Two sizes of ball: a packing with its holes filled
The earlier guides in this rung treated a metal as a pile of identical balls. Ionic crystals are the very next step up: now there are two kinds of ball, big and small, carrying opposite charge. The good news is that the ionic bond is like the metallic one in the way that matters here — it is non-directional. An ion's electrostatic pull reaches equally in every direction, so once again geometry, not chemistry, sets the arrangement. The one new ingredient is the size difference. The recipe writes itself: the large ions (usually the anions) build a close-packed or nearly close-packed framework, and the small ions (usually the cations) tuck into the interstitial holes left between them.
Which holes? That is exactly the question guide 3 answered. The radius ratio r_cation / r_anion decides how many neighbours a cation can hug: a value of 0.225 to 0.414 fits a tetrahedral hole (4 neighbours), 0.414 to 0.732 an octahedral hole (6 neighbours), and 0.732 to 1.0 a cubic site (8 neighbours). Then Pauling's rules layer the chemistry on top: the local charges must balance, and neighbouring polyhedra prefer to share corners rather than edges or faces. Put the geometry and the electrostatics together and a mere handful of arrangements account for most simple ionic solids.
Rock-salt (NaCl): a cation in every octahedral hole
Start with the commonest ionic structure of all. Take an FCC array of the large Cl- anions and drop a Na+ into every one of its octahedral holes. Guide 3 counted exactly one octahedral hole per FCC atom, so filling them all hands you the 1:1 stoichiometry NaCl for free. The coordination is 6:6 — each Na+ touches 6 Cl-, and each Cl- touches 6 Na+. Check the size: with r(Na+) = 1.02 angstrom and r(Cl-) = 1.81 angstrom the radius ratio is 1.02 / 1.81 = 0.56, sitting squarely in the octahedral band of 0.414 to 0.732. The rule and the reality agree.
There is an equally honest second picture: rock salt is simply two interpenetrating FCC lattices, one of Na+ and one of Cl-, shifted from each other by half a cube edge. Count the cell contents to get the density later: 4 Cl- (the FCC positions) and 4 Na+ (12 edge holes times 1/4 plus 1 body-centre hole), so there are Z = 4 formula units per unit cell. Real members of the rock-salt structure flood in — KCl and most alkali halides, the oxides MgO, CaO, FeO and NiO, sulfides like PbS, even nitrides like TiN. Any 1:1 compound with a middling radius ratio tends to land here.
Cesium chloride (CsCl): when the cation grows too big
Now swell the cation. As the radius ratio climbs past 0.732, six neighbours no longer wrap snugly around the cation, and it prefers eight. Enter CsCl. The anions now sit on a simple-cubic array, and a cation drops into the very centre of each cube, touching all 8 corner anions at once: the coordination jumps to 8:8. For CsCl the radius ratio is about 1.70 / 1.81 = 0.94, well inside the cubic 0.732-to-1.0 band. The cell is tiny — Z = 1 formula unit (8 corners times 1/8 gives 1 Cl, plus 1 Cs at the centre).
The CsCl structure is chosen by the big-cation halides CsCl, CsBr and CsI, by ammonium chloride NH4Cl, and by many ordered intermetallics such as beta-brass (CuZn) and NiAl — where 'anion' and 'cation' relax into just 'two kinds of atom'. It is a clean demonstration of how a single number, the radius ratio, can flip a whole structure from 6:6 to 8:8 the moment the cation grows large enough to want more company.
Fluorite (CaF2): filling every tetrahedral hole
What if the chemistry is not 1:1? Take CaF2, with twice as many anions as cations. Build an FCC array of the Ca2+ cations this time, and drop an F- into EVERY tetrahedral hole. Guide 3 counted exactly two tetrahedral holes per FCC atom, so filling all of them gives 2 F- for each Ca2+ — the AB2 formula drops straight out of the geometry, no bookkeeping needed. The coordination is unequal because the chemistry is: each Ca2+ is surrounded by 8 F-, while each F- sits among only 4 Ca2+, giving 8:4.
Counting the cell gives Z = 4 formula units (4 Ca2+ on the FCC framework plus 8 F- in the 8 tetrahedral holes). The fluorite structure houses CaF2 itself, the nuclear-fuel oxides UO2 and ThO2, ceria CeO2, and cubic (stabilized) zirconia ZrO2. Swap the roles — put the anions on the FCC framework and the cations in all the tetrahedral holes — and you get the anti-fluorite structure of the alkali oxides Li2O, Na2O and K2O, the unusual case where small cations outnumber large anions 2:1. Fluorite even has a trick that later rungs will love: it shrugs off missing anions, so UO2 and doped zirconia tolerate oxygen vacancies and conduct ions through them — a whole defect and ionic-conductor story built on this one frame.
Perovskite (ABO3): the workhorse of functional materials
The richest of the four mixes two very different cations. The perovskite structure, prototype CaTiO3, is best pictured octahedron-first. Each small, highly-charged B cation (Ti4+) sits caged in an octahedron of 6 oxygens, and these BO6 octahedra join corner-to-corner in all three directions into an open, cage-like 3D framework. Into the roomy cavity between eight octahedra drops a large A cation (Ca2+), coordinated by a full 12 oxygens. So the coordination is A:12, B:6, and the cell holds Z = 1 formula unit — 1 A at the cube corners, 1 B at the body centre, and 3 O at the face centres.
That corner-sharing is no accident — it is Pauling's third rule made visible. Sharing octahedron faces or edges would drag the highly-charged Ti4+ cations dangerously close and destabilize the crystal, so the structure spreads them as far apart as it can by sharing only corners. Whether the ideal cube actually survives is judged by the Goldschmidt tolerance factor t = (r_A + r_O) / (sqrt(2) times (r_B + r_O)). When t is close to 1 the structure stays cubic; as t drifts away, the rigid octahedra tilt and the cell distorts to take up the slack.
And those tiny distortions are exactly where the magic lives. In barium titanate BaTiO3 the Ti4+ shifts a hair off the octahedron centre, creating a permanent electric dipole — that is ferroelectricity, and it switches on through a displacive transition as the crystal cools. The same one framework, tuned by swapping which ions sit at A and B, gives us ferroelectrics, piezoelectrics, materials with colossal magnetoresistance, high-Tc superconducting cuprates, and the halide perovskites of the newest solar cells. Perovskite is the textbook case of the structure-property relationship: keep the skeleton, change the tenants, and an entire menu of function follows.
Cashing it out: density and the map of structure types
STRUCTURE PROTOTYPE BUILT AS COORD. Z
--------- --------- ------------------------------- ---------- ---
Rock salt NaCl FCC anions, cation in EVERY 6 : 6 4
octahedral hole
CsCl CsCl simple-cubic anions, cation at 8 : 8 1
the cube centre (NOT bcc!)
Fluorite CaF2 FCC cations, anion in EVERY 8 : 4 4
tetrahedral hole
Perovskite CaTiO3 corner-sharing BO6 octahedra, A:12 B:6 1
big A cation in the cavity
Z = formula units per unit cellOnce you know the structure type you know Z and the cell edge, so the density falls out with pure arithmetic — the theoretical density trick from guide 1, now for a two-atom compound. The only twist is that 'mass per cell' now counts both kinds of ion. Watch it work for NaCl.
- Read Z from the structure type: rock salt has Z = 4 formula units of NaCl per cell.
- Add the formula mass: 22.99 (Na) + 35.45 (Cl) = 58.44 g/mol. Mass per cell = 4 times 58.44 / (6.022 times 10^23) = 3.88 times 10^-22 g.
- Get the cell volume from a = 5.64 angstrom = 5.64 times 10^-8 cm, so V = a^3 = 1.79 times 10^-22 cm^3.
- Divide: 3.88 times 10^-22 g / 1.79 times 10^-22 cm^3 = 2.16 g/cm^3 — dead on the measured 2.17 g/cm^3 for table salt.
These four prototypes, plus a few more — spinel AB2O4, which parcels its cations into a mix of tetrahedral and octahedral holes, and zinc-blende and wurtzite waiting in guide 5 — sort thousands of compounds into a handful of structure-type families. But be honest about the radius-ratio rule that carried us here: it is a rough guide, not a law. It predicts the right coordination only about two-thirds of the time. Real ions are not hard spheres, real bonds are never purely ionic, and polarization and covalent character quietly bend the outcome — which is exactly why the covalent diamond and zinc-blende structures of guide 5 need their own treatment. And 'one structure per compound' is itself a simplification: many are polymorphs that switch type with temperature or pressure through a polymorphic transition — zirconia alone runs through three of them on the way from room temperature to its melting point.