How many neighbours does an ion keep?
In the previous guide every ion earned a definite size — an ionic radius that grows or shrinks with its charge and with the crowd around it. Now put those sizes to work on the question that fixes a crystal's shape: pick one cation in the structure and count how many anions are actually touching it. That tally is its coordination number. In ceramics the answers are small whole numbers — silicon sits among four oxygens, magnesium among six, a big zirconium among eight — and each choice writes a different structure.
The count must be consistent from both ends. Stand on an oxygen instead and count the cations reaching it; the two viewpoints are locked together by the formula. In MgO, six oxygens around each Mg2+ forces six Mg2+ around each O2-, simply because there are equal numbers of each. Whatever the number, the touching anions arrange into a tidy cage — four make a tetrahedron, six an octahedron, eight a cube. That cage, the coordination polyhedron, is the crystal's real repeating brick, far more than any single bond is.
The gap in the packing sets the number
Why should magnesium keep exactly six neighbours, and not four or eight? The ionic model answers with plain packing. Anions carry the spare electrons, so they are usually the big marbles; most oxide ceramics are at heart a close-packed stack of anions with the small cations dropped into the gaps between them. Those gaps come in two sizes — a roomier octahedral hole ringed by six anions, and a tighter tetrahedral hole ringed by four.
Which hole a cation takes turns on a single number: the radius ratio r/R, the small marble's radius over the big one's. Picture the cation nestled in a cage of anions that just touch each other. If the cation is too small for that cage, the anions clatter together over its head and the whole arrangement rattles loose — the cation is better off dropping to a smaller cage with fewer neighbours. If it is big enough to prop the anions apart, the roomier cage is stable. So, reaching straight back to guide two, a cation that is larger relative to its anions simply holds on to more neighbours.
Where the cut-offs come from
The boundaries between cages are not measured — they are pure trigonometry. The critical radius ratio of a cage is the smallest cation that can still hold its anions apart: the exact instant the cation touches every anion while the anions just touch one another. Take the octahedron. Slice through its square middle — four anions of radius R at the corners, touching edge to edge, with the cation of radius r at the centre. The square's side is 2R, so its half-diagonal is R times sqrt(2); but that half-diagonal also equals R + r, the cation-anion contact. Setting R times sqrt(2) = R + r gives r/R = sqrt(2) - 1 = 0.414. The same trick gives the tetrahedral limit sqrt(3/2) - 1 = 0.225 and the cubic limit sqrt(3) - 1 = 0.732.
radius ratio r/R coord.no. polyhedron real example ----------------------------------------------------------------- 0.155 - 0.225 3 triangle B3+ in B2O3 0.225 - 0.414 4 tetrahedron Zn2+ in ZnS 0.414 - 0.732 6 octahedron Mg2+ in MgO 0.732 - 1.000 8 cube Ca2+ in CaF2 1.000 and above 12 close-packed (pure metals)
Read the table as a map from one number to a cage. A ratio of 0.225 to 0.414 buys four neighbours; 0.414 to 0.732, six; above 0.732, eight. And it ties straight back to guide two: because an ion swells as its charge or its coordination climbs, the very same element can slide from one band into the next — which is exactly why a good radius table lists a separate value for every coordination number, and why you must pick the matching one before you divide.
A worked example: MgO
- Look up both radii at the right coordination. For MgO, Mg2+ is about 0.072 nm (six-coordinate) and O2- about 0.140 nm.
- Form the ratio: r/R = 0.072 / 0.140 = 0.51.
- Find the band. 0.51 sits inside the 0.414 to 0.732 window — octahedral.
- Read off the prediction: coordination number 6, the cation in an octahedral hole. Each Mg2+ should keep six O2- neighbours.
- Check against reality. In real MgO every Mg2+ does sit in an octahedral hole of a face-centred-cubic O2- array — the rock-salt structure. The rule nailed it.
The same recipe places other ceramics on the map. Push the cation larger, as in ZrO2 or CsCl, and the ratio climbs past 0.732 into eight-fold, cubic coordination; shrink it, as in the zinc-blende form of ZnS, and the ratio falls into the tetrahedral band with only four neighbours. Two numbers, one division, and a table have handed you a candidate crystal structure — which is why the radius ratio is the ionic model's single cleverest trick.
A useful guide, not a law
Beautiful as it is, the radius-ratio rule is only a guide, and it is honest to say where it breaks. It assumes ions are hard, perfectly spherical, and purely ionic — but in guide one we saw that no ceramic bond is fully ionic, and the percent ionic character slides down as the electronegativity difference shrinks. Once a bond turns strongly covalent, the shared electrons care about direction, not just about size, and geometry alone can no longer call the coordination.
There is a mild circularity worth admitting too: the radii you looked up were themselves fitted assuming a coordination number, so the rule is a self-consistent estimate rather than an independent proof — and a big, soft anion beside a small, high-charge cation gets its cloud pulled toward covalent, bending things further. Even so, coordination number is the pivot the rest of this rung turns on. Next guide it feeds Pauling's rules and the electrostatic-valence principle, which balance each anion's charge among the bonds reaching it; and it sets up the through-line to the final guide — that higher-charge, shorter, and therefore stronger bonds are what give a ceramic its high melting point and its hardness.