Rhombohedral cell

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).

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,... 

The equivalent sets of atomic positions (15 atoms) in this rhombohedral cell are ... 

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...

Calcite is a calcium carbonate, CaCOs, with a rhombohedral cell consisting of the large planar CO3 groups, which contain a Ca ion at the center of an... 

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. 

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

These relations are for the particular mutual orientation of the two cells chosen in Fig. 243 (the point D having hexagonal coordinates ). For the alternative orientation in which the rhombohedral cell is rotated 60° so that point D now has hexagonal coordinates J f the relations are ... 

If we wish to have a primitive cell, we must choose the one defined by the vector t3, and two others of equal length related to it by threefold rotation. We then obtain a rhombohedral cell bounded by six rhombuses. 

Figure 16.6. Primitive unit cell of the rhombohedral lattice 3 R. The three fundamental translations a1 a2, a3 are of equal length and make equal angles with e3. Hexagonal nets in four successive layers show how the rhombohedral cell may be constructed.

The primitive rhombohedral cell in Figure 16.6 can be specified by giving the length a ai and the angle a between any pair of the fundamental translation vectors ab a2, 83. Choose ei along the projection of ai in the xy plane 0 is the angle made by ai with e3. 

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).

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.

Another two-dimensional C60 polymer phase is the rhombohedral phase, in which the (111) plane of the original fee C60 lattice is the polymerized plane (Fig. 11). If the unit in the rhombohedral cell is spherical or, at least, has a six-fold symmetry along the axis perpendicular to the plane, the stacking sequence is unique. However, each polymerized C60 unit only has a three-fold symmetry along the axis, and there can be two different ways of stacking, ABC and ACB , where B and C correspond to the polymerized plane with the C60 positions above the B and C sites in Fig. Irrespectively [39]. 

The rhombohedral cell is the space confined by six spheres in closest packing. In this configuration there are twelve points of 5-contact and eight triangular contacts. 

Denote First Layer of Spheres and Dotted Lines the Spheres Lying in the Hollows Formed by the First Layer. T, Ti Are Forms of Tetrahedral Cells R Denotes Rhombohedral Cells. 

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