Figure 1.23 The stacking of rhombus nets vertically above one another to form an orthorhombic lattice with centring on one pair of faces
We can also preserve the symmetry of a diad axis at the intersection of two mirror planes by stacking the rectangular nets in three staggered sequences. These are shown in Figures 1.24a, b and c. The lattices designated A‐centred and B‐centred are not essentially different since they can be transformed into one another by appropriate relabelling of the axes.14 The staggered sequence shown in Figure 1.24c is described by the unit cell shown in Figure 1.19f. It is the orthorhombic body‐centred lattice, symbol I. There is only one possibility for the staggered stacking of rhombus nets. Careful inspection of the right‐hand side of Figure 1.14e shows that the only places in the net where a diad axis lies at the intersection of two perpendicular mirror planes is at points with coordinates (0, 0) and
Figure 1.24 The three possible stacking sequences of rectangular nets. In the three left‐hand diagrams, lattice points in the net at zero level are denoted with dots and those in the net at level z with open circles. The corresponding three‐dimensional views of the arrangements of the nets are shown in the three right‐hand diagrams
Figure 1.25 The staggered stacking of rhombus nets. This form of stacking generates the orthorhombic F Bravais lattice
All of the lattices consistent with 222 – that is, orthorhombic symmetry – are shown in Figures 1.19d, e, f and g. The unit cells are all rectangular parallelepipeds, so that the crystal axes can always be taken at right angles to one another – that is, α = β = γ = 90° – but the cell edges a, b and c may all be different. The primitive lattice P can then be described by a unit cell with lattice points only at the corners, the body‐centred lattice I by a cell with an additional lattice point at its centre and the F lattice by a cell centred on all faces. The A‐, B‐ and C‐centred lattices, shown in Figures 1.24a, 1.24b and 1.23, respectively, are all described by choosing the axes so as to give a cell centred on the (001) face; that is, a C‐centred cell.
We have so far described nine of the Bravais space lattices. All further lattices are based upon the stacking of triequiangular nets of points. The triequiangular net is shown in Figure 1.14c. There are sixfold axes only at the lattice points of the net. To preserve sixfold rotational symmetry in a three‐dimensional lattice, such nets must be stacked vertically above one another so that t3 is normal to t1 and t2. The lattice produced has the unit cell shown in Figure 1.19j. The unique hexagonal axis is taken to lie along the z‐axis so a = b ≠ c, γ = 120° and α and β are both 90°. This lattice (i.e. the array of points) possesses sixfold rotational symmetry and is the only lattice to do so. However, it is also consistent with threefold rotational symmetry about an axis parallel to z. A crystal in which an atomic motif possessing threefold rotational symmetry was associated with each lattice point of this lattice would belong to the trigonal crystal system (Table 1.3).
A lattice consistent with a single threefold rotational axis can be produced by stacking triequiangular nets in a staggered sequence. A unit cell of the triequiangular net of points is shown outlined in Figure 1.26 by the vectors t1, t2 along the x‐ and y‐axes. Axes of threefold symmetry pierce the net at the origin of the cell (0, 0) – at points such as A – and also at two positions within the cell with coordinates
Figure 1.26 The stacking of triequiangular nets of points in a staggered sequence. Lattice points labelled A, B and C belong to nets of lattice points at different heights relative to the plane of the paper
A
