One Recipe, Four Answers
In the first guide of this rung you met the big idea: a ceramic crystal is a close-packed floor of big anions with the small cations dropped into the gaps between them. The AX structures are the simplest possible cast for that play — one cation for every anion, a 1:1 stoichiometry, as in MgO, ZnS, or CsCl. Given the same recipe, why does nature build four different structures? Because two dials are still free to turn: how the anions stack, and which holes the cations choose.
Here is the arithmetic that ties the recipe down. In a close-packed array of N anions there are exactly N octahedral holes and 2N tetrahedral holes — twice as many small tetrahedral gaps as roomier octahedral ones. To end up with equal numbers of cations and anions, you therefore have only two clean choices: fill every one of the N octahedral holes, or fill exactly half of the 2N tetrahedral holes. Those two choices are already most of the AX story.
Which hole a cation lands in is set mostly by its size. A cation that is large next to its anion can shoulder many neighbours around it; a tiny one can only touch a few before the anions start bumping into each other. That is the radius-ratio idea from the first guide, and it sorts the AX structures cleanly by coordination number: a small cation takes 4 neighbours (tetrahedral), a middling one takes 6 (octahedral), and a fat one takes 8 (cubic). Walk that ladder — 4, 6, 8 — and you walk straight through the four structures below.
Rock Salt: Fill Every Octahedral Hole
Start in the middle of the ladder. Take an fcc (cubic close-packed) array of anions and drop a cation into every octahedral hole. Each cation now sits surrounded by 6 anions, and — because there were exactly as many octahedral holes as anions — each anion is likewise ringed by 6 cations: a 6:6 rock-salt structure, named for common salt, NaCl. This is the workhorse of oxide ceramics. MgO, NiO, CoO, MnO, FeO, and CaO all adopt it; so do most transition-metal monoxides. If you remember one ceramic structure, remember this one.
Does the size rule agree? In MgO the octahedral fit is textbook: Mg2+ measures 0.72 A and O2- 1.40 A, a ratio of 0.72 / 1.40 = 0.51, which lands squarely in the 0.41-to-0.73 window for 6-fold coordination. One honest picture to carry: rock salt is really two interpenetrating fcc lattices — one of cations, one of anions, offset by half a cell edge — not a single fcc of one kind of atom. The oxygen scaffold and the magnesium scaffold are each fcc, threaded through each other.
Counting the cell contents is easy and pays off. The cubic unit cell holds 4 anions (8 corners times 1/8 plus 6 faces times 1/2) and 4 cations (12 edges times 1/4 plus 1 body centre), so Z = 4 formula units of MgO per cell. From that alone you can predict the theoretical density: with cell edge a = 0.4213 nm and M(MgO) = 40.30 g/mol, density = Z times M / (N_A times a^3) = (4 times 40.30) / (6.022 x 10^23 times (4.213 x 10^-8 cm)^3) = 3.58 g/cm^3 — the measured value. Guide 5 turns this counting trick into a general tool; here just note that the structure alone fixes the density.
Cesium Chloride: When the Cation Gets Fat
Climb to the top of the ladder. When the cation grows until the radius ratio passes about 0.73, a seventh and eighth anion can crowd in, and 8-fold coordination becomes possible: the cesium chloride structure. Picture a simple-cubic cage of 8 anions with a single big cation sitting dead centre, touching all 8 — an 8:8 arrangement. In CsCl itself, Cs+ (about 1.74 A) over Cl- (1.81 A) gives a ratio near 0.96, comfortably in the 0.73-to-1.0 band for cubic coordination.
Among binary ceramic oxides the CsCl structure is almost a no-show, and for a simple reason: to reach a ratio above 0.73 against the big O2- anion, a cation would have to be enormous, and few are. So this arrangement mostly turns up in the cesium and thallium halides, some ammonium salts, and a class of intermetallics. Still, it is worth knowing as the 8-fold endpoint of the AX ladder — and the same little 8-coordinate cube reappears, tucked inside richer structures like the A-site of perovskite that you will meet two guides from now.
Zinc-Blende and Wurtzite: The Tetrahedral Twins
Now drop to the bottom of the ladder, to the small, tightly bonded cations. When the ratio falls below about 0.41, only 4 anions fit and the cation takes a tetrahedral seat: 4:4 coordination. Here nature offers twins. Fill exactly half the tetrahedral holes of an fcc anion array and you get the zinc-blende structure (sphalerite, ZnS); fill half the tetrahedral holes of an hcp array instead and you get the wurtzite structure (the other ZnS). Locally the two are identical — every cation in a tidy tetrahedron of anions, and every anion in a tetrahedron of cations.
There is an honest wrinkle here. For ZnS the size rule does agree — Zn2+ (0.60 A) over S2- (1.84 A) is 0.33, squarely tetrahedral — but the deeper reason these crystals choose 4-fold coordination is covalency, not size. This family (SiC, GaN, AlN, ZnO, BN) leans toward the covalent end, and covalent bonds are fussy about direction: the sp3-hybrid orbitals point at the corners of a tetrahedron, so directional bonding itself demands 4 neighbours. The purely covalent limit of this same pattern is diamond, and silicon carbide is essentially diamond with every other carbon swapped for silicon.
Reading a Structure with the Radius-Ratio Rule
Put the whole rung's logic into one motion. Given a compound, you want to guess which AX structure it takes; the radius-ratio rule is the quick first pass. It rests on a hard-sphere picture — anions as touching balls, the cation just big enough to keep its neighbours from overlapping — and the geometry of that picture gives the boundaries: 0.22, 0.41, 0.73.
- Look up the ionic radii on one consistent scale (e.g. Shannon), using the value listed for the coordination number you are testing.
- Form the ratio r(cation) / r(anion), always the smaller radius over the larger.
- Read off the coordination band: 0.22-0.41 gives 4 (tetrahedral), 0.41-0.73 gives 6 (octahedral), 0.73-1.0 gives 8 (cubic).
- Match the coordination to the AX structure: 4 to zinc-blende or wurtzite, 6 to rock salt, 8 to cesium chloride.
- Sanity-check against covalency: for a strongly covalent compound (SiC, GaN, the nitrides), expect tetrahedral 4-fold even if the numbers wobble.
The four AX (1:1) ceramic structures structure anion packing cation site (filled) coord examples ----------- --------------- --------------------- ----- -------------------------- rock salt fcc ABCABC... ALL octahedral 6:6 MgO, NiO, CoO, FeO, NaCl CsCl simple cubic cubic body-hole 8:8 CsCl, CsBr (rare in oxides) zinc-blende fcc ABCABC... 1/2 of tetrahedral 4:4 b-SiC, ZnS, cubic GaN wurtzite hcp ABAB... 1/2 of tetrahedral 4:4 AlN, ZnO, GaN, a-ZnS radius-ratio bands: 0.22-0.41 -> 4 | 0.41-0.73 -> 6 | 0.73-1.0 -> 8 an fcc anion cell = 4 anions + 4 octahedral holes + 8 tetrahedral holes fill all 4 octahedral -> rock salt (Z = 4) fill 4 of 8 tetrahedral -> zinc-blende (Z = 4)
Be honest about the rule's reach. Ions are soft, not hard spheres, and bonds are part covalent, so the radius-ratio prediction is right only about half to two-thirds of the time — treat it as a first guess, not a law. Borderline ratios are where it fails most, and strongly directional bonds can override it outright. Even so, you now hold the two levers that build every AX ceramic — anion stacking and hole filling — and can read a density straight off a unit cell. Next we add a second anion: the AX2 structures, fluorite and rutile, where the cation-to-anion ratio flips to 1:2.