Hexagonal space lattice. The primitive translation vector of the hexagonal space lattice may be taken as ā₁ (a) Show that the volume of the primitive cell is (√3/2)a²c. (b) Show that the primitive translations of the reciprocal lattice are 2π 2π + -ŷ; b₂ √√3a a so that the lattice is its own reciprocal, but with a rotation of axes. (c) Describe and sketch the first Brillouin zone of the hexagonal space lattice. b₁ = √3a 2 -x+ y; ā₂ 2 √3a, √³ª +ỹ; â‚= c². -✰ 2 2 2π 2π -x+²y; √√3a a b₂ = ²π 2₂ -2, с b₂ );
Hexagonal space lattice. The primitive translation vector of the hexagonal space lattice may be taken as ā₁ (a) Show that the volume of the primitive cell is (√3/2)a²c. (b) Show that the primitive translations of the reciprocal lattice are 2π 2π + -ŷ; b₂ √√3a a so that the lattice is its own reciprocal, but with a rotation of axes. (c) Describe and sketch the first Brillouin zone of the hexagonal space lattice. b₁ = √3a 2 -x+ y; ā₂ 2 √3a, √³ª +ỹ; â‚= c². -✰ 2 2 2π 2π -x+²y; √√3a a b₂ = ²π 2₂ -2, с b₂ );
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Transcribed Image Text:2. Hexagonal space lattice. The primitive translation vector of the hexagonal space
lattice may be taken as
3a
ā₂
2
(a) Show that the volume of the primitive cell is (√3/2)a²c.
(b) Show that the primitive translations of the reciprocal lattice are
2π 2π
2π
2π
2π
=
= ·x +· -ŷ; b₂
3a a
·+· -ŷ; b₂ = 2,
3a
a
с
so that the lattice is its own reciprocal, but with a rotation of axes.
(c) Describe and sketch the first Brillouin zone of the hexagonal space lattice.
√3a
2
b
=
a₁
x +
2
î+ŷ; ā=c₂.
2
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