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 _______________
Calculating Octahedral Void Radius in FCC Iron
We need to find the radius of the octahedral void in face-centered cubic (FCC) iron ($\gamma$-Fe), given the lattice parameter ($a$).
Key Relationships in FCC Structures
- The relationship between the atomic radius ($r_{atom}$) and the lattice parameter ($a$) in an FCC structure is given by:
$ a = 2\sqrt{2} r_{atom} $
- The radius of an octahedral void ($r_{void}$) relative to the atomic radius ($r_{atom}$) in FCC is approximately:
$ r_{void} \approx 0.414 \times r_{atom} $
Step 1: Calculate Atomic Radius ($r_{atom}$)
Given the lattice parameter $a = 0.3571$ nm. We can rearrange the formula to find the atomic radius:
$ r_{atom} = \frac{a}{2\sqrt{2}} $
Substituting the value of $a$:
$ r_{atom} = \frac{0.3571 \text{ nm}}{2\sqrt{2}} \approx \frac{0.3571 \text{ nm}}{2.8284} \approx 0.12625 \text{ nm} $
Step 2: Calculate Octahedral Void Radius ($r_{void}$)
Now, use the calculated atomic radius to find the void radius:
$ r_{void} = 0.414 \times r_{atom} $
$ r_{void} \approx 0.414 \times 0.12625 \text{ nm} \approx 0.05227 \text{ nm} $
Conclusion
The calculated radius of the octahedral void is approximately 0.05227 nm. This value falls within the expected range of 0.045 nm to 0.06 nm, confirming our calculation for the octahedral void size in the FCC lattice of iron.
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 __________.
- 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)
- If saturation magnetization of iron at room temperature is $1700 \text{ kA m}^{-1}$, the magnetic moment (in $A \text{ m}^2$) per iron atom in the crystal is: _________ $ \times 10^{-23}$
(round off to 1 decimal place).
(Given: Lattice parameter of iron at room temperature = 0.287 nm)
