Rhombohedral-hexagonal cells

The unique axis in the monoclinic unit cell is mosdy taken as the b axis. Rhombohedral unit cells are often specified in terms of a bigger hexagonal unit cell. 

Cl 6c +(u,u,u +m, + m, + m + w, + w, +m) w= This structure has a rhombohedral symmetry but is usually described in terms of a triple-volume hexagonal cell, which makes comparison with the idealized Cdl2 structure simpler. The idealized unit cell is adequate for the purposes of this book. In this representation the anion layers are in cubic closest packing. .. ABC ABC ABC... The metal and nonmetal stacking sequence is... 

Description of a rhombohedral unit cell in terms of the equivalent, triple-primitive, hexagonal cell (see Fig. 3.9). 

Figure 3.9. The relation between a primitive rhombohedral cell and an equivalent tripleprimitive hexagonal cell is shown. The cell orientations have been selected according to the criteria adopted by the International Tables (Hahn 2002).

A detailed example of the alternative descriptions of a given compound, both in terms of its hexagonal unit cell and of the corresponding rhombohedral primitive cell is presented in Chapter 4 the rhombohedral compound Mo6PbSx (the prototype of the family of the so-called Chevrel phases) is described and unit cell constants and atomic positions are listed for its conventional hexagonal cell and for the rhombohedral primitive cell. 

Description of a cubic (primitive, body centred or face centred) unit cell (ac) in terms of the equivalent, primitive rhombohedral, (a,-, a) and triple-primitive hexagonal, cells (ah, ch). See Fig. 3.11. 

In this book, the Pearson symbol will be used throughout and the convention has been adopted indicating in every case the number (ideal or effective) of atoms contained in the chosen unit cell. In the case, therefore, of rhombohedral substances for which the data of the (triple primitive) hexagonal cell are generally reported, the number of atoms is given which is in the hexagonal cell and not the number of atoms in the equivalent rhombohedral cell (Ferro and Girgis 1990). So, for instance,... 

Mo bSg. It is rhombohedral (hR45), space group R3, N.148, with the following atomic positions (45 atoms in the triple primitive hexagonal cell) ... 

The space group is R3, N. 148 with 42 atoms in the (triple primitive) hexagonal cell (14 atoms in the equivalent, primitive, rhombohedral cell). 

Fig. 3 Relationship between the triple hexagonal cell with axes OE, OF, OG to the primitive rhombohedral cell with axes OA, OB, OC...

Fig. 73. Rhombohedral coll (bold lines) with corresponding hexagonal cell (narrow lines) and hexagonal prism (dotted).

It is not easy to determine directly a and a from a powder photograph by the use of this rather unwieldy expression fortunately, however, the atomic arrangement in rhombohedral crystals can always be referred to a larger hexagonal cell (Fig. 73) whose dimensions an and cH are related to those of the rhombohedral cell, an and a, by the relations... 

The procedure is to find the simplest hexagonal indices on the chart already mentioned, to calculate the dimensions of the hexagonal cell, and finally to find the dimensions of the true rhombohedral cell by the above expressions. 

For rhombohedral unit cells, it is best to transform indices to hexagonal indices, and to use the above formula for the spacings of planes. 

Suppose we have rhombohedral indices hR kR lR and wish to transform them to hexagonal indices hH IcH iHlH. The c axis of the hexagonal cell is 00 (through the middle of the rhombohedron). Pass from O to O, first of all directly the phase-difference between waves from O and those from O js In. Now go from O to O by way of rhombohedral axial directions-—for instance, via OD, DK, and KO. Waves from D are hR wavelengths ahead of those from 0, those from K are k.R wavelengths ahead of those from D, and those from O are lR wavelengths ahead of those from K. The total is h R- rk R -l r- Thus lH = hRA-kRA lR ... 

Fig. 243. Relations between rhombohedral and hexagonal indices. ODKEO LFJ, rhombohedral cell OAGBO A G B, hexagonal cell.

The trigonal system can be considered as a subdivision of the hexagonal unit. On this basis there would be only six different crystal systems, but conventionally, the trigonal system (also called the rhombohedral system) is retained separately. Figure 2.1 shows two rhombohedral cells within a hexagonal cell. 

Figure 2.1. Two rhombohedral cell are shown within a hexagonal cell (lighter lines).

The bismuth triiodide crystals are hexagonal (or rhombohedral), with six molecules in the hexagonal cell, C, R 3, a0 = 7.498, and... 

Figure B.l (a) The relationship of a hexagonal cell to trigonal (rhombohedral) cells. (b) the 60° rhombohedral cell related to a face-centered cubic cell.

The rhombohedral unit cells for rhodium and iridium trifluorides (44) contain two formula units. The structure can be related to the first structure type by considering anion positions, which here correspond to a hexagonal, close-packed array. There are no vacant anion sites and the cations occupy one-third of the octahedral holes. This leads to M—F—M angles of 132°, characteristic for filling adjacent, octahedral holes in a hexagonal close-packed lattice. Alternatively, the structure can be described as a linking of octahedra through all corners, but the octahedra are now tilted with respect to each other. 

Bravais showed in 1850 that all three-dimensional lattices can be classified into 14 distinct types, namely the fourteen Bravais lattices, the unit cells of which are displayed in Fig. 9.2.3. Primitive lattices are given the symbol P. The symbol C denotes a C face centered lattice which has additional lattice points at the centers of a pair of opposite faces defined by the a and b axes likewise the symbol A or B describes a lattice centered at the corresponding A or B face. When the lattice has all faces centered, the symbol F is used. The symbol I is applicable when an additional lattice point is located at the center of the unit cell. The symbol R is used for a rhombohedral lattice, which is based on a rhombohedral unit cell (with a = b = c and a = ft = y 90°) in the older literature. Nowadays the rhombohedral lattice is generally referred to as a hexagonal unit cell that has additional lattice points at (2/3,1 /3, /s) and (V3,2/3,2/3) in the conventional obverse setting, or ( /3,2/3, ) and (2/3, /3,2/3) in the alternative reverse setting. In Fig. 9.2.3 both the primitive rhombohedral (.R) and obverse triple hexagonal (HR) unit cells are shown for the rhombohedral lattice. 

As a concrete example, consider a rhombohedral lattice and the relationship between the primitive rhombohedral unit cell (in the conventional obverse setting) and the associated triple-sized hexagonal unit cell, as indicated in Fig. 9.2.5. 

Rhombohedral unit cell based on hexagonal lattice. 

Nominal Upper and Dimensions of Equivalent Hexagonal Cell, A. (Rhombohedral) Dimensions of Cubic Cell,... 

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