1
Theory of Solids HW5 Solution
By Qifeng Shan
Ashcroft and Mermin’s book
44
(a) Prove that the WignerSeitz cell for any twodimensional Bravais lattice is
either a hexagon or a rectangle.
(b) Show that the ratio of the lengths of the diagonals of each parallelogram face
of the WignerSeitz cell for the facecentered cubic lattice (Figure 4.16) is
√
2:1.
(c) Show that every edge of the polyhedron bounding the WignerSeitz cell of the
body centered cubic lattice is
√
2/4 times the length of the conventional cubic cell.
(d) Prove that the hexagonal faces of the bcc WignerSeitz cell are all regular
hexagons. (Note that the axis perpendicular to a hexagonal face passing through its center
has only three fold symmetry, so this symmetry alone is not enough.)
Solution:
(a)
For a 2D Bravais lattice, there are only five types of Bravais lattice: (1), Square
lattice; (2), rectangular lattice; (3), hexagonal lattice; (4), oblique lattice; (5),
centered rectangular lattice. It is very obvious that the WignerSeitz cells for the
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 Fall '09
 Shan
 Geometry, Solid State Physics, Cubic crystal system, Qifeng Shan Ashcroft, WignerSeitz cell

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