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12 Solids and Modern Materials Visualizing Concepts 12.1 When choosing a unit cell, remember that the environment of each lattice point must be identical and that unit cells must tile to generate the complete two-dimensional lattice. For a given structure, there are often several ways to draw a unit cell. We will select the unit cell with higher symmetry (more 90° or 120° angles) and smaller area (a X b). Two- (i) (ii) dimensional structure (a) unit cell (b) lattice Square Square (c) cell contents One black, one white One black 12.2 (a) Clearly, the structure is close packed. The question is: cubic or hexagonal? This is a side view of a close packed array, like the one in Figure 12.13. The key is the arrangement of the third row relative to the first. Looking at any three rows of cannon balls, there is a ball in the third row directly above (at the same horizontal position as) one in the first row. This is an ABABAB pattern and the structure is hexagonal close packed. (b) CN = 12, regardless of whether the structure is hexagonal or cubic close packed. 12.3 (a) There is a Re atom (gray sphere) at each corner of the unit cell: (8 X 1/8 = 1). There is an o atom (red sphere) in the middle of each cell edge: (12 1/4 = 3). There are 1 Re and 30 atoms per unit cell, for an empirical formula of (b) Each cell edge goes through two half Re atoms (at the corners) and one full atom (centered on the edge). The length of an edge, a, is then a = Re + = 2(0.70 Å) + 2(1.26 Å) = 3.92 Å (c) The density of a crystalline solid is the mass of the unit cell contents divided by the unit cell volume. There is one ReO₃ unit in each primitive cubic unit cell. The unit cell volume is (3.92 Å)³. 1 (3.92 ReO₃ Å)³ unit X cm)³ X 234.205 units Check. The units of density are correct. 334