Five Rules, One Big Idea
The earlier guides in this rung handed you three tools. Guide 1 showed that a ceramic bond is a blend of ionic and covalent character, set by the electronegativity difference. Guide 2 showed that ions have sizes that swell or shrink with charge and coordination. Guide 3 showed how the radius ratio predicts how many anions pack around a cation. Linus Pauling gathered all of this into five rules for ionic crystals — a short checklist for guessing, and sanity-checking, the structure of a ceramic using nothing but charges, sizes, and counts.
One idea drives all five: an ionic crystal is a three-dimensional balancing act. Like charges repel, opposite charges attract, so the ions settle into whatever arrangement neutralizes charge as locally as possible and sits at the lowest total energy. Every rule below is just a consequence of that single demand — think of them as the ionic model turned into practical bookkeeping, the atomic logic behind every structure in the next rung.
Rule 1 — The Coordination Polyhedron
Rule 1 — the coordination polyhedron. Around every cation sits a polyhedron of anions: the anions are the corners, the cation is at the centre. Two facts fix that polyhedron. First, the cation-anion distance is simply the sum of the two ionic radii you learned to look up in guide 2. Second, the number of anion corners — the coordination number — is set by the radius ratio of guide 3. So rule 1 is really guide 3's radius-ratio rule, promoted to Pauling's opening move.
Make it concrete with our running example, MgO. The radius ratio r(Mg2+)/r(O2-) is about 0.072 nm / 0.140 nm = 0.51, which lands squarely in the octahedral window of 0.414 to 0.732. So each Mg2+ drops into an octahedral hole of the close-packed O2- array, keeping coordination 6 — six oxygens at the corners of an octahedron. Do the same for the whole crystal and you have built the rock-salt structure, the simplest ceramic structure of all.
Now picture the whole crystal as space packed full of these polyhedra — octahedra, tetrahedra, cubes — fitted together corner to corner. That mental picture is the key: Pauling's remaining rules are not about single ions at all, but about how neighbouring polyhedra are allowed to join. Everything that follows is a rule about sharing.
Rule 2 — The Electrostatic-Valence Principle
Rule 2 — the electrostatic-valence principle, also called local electroneutrality, is the workhorse of the whole set. Give each cation-anion bond a bond strength s = (cation charge) / (its coordination number) — the cation's charge shared out equally among all the bonds it makes. The electrostatic-valence principle then says: the strengths of all the bonds arriving at a given anion must add up to that anion's own charge. Charge is balanced not just overall, but locally, at every single ion.
- For each cation, compute its bond strength s = z / CN — its charge divided by its coordination number.
- Pick an anion and list every cation that touches it (that count is the anion's own coordination number).
- Add up the bond strengths s arriving from those neighbours.
- Check the total: it should equal the anion's charge magnitude. If it does, charge is locally balanced; if the numbers only work for one particular anion coordination, you have just predicted that coordination.
Watch it work. In MgO, Mg2+ has s = 2/6 = 1/3, and each O2- is touched by six Mg2+, so the strengths sum to 6 times 1/3 = 2 — exactly the charge on O2-. In rutile, TiO2, Ti4+ has s = 4/6 = 2/3, and each oxygen is shared by three Ti, giving 3 times 2/3 = 2 again; that balance is why the rutile structure uses three-coordinate oxygen. The prettiest case is silica: Si4+ sits in a SiO4 tetrahedron with coordination 4, so s = 4/4 = 1. To reach 2, each oxygen must be shared by exactly two tetrahedra — 2 times 1 = 2. Rule 2 alone forces every oxygen in silica to bridge two tetrahedra, so the tetrahedra can only share corners. The entire silicate world of chains, sheets, and frameworks falls out of that one sum.
Rules 3 & 4 — Why Edges and Faces Are Bad News
Two polyhedra can meet in three ways: sharing a single corner (one anion), an edge (two anions), or a face (three or more). Rule 3 says sharing edges, and especially faces, lowers a structure's stability — and the reason is pure electrostatics. As you go from corner to edge to face, the two central cations are dragged closer together, and because both are positive, their mutual repulsion climbs steeply. Corner-sharing keeps the cations comfortably far apart; face-sharing jams them almost on top of each other.
How two coordination polyhedra join, and how close it pulls their two central (+) cations together: CORNER-share 1 shared anion (+)---O---(+) d = 1.00 most stable EDGE-share 2 shared anions (+)==O==(+) d ~ 0.58 less stable FACE-share 3+ shared anions (+)#O#(+) d ~ 0.33 least stable (d = cation-cation distance for tetrahedra, corner-share set to 1.00; closer cations -> stronger + / + repulsion -> lower stability) High-charge, low-CN cations feel this most: Si4+ in an SiO4 tetrahedron never shares an edge or face -> silica only corner-shares.
Rule 4 sharpens this: the penalty for edge- and face-sharing is worst for cations of high charge and low coordination number, because a big charge squeezed into a small polyhedron gives the strongest repulsion. Si4+ (charge +4, coordination just 4) is the extreme case, which is exactly why SiO4 tetrahedra never share edges or faces — the same conclusion rule 2 already forced. Larger, more spread-out cations are more forgiving: rutile's TiO2 octahedra do share some edges, and a handful of dense oxides even tolerate face-sharing — but always at a real cost in stability.
Rule 5, the Limits, and the Through-Line
Rule 5, the rule of parsimony, is the gentlest of the five: the number of essentially different kinds of building unit in a crystal tends to be small. Nature would rather reuse one or two kinds of polyhedron over and over than invent a dozen. It is more a tidy observation than a sharp prediction, and it is the rule most often stretched in complex ceramics.
Be honest about the limits. Pauling's rules are guidelines for dominantly ionic crystals; modern surveys of thousands of oxides find that only a minority obey all five at once, with rule 2, the electrostatic-valence rule, by far the most reliable and the edge/face and parsimony rules softer tendencies. They bend where bonding turns strongly covalent and directional — the SiC end of guide 1's spectrum — where lone pairs push a cation off-centre, or where the very idea of a fixed ionic radius gets fuzzy. Use them as a compass, not a ruler.
Even so, rule 2's simple bookkeeping is the ancestor of the modern bond-valence method and of the full energy accounting behind a crystal's lattice energy, where the Madelung constant sums up every attraction and repulsion in the lattice. And it carries straight into the last guide of this rung: bonds that are stronger, higher in charge, and shorter store more electrostatic energy, so they take more heat to break and resist deformation more — higher melting points and greater hardness. Pauling's rules are the atomic grammar of ceramic structure; guide 5 reads the properties straight off it.