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HW03 Soln - Problem 3.2 J F Shackelford Introduction to...

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3.2 What would be an equivalent two-dimensional point lattice for the area-centered hexagon? SOLUTION The two-dimensional point lattices are described in Example 3.1 (p. 61 of the text). There are 5 of them, and all are sketched at the top of page 62. The Note at the bottom of p. 61 explains that there may be some degeneracies in these constructions, such as an area-centered square lattice being equivalent to a simple square lattice in another orientation. This problem asks you to find an equivalent to the area-centered hexagon, reproduced here from the sketch labeled (v) on p. 62. As the outlines of the unit cells show, each cell contains 2 lattice points, one (1) fully contained within the cell and six at the perimeter shared six ways, (6 x = 1). An equivalent point lattice can be found be constructing one with a unit cell that contains only a single lattice point per unit cell, known as a "primitive" cell. Begin by "erasing" all of the lines between lattice points above and draw new lines in different unit cell configurations. There are several options, such as a rhombus, Problem 3.2 J. F. Shackelford, Introduction to Materials Science for Engineers , 7 th Edition, Prentice Hall, New Jersey (2009) Problem 3.2 Solution Professor R. Gronsky page 1 of 2
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or a parallelogram. The only requirement is that the unit cell must have parallel opposing sides , so that the unit cell can be translated along its basis vectors to generate the entire lattice with no gaps, and no over- laps. Consequently, triangles are excluded from the definition of a unit cell. Problem 3.2 J. F. Shackelford, Introduction to Materials Science for Engineers , 7 th Edition, Prentice Hall, New Jersey (2009) Problem 3.2 Solution Professor R. Gronsky page 2 of 2
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3.10 Calculate the APF of 0.74 for hcp metals. SOLUTION Begin with the definition. The atomic packing factor (APF) is defined on p. 62 of the text as a ratio of the volume of the atoms contained within a single unit cell to the volume of that unit cell. A sample APF calculation is presented in Example 3.6 of the text on page 78. The hexagonal close-packed (hcp) structure is shown in Figure 3.6 of the text (p. 64) to have a "simple" hexagonal unit cell, which by definition is a "primitive" cell containing only 1 lattice point, and a 2 atom motif. The caption to the same figure indicates that the total number of at- oms contained within the unit cell is 2: 4 corner atoms contributing 1/6 of their volume, 4 more corner atoms contributing 1/12 of their volume and 1 full atom in the interior of the cell.
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