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midterm_01

midterm_01 - Solutions to midterm 1(141A Sp 07 1 These...

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Unformatted text preview: Solutions to midterm 1 (141A Sp 07) 1 These solutions are a little terse. You would be expected to explain fully in your answer. 1. (a) ~ a 1 = d 3 x ~ a 2 = d 3 1 2 x + 3 2 y ! See figure 1. (b) Two atoms in a basis, at ~ 1 = ~ 0, ~ 2 =- 1 3 ~ a 1 + 2 3 ~ a 2 . (c) We choose our reciprocal lattice vectors such that ~ b i ~ a j = 2 ij Then ~ b 1 = 4 3 d 3 2 x- 1 2 y ! ~ b 2 = 4 3 d y (d) See figure 1. Using geometry, you can show that the area is 8 2 / (3 3 d 2 ). 2. (a) In the conventional unit cell, the corner atoms contribute 1 atom. The face atoms give 6 / 2, since they each belong to two unit cells. Therefore there are 4 atoms in the conventional unit cell, and so the volume per atom is (5 . 63 3 / 4) 3 = 44 . 6 3 . (b) See figure 8a on page 96 of ISSP. Theres one longitudinal acoustic, two transverse acoustic, one longitudinal optical, two transverse optical modes....
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midterm_01 - Solutions to midterm 1(141A Sp 07 1 These...

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