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EMS162_2011_Lecture4

# EMS162_2011_Lecture4 - Rotation Axes N=2 N=3 N=4 = N=6...

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1 1/13/2011 EMS 162 - Winter 2010 33 Yayoi Takamura N=6 Rotation Axes N=2 N=3 N=4 N ° = 360 ϕ Axis is a line perpendicular to the board 1/13/2011 EMS 162 - Winter 2010 34 Yayoi Takamura Rotation Axes N ° = 360 ϕ Axis is a line in plane of the board or

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2 1/13/2011 EMS 162 - Winter 2010 35 Yayoi Takamura Pentagons 1/13/2011 EMS 162 - Winter 2010 36 Yayoi Takamura Crystallographic Restriction 1. Consider lattice pts A and A’ separated by distance, t . 2. Perform a rotation upon A to rotate by an angle, α . A new position, B is created from A’. 3. Perform the same rotation by an angle – α upon A’ to create a new position B’. Now B and B’ are separated by a distance, t’. Due to the translational symmetry of the lattice, t’ must be an integer multiple of t , or t’=mt where m =integer. 4. Therefore t’= mt= t - 2tcos( α ) = t(1-2cos( α )) m=1-2cos( α ) |cos( α )| = |(1-m)/2|

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