When A and B Differ: Filling Only Some of the Holes
The last two guides handed you a powerful habit: read a ceramic as a close-packed stack of anions with the small cations dropped into the holes. Rock salt filled every octahedral hole of an FCC oxygen array; fluorite filled every tetrahedral hole; rutile filled half the octahedral holes of a distorted stack. In each of those the cations were all the same, so the accounting was simple. Now we meet the structures that crown ceramic engineering — where the holes are filled only partly, and only in a chosen pattern.
The reason is simple bookkeeping. Charge must balance across the whole crystal, so the number of cations is pinned by their charge, not by how many holes happen to be available. In a sesquioxide like Al2O3 there are only 2 cations for every 3 oxygens, and in an oxide carrying two different cations like BaTiO3 or MgAl2O4 the counts are stranger still. Whenever the cations cannot fill every hole, two new questions appear: what fraction of the holes is occupied, and — because it sets the whole architecture — which holes.
Corundum: The Sesquioxide of Alumina
Start with alumina, Al2O3, whose structure is named corundum after the mineral. Here the oxygens stack in a hexagonal close-packed array — the ABAB stack rather than the FCC one — and the aluminums slot into octahedral holes. But charge balance forbids filling them all: with 2 Al3+ for every 3 O2-, and one octahedral hole per oxygen, the aluminums can occupy only two-thirds of the octahedral holes. The corundum structure is precisely that ordered two-thirds filling.
Read off the coordination and the personality follows. Each aluminum sits among 6 oxygens (octahedral), and each oxygen is shared among 4 aluminums. Those short, strong, high-charge Al3+-O2- bonds, packed dense and rigid, are why alumina is so hard (Mohs 9, just under diamond), chemically inert, and melts near 2054 degrees C. The same corundum framework houses a whole family — Cr2O3, and hematite (alpha-Fe2O3) — and it is the canvas for gemstones: pure Al2O3 is colourless sapphire, a trace of Cr3+ makes ruby, and iron-plus-titanium makes blue sapphire.
One honest wrinkle: because only two-thirds of the holes are filled, nature must choose a repeating pattern of filled and empty ones, and it does so to keep the aluminums as far apart as possible. Where two filled octahedra share a face, the two Al3+ repel and pucker away from each other, so the real oxygen stack is a slightly puckered HCP rather than a perfect one. Partial filling always brings this question of ordering — a theme that returns in spinel.
Perovskite: A Cage with a Marble
Perovskite, the ABO3 family named after the mineral CaTiO3, is the structure behind a huge slice of electronic ceramics. Picture the ideal cubic cell of BaTiO3: a big Ba2+ at each corner, a small Ti4+ at the body centre, and an O2- at the centre of each face. Read it as holes-in-a-lattice and it is elegant — the Ba2+ and the oxygens together build a close-packed array, and the tiny Ti4+ drops into one quarter of its octahedral holes, choosing only the holes ringed entirely by oxygens. So Ti4+ is 6-coordinate in an oxygen octahedron, while the roomy Ba2+ is 12-coordinate. The perovskite structure is a 3-D framework of corner-sharing TiO6 octahedra with Ba parked in the cavities between them.
Here is where perovskite earns its fame. In BaTiO3 the octahedral cavity is just slightly too big for Ti4+, so below about 130 degrees C the titanium cannot sit still at the centre — it settles off-centre, like a marble resting in one of two dimples on either side of the cage. That tiny displacement pulls positive and negative charge apart and gives every cell a built-in electric dipole, and because the marble can be pushed from one dimple to the other by a field, the whole crystal becomes switchable: this is ferroelectricity. It is exactly why barium titanate has a colossal dielectric constant and fills the multilayer capacitors inside almost every phone.
Spinel: Two Cations, Two Kinds of Hole
Spinel, the AB2O4 family named after gem-quality MgAl2O4, is the most intricate filling of the three — and the one that finally uses both kinds of hole at once. The oxygens sit in an FCC array, and now the two different cations sort themselves by size and charge: the smaller Mg2+ takes tetrahedral holes and the Al3+ takes octahedral holes. Counting is the payoff. The spinel unit cell packs 8 formula units — 32 oxygens, and with them 32 octahedral and 64 tetrahedral holes. Into those, 8 Mg2+ fill one-eighth of the tetrahedral holes and 16 Al3+ fill one-half of the octahedral holes.
That neat picture — A in tetrahedral, B in octahedral — is called a normal spinel, but nature keeps a twist. In an inverse spinel the A2+ cations swap into octahedral sites and push half of the B3+ out into the tetrahedral ones, and real crystals sit anywhere along that scale. The swap matters enormously when the cations are magnetic: in the ferrites (spinels like magnetite Fe3O4 and NiFe2O4), the magnetic moments on the tetrahedral and octahedral sites point opposite ways, so which cation lands where decides the net magnetization. This is why the soft ferrites that core transformers and inductors are, structurally, just spinels with the site distribution tuned.
Notice the echo of corundum: once cations only partly fill the available holes, which holes they choose — and how orderly that choice is — becomes a design knob in its own right. Dense MgAl2O4 spinel is a prized refractory and, when pore-free, a transparent ceramic tough enough for armour windows. Same close-packed oxygens as alumina; a different filling recipe; a different material.
Reading a Structure — and Weighing Its Cell
structure formula anion array cations fill example -------------------------------------------------------------------------------------- rock salt AX FCC all octahedral (1/1) MgO corundum A2O3 HCP 2/3 of octahedral Al2O3 spinel AB2O4 FCC 1/8 tetrahedral + 1/2 octahedral MgAl2O4 perovskite ABO3 (A+O) c.p. 1/4 octahedral (oxygen-only holes) BaTiO3
Reading a structure this way does more than name it — it hands you the exact contents of one unit cell, and from those you can compute a ceramic's theoretical density: the density it would have with zero pores and zero flaws. The formula is just mass over volume for one cell: density = (Z times M) / (N_A times V_cell), where Z is the number of formula units in the cell, M is the mass of one formula unit, N_A is Avogadro's number, and V_cell is the cell volume. Spinel, whose cell we just counted, is a perfect test case.
- Count the cell contents. From the reading above, one spinel cell holds Z = 8 formula units of MgAl2O4 (32 O2-, 8 Mg2+, 16 Al3+).
- Add the mass of one formula unit: M = 24.31 + 2 times 26.98 + 4 times 16.00 = 142.27 g/mol.
- Get the mass inside one cell: Z times M / N_A = 8 times 142.27 / (6.022 times 10^23) = 1.89 times 10^-21 g.
- Get the cell volume from the lattice parameter a = 0.808 nm = 8.08 times 10^-8 cm: V_cell = a^3 = 5.28 times 10^-22 cm^3.
- Divide: density = 1.89 times 10^-21 / 5.28 times 10^-22 = 3.58 g/cm^3 — spot on the measured value for dense spinel. Any real, porous body weighs less, and the ratio of the two is its percent of theoretical density.
One last idea threads all of this together and opens the next guide. Nothing forces a composition to keep a single structure: change the temperature or the pressure and the same atoms can re-pack into an entirely different arrangement. Silica flips between quartz, tridymite, and cristobalite; zirconia shifts among monoclinic, tetragonal, and cubic forms; titania appears as rutile or anatase. This is polymorphism, and each switch means a jump in density and often a change in volume — the very effect that will drive the phase transformations, thermal-expansion surprises, and toughening tricks waiting in the rest of the ladder.