From a single ionic bond to a whole crystal
On the bonding rung you learned that an ionic bond is just the electrostatic pull between oppositely charged ions, with Na giving an electron to Cl. But a salt is not a molecule. There is no such thing as a discrete "NaCl unit" floating free — the attraction has no direction and no saturation, so each Na+ pulls on every Cl- it can reach and vice versa. The only way to satisfy that pull everywhere at once is to assemble a vast, repeating three-dimensional array of ions: a crystal. This rung is about how that array is built, and this guide lays the geometric foundation the whole rung rests on.
To make the geometry tractable we adopt the ionic model, a deliberate idealization: picture every ion as a hard, incompressible sphere carrying a whole-number charge spread evenly over its surface, like a tiny charged ball bearing. Cations are small spheres (they lost electrons and shrank); anions are big spheres (they gained electrons and puffed out). The forces are purely electrostatic and act in all directions equally. That is the entire model — charged marbles that attract opposites, repel likes, and refuse to overlap. Everything else in this guide is just working out how such marbles pack.
Stacking marbles: hexagonal and cubic close packing
Start with just the big spheres — usually the anions — and ask the most efficient way to fill space with identical balls. Pour marbles into a tray and shake: each one drops into the dimple between three of its neighbors, giving one layer where every sphere touches six others in a perfect hexagonal pattern. This is the densest a single layer can be. The interesting choice comes with the next layer, which again sits in the dimples of the first. The result is close packing, and it fills almost 74 percent of all space with solid sphere — the tightest any arrangement of equal spheres can achieve.
The first layer is layer A. The second layer, B, nestles into half of A's dimples. The whole story is then decided by where the third layer goes. If the third layer sits directly above the A spheres, you get the stacking ABABAB — hexagonal close packing (hcp). If instead the third layer slides into the still-empty set of dimples, a new position C, you get ABCABC — cubic close packing (ccp), which is the same as the face-centered cubic lattice you may have heard named. Both fill 74 percent of space and give every sphere twelve nearest neighbors; they differ only in the rhythm of the stack.
view from the side, layers stacked upward: hcp (ABAB...) ccp (ABCABC...) ---- A ---- A ==== B ==== B ---- A <- repeats :::: C <- third site, new ==== B ---- A <- only now repeats both: 74% filled, each sphere touches 12 others (coordination number = 12)
The holes between the spheres
Even the tightest packing leaves 26 percent of space empty, and that empty space is the secret to the whole subject. The gaps are not random; they come in exactly two well-defined shapes, the tetrahedral and octahedral holes. A tetrahedral hole is the small pocket where a sphere in one layer rests in the dimple of three spheres below it: four spheres surround the gap, their centers marking the corners of a tetrahedron. An octahedral hole is the larger cavity formed where three spheres in one layer and three in the next are staggered: six spheres surround it, centered on the corners of an octahedron.
Now the bookkeeping that makes everything click. In any close-packed array of N spheres there are exactly N octahedral holes and 2N tetrahedral holes — twice as many tetrahedral holes as there are spheres, and as many octahedral holes as spheres. And they differ in size: the octahedral hole, ringed by six spheres, is roomier than the tetrahedral hole, ringed by only four. So a packing of large anions comes pre-supplied with a precise, countable menu of small and smaller pockets, ready to receive cations.
Filling holes builds real structures
Let us actually build some salts. Take a cubic close packing of chloride ions and drop a sodium ion into every octahedral hole. The counts say the ratio is 1:1, giving NaCl, and each Na+ ends up surrounded by six Cl- and each Cl- by six Na+ — six-coordinate, or "6:6". That is the rock-salt structure, shared by a huge family from MgO to most alkali halides. The very same anion packing, but now with cations in the tetrahedral holes, gives a different world.
If cations fill half the tetrahedral holes of a ccp anion array, the ratio is 1:1 again but each ion is now four-coordinate — that is zinc blende, ZnS, the structure also adopted by diamond-like semiconductors. Fill every tetrahedral hole instead and the ratio becomes 2:1: that is the fluorite structure of CaF2, with eight-coordinate calcium and four-coordinate fluoride. Antifluorite simply swaps the roles, packing the anions and filling holes with cations, as in Na2O. One packing, a choice of which holes and how many, and the structures of much of the ionic world drop out.
- Decide which ion does the packing — almost always the larger one, usually the anion — and choose ccp or hcp for it.
- Count the holes: N octahedral and 2N tetrahedral per N packing spheres.
- Place the smaller ion into the appropriate holes, filling all or a fixed fraction of them.
- Read off the formula and coordination numbers directly from the ratio of filled holes to packing spheres.
Which hole? Radius ratios, and where the model breaks
Why does NaCl choose the roomy octahedral holes while ZnS settles for cramped tetrahedral ones? The ionic model gives a clean geometric answer through the radius-ratio rules. A cation wants the highest coordination number it can manage, because touching more anions lowers the energy — but only as long as it actually touches them. If the cation is too small for an octahedral hole, it rattles around without contacting the surrounding anions, which then close in on each other; the structure switches to a smaller hole with fewer neighbors where the cation again fits snugly. The deciding number is the ratio of the cation radius to the anion radius.
Simple trigonometry on touching spheres gives the cutoffs: roughly, a ratio above 0.73 favors eight-coordinate cubic packing (the cesium-chloride type), between about 0.41 and 0.73 favors six-coordinate octahedral holes (rock salt), and between about 0.22 and 0.41 favors four-coordinate tetrahedral holes (zinc blende). This is genuinely useful as a first guess — but be warned, it is the weakest link in the chapter. The rules assume perfect hard spheres, so they fail whenever bonding is partly covalent (ZnS is six-coordinate by the ratio yet is observed four-coordinate, because Zn-S bonds are directional and covalent) or whenever the anion is soft and polarizable. Treat radius ratios as a hint, not a verdict.
Three honest caveats round this out. First, an ion has no sharp edge, so an ionic radius is not a fixed physical length — it is a value back-calculated from measured spacings by splitting the cation-anion distance, and it shifts with coordination number; the tables you use already build in a convention. Second, neither hcp nor ccp is automatically more stable; subtle energy differences and the directional pull of partial covalency tip the balance, which is why some compounds adopt one and close cousins the other. Third, packing geometry tells you the shape but not the binding energy: how strongly the lattice actually holds together is the job of lattice enthalpy and the Born-Haber cycle, the very next steps on this rung. Geometry first, energy next.