Question

The effective number of atoms in a body-centered cubic (BCC) unit cell is

• A. 1
• B. 2
• C. 4
• D. 8

Hint:

Remember the effective number of atoms for common cubic lattices:
- Simple Cubic (SC) = 1
- Body-Centered Cubic (BCC) = 2
- Face-Centered Cubic (FCC) = 4

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the effective (or net) number of atoms contained within a single body-centered cubic (BCC) unit cell.

Step 2: Key Formula or Approach:
The net number of atoms (\( N_e \)) in a cubic unit cell can be calculated using the formula:
\[ N_e = \frac{N_c}{8} + \frac{N_f}{2} + N_i \]
where:
\( N_c \) is the number of atoms located at the corners,
\( N_f \) is the number of atoms located at the faces, and
\( N_i \) is the number of atoms located entirely inside the cell (interior/center).

Step 3: Detailed Explanation:

• BCC Atom Distribution: In a body-centered cubic unit cell, the atoms are arranged as follows:
- There are \( 8 \) atoms located at the \( 8 \) corners of the cube (\( N_c = 8 \)).
- There is \( 1 \) atom located at the center of the unit cell (\( N_i = 1 \)).
- There are no atoms on the faces (\( N_f = 0 \)).

• Calculating the Net Number of Atoms:
- Each corner atom is shared equally among \( 8 \) adjacent unit cells. Thus, only \( 1/8 \) of each corner atom belongs to a given unit cell.
- The central atom is completely contained within the single unit cell, contributing a value of \( 1 \).
Applying these values to the formula:
\[ N_e = \left( 8 \times \frac{1}{8} \right) + 1 = 1 + 1 = 2 \]


Step 4: Final Answer:
Therefore, the effective number of atoms in a BCC unit cell is 2, matching Option (B).