3 The structures of simple solids
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Answer We need to identify the displacements that, when applied to the entire cell, result in every atom
arriving at an equivalent location (same atom type with the same coordination environment). In this case, the displacements (0, +½, +½), (+½, +½, 0), and (+½, 0, +½), where +½ in the x , y , or z coordinate represents a translation along the appropriate cell direction by a distance of a /2, b /2, or c /2 respectively, have this effect. For example, starting at the labelled Zn 2 + ion towards the near bottom left-hand corner of the unit cell (the origin), which is surrounded by four S 2 −
the translation ( +½, 0, +½), we arrive at the Zn 2 +
cell, which has the same tetrahedral coordination to sulfur. Identical translational symmetry exists for all the ions in the structure. These translations correspond to those of the face-centred lattice, so the lattice type is F.
(b) Fractional atomic coordinates and projections Key point: Structures may be drawn in projection, with atom positions denoted by fractional coordinates. The position of an atom in a unit cell is normally described in terms of fractional coordi- nates , coordinates expressed as a fraction of the length of a side of the unit cell. Thus, the position of an atom, relative to an origin (0,0,0), located at xa parallel to a , yb parallel to b , and zc parallel to c is denoted ( x , y , z ), with 0 ≤ x , y , z ≤ 1. Three-dimensional representa- tions of complex structures are often diffi cult to draw and to interpret in two dimensions. A clearer method of representing three-dimensional structures on a two-dimensional sur- face is to draw the structure in projection by viewing the unit cell down one direction, typically one of the axes of the unit cell. The positions of the atoms relative to the projec- tion plane are denoted by the fractional coordinate above the base plane and written next to the symbol defi ning the atom in the projection. If two atoms lie above each other, then both fractional coordinates are noted in parentheses. For example, the structure of body- centred tungsten, shown in three dimensions in Fig. 3.8 a, is represented in projection in Fig. 3.8 b. E X A MPLE 3. 2 Drawing a three-dimensional representation in projection Convert the face-centred cubic lattice shown in Fig. 3.5 into a projection diagram.
perpendicular to one of its faces. The faces of the cubic unit cell are square, so the projection diagram viewed from directly above the unit cell is a square. There is a lattice point at each corner of the unit cell, so the points at the corners of the square projection are labelled (0,1). There is a lattice point on each vertical face, which projects to points at fractional coordinate ½ on each edge of the projection square. There are lattice points on the lower and on the upper horizontal face of the unit cell, which project to two points at the centre of the square at 0 and 1, respectively, so we place a fi nal point in the centre of a square and label it (0,1). The resulting projection is shown in Fig. 3.9 . Download 207.23 Kb. Do'stlaringiz bilan baham: |
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