Structure from S values

We compare this sequence with the allowed reflections for different cubic structures:

• Simple Cubic (SC): All $(hkl)$ are allowed. Possible $S$ values: 1, 2, 3, 4, 5, 6, 8,... The observed sequence starts with $S=3$, but $S=1$ or $S=2$ would be expected first if allowed. This doesn't match.
• Body-Centered Cubic (BCC): Allowed reflections require $(h+k+l)$ to be even. Possible $S$ values: 2 (110), 4 (200), 6 (211), 8 (220), 10 (310),... The sequence must start with $S=2$. Our observed first peak corresponds to $S=3$. This doesn't match.
• Face-Centered Cubic (FCC): Allowed reflections require $(hkl)$ to be either all odd or all even. Possible $S$ values: 3 (111), 4 (200), 8 (220), 11 (311), 12 (222), 16 (400),... The observed sequence (3, 4, 8, 11) perfectly matches the allowed $S$ values for FCC.

Conclusion

Based on the analysis of the diffraction peak data and the calculated $h^2+k^2+l^2$ values, the crystal structure of the metal is face-centered cubic (fcc).