Hexagonal and rhombohedral systems

Abstract

Fig. 24 and Tables 5, 6 and 7 provide for calculations and stereographic projections for hexagonal crystals over a wide range of axial ratios.† Rhombohedral lattices referring to these hexagonal axes include the three cubic lattices with the axial ratios given in Table 5. In Fig. 24 (0001) = (00.1) hexagonal or (111) rhombohedral (cubic) is the plane of projection. The four-index system is used in the diagrams to emphasize the difference between the H and R axes but only three indices are desirable for transformations and calculations of angles etc. (see §9.2). The following notes may be of help when using Fig. 24 and Tables 5, 6 and 7:

• The rhombohedral faces have poles which fall in the position shown at equal angles ρ ₂ from the threefold axis.

• Half-way between any pair of these planes, e.g. (100) and (101), is located the corresponding (110). This provides the angle λ/2, which in Fig. 24 is measured along the great circle.

• The {110} poles are at an angle ρ ₁ from the centre.

• It is clear that a knowledge of any one of the three angles ρ ₁,ρ ₂ or λ is sufficient to determine the other two and to establish the stereographic projection completely for either a rhombohedral or a hexagonal lattice.

• The angles are functions of the hexagonal axial ratio c\a and are deduced from the general formulae given below.