Unformatted text preview: a ' + a " = n a , where n is an integer (positive, negative, or zero). Taking the dot product of this equation with a yields 2 a 2 cos = na 2 . Thus, cos = 0, 1/2, or 1. The only possible values of are 0 , 60°, 90°, and their multiples. In 3D, consider an arbitrary lattice vector R that is not parallel to the rotation axis. Rotate it by to generate R '; the difference R '  R is a nonzero lattice vector perpendicular to the rotation axis. It follows that one can always find a line of lattice points perpendicular to the rotation axis; let a be the primitive (shortest) lattice vector for this 1D lattice. The rest of the proof is the same as the 2D case. T.C. Chiang...
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This note was uploaded on 03/21/2012 for the course PHYS 560 taught by Professor Flynn during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Flynn
 Physics

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