hw2.solution - Solutions of Problem set 21 PHYS 361(Solid...

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Solutions of Problem set # 21 PHYS 361 (Solid State Physics), Autumn 2009 1. Scattering from a honeycomb lattice (a) primitive vector ~ a 1 = 3 b ˆ x , ~ a 2 = 3 2 b ˆ x + 3 2 b ˆ y ; (b) reciprocal lattice vectors ~ b 1 = 4 π 3 b ( 3 2 ˆ x - 1 2 ˆ y ), ~ b 1 = 4 π 3 b ˆ y ; (c) draw the reciprocal lattice, and please label those points with different distances w.r.t some reference point. Up to the 4th smallest reciprocal lattice vector, the 4 length are d, 3 d, 2 d, 7 d , where d = 4 π 3 b ; (d) use geometrical structure factor: choose basis for this lattice ~ d 1 = 0 and ~ d 2 = 3 2 b ˆ x + 1 2 b ˆ y , then the geometrical structure factor S ~ K = e i ~ K · ~ d 1 + e i ~ K · ~ d 2 where ~ K = n 1 ~ b 1 + n 2 ~ b 2 = 4 π 3 b ( 3 2 n 1 ˆ x + ( n 2 - 1 2 n 1 y ) thus S ~ K = 1 + e i · 2 3 ( n 1 + n 2 ) π and | S ~ K | 2 = 4 , ( n 1 + n 2 ) mod 3 = 0; 1 , ( n 1 + n 2 ) mod 3 = 1 , 2 the intensities of diffracted waves is proportional to | S ~ K | 2 , and from c) (express those points in c) in the form of ~ K = n 1 ~ b 1 + n 2 ~ b 2 )you can find that the points that have 2nd smallest reciprocal lattice vectors have a relative intensity of 4 while other points in c) have 1; (e) use the Ewald construction for the rotating-crystal method (see Figure 6.9 of
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