The Rhombohedral Lattice

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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rhombohedral unit cell The rhombohedral (R) cell presents special problems in visualization. To bring out the hexagonal nature of its lattice, and the relationship between rhombohedral crystals and ordinary trigonal or hexagonal crystals, the R cell is often represented as at right, with two interior points equally spaced along a diagonal. It is the only Bravais lattice cell with more than one interior point.
rhombohedral unit cell At left is a side view of a rhombohedron, which can be considered a cube stretched or flattened along one diagonal axis. A rhombohedron has six identical faces, all rhombuses. At right is a top view showing the three-fold symmetry of a rhombus. Red lines denote hidden edges.
rhombohedral unit cell The vertices of a rhombus fall onto equally-spaced planes as shown at left. At right is a top view.
rhombohedral lattice Here we see a layer of rhombohedra viewed from the top.
rhombohedral lattice Remember, the edges of a unit cell aren't physically real, only the points within the cell (the atoms) are. We may find it convenient sometimes to draw unit cells with corners not coinciding with atoms, especially if it brings out the symmetry better, but we can always construct a lattice with atoms at the corners of the unit cells.

Here we remove the rhombohedron edges and construct a new set of edges defining 60-120 rhombuses. We see that they enclose two equally-spaced points on a line sloping down from one corner to the diagonally-opposite corner below. Thus the two cells define the same lattice.

rhombohedral unit cell Here we see how rhombohedra relate to the R-cell. Blue points are points of the R-cell, purple points are in front of it, green ones behind, and gray in the same plane but outside the R-cell.

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Created 18 September 1998, Last Update 18 September 1998

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