Question

The angle between [111] and [001] directions in a cubic crystal is:

• A. \(54.74^\circ\)
• B. \(45^\circ\)
• C. \(35.26^\circ\)
• D. \(60^\circ\)

Hint:

For directional angles in cubic crystals, always use the dot product formula. Knowing standard angles like 54.74◦for [111] and [001] can save time.

Answer 1

The Correct Option is A

The angle \(\theta\) between two directions \([h_1k_1l_1]\) and \([h_2k_2l_2]\) in a cubic crystal is given by:
\[\cos \theta = \frac{h_1h_2 + k_1k_2 + l_1l_2}{\sqrt{h_1^2 + k_1^2 + l_1^2} \cdot \sqrt{h_2^2 + k_2^2 + l_2^2}}\]
For \([111]\) and \([001]\):
\[\cos \theta = \frac{1 \cdot 0 + 1 \cdot 0 + 1 \cdot 1}{\sqrt{1^2 + 1^2 + 1^2} \cdot \sqrt{0^2 + 0^2 + 1^2}} = \frac{1}{\sqrt{3}}\]
\[\theta = \cos^{-1} \left( \frac{1}{\sqrt{3}} \right) \approx 54.74^\circ\]