Question

A diffraction pattern is obtained from a powdered sample of a pure element, which has FCC crystal structure. If \(x\) and \(y\) are the Bragg angles of the first and the third peaks, respectively, then the ratio \(\dfrac{\sin y}{\sin x}\) is (rounded off to one decimal place).

Hint:

In cubic systems, \(\sin^2 \theta\) is proportional to \(h^2+k^2+l^2\). For FCC, remember only “all odd or all even” reflections are allowed.

Answer 1

Correct Answer: 1.4

Step 1: Bragg’s law.
Bragg’s law states: \[ n\lambda = 2 d \sin \theta \] where \(d\) is the interplanar spacing, \(\theta\) is the Bragg angle, and \(\lambda\) is the wavelength. Since the wavelength is constant, \(\sin \theta \propto \frac{1}{d}\).

Step 2: Interplanar spacing for cubic lattice.
For a cubic crystal, \[ d_{hkl} = \frac{a}{\sqrt{h^2+k^2+l^2}} \] Thus, \[ \sin \theta \propto \sqrt{h^2+k^2+l^2} \]

Step 3: Allowed reflections in FCC.
FCC reflections are allowed only if \(h,k,l\) are all odd or all even. So, the first few reflections are: - (111): \(h^2+k^2+l^2 = 3\) - (200): \(h^2+k^2+l^2 = 4\) - (220): \(h^2+k^2+l^2 = 8\) Hence, the first peak corresponds to (111), the second to (200), and the third to (220).

Step 4: Ratio of \(\sin \theta\).
\[ \frac{\sin y}{\sin x} = \frac{\sqrt{8}}{\sqrt{3}} = \sqrt{\frac{8}{3}} \] \[ \sqrt{\frac{8}{3}} = \sqrt{2.6667} \approx 1.633 \] Final Answer:
\[ \boxed{1.6} \]