Radius Ratio Rule: Ionic Coordination Geometry

Ion Radii

Cation radius r⁺ 95 pm

Anion radius r⁻ 181 pm

Coordination Number

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Real-Ion Presets

Live Readout

Radius ratio r⁺/r⁻

0.525

Coordination no.

Geometry

Octahedral

Anion–anion contact (need ≥ 100%)

120%

How it works

Pauling's radius-ratio rule predicts how many anions can pack around a central cation while every anion still touches the cation, without the anions themselves overlapping. For a coordination polyhedron whose anions sit at cation–anion distance d = r⁺ + r⁻, the nearest-neighbour anion–anion distance is k·d for a geometry-dependent constant k (√3 for trigonal, √(8/3) for tetrahedral, √2 for octahedral, 2/√3 for cubic, 1 for cuboctahedral). Anions first touch each other exactly when k·d = 2r⁻, which gives the critical ratio:

r⁺/r⁻ (critical) = (2 − k) / k

r⁺/r⁻ range	CN	Geometry
0.155 – 0.225	3	Trigonal planar
0.225 – 0.414	4	Tetrahedral
0.414 – 0.732	6	Octahedral
0.732 – 1.000	8	Cubic
> 1.000	12	Cuboctahedral (ccp/hcp)

Below a geometry's critical ratio the cation is "rattling" — too small to keep the anions from touching each other — so the crystal adopts the next lower coordination number instead. The simulator computes the nearest-neighbour anion–anion distance directly from the actual 3D vertex positions of the selected polyhedron (not from the table above), so dragging r⁺ or r⁻ past a threshold visibly makes neighbouring anion spheres overlap and turn red — exactly the geometric contact violation the rule is built on.

Real ions: Na⁺ (95 pm) / Cl⁻ (181 pm) gives ratio ≈ 0.52 → octahedral CN 6, the actual rock-salt structure. Cs⁺ (167 pm) / Cl⁻ (181 pm) gives ≈ 0.92 → cubic CN 8, matching real CsCl. Zn²⁺ (74 pm) / S²⁻ (184 pm) gives ≈ 0.40 → tetrahedral CN 4, matching zinc blende.

Octahedral · CN 6

Cation (r⁺) Anion (r⁻) Anion–anion overlap

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