How many atoms are there in a face-centered cubic unit cell? 1. 4 2. 6 3. 10 4. 12 5. 8
The Correct Answer and Explanation is:
The correct answer is 4.
In a face-centered cubic (FCC) unit cell, the arrangement of atoms can be understood as follows:
- Corners: A cube has 8 corners, and in an FCC unit cell, each corner atom is shared by 8 adjacent unit cells. Since each corner atom contributes 1/8 of its total volume to the unit cell, the total contribution from all 8 corner atoms is: 8×18=1 atom.8 \times \frac{1}{8} = 1 \text{ atom}.8×81=1 atom.
- Faces: Each face of the cube contains an atom at its center. These face-centered atoms are shared between two adjacent unit cells. So, each face-centered atom contributes 1/2 of its total volume to the unit cell. Since the FCC unit cell has 6 faces, the total contribution from all 6 face-centered atoms is: 6×12=3 atoms.6 \times \frac{1}{2} = 3 \text{ atoms}.6×21=3 atoms.
Thus, the total number of atoms in one FCC unit cell is the sum of the atoms from the corners and the faces:1 (from corners)+3 (from faces)=4 atoms.1 \text{ (from corners)} + 3 \text{ (from faces)} = 4 \text{ atoms}.1 (from corners)+3 (from faces)=4 atoms.
Why This Structure is Efficient:
The FCC structure is one of the most efficient ways to pack atoms in a solid. It allows for the highest packing efficiency of any crystal structure, with about 74% of the volume occupied by atoms. The remaining space is occupied by voids, which are small gaps between the atoms. This efficient packing makes FCC crystals ideal for metals like aluminum, copper, gold, and lead, which all adopt this structure due to the close packing of atoms.
In summary, the face-centered cubic unit cell contains 4 atoms: 1 from the corners and 3 from the faces.
