To calculate the volume of a unit cell for face-centered cubic (FCC) and body-centered cubic (BCC) structures, we first need to understand the geometry of these unit cells.
Volume of the Unit Cell
The volume of a unit cell can be determined using the formula:
Volume = a³
where a is the edge length of the unit cell.
For FCC:
In an FCC structure, the atoms are located at each corner of the cube and in the centers of each face. The relationship between the edge length a and the atomic radius r is given by:
a = 2√2 * r
Thus, the volume of the FCC unit cell is:
Volume (FCC) = (2√2 * r)³
This can be simplified to:
Volume (FCC) = 16√2 * r³
For BCC:
In a BCC structure, there is an atom at each corner and one atom in the center of the cube. The relationship between the edge length a and the atomic radius r in BCC is:
a = 4r / √3
Thus, the volume of the BCC unit cell is:
Volume (BCC) = (4r / √3)³
This can be simplified to:
Volume (BCC) = 64r³ / 3√3
Volume of Atoms Within a Unit Cell
The volume occupied by the atoms within a unit cell is calculated by considering how many atoms are in each unit cell and the volume of a single atom.
For FCC:
In an FCC structure, there are a total of 4 atoms per unit cell (1 from each corner (1/8 × 8) + 1 from each face (1/2 × 6)). The volume of one atom can be calculated using the formula for the volume of a sphere:
Volume of one atom = (4/3)πr³
The total volume of atoms within the FCC unit cell is:
Total Volume (FCC) = 4 * (4/3)πr³ = (16/3)πr³
For BCC:
In a BCC unit cell, there are 2 atoms (1 from each corner (1/8 × 8) + 1 from the center). The volume of atoms within a BCC unit cell is:
Total Volume (BCC) = 2 * (4/3)πr³ = (8/3)πr³
Conclusion
To summarize:
- FCC Unit Cell Volume: 16√2 * r³, Total Volume of Atoms: (16/3)πr³
- BCC Unit Cell Volume: 64r³ / 3√3, Total Volume of Atoms: (8/3)πr³