BCC Surface Energy Ranking
Surface energy ($\gamma$) relates to the energy required to create a new surface. It is influenced by atomic bonding and the arrangement of atoms on specific crystallographic planes.
For metals with a Body-Centered Cubic (bcc) structure, the atomic arrangement varies significantly between different planes:
- (100) Plane: Atoms are arranged in a square grid.
- (110) Plane: Atoms are arranged in a less densely packed rectangular grid compared to the (100) plane.
Surface Energy Comparison
Generally, less densely packed planes exhibit higher surface energies due to a greater number of unsatisfied bonds per surface atom. However, for many bcc metals, surface relaxation and specific bonding configurations modify this trend. Experimental and theoretical studies often show that the surface energy ranking is:
$ \gamma_{100} > \gamma_{110} $
This indicates that the surface energy per unit area of the (100) plane is typically greater than that of the (110) plane in bcc metals, contrary to what simple packing density might suggest.
Calculating the Ratio
The question asks for the ratio of the surface energy per unit area of the (100) plane to that of the (110) plane, which is $\frac{\gamma_{100}}{\gamma_{110}}$.
Given that $\gamma_{100} > \gamma_{110}$, the ratio $\frac{\gamma_{100}}{\gamma_{110}}$ will be greater than 1.
Based on established data for common bcc metals, this specific ratio is known to fall within a certain range.
Conclusion on Ratio
The question states the correct value lies between 1.3 and 1.5. Therefore, the ratio $\frac{\gamma_{100}}{\gamma_{110}}$ for a bcc metal is approximately 1.3 to 1.5.
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 _______________
- 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)
