Compare planar densities (Section 3.11 and Problem 3.54) for the (100), (110), and (111) planes for FCC.

(a) Compare planar densities (Section 3.11 and Problem 3.54) for the (100), (110), and (111) planes for FCC.

(b) Compare planar densities (Problem 3.55) for the (100), (110), and (111) planes for BCC.

The Correct Answer and Explanation is:

To compare the planar densities for the (100), (110), and (111) planes in FCC and BCC crystal structures, we calculate the number of atoms per unit area on each of these planes.

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(a) Planar Densities for FCC:

FCC (Face-Centered Cubic) has atoms at each corner and at the center of each face.

Let’s use the atomic radius RRR and lattice parameter a=22Ra = 2\sqrt{2}Ra=22​R.

1. (100) plane:
   • Atoms per plane: 1 face-centered atom (½ in the plane) + 4 corners (¼ atom per corner) = 1+4×14=21 + 4 \times \frac{1}{4} = 21+4×41​=2
   • Area of (100) plane: a2a^2a2
   • Planar density: 2a2\frac{2}{a^2}a22​
2. (110) plane:
   • Atoms per plane: 2 face-centered atoms (each shared) + 2 corners (½ atom total) = 2+0.5=2.52 + 0.5 = 2.52+0.5=2.5
   • Area of plane: a×a2=a22a \times a\sqrt{2} = a^2\sqrt{2}a×a2​=a22​
   • Planar density: 2.5a22\frac{2.5}{a^2\sqrt{2}}a22​2.5​
3. (111) plane:
   • Atoms: 3 face-centered atoms (each shared among 2 planes) + 3 corners (each shared among 6 planes) = 3×12+3×16=1.5+0.5=23 \times \frac{1}{2} + 3 \times \frac{1}{6} = 1.5 + 0.5 = 23×21​+3×61​=1.5+0.5=2
   • Area of plane: 32a2\frac{\sqrt{3}}{2} a^223​​a2
   • Planar density: 2(3/2)a2=43a2\frac{2}{(\sqrt{3}/2) a^2} = \frac{4}{\sqrt{3} a^2}(3​/2)a22​=3​a24​

Conclusion (FCC):

• Highest planar density: (111)
• Then: (100)
• Lowest: (110)

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(b) Planar Densities for BCC:

BCC (Body-Centered Cubic) has atoms at the corners and one atom in the center.

Let’s use a=4R3a = \frac{4R}{\sqrt{3}}a=3​4R​

1. (100) plane:
   • Atoms: 4 corners (¼ atom) = 1 atom
   • Area = a2a^2a2
   • Planar density = 1a2\frac{1}{a^2}a21​
2. (110) plane:
   • Atoms: 4 corners (¼) + 1 body-centered atom on the plane = 1+1=21 + 1 = 21+1=2
   • Area = a22a^2 \sqrt{2}a22​
   • Planar density = 2a22\frac{2}{a^2\sqrt{2}}a22​2​
3. (111) plane:
   • Atoms: 3 corners (1/6 each) + portion of body-centered atom (depends on plane cut) ≈ 1.5 atoms
   • Area = 32a2\frac{\sqrt{3}}{2} a^223​​a2
   • Planar density = 1.5(3/2)a2=33a2=3a2\frac{1.5}{(\sqrt{3}/2) a^2} = \frac{3}{\sqrt{3} a^2} = \frac{\sqrt{3}}{a^2}(3​/2)a21.5​=3​a23​=a23​​

Conclusion (BCC):

• Highest: (110)
• Then: (111)
• Lowest: (100)

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Summary Table

Plane	FCC Planar Density (Relative)	BCC Planar Density (Relative)
(100)	Moderate	Lowest
(110)	Lowest	Highest
(111)	Highest	Moderate