Question

Consider a crystal that has a basis of one atom. Its primitive vectors are \( \vec{a}_1 = a\hat{i} \), \( \vec{a}_2 = a\hat{j} \), \( \vec{a}_3 = \frac{a}{2}(\hat{i} + \hat{j} + \hat{k}) \), where \( \hat{i}, \hat{j}, \hat{k} \) are the unit vectors in the \( x \), \( y \), and \( z \) directions of the Cartesian coordinate system and \( a \) is a positive constant. Which one of the following is the correct option regarding the type of the Bravais lattice?

• A. It is BCC and the volume of the primitive cell is \( \frac{a^3}{2} \)
• B. It is FCC and the volume of the primitive cell is \( \frac{a^3}{4} \)
• C. It is BCC and the volume of the primitive cell is \( \frac{a^3}{8} \)
• D. It is FCC and the volume of the primitive cell is \({a^3}\)

Hint:

For BCC lattices, the volume of the primitive cell can be found using the cross product of the primitive vectors. The BCC structure has one atom at the center and eight atoms at the corners (each shared by 8 cells).

Answer 1

The Correct Option is A

The given primitive vectors represent a body-centered cubic (BCC) lattice. To verify this, let's examine the structure. 
The primitive vectors are given as:
\[ \vec{a}_1 = a\hat{i}, \quad \vec{a}_2 = a\hat{j}, \quad \vec{a}_3 = \frac{a}{2}(\hat{i} + \hat{j} + \hat{k}). \]
The BCC lattice is characterized by one atom at the corner and one at the center of the unit cell. To calculate the volume of the primitive cell, we can use the scalar triple product:
\[ \textbf{Volume of the primitive cell} = \left| \vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3) \right|. \]
First, calculate the cross product \( \vec{a}_2 \times \vec{a}_3 \):
\[ \vec{a}_2 \times \vec{a}_3 = a\hat{j} \times \frac{a}{2}(\hat{i} + \hat{j} + \hat{k}) = \frac{a^2}{2}(\hat{i} - \hat{k}). \]
Now, calculate the dot product with \( \vec{a}_1 = a\hat{i} \):
\[ \vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3) = a\hat{i} \cdot \frac{a^2}{2}(\hat{i} - \hat{k}) = \frac{a^3}{2}. \]
Thus, the volume of the primitive cell is \( \frac{a^3}{2} \). Therefore, the correct answer is option (A).