763628S CONDENSED MATTER PHYSICS Problem Set 1 Spring 2012

1. Honeycomb lattice(a) Verify that the construction

a1 = a

(√3

2,1

2

), a2 = a

(√3

2,−1

2

)

v1 = a

(1

2√3, 0

), v2 = a

(− 1

2√3, 0

),

is correct so that the distance between all neighboring points isidentical.

(b) Sketch the neighborhoods of two particles in the honeycomblattice which are not identical, and describe the rotation thatwould be needed to make them identical.

2. Hexagonal lattice(a) The hexagonal lattice may be viewed as a special case of the

centered rectangular lattice. Find the ratio c/a for which thecentered rectangular lattice would become hexagonal.

(b) Enumerate the symmetries of the hexagonal lattice and comparethem with the symmetries of the centered rectangular lattice.

3. Allowed symmetry axesConsider two-dimensional Bravais lattice.(a) Try to justify the following: if the lattice is left invariant after

rotation around an arbitrary point in the plane, it is left invariantafter rotation around every lattice point.

(b) Consider two nearest-neighbor points in the lattice and choosethe primitive vector a1 to be between them. List the alternativeswe have for choosing a2.

(c) Assume that the lattice is left invariant after rotation by angle θ.Construct a2 from a1 using this rotation. Find the condition thatthe image of a1 under rotation by −θ is a lattice point. Usingthis, show that the only allowed axes are one-, two-, three-, four-,and sixfold.