Problem_5-fold_rotation_axis_in_crystals

Problem_5-fold_rotation_axis_in_crystals - a ' + a...

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Crystallographic restriction theorem See: http://en.wikipedia.org/wiki/Crystallographic_restriction_theorem Problem: Consider rotational symmetry operations of a 2D periodic lattice with the rotation axis perpendicular to the lattice plane. Show that only rotations that are multiples of 60° and 90° can be symmetry operations. Thus, 2, 3, 4, and 6-fold symmetries are allowed, but not 5-fold. This theorem also holds for 3D lattices. Assume that a 2D lattice is invariant under rotation of angle . Let a be a primitive lattice vector. Rotate a by angle to generate a ', and rotate a by angle - to generate a ". Both a ' and a " must be lattice vectors, and so is a ' + a ", which points in either the same or the opposite direction as a . Since a is primitive (shortest along its direction),
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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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