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Unformatted text preview: Prob. 1.1 Find atoms/cell and nearest neighbor distance for sc, bcc, and fcc lattices. sc: atoms/cell = 8 ⋅ 1 = 1 8 nearest neighbor distance = a
sc lattice a 2 bcc: atoms/cell = 8 ⋅ 1 + 1 = 2 8
a⋅ 3 nearest neighbor distance = 2 bcc lattice a 2 a 2 √3 a 2 √2 fcc: atoms/cell = 8 ⋅ 1 + 6 ⋅ 1 = 4 8 2 nearest neighbor distance =
a⋅ 2 2 fcc lattice a 2 a 2 √2 a 2 Prob. 1.2 Draw cubes showing four {111} planes and four {110} planes. {111} planes {110} planes Prob. 1.3 Find fraction occupied for sc, bcc, and diamond lattices. sc: atoms/cell = 8 ⋅ 1 = 1 8 nearest neighbour = a
a radius = 2
3 sc lattice a 4π ⎛ a ⎞ π ⋅ a3 atom sphere volume = ⋅⎜ ⎟ = 3 ⎝2⎠ 6 unit cell volume = a 3
1⋅ π ⋅ a3 6 = π = 0.52 3 a 6 fraction occupied = bcc: atoms/cell = 8 ⋅ 1 + 1 = 2 8 nearest neighbour =
a⋅ 3 2 bcc lattice a radius =
3 a⋅ 3 4 4π ⎛ a ⋅ 3 ⎞ π ⋅ 3 ⋅ a3 atom sphere volume = ⋅⎜ ⎟= 3⎜ 4⎟ 16 ⎝ ⎠ unit cell volume = a 3 2⋅ fraction occupied = π ⋅ 3 ⋅ a3 π⋅ 3 16 = = 0.68 3 a 8
a 2 a 2 √3 a 2 √2 diamond: atoms/cell = 4 (fcc) + 4 (offset fcc) = 8 nearest neighbour =
a⋅ 3 4 atom radius =
3 a⋅ 3 8 diamond lattice a/2 4π ⎛ a ⋅ 3 ⎞ π ⋅ 3 ⋅ a3 atom sphere volume = ⋅⎜ = ⎟ 3⎜8⎟ 128 ⎝ ⎠ unit cell volume = a 3 π ⋅ 3 ⋅ a3 8⋅ π⋅ 3 128 fraction occupied = = = 0.34 3 a 16
a 4 a 4 √3 a 4 √2 ...
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 Spring '08
 SanjayBanerjee
 nearest neighbor distance

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