Rhombohedral Bravais Lattice

This page is reference material.

hR

A rotation axis of order 3 along the body-diagonal of the unit cell (shown as a dashed line) constrains all of the sides to be of equal length and all of the angles to be equal, as shown above. A unit cell with these axes is referred to as primitive rhombohedral.

However, instead of choosing a primitive rhombohedral unit cell to represent this Bravais lattice, it is normal to choose an equivalent rhombohedrally-centred hexagonal cell as shown below. Thus the standard notation for this Bravais lattice type is hR, although a more logical notation would be rP, which indeed is sometimes used with respect to the above unit cell.

hR

This figure shows the rhombohedrally-centred hexagonal cell (solid lines) and its relationship to the primitive rhombohedral cell (dotted lines); the two-middle points on the dashed line correspond to the rhombohedral lattice centring.

Both these unit cells apply only to the rhombohedral subset of the trigonal crystal system. Note that the volume of the R-centred hexagonal cell is three times that of the primitive rhombohedral cell.

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