Rhombohedral axes

Rhomobohedral is a subset of the trigonal system in which the unit cell can be chosen on either hexagonal or rhombohedral axes. 

It is often more convenient to refer some trigonal crystals to a rhombo-hedral oell which has three equal axes making equal angles not 90° with each other. The three equal rhombohedral axes are equally inclined to the c axis of the hexagonal-type cell. 

To check this by a specific example, consider the 431 rhombohedral reflection. To find the lH index in terms of rhombohedral indices, we must travel from O to O by way of rhombohedral axes, that is, via OD, DK, and KO it is evident from Fig. 243 b that from O to D we cross four planes (hn), from D to K three planes (kR), and from AT to O one plane (lR), so that in all eight planes are crossed (hRA-kR- lR). For hll9 we have to go from O to A to get there by way of rhombohedral axial directions, the simplest course is via OD and DA from O to D four planes are crossed, while from D to A three planes are crossed, but these have to be subtracted because we are returning towards the original plane through 0 (or, in other... 

Consider now the reverse transformation, from hexagonal to rhombohedral indices. It is necessary to go from the origin to some point, first by way of rhombohedral axes and fhen by way of hexagonal axes. Such a point is M if we go. there directly, the distance is 36, and thus the phase-difference between waves from O and Mf is 3fcjj. Now go from O to M by way of hexagonal axes—for instance, via B and M. From OtoB the phase-difference is Jcf, from B to M it is —hH (since we are going in the negative aH direction), and from M to M it is lH the total is kH—ha+ln. Thus %kR -- kH—hH+lH. 

Certain trigonal crystals may also be referred to rhombohedral axes, the unit cell beaig a rhombohedron defined by cell edge a and irteraxial angle a (j 90°)... 

Here we use hexagonal axes description of the R3c space group. The transformation to monoclinic and rhombohedral axes can be found in Ref. [5]. 

In the alternative reverse setting (seldom used), the rhombohedral axes are rotated by 60° in an anti-clockwise sense about the c axis of the hexagonal unit cell. The equivalent lattice points are then (0 0 0), (Vs, 2k, Vs), and (2/s, Vs, 2/s). The reader can readily deduce the vectorial and matrix relationships for the two sets of axes and verify that Vr/Vh remains unchanged at 1/3. 

Relationship between the rhombohedral axes aR, bR, cR (obverse or positive, or direct setting) and the hexagonal axes aH, bH, cH [4, pp. 20-21). 

If one wishes to know the indices HK L), referred to hexagonal axes, of a plane whose indices ihkl), referred to rhombohedral axes, are known, the following equations may be used ... 

When the pattern of a rhombohedral substance has been so indexed, i.e., with reference to hexagonal axes, and the true nature of the lattice determined, we usually want to know the indices hkl) of the reflecting planes when referred to rhombohedral axes. The transformation equations are... 

The trigonal system has three axes of equal length, each enclosing equal angles a (= / = y), forming a rhombohedron. The axes are called rhombohedral axes, while the system name trigonal refers to the presence of a three-fold axis of... 

For visualization, it is more convenient to use hexagonal rather than rhombohedral axes (Figure 2.15). The transformation from rhombohedral to hexagonal axes is ... 

Since 3m and 6/mmm are included in the same point lattice, they are sometimes subsumed into the hexagonal crystal family. So there are seven crystal systems but six crystal families. Note further that rhombohedral symmetry is a special case of centering (/J-centering) of the trigonal crystal system and offers two equivalent possibilities for selecting the cell parameters hexagonal or rhombohedral axes (see Table 1.3-4 again). 

Symbols. P, primitive C, end centered F, face centered /, body centered R, primitive in rhombohedral axes a, b, c, glide planes n, number of rotation axes. For symmetry elements (e.g., 1,2/m), see Figure 20.13. 

The structures of the two rhombohedral forms of elemental boron (Table 5) are of interest in illustrating what can happen when icosahedra are packed into an infinite three-dimensional lattice. In these rhombohedral structures the local symmetry of a Bj2 icosahedron is reduced from //, to Dsd because of the loss of the 5-fold rotation axis when packing icosahedra into a crystal lattice. The 12 vertices of an icosahedron, which are all equivalent under //, local symmetry, are split under iXd local symmetry into two nonequivalent sets of six vertices each (Figure 19a). The six rhombohedral vertices (labeled R in Figure 19a) define the directions of the rhombohedral axes. The six equatorial vertices (labeled E in Figure 19a) lie in a staggered belt around the equator of the... 

The trigonal crystal system can be defined by hexagonal or rhombohedral axes, denoted by (H) and (R), respectively. 

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