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12 Modern Materials Solutions to Exercises 12.16 In amorphous silica (SiO₂) the regular structure of quartz is disrupted; the loose, disordered structure, has many vacant "pockets" throughout. There are fewer SiO₂ groups per volume in the amorphous solid; the packing is less efficient and less dense. 12.17 Analyze. Given two two-dimensional structures, draw and describe the unit cells and lattice vectors. Plan. When choosing a unit cell, the environment of each lattice point must be identical and the 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. The radii of A and B are equal. Solve. Two-dimensional (i) (ii) structure (a) unit cell = 90°, a = b = a=b (c) lattice type square hexagonal 12.18 Two-dimensional (i) (ii) structure (a) unit cell = 90°, a = b = 90°, a # b (c) lattice type square rectangular 12.19 Plan. Refer to Figure 12.6 to find geometric characteristics of the seven three dimensional primitive lattices. (a) Orthorhombic, a # b # c, A rectangular base has unequal side lengths and an angle of 90°. The third lattice vector is perpendicular to the first two and a different length, so all angles are 90° and all vector lengths are unequal. (b) Tetragonal, a = b # c, The only difference between the two lattices is that the second has a square (rather than rectangular) base. This means that two of the edge lengths are equal but the third is not, and all angles remain 12.20 (a) If the base of the unit cell is an arbitrary parallelogram, a # b and the angle between them does not equal If is perpendicular to the other two lattice vectors, the angles between c and a, as well as the angles between c and b, are 90°; a # b # c, α = = 90°, # The "standard" monoclinic cell is a # b # c, α = = 90°, # The unit cell described here is "non-standard", but can be 337