# 633 now note that the ratio of the burgers vectors

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Unformatted text preview: includes x, a and c/2 as its sides permits the side lengths to be summed as a2 = x2 + (c/2)2 Substituting, (c/2) = a − 2 2 and solving, ￿ a/2 cos 30◦ ￿2 c/a = 1.633 Now note that the ratio of the Burgers vectors along the two directions of interest is exactly equal to the c/a ratio. |b|[0001] = c a |b|[11¯ 20] Consequently the relative dislocation energies differ by a smaller factor, |b|2 [0001] = (1.633)2 = 2.67 |b|2 [11¯ 20] Problem 4.23 Solution Professor R. Gronsky page 3 of 3 Problem 4.25 J. F. Shackelford, Introduction to Materials Science for Engineers, 7th Edition, Prentice Hall, New Jersey (2009) 4.25 Determine the grain-size number, G, for the microstructure shown in Figure 4-18. (Keep in mind that the precise answer will depend on your choice of an area of sampling.) nλ = 2dhkl sin θ SOLUTION a dhkl = √ 2 + k 2 + l2 h Using the procedure outlined in Figure 4-20 (p. 116 of the text) for the calculation of the ASTM grain size number (G), begin by drawing a circle of known dimensions in the center of micrograph shown in Figure 4-18, which was recorded at the required magnification of 100× (NOTE: nλ ￿ 2 2 2 sin θ = a micrograph recorded at a different magnificationh + k require a magnification correction). would + l 2a The enclosing circle shown here has a diameter of 2.25 inches. ρatoms = 2.33 Mg 1 mol 106 g 0.6023 × 102 4 atoms × × × = 5.00×1028 atoms/m3 3 m 28.09 g Mg mol ρvacancies = 1 × 107 vacancies/atom × 5.00 × 1028 atoms/m3 |b111 | = |b110 | = √ 3a 2 2a √ |b001 | = a |b110 |2 2 circle and 39 cut by the circumference, which are 8 There are 77 grains completely enclosed by thea2 = 3 a2 = = 2.67 2 of 96.5 grains. The area density (N) of grains is next counted as 39/2 or 19.5 grains, forba total | 111 | 3 4 calculated as follows: N= 96.5 grains grains = 24.3 2 in2 π (2.25/2) in2 Problem 4.25 Solution Professor R. Gronsky page 1 of 2 |b001 | = a Problem 4.25 4 |b110 Materials Science for Engineers, 7th Edition, Prentice Hall, New Jersey (2009) 2a 8 J. F. Shackelford, Intr...
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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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