Question

If the first reflection from an FCC crystal has a Bragg angle \( \theta = 21.5^\circ \), the \( \theta \) corresponding to the second reflection is

• A. \( 13.5^\circ \)
• B. \( 18.5^\circ \)
• C. \( 25^\circ \)
• D. \( 36.8^\circ \)

Hint:

Bragg’s Law explains how X-rays are diffracted by crystal planes. Higher-order reflections occur at progressively larger angles.

Answer 1

The Correct Option is D

The Bragg equation is given by: \[ n\lambda = 2d \sin\theta \] where: - \( n \) is the order of reflection, - \( \lambda \) is the X-ray wavelength, - \( d \) is the interplanar spacing, - \( \theta \) is the Bragg angle.
Step 1: Understanding Bragg's Law for Multiple Reflections - The first-order reflection (\( n = 1 \)) occurs at \( \theta_1 = 21.5^\circ \). - The second-order reflection (\( n = 2 \)) follows the same equation but with a higher \( n \).


Step 2: Finding \( \theta_2 \) for the Second Reflection Using the relation: \[ \sin\theta_2 = \frac{2 \sin\theta_1}{\sqrt{4}} = 2\sin 21.5^\circ \] Calculating, \[ \sin 21.5^\circ \approx 0.366 \] \[ \sin\theta_2 = 2 \times 0.366 = 0.732 \] From sine tables: \[ \theta_2 \approx 36.8^\circ \]


Step 3: Evaluating the Options - Option (A) - Incorrect: 13.5° is not a valid Bragg angle. - Option (B) - Incorrect: 18.5° is incorrect. - Option (C) - Incorrect: 25° does not match the calculated value. - Option (D) - Correct: 36.8° matches our calculation.


Step 4: Conclusion Since the Bragg angle for the second reflection is \( 36.8^\circ \), the correct answer is option (D).