Fluorite 5: holes in the structure

We can think of the fluoride ions occupying holes in the face-centred cubic array of calcium ions. Inspection of any of the fluoride ions in the unit cell shows that its four calcium neighbours lie on the corners of a tetrahedron with the fluoride ion at its centre. This is shown in the image to the left, in which a red tetrahedral wireframe is drawn around the neighbouring calcium ions around the top left fluoride ion.

Ratio of holes to calcium ions:

Cubes can be divided into eight smaller cubes, called octants, four in the top half and four in the bottom half of the larger cube. The eight fluoride ions in the unit cell in all the tetrahedral holes in the face-centred cubic lattice of calcium ions are divided up one per octant. Since these ions are entirely inside the unit cell, and the cell occupancy of calcium ions is 4 (see last page), then the ratio of tetrahedral holes to face-centred cubic ions is 2:1.

Holes in the close-packed layers:

If you rotate the image so that the cube is standing on a corner with the body diagonal running vertically you can see how the tetrahedral holes lie in relation to the close-packed layers of calcium ions (they are not, strictly, close-packed as the calcium ions are not touching, but they occupy the lattice coordinates associated with close-packing). The close-packed calcium layers are horizontal when the cube is in this orientation. The holes can be seen to be nested above the centre of an equilateral triangle of calcium ions and beneath another calcium ion. This is a nice illustration of the three-fold symmetry found in tetrahedra.

Go to page 6 to focus on the tetrahedral hole between the close-packed layers.

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