# Problem 370 solution professor r gronsky page 1 of 2

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Unformatted text preview: alues of θ are those having the smallest values of h2 + k2 + l2. Problem 3.70 Solution Professor R. Gronsky page 1 of 2 Problem 3.70 J. F. Shackelford, Introduction to Materials Science for Engineers, 7th Edition, Prentice Hall, New Jersey (2009) But there is one more important consideration in predicting diffraction. Sometimes diffraction results in destructive interference, rather than the constructive interference. Under conditions of destructive interference, no peaks are seen. This phenomenon is described by the so-called “reflection rules” listed in Table 3.4 (also known as the “structure factor rules”) at the bottom of p. 94. The rules predict the values of h, k, and l for forbidden reflections (diffraction not observed) and allowed reflections (diffraction is observed). Different crystal structures produce different diffraction patterns, and peaks are indexed according to these rules. So, because the sample is a BCC crystal, it will show diffraction peaks from families of (hkl) planes ONLY when the sum of the indices (h + k + l) is an even number, as indicated in Table 3.4. Combining these requirements yields the answer, as developed in the following table. hkl 001 011 111 002 012 112 h2 + k2 + l2 1 2 3 4 5 6 h+k+l ODD EVEN ODD EVEN ODD EVEN Allowed? NO YES NO YES NO YES Note that the indices on many of these peaks have degeneracies. A peak at labeled 011 can equivalently be labeled 101 or 110 because the associated families of planes (011), (101), and (110) have equivalent d-spacings. For a sample in powder form, (as in Figure 3.33) all families of planes with the same d-spacing contribute to each observed peak. Conclusion: The first three diffraction peaks for a bcc crystal are: 011, 002, and 112. Problem 3.70 Solution Professor R. Gronsky page 2 of 2 Problem 4.19 J. F. Shackelford, Introduction to Materials Science for Engineers, 7th Edition, Prentice Hall, New Jersey (2009) 4.19 Calculate the density of Schottky pairs (in m–3)in MgO if the fraction of vacant lattice sites is 5 × 10–6. (The density of MgO is 3.60 Mg/m3). SOLUTION Recall from page 108 of the text that a “Schottky pair” in an ionic solid is a vacant ani...
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## This note was uploaded on 01/21/2011 for the course E 45 taught by Professor Gronsky during the Fall '08 term at University of California, Berkeley.

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