Oranges in a Crate, Marbles in the Gaps
You already know from the bonding rung that most oxide and halide ceramics behave, to a good first approximation, as if they were held together by pure electrostatics — the ionic model, atoms as charged spheres. Two facts about those spheres now do almost all the work. First, the anion (O2-, F-, Cl-) is big — the oxygen ion is about 140 pm in radius — because it has gained electrons and puffed up. Second, the cation (Mg2+, Al3+, Ti4+) is small — Mg2+ is only about 72 pm — because it has lost electrons and shrunk. So the crystal is mostly a stack of big negative balls, with tiny positive balls hiding among them.
The big idea of this whole rung follows in one line: the large anions stack in the most space-saving way they can, like oranges packed in a crate, and the small cations drop into the gaps left over. That efficient stacking is close packing. There are two common ways to do it — cubic close packing (an FCC arrangement, layers stacked ABCABC) and hexagonal close packing (HCP, stacked ABAB) — and both fill about 74 percent of space with spheres. The other 26 percent is not solid: it is a regular network of empty pockets, and those pockets are where the cations live.
Two Kinds of Holes, and How Many
Stack close-packed layers and exactly two shapes of hole appear between them. A tetrahedral hole is the small pocket where a sphere nestles into the dimple between three spheres in the next layer — four anions surround it, one at each corner of a tetrahedron, so a cation sitting there has coordination number 4. An octahedral hole is the roomier pocket ringed by six anions at the corners of an octahedron, giving coordination 6. The octahedral pocket is the bigger of the two, so it takes a bigger cation.
The counting is fixed and worth memorizing. In an array of N close-packed anions there are exactly N octahedral holes and 2N tetrahedral holes — twice as many tetrahedral pockets, but each one smaller. That inventory is set purely by the packing; it does not care what compound you are making. What the compound then decides is the fraction of those holes the cations actually occupy — and, as the next sections show, that single choice is what tells rock salt apart from spinel.
Radius Ratio: Which Hole a Cation Fits
Which hole a given cation chooses is mostly a question of size. The rule of thumb is the radius ratio, r_cation divided by r_anion. A cation is happiest in the largest hole in which it still touches all its anion neighbors at once: big enough to prop the surrounding anions apart, not so small that it rattles around loose. Work out the geometry of touching spheres and you get sharp cut-off values — the critical radius ratios — where the preferred coordination jumps from one value to the next.
- Look up the ionic radii for the cation and the anion in the right coordination (radii grow with coordination, so use a consistent table such as Shannon's).
- Divide: radius ratio = r_cation / r_anion. For a small cation sitting in a big-anion array it is always less than 1.
- Compare with the cut-offs: 0.155-0.225 favours 3-fold (triangular), 0.225-0.414 favours 4-fold (tetrahedral), 0.414-0.732 favours 6-fold (octahedral), and 0.732-1.0 favours 8-fold (cubic).
- Read off the coordination, then match it to a structure — 6-fold in an FCC anion array with every octahedral hole filled is rock salt.
Try it on MgO. With Mg2+ near 72 pm and O2- near 140 pm, the ratio is about 0.51, which lands in the 0.414-to-0.732 band: 6-fold, octahedral. And indeed each Mg2+ sits in an octahedral hole of an FCC oxygen array, coordination 6 — that is exactly the rock-salt structure. Be candid about the rule's limits, though: it is a geometric guide, not a law, and roughly a third of compounds disobey it because real bonds are partly covalent, ions polarize and squish, and radii themselves shift with coordination. Use it to form an expectation, then let the measured structure have the final word.
Stoichiometry Picks the Pattern
The packing hands you an inventory of holes; the chemical formula, the stoichiometry, decides how many to fill so that positive and negative charge balance exactly. That is the whole game. An AX compound with one cation per anion, like MgO, must fill enough holes to place one cation for each anion: fill all N octahedral holes of an FCC array and you get rock salt (coordination 6:6); fill half the 2N tetrahedral holes instead and you get zinc-blende, or on an HCP array wurtzite (coordination 4:4). Same 1:1 formula, three different fillings.
Change the formula and you simply change the fraction. An AX2 oxide like rutile TiO2 has half as many cations as anions, so its Ti4+ fills only half the octahedral holes; fluorite ZrO2 and CaF2 take the roomier 8-fold route instead. A sesquioxide A2X3 such as corundum, Al2O3, fills two-thirds of the octahedral holes of an HCP oxygen array. Step up to two cation species and you reach the showcase structures: perovskite ABO3 (BaTiO3), where a small B cation fills a quarter of the octahedral holes while a large A cation joins the close packing itself, and spinel AB2O4 (MgAl2O4, the ferrites), which fills one-eighth of the tetrahedral and half of the octahedral holes. Every one of them is nothing more than an anion array plus a rule for which holes to fill.
STRUCTURE FORMULA ANION ARRAY CATIONS FILL... CN (cat:an) ------------ ------- ------------ -------------------------- ----------- rock salt AX FCC ALL octahedral holes 6 : 6 zinc-blende AX FCC 1/2 of tetrahedral holes 4 : 4 wurtzite AX HCP 1/2 of tetrahedral holes 4 : 4 CsCl * AX simple cubic the one cubic hole 8 : 8 fluorite AX2 FCC (cations) anions in ALL tet holes 8 : 4 antifluorite A2X FCC ALL tetrahedral holes 4 : 8 rutile AX2 ~HCP oxygen 1/2 of octahedral holes 6 : 3 corundum A2X3 HCP 2/3 of octahedral holes 6 : 4 perovskite ABO3 FCC (A + O) B in 1/4 octahedral holes B:6 A:12 spinel AB2O4 FCC 1/8 tet + 1/2 octahedral mixed * CsCl is NOT close-packed -- the honest exception to the whole picture.
One member refuses to play along, and it is worth knowing why. In cesium chloride, CsCl, the cation is nearly as big as the anion (ratio about 0.94), so the octahedral hole is far too tight; Cs+ demands 8 neighbors. The anions oblige by giving up close packing altogether and sitting on a simple cubic lattice, with the cation in the middle of the cube — coordination 8:8. It is the exception that proves the rule: when a cation grows too big for any close-packed hole, the packing itself reorganizes to feed its appetite.
From Unit Cell to Density — and One Recipe, Many Crystals
Once you know the recipe you know the unit cell, and once you know the unit cell you can compute the crystal's theoretical density — the density a perfectly dense, pore-free single crystal would have. The formula is just mass over volume for one repeating box: density = (n times M) / (V times N_A), where n is the number of formula units in the cell, M the formula mass, V the cell volume, and N_A Avogadro's number (6.022 x 10^23 per mol).
Do it for MgO. Its rock-salt cell holds n = 4 formula units, the formula mass is M = 40.3 g/mol, and the cell edge is a = 0.421 nm, so V = a^3 = 7.46 x 10^-23 cm^3. Then density = (4 times 40.3) / (7.46 x 10^-23 times 6.022 x 10^23) = 161.2 / 44.9, about 3.59 g/cm^3 — right on top of the measured value. This number becomes the yardstick for processing: a fired ceramic is graded as a percent of theoretical density, and the whole art of firing is chasing the last few percent by closing pores, a story the sintering rung will pick up.
Finally, the twist that will drive much of what comes later: the same composition can adopt more than one of these structures depending on temperature and pressure — polymorphism. Silica runs through quartz, tridymite, and cristobalite as it heats; titania comes as rutile, anatase, and brookite; and zirconia climbs from monoclinic to tetragonal to cubic on the way up, snapping back with a volume change on the way down. That last transformation is no curiosity — it cracks pure zirconia on cooling, and taming it is exactly what makes toughened zirconia possible. Polymorphism is where structure stops being a static catalogue and starts to move, which is where the phase-transformation rungs begin.