The Rhombohedral-Hexagonal Transformation

The lattice of points shown in Fig. A4-1 is rhombohedral, that is, it possesses the symmetry elements characteristic of the rhombohedral system. The primitive rhombohedral cell has axes ai(R), ajCR), and a3(R). The same lattice of points, however, may be referred to a hexagonal cell having axes a,(H), ajCH), and c(H). The hexagonal cell is no longer primitive, since it contains three lattice points per unit cell (at 0 0 0, f and f f), and it has three times the volume of the rhombohedral cell. 

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  

the (001) face of the rhombohedral cell (shown shaded in the figure) has indices (OT 1) when referred to hexagonal axes. 

Since a rhombohedral lattice may be referred to hexagonal axes, it follows that the powder pattern of a rhombohedral substance can be indexed on a hexagonal Hull-Davey or Bunn chart. How then can we recognize the true nature of the lattice From the equations given above, it follows that 

If the lattice is really rhombohedral, then k is an integer and the only lines appearing in the pattern will have hexagonal indices HK L) such that the sum -H- rK+ L) is always an integral multiple of 3. If this condition is not satisfied, the lattice is hexagonal. 

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