The atomic packing factor (APF) is a way of expressing how densely particles (atoms) are packed in a crystal structure. For face-centered cubic (FCC) and body-centered cubic (BCC) crystal structures, we can calculate the APF using their respective arrangements of atoms.
Atomic Packing Factor for FCC
In an FCC structure, each unit cell has atoms at each of the corners and in the center of each face. There are a total of 4 atoms per unit cell:
- 1 atom from each of the 8 corner positions (1/8 * 8 = 1 atom)
- 1 atom from each of the 6 face positions (1/2 * 6 = 3 atoms)
This contributes to a total of 4 atoms for the FCC unit cell.
The volume occupied by these atoms can be calculated using the radius (r) of the atoms. The volume of a single atom (assumed to be spherical) is:
Volume of one atom = (4/3)πr³
So, the total volume of the 4 atoms is:
Total volume of atoms = 4 * (4/3)πr³ = (16/3)πr³
For the FCC unit cell, the edge length (a) is related to the atomic radius as follows:
a = 2√2r
Therefore, the volume of the unit cell is:
Volume of unit cell = a³ = (2√2r)³ = 16√2r³
Now, the APF can be calculated as:
APF = (Volume of atoms in unit cell) / (Volume of unit cell)
APF = ((16/3)πr³) / (16√2r³) = (π/3√2) ≈ 0.74
Atomic Packing Factor for BCC
In a BCC structure, there are atoms at each of the 8 corners and one atom completely inside the center of the cell, giving a total of 2 atoms per unit cell:
- 1 atom from each of the 8 corners (1/8 * 8 = 1 atom)
- 1 atom from the center (1 atom)
The volume occupied by these atoms is:
Total volume of atoms = 2 * (4/3)πr³ = (8/3)πr³
For a BCC unit cell, the edge length (a) relates to the radius as follows:
a = 4r/√3
Hence, the volume of the unit cell is:
Volume of unit cell = a³ = (4r/√3)³ = 64r³/3√3
Now, we can calculate the APF:
APF = (Volume of atoms in unit cell) / (Volume of unit cell)
APF = ((8/3)πr³) / (64r³/3√3) = (2π/64√3) ≈ 0.68
To summarize, the APF for FCC is approximately 0.74, indicating a higher packing efficiency compared to BCC, which has an APF of approximately 0.68. This difference arises from the varying arrangement of atoms in these two crystal structures.