HW04 Soln

In metals a burgers vector is always a lattice

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Unformatted text preview: ) dislocation configuration in a crystalline lattice, it is simplest to find those dislocations with the shortest Burgers vectors. In metals, a Burgers vector is always a lattice translation vector along the closest-packed direction, as described on p. 110 of the text. This problem features hexagonal close-packed (HCP) metals. As described in Chapter 3 of the text, the HCP structure is described by a simple hexagonal lattice and a two-atom motif. The Burgers vectors connecting lattice points along the reference [11¯ direction and the requested 20] [1¯ direction are shown here. 100] !# $%& b[1¯ 100] b[11¯ 20] !" !! Problem 4.23 Solution Professor R. Gronsky page 1 of 3 Problem 4.23 J. F. Shackelford, Introduction to Materials Science for Engineers, 7th Edition, Prentice Hall, New Jersey (2009) (a) From the sketch it is shown that the Burgers vectors along the reference [11¯ direction and 20] the requested [1¯ direction are orthogonal, and form the opposite and adjacent sides of a 30°100] 60°-90° triangle. The ratio of their magnitudes is therefore readily calculated by a trigonometic relationship. |b|[1¯ 100] |b|[11¯ 20] = tan 60◦ = √ 3 Consequently, |b|2 [1¯ 100] |b|2 [11¯ 20] =3 so the dislocation energies differ by a factor of 3. (b) This calculation employs the c/a ratio for HCP structures. It also requires another projection of the hexagonal close packed structure to assess the geometry in more detail, and in this instance, the motif is a significant contributor to the calculation. The three interior atoms sitting on top of the basal plane contribute to a tetrahedral arrangement of equal-sized atoms with their basal plane neighbors. The height of the mid-layer atom above the basal plane (solid triangle) is shown with another triangle construction below. One side is a because the atoms also touch along this direction. ! " #$% Problem 4.23 Solution Professor R. Gronsky page 2 of 3 Problem 4.23 J. F. Shackelford, Introduction to Materials Science for Engineers, 7th Edition, Prentice Hall, New Jersey (2009) The triangle that...
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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 Berkeley.

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