HW-6 - 1 Theory of Solids HW-6 Solution By Qifeng Shan...

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Unformatted text preview: 1 Theory of Solids HW-6 Solution By Qifeng Shan Ashcroft and Mermin’s book 5-1 (a) Prove that the reciprocal lattice primitive vectors defined in (5.3) satisfy: 3 2 1 3 3 2 1 2 a a a b b b . (5.15) (Hint: Write b 1 (but not b 2 or b 3 ) in terms of the a i , and use the orthogonality relations (5.4).) (b) Suppose primitive vectors are constructed from the bi in the same manner (Eq. (5.3)) as the bi are constructed from the ai. Prove that these vectors are just the ai themselves; i.e., show that 1 3 2 1 3 2 2 a b b b b b . (5.16) (Hint: Write b 3 in the numerator (but not b 2 ) in terms of the ai, use the vector identity A × ( B × C ) = B ( A · C ) C ( A · B )), and appeal to the orthogonality relations (5.4) and the result (5.15) above.) (c) Prove that the volume of a Bravais lattice primitive cell is , 3 2 1 a a a v (5.17) were the ai are three primitive vectors. (In conjunction with (5.15) this establishes that the were the ai are three primitive vectors....
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HW-6 - 1 Theory of Solids HW-6 Solution By Qifeng Shan...

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