HW-7 - Theory of Solids HW-7 Solution By Qifeng Shan...

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1 Theory of Solids HW-7 Solution By Qifeng Shan Ashcroft and Mermin’s book 6-1 Power specimen of that one sample is face-centered cubic, one is body-centered cubic and one has the diamond structure. The approximate positions of the first four diffraction rings in each case are (see Figure 6.13): Values of φ for samples A B C 42.2 o 28.8 o 42.8 o 49.2 o 41.0 o 73.2 o 72.0 o 50.8 o 89.0 o 87.3 o 59.6 o 115.0 o (a) Identify the crystal structures of A, B, and C. (b) If the wavelength of the incident X-ray beam is 1.5A, what is the length of the side of the conventional cell in each other? (c) If the diamond structure were replaced by a zincblende structure with a cubic unit cell of the same side, at what angles would the first four rings now occur? Solution: (a) For the powder method, the relation for incident and diffracted wave vector is shown in Figure 6.10:
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2 The diffraction vector satisfies the following relation: 2 1 sin 2 k K (1) Since it is an elastic scattering, k is same for the incident and diffracted beam. From Eq. (1), keeping k the same, the ratio of the magnitude of reciprocal lattice vector is the same as the ration of sin ( φ /2). Then, we have: 4 3 2 1 4 3 2 1 2 1 sin : 2 1 sin : 2 1 sin : 2 1 sin : : : K K K K (2) Thus, for A: 917 . 1 : 633 . 1 : 156 . 1 : 1 : : : 4 3 2 1 K K K K (3) For B: 998 . 1 : 724 . 1 : 408 . 1 : 1 : : : 4 3 2 1 K K K K (4) For C: 311 . 2 : 921 . 1 : 634 . 1 : 1 : : : 4 3 2 1 K K K K (5)
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HW-7 - Theory of Solids HW-7 Solution By Qifeng Shan...

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