A rhombohedral unit cell is a primitive crystallographic building block containing one lattice point. The distorted shape of the rhombohedron makes it difficult to visualise and manipulate. You can visualise a lattice comprising these complex rhombohedral unit cells as a much simpler system consisting of hexagonal unit cells. Hexagonal cells are not primitive, have three times the volume of the rhombohedral cells, and contain three lattice points but are easier to visualise than rhombohedral cells. You can apply the hexagonal frame of reference to atom positions in a rhombohedral crystal to convert the rhombohedral coordinates to the equivalent hexagonal coordinates.
Represent the three rhombohedral coordinates by the symbols Rx, Ry, and Rz, and the three hexagonal coordinates by the symbols Hx, Hy, and Hz. Write a set of crystallographic position coordinates as Rx,Ry,Rz or Hx,Hy,Hz.
Convert the rhombohedral x-coordinate (Rx) to the equivalent hexagonal x-coordinate (Hx) by applying the formula: Hx = (2 x (Rx) -- Ry -- Rz)/3. For example, an atom positioned at 1,0,0 in a rhombohedral unit cell will have a rhombohedral x-coordinate of Rx = 1. The equivalent Hx coordinate is: (2 x (Rx) -- Ry -- Rz)/3 = (2 x (1) -- 0 -- 0)/3 = 2/3.
Change the rhombohedral y-coordinate (Ry) to the equivalent hexagonal y-coordinate (Hy) by applying the formula: Hy = (Rx + Ry -- (2 x Rz))/3. For example, an atom positioned at 1,0,0 in a rhombohedral unit cell will have a rhombohedral y-coordinate of Ry = 0. The equivalent Hy coordinate is: (Rx + Ry -- (2 x Rz))/3 = (1 + 0 -- (2 x 0))/3 = 1/3.
Transform the rhombohedral z-coordinate (Rz) to the equivalent hexagonal z-coordinate (Hz) by applying the formula: Hz = (Rx + Ry + Rz)/3. For example, an atom positioned at 1,0,0 in a rhombohedral unit cell will have a rhombohedral z-coordinate of Rz = 0. The equivalent Hz coordinate is: (Rx + Ry + Rz)/3 = (1 + 0 + 0)/3 = 1/3.
Combine the values for Hx, Hy and Hz and write them as a set of position coordinates for the hexagonal cell: Hx,Hy,Hz. For example, an atom positioned at 1,0,0 in a rhombohedral unit cell will be positioned at 2/3,1/3,1/3 in a hexagonal unit cell.
You can convert atom coordinates from rhombohedral to hexagonal by applying a transposition of the transformation matrix to the original coordinates. The rhombohedral lattice point position of 0,0,0 allows you to convert to hexagonal coordinates without any translation. Note, however, that rhombohedral lattice points can correspond with three positions 0,0,0 or 2/3,1/3,1/3 or 1/3,2/3,2/3 in the hexagonal cell, as the rhombohedral cell is only a third of the size of the hexagonal cell. To generate a complete set of possible coordinates in the much larger hexagonal cell, individually add each of these three hexagonal unit cell lattice point coordinates to the converted hexagonal coordinates.