Imagine you are trying to fit a small marble (the cation) into a cavity formed by larger marbles (the anions). If the small marble is too small, it will rattle around — the structure is unstable because the cation cannot touch all the surrounding anions. If it is just the right size, it touches every neighbouring anion perfectly. If it is too large, it will push the anions apart, forcing a different arrangement.
That is the entire physical idea behind the Radius Ratio Rule. The ratio rcation/ranion determines how many anions can pack around a cation — the coordination number — and therefore the geometry of the crystal.
The Precise Statement
For an ionic crystal, the radius ratio is defined as
ρ=ranionrcation
The rule states: For a given coordination number and geometry, there is a minimum value of ρ below which the cation is too small to remain in contact with all the surrounding anions. Above that minimum, the geometry is stable.
The key idea is cation–anion contact. In a stable ionic crystal, the cation should touch as many anions as possible. If ρ falls below the critical value, the cation loses contact with some anions, and the structure becomes unstable — it will adopt a lower coordination number instead.
Important
The radius ratio rule is a geometric criterion, not a perfect predictor. It works best for purely ionic compounds where ions behave like hard spheres. Covalent character, polarisation, and temperature can cause deviations.
The Critical Limits and Geometries
Here are the most important ranges you need for exams:
Radius ratio (ρ)
Coordination number
Geometry
Example
0.155 – 0.225
3
Triangular planar
B2O3
0.225 – 0.414
4
Tetrahedral
ZnS (sphalerite)
0.414 – 0.732
6
Octahedral
NaCl
0.732 – 1.000
8
Cubic (body-centred)
CsCl
1.000
12
Hexagonal close-packed / cubic close-packed
(metallic, not ionic)
Note
The lower limit for each geometry is derived from pure geometry. For example, in an octahedral arrangement, the critical ratio comes from the condition that the cation touches all six anions simultaneously. The calculation uses the diagonal of a square face.
Where Do These Numbers Come From? (The Octahedral Case)
Take an octahedral hole. Four anions lie in a square plane, with the cation at the centre. The diagonal of the square is 2ranion+2rcation. But the diagonal of a square of side 2ranion is 22ranion. Equating:
Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.
CBSE 2025Set ANNUAL1 markMCQ
Q.In calcium fluoride, having the flurite structure, the coordination number of Ca2+ ion and F− ion are :
(a) 8 and 4
(b) 4 and 2
(c) 4 and 8
(d) 6 and 6
›Reveal solutionSolution
The fluorite structure of CaF2 places Ca2+ ions in a face-centred cubic arrangement with 8-fold coordination to F−, while the smaller, twice-as-numerous F− ions occupy all the tetrahedral voids with 4-fold coordination to Ca2+ — the stoichiometry CaF2 (1 Ca : 2 F) is exactly consistent with an 8:4 coordination ratio.
In the fluorite structure, the larger Ca2+ ions constitute a face-centred cubic (ccp) lattice. The smaller F− ions occupy all the tetrahedral voids of this ccp lattice (there are 2 tetrahedral voids per ccp lattice point, matching the 1:2 stoichiometry of CaF2). Each Ca2+ ion is then surrounded by 8 F− ions (cubic/8-coordination), while each F− ion, sitting in a tetrahedral hole, is surrounded by only 4 Ca2+ ions. This …
Q.Answer in one word: Write the radius ratio for tetrahedral void.
›Reveal solutionSolution
A cation occupying a tetrahedral void must have a radius ratio (r⁺/r⁻, cation/anion) between about 0.225 and 0.414 for the smaller ion to just touch all four surrounding larger ions without them overlapping.
In close-packed structures, a tetrahedral void is the small gap surrounded by 4 spheres arranged tetrahedrally. Geometric analysis of this arrangement shows that a smaller sphere fits snugly in this void (touching all 4 neighbours) when its radius is between 0.225 and 0.414 times the radius of the larger (anion) spheres.
Q.The number of chloride ions that surrounds the central Na+ ion in NaCl crystal is :
(a) 6
(b) 8
(c) 4
(d) 12
›Reveal solutionSolution
In the NaCl crystal, each Na+ ion is surrounded by 6 Cl− ions in an octahedral arrangement (6:6 coordination).
Sodium chloride adopts the rock-salt structure: the larger Cl− ions form a cubic close-packed (face-centred cubic) lattice, and the smaller Na+ ions occupy all the octahedral voids of this lattice. An octahedral void, by definition, is surrounded by exactly 6 nearest neighbours arranged at the vertices of an octahedron around the central ion.