Question

The angle between \((111)\) and \((001)\) directions in a cubic crystal is:

• A. \(\cos^{-1}\left(\dfrac{2}{\sqrt{3}}\right)\)
• B. \(\cos^{-1}(\sqrt{3})\)
• C. \(\cos^{-1}\left(\dfrac{\sqrt{3}}{2}\right)\)
• D. \(\cos^{-1}\left(\dfrac{1}{\sqrt{3}}\right)\)

Hint:

Use the direction cosine formula to find angle between crystallographic directions.

Answer 1

The Correct Option is D

Concept:
For cubic crystals, angle between two directions \([u_1v_1w_1]\) and \([u_2v_2w_2]\) is: \[ \cos\theta=\frac{u_1u_2+v_1v_2+w_1w_2}{\sqrt{u_1^2+v_1^2+w_1^2}\sqrt{u_2^2+v_2^2+w_2^2}} \]

Step 1: Write the directions. \[ [111]\quad \text{and}\quad [001] \] \[ u_1=1,\ v_1=1,\ w_1=1 \] \[ u_2=0,\ v_2=0,\ w_2=1 \]

Step 2: Substitute values. \[ \cos\theta=\frac{1(0)+1(0)+1(1)}{\sqrt{1^2+1^2+1^2}\sqrt{0^2+0^2+1^2}} \] \[ \cos\theta=\frac{1}{\sqrt{3}\times1} \] \[ \cos\theta=\frac{1}{\sqrt{3}} \] \[ \theta=\cos^{-1}\left(\frac{1}{\sqrt{3}}\right) \] \[ \therefore \text{Correct Answer is (D)} \]