Question
For a fcc unit cell, the ratio of the number of tetrahedral voids to the number of atoms is
The correct answer is
$2:1$
Understanding FCC Unit Cells and Voids
This explanation focuses on determining the ratio of tetrahedral voids to atoms within a face-centered cubic (fcc) unit cell.
Calculating Atoms in an FCC Unit Cell
An fcc unit cell contains atoms positioned at the corners and the center of each face.
- Corner atoms: An fcc cell has 8 corners, and each corner atom contributes \frac{1}{8} to the unit cell. Total corner contribution = 8 \times \frac{1}{8} = 1 atom.
- Face atoms: An fcc cell has 6 faces, and each face-centered atom contributes \frac{1}{2} to the unit cell. Total face contribution = 6 \times \frac{1}{2} = 3 atoms.
- Total atoms: The total number of atoms in an fcc unit cell is the sum of corner and face contributions: 1 + 3 = 4 atoms.
Determining Tetrahedral Voids
For any given crystal lattice, the number of tetrahedral voids is twice the number of atoms.
- Number of tetrahedral voids = 2 \times (\text{Number of atoms})
- Number of tetrahedral voids = 2 \times 4 = 8 voids.
Calculating the Ratio
The question requires the ratio of tetrahedral voids to the total number of atoms in the fcc unit cell.
- Ratio = (Number of tetrahedral voids) : (Number of atoms)
- Ratio = 8 : 4
- Simplifying this ratio gives 2 : 1.
Conclusion
Therefore, the ratio of tetrahedral voids to atoms in an fcc unit cell is 2:1.
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Important Questions from Crystal Structure Density Atomic Packing Factor
- Match the crystal systems in Column I with the corresponding axial lengths (a, b, c) and interaxial angles ($\alpha$, $\beta$, $\gamma$) provided in Column II
Column I Column II (P) Tetragonal (1) $a \neq b \neq c$, $\alpha = \beta = \gamma = 90^\circ$ (Q) Rhombohedral (2) $a = b \neq c$, $\alpha = \beta = \gamma = 90^\circ$ (R) Orthorhombic (3) $a \neq b \neq c$, $\alpha = \gamma = 90^\circ \neq \beta$ (S) Monoclinic (4) $a = b = c$, $\alpha = \beta = \gamma \neq 90^\circ$ - The coordination number for an octahedral site in pure copper is __________.
The lattice parameter of face-centered cubic iron ($\gamma$-Fe) is 0.3571 nm. The radius (in nm) of the octahedral void in $\gamma$-Fe is _______________
- For a bcc metal the ratio of the surface energy per unit area of the (100) plane to that of the (110) plane is ________
- Pure iron transforms from body centered cubic (BCC) to face centered cubic (FCC) crystal structure at $912 \text{ °C}$. If the lattice parameter of the BCC phase is $0.293 \text{ nm}$ and that of the FCC phase is $0.363 \text{ nm}$, the associated volume change is ________ (in % to one decimal place)
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