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homework1 - x,y about an axis through the origin that forms...

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Physics 481: Condensed Matter Physics - Homework 1 due date: Jan 21, 2011 Problem 1: Honeycomb lattice (10 points, Marder - Problem 1.1 ) The honeycomb lattice can be constructed by starting from the hexagonal Bravais lattice with primitive vectors ~a 1 = a ( 3 / 2 , 1 / 2) and ~a 2 = a ( 3 / 2 , - 1 / 2) where a is the lattice constant. Each lattice point is then decorated with basis particles at relative positions ~v 1 = a ( 3 / 6 , 0) and ~v 2 = a ( - 3 / 6 , 0) a) Verify that this construction leads to a regular honeycomb lattice (the distances between all neighboring points are identical, and all internal angles are 120 ). b) Sketch the neighborhoods of two particles in the honeycomb lattice which are not equivalent, and describe the rotation that would be needed to make them identical. Problem 2: General reflection matrix in two dimensions (15 points) Consider the reflection of a lattice point ~
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Unformatted text preview: x,y ) about an axis through the origin that forms an angle φ with the x-axis. Show that the matrix form of this operation is cos(2 φ ) sin(2 φ ) sin(2 φ )-cos(2 φ ) ! Hint: An efficient way of finding the matrix consists in rotating the lattice such that the reflection axis coincides with the x-axis, performing the reflection, and rotating back! Problem 3: Allowed rotation axes (15 points, Marder - Problem 1.4) Prove that the only allowed rotation axis in a two-dimensional Bravais lattice are twofold, threefold, fourfold, and sixfold! To this end, consider the images of the lattice point ( a, 0) under rotations around the origin by angles φ and-φ . Both must be in the Bravais lattice! From these conditions derive a simple expression that implicitly specifies all possible φ ....
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