Radius Ratio Rules

Radius ratio reasoning survives as a fast, back-of-envelope first check in mineralogy, solid-state chemistry, and materials design, even though modern work increasingly relies on full lattice-energy calculations, density-functional theory, and empirical bond-valence methods for quantitative prediction. In mineralogy, it remains a teaching and prospecting tool for anticipating which coordination environment a cation will adopt when substituting into a silicate or oxide lattice — relevant, for instance, to predicting trace-element partitioning between mantle minerals and melts, where a slightly-too-large or slightly-too-small trace cation for a given lattice site controls whether it is incorporated or excluded (a phenomenon geochemists exploit to interpret magma evolution and even to date rocks via partition-coefficient systematics). In materials chemistry, the Goldschmidt tolerance factor — a direct generalization of the radius ratio idea to perovskite ABX₃ compounds — is used routinely today to screen candidate halide and oxide perovskites for photovoltaics (including hybrid organic–inorganic perovskite solar cells, where the "A-site" cation size, whether Cs⁺, methylammonium, or formamidinium, is chosen partly using tolerance-factor arguments to predict whether a stable, undistorted perovskite phase will form). Battery materials science uses radius-ratio-style reasoning to anticipate whether a given mobile cation (Li⁺, Na⁺, or larger alkali and multivalent ions) will fit comfortably into octahedral or tetrahedral sites of a host lattice, which in turn affects ionic mobility and hence charge/discharge rates in solid electrolytes and intercalation cathodes. In solid-state synthesis more generally, radius-ratio and tolerance-factor screening are embedded in modern high-throughput computational materials discovery pipelines as a cheap geometric pre-filter, applied to enormous candidate compound lists before more expensive first-principles stability calculations are run — an economically important role even though, as this article has stressed, the rule's absolute predictive accuracy for coordination number is limited. Finally, in inorganic pedagogy the rule remains the standard bridge from simple ionic-radius tables to the real structures of Rock Salt (NaCl) Structure and Cesium Chloride Structure, precisely because it converts an otherwise abstract three-dimensional packing argument into one arithmetic ratio students can compute themselves.

Ask the tutor: How would the critical radius ratio change if we required contact along the octahedron's axial (rather than equatorial) direction instead — and why does the equatorial constraint dominate?

Picture an anion lattice as a set of touching (or nearly touching) hard spheres, and ask: how large a cation can sit in the hole between them while still touching all the surrounding anions? If the cation is very small, it can only touch a few anions before the anions themselves would have to interpenetrate to stay in contact with a cation that small — geometrically forbidden for hard spheres. As the cation grows, more anions can be accommodated in contact with it without anion–anion overlap, so higher coordination numbers become geometrically permissible. This is the entire logical chain behind the radius ratio rule: coordination number is limited from below by cation size, because a coordination polyhedron only remains "closed" (anions touching the cation, and not forced to overlap each other) above a critical ratio. The three benchmark geometries are the tetrahedral hole (CN 4, as in ZnS/wurtzite and sphalerite), the octahedral hole (CN 6, as in NaCl and the rutile/rock-salt family), and the cubic hole (CN 8, as in CsCl). Each corresponds to a regular polyhedron of anions with the cation at the geometric center, and each has a well-defined critical ratio at which the anions just touch each other while also touching the cation. Above the critical ratio, anions are pushed slightly apart (which is fine — spheres need not touch, they only must not overlap); below it, the anions would have to overlap to stay in contact with the cation, so the structure is unstable and the cation "rattles," typically triggering a collapse to a lower coordination number where a stable, touching arrangement exists. It is essential to see this as a worst-case (limiting) contact condition, not as an energetic optimum. Real ionic solids do not sit exactly at critical ratios; they sit wherever the total lattice energy — Madelung electrostatics, Born repulsion, polarization, and any covalent contribution — is minimized. The radius ratio rule only tells you the geometric floor. This is why the rule correctly excludes CN 8 for very small cations but frequently fails to correctly select between CN 6 and CN 4 when a ratio lies comfortably inside the "should be octahedral" range yet the compound crystallizes tetrahedrally (ZnS being the classic case, discussed below) because directional covalent bonding, not sphere-packing, controls the true energetic minimum.

The second useful mental picture is a "phase line" for $\rho$: as the ratio increases continuously from 0 to 1, the maximum stable coordination number increases in discrete jumps at the critical values 0.155 (CN 3, triangular), 0.225 (CN 4, tetrahedral), 0.414 (CN 6, octahedral), 0.732 (CN 8, cubic), and 1.0 (CN 12, close-packed, equal spheres). Real compounds are then classified by which bracket their measured $\rho$ falls into — with the important caveat, elaborated later, that many real structures adopt the coordination number for the next bracket down from what pure geometry allows, because covalency prefers lower, more directional coordination even when geometry would tolerate more neighbors.

The governing relationship is simply the definition of the ratio itself, together with the trigonometric contact conditions for each polyhedron:

where: $r_+$ is the cation (Shannon/Pauling) ionic radius, $r_-$ is the anion ionic radius, both taken for the same coordination number and consistent radius scale, and $\rho$ is dimensionless.

where: this is the ratio at which a cation just touches four anions at the corners of a regular tetrahedron while adjacent anions are simultaneously in mutual contact along the tetrahedron edge.

where: this is the ratio for a cation at the center of a regular octahedron of anions, contact occurring along the octahedron's face diagonal direction (equivalently, along the square formed by four coplanar anions).

where: this is the ratio for a cation at the body center of a cube of anions, contact occurring along the cube's space diagonal.

Reading: each critical ratio is the minimum $\rho$ for which that coordination geometry is geometrically stable (no anion–anion overlap). Below it, the polyhedron would require anions to interpenetrate, so nature drops to a smaller, more compact coordination number where contact is possible without overlap. Above it, the cation is "too big" for that polyhedron in the sense that it pushes the anions apart from mutual contact — but this is not forbidden; it simply means the next larger coordination number becomes geometrically available and often energetically preferred (a bigger CN generally lowers electrostatic energy further, so real ions with $\rho$ comfortably above a critical value often do adopt the higher CN).

Example 1 — Foundational: classifying NaCl from tabulated radii. Using Shannon effective ionic radii for six-coordination (the self-consistent CN to use, since NaCl is six-coordinate and radii are CN-dependent), $r(\mathrm{Na^+}, \text{CN}6) = 1.02\ \text{Å}$ and $r(\mathrm{Cl^-}, \text{CN}6) = 1.81\ \text{Å}$. The ratio is

This falls between 0.414 and 0.732, so the rule predicts octahedral (CN 6) coordination — exactly the observed rock-salt structure, which anticipates the fuller treatment in Rock Salt (NaCl) Structure. As a limiting-case sanity check: if the cation radius were shrunk to exactly the CN-6 critical value with the same anion, $r_+ = 0.414 \times 1.81\ \text{Å} = 0.749\ \text{Å}$, right at the boundary the structure would be geometrically marginal for octahedral coordination — and indeed smaller alkali-halide-like ions with $\rho$ below 0.414 (relative to a given anion) are known to adopt lower coordination in appropriate structure types.

Example 2 — Applied: MgO versus the anomalous case of ZnS. Take MgO: $r(\mathrm{Mg^{2+}}, \text{CN}6) = 0.72\ \text{Å}$, $r(\mathrm{O^{2-}}, \text{CN}6) = 1.40\ \text{Å}$.

This lies in the 0.414–0.732 bracket, predicting (correctly) the rock-salt structure with CN 6 for both ions — MgO is indeed isostructural with NaCl. Now contrast this with zinc sulfide. Using $r(\mathrm{Zn^{2+}}, \text{CN}4) = 0.60\ \text{Å}$ and $r(\mathrm{S^{2-}}, \text{CN}4) = 1.84\ \text{Å}$ (radii tabulated at the coordination number actually observed),

Using CN-6 radii instead ($r(\mathrm{Zn^{2+}},\text{CN}6) = 0.74\ \text{Å}$, $r(\mathrm{S^{2-}},\text{CN}6)\approx 1.84$ Å, since S²⁻ radii are fairly stable across the accessible range) gives $\rho \approx 0.40$, still inside the "should be octahedral" 0.414 bracket by only a hair, or just below it depending on the tabulation used — placing ZnS right at the tetrahedral/octahedral boundary. Yet both sphalerite and wurtzite polymorphs of ZnS are tetrahedrally coordinated (CN 4) in reality, not octahedral. This is the textbook illustration that the radius ratio rule, built purely on ionic hard-sphere contact, cannot capture the substantial covalent/directional character of the Zn–S bond (Zn²⁺ is a borderline-soft, polarizing cation and S²⁻ is a large, polarizable soft anion — Fajans' rules territory), which favors tetrahedral $sp^3$-like bonding regardless of what pure sphere-packing would allow.

Example 3 — Exam/competition level: back-calculating an unknown radius from an observed structure change. A hypothetical binary ionic solid MX crystallizes in the NaCl structure at room temperature and pressure with a measured cation radius $r_+ = 0.90\ \text{Å}$ and anion radius $r_- = 1.95\ \text{Å}$. Under high pressure, the compound is observed to transform to the CsCl structure. Estimate the minimum fractional decrease in anion radius (equivalently, the minimum compressibility of the anion sublattice relative to the cation) that would be needed to drive this transition through the radius-ratio mechanism alone, assuming the cation is incompressible. First compute the ambient ratio:

This is comfortably within the NaCl-favored bracket (0.414–0.732), consistent with the starting structure. For the CsCl structure to become geometrically favored via the radius-ratio criterion, we need $\rho \geq 0.732$. Since the cation is assumed incompressible, the anion radius must shrink to $r_-'$ such that

Solving,

The fractional decrease required in the anion radius is

i.e. roughly a 37% reduction in anion radius. This is physically unreasonable for a simple pressure-induced radius change (anion compressibilities under accessible laboratory pressures rarely exceed a few percent), which is itself an instructive result: it shows that real pressure-induced NaCl → CsCl transitions (well documented, e.g., in CsCl itself at very high pressure, or conceptually invoked for alkali halides) are not driven by anions physically "shrinking" by tens of percent. Instead they arise because the whole lattice energy landscape shifts with pressure — the CsCl structure has a shorter, more efficient packing (higher Madelung constant per formula unit is not the whole story, but higher density) that becomes favorable once $PV$ contributions to the Gibbs energy are included — and the radius-ratio picture only crudely tracks this by pretending it is a pure hard-sphere size effect. The exercise demonstrates both how to manipulate the ratio algebraically and where its physical content runs out: geometry can rationalize which polyhedra are allowed, but the actual pressure- or temperature-driven phase transition is an energetic competition the ratio alone does not compute.

Several systematic limitations deserve explicit attention:

• Radii are not rigid. Ionic radii are themselves empirical constructs (see Ionic Radii and Their Determination), derived by partitioning observed interatomic distances using a chosen reference scale (Shannon, Pauling, etc.), and they depend on coordination number, spin state, and even which anion partners them. Using CN-6 radii to predict CN-4 behavior mixes apples and oranges.
• Covalency invalidates the hard-sphere picture entirely. Directional, partially covalent bonds (as in ZnS, CdS, many II–VI and III–V semiconductors) favor specific bond angles (109.5° for $sp^3$) rather than maximal ionic contact, so the tetrahedral structure can be preferred even when $\rho$ sits inside the "octahedral" bracket.
• Polarization and the Fajans' rules regime. Small, highly charged cations polarize large, soft anions, distorting the electron cloud away from spherical symmetry — again breaking the hard-sphere assumption that underlies every critical ratio derived above.
• Jahn–Teller and crystal-field effects for transition-metal cations (e.g., Cu²⁺, Mn³⁺) distort "expected" octahedra into tetragonally elongated or compressed shapes not captured by an isotropic radius.
• The rule is asymmetric in reliability: it is a much better lower bound (predicting which coordination numbers are geometrically excluded) than an upper bound or unique predictor (many compounds with $\rho$ comfortably inside a bracket still adopt a different structure for energetic/covalent reasons).
• Radius ratio says nothing about lattice energetics such as Madelung constants, Born exponents, or the actual thermodynamic stability ranking between competing structure types — for that, one must go to full lattice-energy calculations, not simply to $r_+/r_-$.

The radius ratio approach is usually associated with Victor Moritz Goldschmidt (1888–1947), the Swiss-born, Norwegian-based mineralogist and geochemist often called the father of modern crystal chemistry and geochemistry. Goldschmidt systematically tabulated ionic radii from X-ray-determined interatomic distances in minerals during the 1920s, building one of the first comprehensive self-consistent radius scales, and used ratios of these radii to rationalize which structure type a given mineral composition would adopt — laying essential groundwork for what became the modern radius-ratio rules taught alongside Linus Pauling's later refinements and "rules" for ionic crystals (Pauling's rules for coordination polyhedra, published in the early 1930s, formalized much of the same geometric reasoning into general principles for complex ionic structures). Goldschmidt's tolerance factor for perovskites, still used today to predict whether ABX₃ compounds will adopt cubic, tilted, or non-perovskite structures, is a direct descendant of the same size-ratio logic applied to a more complex structure type.