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HW04 Soln

# HW04 Soln - Problem 3.70 J F Shackelford Introduction to...

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3.70 The diffraction peaks labeled in Figure 3.33 correspond to the reflection rules for an FCC metal ( h , k , l unmixed, as shown in Table 3.4). What would be the hkl indices for the three lowest diffraction-angle peaks for a BCC metal? SOLUTION First note an error in the problem statement in the text, which is corrected above. Diffraction in- dices are indicated by hkl with NO adornments, that is, no enclosing parentheses or brackets, by the convention of the International Union of Crystallography, the professional society charged with such standards (see http://it.iucr.org/Ab/ch1o1v0001/ch1o1.pdf ). A typical "diffraction pattern" is shown in Figure 3-33, which shows diffraction "peaks" or "re- flections" occurring at different angles ( θ ) from a crystal, in this case, aluminum. The positions of the peaks, denoted by an angle measurement on the x -axis, are established by the symmetry of the crystalline lattice associated with the sample. Crystals with different lattices (say, FCC vs BCC) produce different diffraction patterns, and diffraction is the experimental method by which such structural distinctions are made. The governing mathematical expression relating diffraction angles to the families of ( hkl ) planes responsible for diffraction is known as Bragg’s Law. It appears as equation (3.5) in the text on p. 93, where λ is the wavelength of the radiation used for the diffraction experiment. Furthermore, the d -spacings are related to the lattice constant ( a ) of a cubic lattice (FCC or BCC) by equation (3.6) of the text, found on p. 94, Combining these two equations generates an expression relating the diffraction angle θ to the hkl indices of the diffraction peaks, which indicates that the peaks occurring at the smallest values of θ are those having the smallest values of h 2 + k 2 + l 2 . n λ = 2 d hkl sin θ d hkl = a h 2 + k 2 + l 2 sin θ = n λ 2 a h 2 + k 2 + l 2 Problem 3.70 J. F. Shackelford, Introduction to Materials Science for Engineers , 7 th Edition, Prentice Hall, New Jersey (2009) Problem 3.70 Solution Professor R. Gronsky page 1 of 2

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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 “re- flection 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
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