PS2_solution

# PS2_solution - MECE 6700 Problem Set 2 Due on Oct. 12th...

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MECE 6700 James Hone, Fall 2011 Problem Set 2 Due on Oct. 12 th 1 . Determine the real-space unit vectors C h and T , and the reciprocal lattice vectors K 1 and K 2 for the following tubes: (a) amchair (5,5) (b) zigzag (6,0) (c) zigzag (7,0) (d) chiral (6,2) (e) chiral (5,2) Determine the radius of each tube, and the number of atoms in the unit cell. Solution The real-space unit vector C h is deﬁned as C h = n~a 1 + m~a 2 , where ~a 1 = a 2 ( 3 ˆ i + ˆ j ) ~a 2 = a 2 ( 3 ˆ i - ˆ j ) a = lattice constant = 2 . 49 ˚ A . The translational vector T is deﬁned as T = t 1 ~a 1 + t 2 ~a 2 , where t 1 = 2 m + n d R t 2 = - m + 2 n d R d R = Greatest Common Divisor of 2 m + n,n + 2 m. The reciprocal lattice vectors K 1 and K 2 are deﬁned as K 1 = 1 N ( - t 2 ~ b 1 + t 1 ~ b 2 ) K 2 = 1 N ( m ~ b 1 - n ~ b 2 ) , where N = number of hexagons in the unit cell = 2( n 2 + m 2 + mn ) d R Page 1 of 6

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MECE 6700 James Hone, Fall 2011 ~ b 1 = 2 π a ( 1 3 ˆ i + ˆ j ) ~ b 2 = 2 π a ( 1 3 ˆ i - ˆ j ) . The radius of the nanotube
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## This note was uploaded on 02/05/2012 for the course APMAE 4200 taught by Professor R during the Fall '11 term at Columbia.

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PS2_solution - MECE 6700 Problem Set 2 Due on Oct. 12th...

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