# HW-5 - Theory of Solids HW-5 Solution By Qifeng Shan...

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1 Theory of Solids HW-5 Solution By Qifeng Shan Ashcroft and Mermin’s book 4-4 (a) Prove that the Wigner-Seitz cell for any two-dimensional 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 Wigner-Seitz cell for the face-centered cubic lattice (Figure 4.16) is 2:1. (c) Show that every edge of the polyhedron bounding the Wigner-Seitz 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 Wigner-Seitz 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 2-D 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 Wigner-Seitz cells for the

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## This note was uploaded on 10/04/2011 for the course PHYS 4720 taught by Professor Shan during the Fall '09 term at Rensselaer Polytechnic Institute.

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HW-5 - Theory of Solids HW-5 Solution By Qifeng Shan...

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