Hw04 soln

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Unformatted text preview: Problem 3.70 J. F. Shackelford, Introduction to Materials Science for Engineers, 7th Edition, Prentice Hall, New Jersey (2009) 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 indices 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 "reflections" 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, nλ = 2dhkl sin θ where λ is the wavelength of the radiation used for the diffraction experiment. Furthermore, the a dhknλ (a) of sin θ d-spacings are related to the lattice constant =√2d a cubic lattice (FCC or BCC) by equation l= h2hkl k 2 + l2 + (3.6) of the text, found on p. 94, nλ = 2dhkl a θ sin dhkl = √ h2 + k 2 + l 2 a dhkl an √ Combining these two equations generates = expression+ l2 2 + k 2 relating the diffraction angle θ to the hkl h indices of the diffraction peaks, sin θ = nλ ￿ 2 h + k 2 + l2 2a which indicates that the peaks occurring at the smallest v...
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