Question:
In a face-centered cubic (fcc) lattice, the number of atoms per unit cell is
In a face-centered cubic (fcc) lattice, the number of atoms per unit cell is
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Remember that in crystal structures, atoms at the corners and faces are shared among multiple unit cells, which affects their contribution to a single unit cell.
Updated On: Jun 3, 2026
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The Correct Option is A
Solution and Explanation
Step 1: Concept
In a face-centered cubic (fcc) lattice, atoms are located at each corner of the cube and also at the center of each face. This arrangement maximizes the packing efficiency.
Step 2: Meaning
The number of atoms per unit cell refers to how many atoms contribute to that specific volume in the crystal structure.
Step 3: Analysis
Consider an fcc unit cell: Each corner atom is shared among 8 adjacent unit cells. Each face-centered atom is shared among 2 adjacent unit cells. For corners, each atom contributes \( \frac{1}{8} \) of its volume to one unit cell. Since there are 8 corners in a cube, the total contribution from all corner atoms is: \[8 \times \frac{1}{8} = 1\] For face-centered atoms, each atom contributes \( \frac{1}{2} \) of its volume to one unit cell. Since there are 6 faces in a cube, the total contribution from all face-centered atoms is: \[6 \times \frac{1}{2} = 3\] Adding these contributions together gives the total number of atoms per unit cell: \[1 + 3 = 4\]
Step 4: Conclusion
Thus, an fcc lattice has 4 atoms per unit cell.
Final Answer: (A)
In a face-centered cubic (fcc) lattice, atoms are located at each corner of the cube and also at the center of each face. This arrangement maximizes the packing efficiency.
Step 2: Meaning
The number of atoms per unit cell refers to how many atoms contribute to that specific volume in the crystal structure.
Step 3: Analysis
Consider an fcc unit cell: Each corner atom is shared among 8 adjacent unit cells. Each face-centered atom is shared among 2 adjacent unit cells. For corners, each atom contributes \( \frac{1}{8} \) of its volume to one unit cell. Since there are 8 corners in a cube, the total contribution from all corner atoms is: \[8 \times \frac{1}{8} = 1\] For face-centered atoms, each atom contributes \( \frac{1}{2} \) of its volume to one unit cell. Since there are 6 faces in a cube, the total contribution from all face-centered atoms is: \[6 \times \frac{1}{2} = 3\] Adding these contributions together gives the total number of atoms per unit cell: \[1 + 3 = 4\]
Step 4: Conclusion
Thus, an fcc lattice has 4 atoms per unit cell.
Final Answer: (A)
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