Halite 3: the unit cell of halite

The unit cell of halite is shown to the left. The chloride ions are shown in green and the sodium ions in grey. The chloride ions adopt a face-centred cubic arrangement, ie the ions occupy the eight corners of the cubic unit cell and the centres of the six faces; this is the same arrangement as seen in cubic close-packing (except the chloride ions are not close-packed on account of the sodium ions between them). The sodium ions occupy the centres of the 12 cube edges and the body-centre of the cube. So altogether there are 27 ions involved with the unit cell, but the cell occupancy is less than this since many of the ions are only partially inside the cell.

Cell occupancy:

The cell occupancy of chloride ions = (8 x 1/8) + (6 x 1/2) = 4.

The cell occupancy of sodium ions = (12 x 1/4) + 1 = 4.

The equal cell occupancy reflects the 1:1 stoichiometry of sodium and chlorine in the formula of sodium chloride.

Coordination numbers:

The coordination number of sodium is 6, ie it has six nearest neighbours. You can see that the six nearest neighbours of the sodium in the body-centre of the unit cell are the chloride ions in the centre of the six faces of the cube. The coordination number of chloride is also 6. This is less easy to observe in the unit cell but will be apparent in the structure in three pages' time. Note that the nearest neighbours of an ion are the counter ions of opposite change. This is how the structure maximises the attractive ionic forces between ions of opposite charge.

Calculating ionic bond energies:

If we know the size of the ions, the charges on them, and the Avogadro number we ought to be able to calculate the molar ionic bond energy. To do this we need to sum over all the ionic interactions in the crystal. As well the attraction of ions to their six nearest neighbours there are also longer range interactions with other ions in the crystal. Inspection of the central sodium ion in the unit cell reveals that there are also:

• 12 next-nearest neighbours: these are the sodium ions at the centres of the 12 edges of the cube. Interactions with these ions will be repulsive.
• 8 next-next-nearest neighbours: these are the chloride ions at the eight corners of the cube. Interactions with these ions will be attractive.

By considering the shells of next-nearest neighbours and calculating (by simple trigonometry) their distance from the central ion, it is possible to write a series of these interactions. This series converges to give the so-called Madelung structure constant for the crystal structure.

Go to page 4 to consider hole occupancy in the lattice.

Other elements/compounds with a cubic unit cell:

CdCl₂, Li₃B, ReO₃, CO₂, N₂, NH₃, Ne, SiF₄, Ag₂O, La₂O₃, XeF₆, Cu₃Au, NaTl, brass (Cu₃Zn₂), perovskite (CaTiO₃), magnetite (Fe₃O₄), spinel (MgAl₂O₄), pyrite (FeS₂), galena (PbS), YAG (yttrium aluminium garnet, Y₃Al₅O₁₂), elpasolite (K₂NaAlF₆), phosphomolybdate ((NH₄)₃Mo₁₂O₃₆.PO₄.21H₂O), sal ammoniac (NH₄Cl), uraninite (UO₂), chengdeite (Ir₃Fe), linnaeite (Co₃S₄) and senarmontite (Sb₂O₃).

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