Many structures are based upon the close-packing of atoms or ions into hexagonal layers. In a close-packed structure, each atom or ion is surrounded by six others, resulting in very efficient packing. These hexagonal layers, in turn, may be packed in two different ways, giving rise either to a hexagonal close-packed structure, or a cubic close packed structure.

In one type of structure the hexagonal layers are stacked in an ABC fashion, so that the fourth layers lies immediately over the first, the fifth layer lies immediately over the second, and so forth.. This is emphasized by the vertical lines, which connect the first and fourth layers. The ABC stacking of hexagonal layers gives rise to the face-centered cubic structure, and is discussed more thoroughly in a separate section

Hexagonal layers may also stack in an ABAB fashion, as shown in this illustration. Note that the two red layers lie immediately above each other, as do the two blue layers. The longer vertical lines connecting the layers are shown for clarity.Whereas the satking of layers in an ABC gives rise to a face-centered cubic structure, the stacking of layers in an AB fashion gives rise to a hexagonal structure.

Regardless of whether hexagonal layers are stacked in an AB or ABC fashion, there exist two types of spaces or holes between the layers. One type of space is called an octahedral hole, and is formed between three atoms in one layer and three atoms in the layer immediately above or underneath. Although it takes six spheres to form an octahedron, the name is derived from the fact that the resulting shape has eight sides. 

A second type of space which can exist between stacked hexagonal layers is called a tetrahedral hole. Tetrahedral holes are formed between three atoms in one layer and a single atom immediately above or underneath. 

There exists two different types of spaces or holes within a series of stacked hexagonal layers. This illustration shows in green the location of two octahedral holes. The term octahedral is derived from the fact that the resulting space has eight sides. T

Many ionic compounds adopt a structure in which one set of ions forms a face-centered cubic array and the other set of ions reside within the octahedral holes. The rock salt (halite) structure is one example

The second type of space that can exist between stacked hexagonal layers is called a tetrahedral hole. This illustration shows in green the location of four tetrahedral holes in the two unit cells shown.. A tetrahedral hole is formed by three atoms in one hexagonal layer and a single atom in the layer above or beneath.

Many ionic compounds adopt a structure in which one type of ion forms a face-centered cubic array and the other set of ions resides in the octahedral holes. There are 2N tetrahedral holes for every N atoms, so this structure works well for compounds that have 1:2 stoichiometry. 

This structure shows the halite, or rock-salt structure. The white spheres represent the larger chloride ions and the smaller red spheres represent the sodium ions. Im this strcture the chloride ions form a face-centered cubic array, with the sodium ions residing inside the octahedral holes. While the halite structure is almost always drawn to illustrate its cubic nature, recall from the previous discussion that the face-centered cubic array can be alternately viewed as a series of ABC-stacked hexagonal layers.

In this illustration, eight of the octahedral holes are outlined in green. For the sake of clarity, not all of the octahedral holes are outlined. Each of the sodium ions, represented by the red spheres, resides within an octahedral hole. Consider the stoichiometry of this cell for a moment. Each of the white spheres forming the face-centered cubic array resides 1/8 inside a single cell and each of the six atoms on the six faces of the cell lie 1/2 inside the cell. Therefore a total of four sodium ions lie inside a single cell. Now consider the chloride ions. One lies at the center and is entirely contained within the cel.l. The remaining twelve each lie 1/4 within the cell. Therefore a total of four chloride ions lie inside the cell. With four sodium ions and four chlroide ions inside the cell, the stoichiometry is indeed 1:1.

The face-centered array formed by the chloride ions results in a series of octahedral and tetrahedral holes. The octahedral holes, as mentioned above, are filled by the sodium ions. The smaller tetrahedral holes, shown here, remain unfilled. You may ask, why do the sodion ions reside exclusively inside the octahedral holes? This is most likely a size issue; the octahedral holes are larger than the tetrahedral holes.

Consdier the stoichiometry of single unit cell. Each of the corner calcium ions is 1/8 inside the cell; since there are eight corners these add up to one ion inside the cell. There are six faces to a sigle cell, each with a calcium ion one-half inside the cell. Therefore a single cell contains four four calcium ions. A single cell also contans eight fluoride ions, each one located entirely within the unit cell. Since there four calcium ions and eight fluoride ions inside the cell, the 1:2 stoichiometry is maintained.

In the fluorite structure, the fluorude ions reside within the tetrahedral holes formed by the face-centered cubic array of calcium ions, and the octahedral holes are vacant. In this illustration the green cylinders outline eight of the vacant octahedral holes.

This illustration shows the location of the tetrahedral holes in the fluorite structure. Consdier why the fluroide ions would reside in the tetrahedral holes rather than the octahedral holes. The most obvious answer to this question is, of course, stoichiometry. There are two fluoride ions for every one calcium ion, and since an array of N atoms results in the formation of N octahedral holes, there would simply not be enough spaces for all fluoride ions. If the ions were reversed, with the fluoride ions forming the face-centered cubic array, there would be enough calcium ions to fill only 1/4 of the tetrahedral holes or 1/2 of the octahedral holes; this would be terribly inefficient.

Technically, the descriptions of the fluoriute structure given above are inaccurate in the sense that becasue the fluoride ions are in fact larger than the calcium ions, they therefore do not "fit inside" the tetrahedral holes. As can bee seen here, the calcium ions form a sort of "expanded" face-centered cubic structure and do not physically touch each other. Nevertheless this does represent the most efficient packing arrangement.