The_Cambridge_Handbook_of_Physics_Formulas

6.3 Crystalline structure
127
Cubic lattices
lattice primitive (P) body-centred (I) face-centred (F)
lattice parameter a a a
volume of conventional cell a 3 a 3 a 3
lattice points per cell 1 2 4
1st nearest neighbours a 6 8 12
1st n.n. distance a a √ 3/2 a/ √ 2
2nd nearest neighbours 12 6 6
2nd n.n. distance a √ 2
packing fraction b π/6
a
√
3π/8
a
√
2π/6
reciprocal lattice c P F I
a 1 = aˆx a 1 = a 2 (ŷ +ẑ − ˆx) a 1 = a 2
(ŷ +ẑ)
primitive base vectors d a 2 = aŷ a 2 = a 2 (ẑ + ˆx−ŷ) a 2 = a 2
(ẑ + ˆx)
a 3 = aẑ a 3 = a 2 (ˆx+ŷ −ẑ) a 3 = a 2 (ˆx+ŷ)
a Or “coordination number.”
b For close-packed spheres. The maximum possible packing fraction for spheres is √ 2π/6.
c The lattice parameters for the reciprocal lattices of P, I, and F are 2π/a, 4π/a, and 4π/a respectively.
d ˆx, ŷ, and ẑ are unit vectors.
Crystal systems a
system symmetry unit cell b lattices c
triclinic none
a ≠ b ≠ c;
α ≠ β ≠ γ ≠90 ◦ P
monoclinic one diad ‖ [010]
orthorhombic
three orthogonal diads
tetragonal one tetrad ‖ [001]
trigonal d one triad ‖ [111]
hexagonal one hexad ‖ [001]
a ≠ b ≠ c;
α = γ =90 ◦ , β ≠90 ◦ P, C
a ≠ b ≠ c;
α = β = γ =90 ◦ P, C, I, F
a = b ≠ c;
α = β = γ =90 ◦ P, I
a = b = c;
α = β = γ