Consider a face-centered cubic unit cell as shown here. How many corner atoms are shown in this image? What fraction of each corner atom is inside the boundaries of the cube? How many face atoms are shown in this image? What fraction of each face atom is inside the boundaries of the cube? If you sum all the fractions of atoms, how many atoms are actually inside a face-centered cubic unit cell?
The correct answer and explanation is:
In a face-centered cubic (FCC) unit cell:
- Corner Atoms: There are 8 corner atoms in a cubic unit cell.
- Each corner atom is shared among 8 adjacent unit cells, meaning only 1/8 of each corner atom is inside a single unit cell.
- Face Atoms: There are 6 face atoms in the FCC unit cell.
- Each face atom is shared between 2 adjacent unit cells, meaning 1/2 of each face atom is inside the boundaries of the unit cell.
- Total Atoms Inside the FCC Unit Cell:
- From the corner atoms: 8×18=18 \times \frac{1}{8} = 1.
- From the face atoms: 6×12=36 \times \frac{1}{2} = 3.
Adding these contributions, the total number of atoms inside the FCC unit cell is:
1+3=4 atoms.1 + 3 = 4 \text{ atoms.}
Explanation:
An FCC unit cell is highly symmetric and efficient in packing. It consists of atoms at each of the 8 corners of the cube and 1 atom at the center of each of the 6 faces.
- Corner Atoms: Each corner atom is shared equally by 8 unit cells because it sits at the intersection of 8 cubes. Thus, only 18\frac{1}{8} of a corner atom belongs to a single unit cell.
- Face Atoms: A face atom lies at the center of a face and is shared between 2 adjacent unit cells. Thus, 12\frac{1}{2} of each face atom contributes to a single unit cell.
Summing the contributions of corner and face atoms gives the total number of atoms effectively inside the FCC unit cell. This calculation demonstrates the high packing efficiency of FCC structures, which have a packing efficiency of about 74%. FCC structures are common in metals like aluminum, copper, and gold, making them crucial in material science.