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, \( \frac{\sin y}{\sin x} \), is (rounded off to one decimal place)........... 

Hint:

Understanding the relationship between diffraction peaks and Miller indices is key to calculating the ratio of Bragg angles in X-ray diffraction.

Answer 1

For FCC crystals, the Bragg angle \( x \) corresponds to the first diffraction peak and \( y \) corresponds to the third diffraction peak. The diffraction pattern for FCC crystals follows the relationship between the Miller indices and the diffraction angle. For the first and third peaks, the Miller indices are (111) and (333), respectively.
The general equation for Bragg’s law is:
\[ n\lambda = 2d \sin \theta \] Where:
- \( n \) is the order of the diffraction,
- \( \lambda \) is the wavelength of the incident radiation,
- \( d \) is the interplanar spacing, and
- \( \theta \) is the Bragg angle.
Now, the relationship between the Bragg angle \( x \) and \( y \) for the FCC crystal is as follows:
- For the first peak, \( n = 1 \) and \( d_1 = \frac{a}{\sqrt{3}} \),
- For the third peak, \( n = 3 \) and \( d_3 = \frac{a}{\sqrt{9}} \).
From this, we know that the ratio of the sine of the angles is:
\[ \frac{\sin y}{\sin x} = \frac{\sqrt{3}}{1} = 1.73 \] Thus, the ratio lies between 1.4 and 1.8.