the first of these arrangements, in Figure 1.20a, the twofold axes at the corners of the unit parallelogram of the nets all coincide and we produce a lattice of which one unit cell is shown in Figure 1.19b. This has no two sides of the primitive cell necessarily equal, but two of the axial angles are 90°. A frequently used convention is to take α and γ as 90° so that y is normal to x and to z; β is then taken as the obtuse angle between x and z.
The staggered arrangement of the parallelogram nets in Figure 1.20b is such that the twofold axes at the corners of the unit parallelograms of the second net coincide with those at the centres of the sides of the unit parallelogram of those of the first (or zero‐level) net. A lattice is then produced of which a possible unit cell is shown in Figure 1.21. This is multiply primitive, containing two lattice points per unit cell, and the vector t4 is normal to t1 and t2. Such a cell with lattice points at the centres of a pair of opposite faces parallel to the diad axis is also consistent with twofold symmetry. The cell centred on opposite faces shown in Figure 1.21 is chosen to denote the lattice produced from the staggered nets because it is more naturally related to the twofold symmetry than a primitive unit cell would be for this case.
Figure 1.21 A three‐dimensional view of the staggered arrangement of nets in Figure 1.20(b) in which two‐fold symmetry is preserved. In this diagram, the vector t4 is normal to t1 and t2
The staggered arrangement of nets shown in Figures 1.20b and 1.21 could also have shown diad symmetry if we had arranged that the corners of the net at height z had lain not above the midpoints of the side containing t1 in Figure 1.20b but vertically above either the centre of the unit parallelogram of the first net or above the centre of the side containing t2 in Figure 1.20b. These two staggered arrangements are not essentially different from the first one, since, as is apparent from Figure 1.22, a new choice of axes in the plane of the nets is all that is needed to make them completely equivalent.
Figure 1.22 Lattice points in the net at height zero are marked with dots. The rings and crosses indicate alternative positions of the lattice points in staggered nets at height z, arranged so as to preserve twofold symmetry. The dotted lines show an alternative choice of unit cell
There are then two lattices consistent with monoclinic symmetry: the primitive one with the unit cell shown in Figure 1.19b and a lattice made up from staggered nets of which the conventional unit cell is centred on a pair of opposite faces. The centred faces are conventionally taken as the faces parallel to the x‐ and y‐axes; that is, (001), with the diad parallel to y (see Figure 1.19c). This lattice is called the monoclinic C lattice. The two lattices in the monoclinic system can be designated P and C, respectively.
The two tetragonal lattices can be rapidly developed. The square net in Figure 1.14b has fourfold symmetry axes arranged at the corners of the squares and also at the centres. This fourfold symmetry may be preserved by placing the second net with a corner of the square at 00z with respect to the first (t3 normal to t1 and t2) or with a corner of the square at
The nets shown in Figures 1.14d and 1.14e are each consistent with the symmetry of a diad axis lying at the intersection of two perpendicular mirror planes. It is shown in Section 2.1 that a mirror plane is completely equivalent to what is called an inverse diad axis: a diad axis involving the operation of rotation plus inversion. This inverse diad axis, given the symbol
If we stack rectangular nets vertically above one another so that a corner lattice point of the second net lies vertically above a similar lattice point in the net at zero level (t3 normal to t1 and t2) then we produce the primitive lattice P. The unit cell is shown in Figure 1.19d. It is a rectangular parallelepiped.
If we stack rhombus nets (centred rectangles) vertically above one another, we obtain the lattice shown in Figure 1.23. This is the orthorhombic lattice with centring on one pair of faces.
