derive planar density expressions for BCC (100) and (110) planes interms of the atomic radius R.(b) Compute and compare planar density values for these same two planes for molybdenum.
The Correct Answer and Explanation is :
Planar Density Expressions for BCC (100) and (110) Planes
Planar Density Formula:
Planar density (PD) is given by:
[
PD = \frac{\text{Number of atoms centered on the plane}}{\text{Area of the plane}}
]
1. BCC (100) Plane
- In a BCC (Body-Centered Cubic) structure, each unit cell contains 2 atoms.
- The (100) plane contains atoms at the four corners of a square.
- Each corner atom contributes 1/4th of an atom to the plane.
- The central atom in the unit cell does not lie on this plane.
- The lattice parameter (( a )) in terms of atomic radius (( R )) for a BCC structure is: [
a = \frac{4R}{\sqrt{3}}
] - The area of the (100) plane is: [
A_{100} = a^2 = \left(\frac{4R}{\sqrt{3}}\right)^2 = \frac{16R^2}{3}
] - The number of atoms per (100) plane: [
N_{100} = 1 (4 \times 1/4)
] - Planar Density for BCC (100) Plane: [
PD_{100} = \frac{1}{\frac{16R^2}{3}} = \frac{3}{16R^2}
]
2. BCC (110) Plane
- The (110) plane contains atoms from two full corner atoms and one additional atom at the center.
- The area of the (110) plane is: [
A_{110} = a \times \frac{a}{\sqrt{2}} = \frac{4R}{\sqrt{3}} \times \frac{4R}{\sqrt{3}\sqrt{2}}
] [
A_{110} = \frac{16R^2}{3\sqrt{2}}
] - The number of atoms per (110) plane: [
N_{110} = 2 \times \frac{1}{2} + 1 = 2
] - Planar Density for BCC (110) Plane: [
PD_{110} = \frac{2}{\frac{16R^2}{3\sqrt{2}}} = \frac{3\sqrt{2}}{8R^2}
]
(b) Compute and Compare Planar Density for Molybdenum
- Given atomic radius of Mo: ( R = 0.1363 ) nm.
BCC (100) Plane:
[
PD_{100} = \frac{3}{16(0.1363)^2} = \frac{3}{16(0.01858)}
]
[
PD_{100} = 10.11 \times 10^{18} \text{ atoms/m}^2
]
BCC (110) Plane:
[
PD_{110} = \frac{3\sqrt{2}}{8(0.1363)^2} = \frac{3(1.414)}{8(0.01858)}
]
[
PD_{110} = 12.00 \times 10^{18} \text{ atoms/m}^2
]
Comparison:
- BCC (110) plane has a higher planar density than BCC (100) plane.
- This indicates that (110) planes are more closely packed than (100) planes in BCC structures.
Now, I will generate an image illustrating the (100) and (110) planes in a BCC structure.
The image illustrates the (100) and (110) planes in a Body-Centered Cubic (BCC) structure, showing their atomic arrangements and differences in planar density. Let me know if you need any modifications or additional explanations!