Goldschmidt Tolerance Factor: Formula and Examples

Q: Why √2?

In the ideal cubic perovskite the A–X distance is half the face diagonal of the cubic cell and the B–X distance is half the cell edge, so A–X = √2 × B–X. The tolerance factor measures departure from that ratio.

Q: What is Bartel’s τ and is it better?

τ is a tolerance factor found by data-driven search (Bartel et al., 2019) that also uses the A-site oxidation state. On their set of 576 ABX₃ compounds it classified perovskite versus non-perovskite correctly about 92% of the time, compared with 74% for t.

Q: Does it work for double perovskites?

Approximately: use the average radius of the B-site cations (and A-site cations if mixed). Ordering and size mismatch add effects the single number cannot capture.

What it is

Perovskites have the formula ABX₃: a small B cation sits at the centre of an octahedron of X anions, and a large A cation sits in the cavity between eight corner-sharing octahedra. In the ideal cubic structure the geometry fixes a relationship between the A–X and B–X distances: the A–X distance should equal √2 times the B–X distance.

In 1926 Victor Moritz Goldschmidt expressed how far real ions depart from that ideal as the tolerance factor t. When t ≈ 1 the ions fit the cubic cell. When the A cation is too small (t < 1) the octahedra tilt to shrink the cavity, giving orthorhombic or rhombohedral perovskites. When the A cation is too large (t > 1) the structure often switches to hexagonal stacking with face-sharing octahedra, or the B cation shifts off-centre.

Because it needs only three radii, the tolerance factor is one of the cheapest structure-prediction tools in materials science and is widely used to pre-screen perovskite oxides and halides for photovoltaics, solid oxide fuel cells, ferroelectrics and catalysis.

Schematic diagram: Goldschmidt tolerance factor — At a glance: Goldschmidt tolerance factor. Schematic, not to scale.

Why it matters for R&D decisions

Perovskite chemistry spans thousands of possible A/B/X combinations. A tolerance-factor screen removes combinations that cannot form a perovskite before you compute or synthesise them, and predicts the distortions (tilts, off-centring) that control band gaps, ionic conductivity and ferroelectricity.

The formula

t = (r_A + r_X) / (√2 · (r_B + r_X))

r_A: Ionic radius of the A-site cation (usually 12-coordinate)
r_B: Ionic radius of the B-site cation (6-coordinate)
r_X: Ionic radius of the anion (6-coordinate in Shannon’s table)
μ = r_B / r_X: Octahedral factor; below the geometric limit of ~0.41 (√2 − 1) the B cation is too small for a stable BX₆ octahedron. Empirical cut-offs are slightly higher: ~0.43 in Bartel et al. (2019), ~0.44 for halides in Li et al. (2008)

Bartel et al. (2019) proposed τ = r_X/r_B − n_A (n_A − (r_A/r_B) / ln(r_A/r_B)), where n_A is the oxidation state of A; τ < 4.18 predicts perovskite and classified known compounds more accurately than t.

How to apply it, step by step

1
Fix oxidation states
Choose charges that balance the formula (for example A²⁺B⁴⁺O₃ or A³⁺B³⁺O₃, or A⁺B²⁺X₃ for halides). The radii depend on them.
2
Look up radii for the right coordination
Use Shannon radii: A in 12-fold coordination, B in 6-fold, X in 6-fold. If a 12-coordinate value is not tabulated, note the substitute you used.
3
Compute t and the octahedral factor
Calculate both. A promising candidate typically has t between ~0.8 and ~1.0 and μ above ~0.41–0.44.
4
Interpret the range
Use the table below to anticipate cubic, tilted or hexagonal structures — and therefore which properties to expect.
5
Confirm with a better classifier or calculation
For borderline cases, compute Bartel’s τ and check hull distance in computed databases before committing to synthesis.

Worked examples

Example 1

SrTiO₃ — the ideal cubic perovskite

Shannon radii: Sr²⁺ (CN 12) = 1.44 Å, Ti⁴⁺ (CN 6) = 0.605 Å, O²⁻ (CN 6) = 1.40 Å.

01Numerator: r_A + r_X = 1.44 + 1.40 = 2.84 Å.
02Denominator: √2 × (0.605 + 1.40) = 1.414 × 2.005 = 2.835 Å.
03t = 2.84 / 2.835 ≈ 1.00.
04Octahedral factor μ = 0.605 / 1.40 ≈ 0.43 — above the ~0.41 geometric limit, though close to the empirical cut-offs.

RESULTt ≈ 1.00: SrTiO₃ is cubic at room temperature, as observed.

A perfect fit gives the undistorted cubic structure.

Example 2

CaTiO₃ and BaTiO₃ — too small and too big

Swap the A cation while keeping Ti⁴⁺ and O²⁻: Ca²⁺ (CN 12) = 1.34 Å, Ba²⁺ (CN 12) = 1.61 Å.

01CaTiO₃: t = (1.34 + 1.40) / 2.835 = 2.74 / 2.835 ≈ 0.97.
02Ca²⁺ is slightly small for the cavity, so the TiO₆ octahedra tilt; CaTiO₃ is orthorhombic at room temperature.
03BaTiO₃: t = (1.61 + 1.40) / 2.835 = 3.01 / 2.835 ≈ 1.06.
04Ba²⁺ is slightly large, stretching the cell so that Ti⁴⁺ can shift off-centre; BaTiO₃ is tetragonal and ferroelectric at room temperature.

RESULTSmall changes in t around 1 correlate with octahedral tilting (t < 1) or B-site off-centring (t > 1).

The tolerance factor predicts not only whether a perovskite forms but which distortion — and therefore which properties — to expect.

Example 3

CsPbI₃ — a pass that is not the whole story

A halide perovskite candidate for solar cells. Shannon radii: Cs⁺ (CN 12) = 1.88 Å, Pb²⁺ (CN 6) = 1.19 Å, I⁻ (CN 6) = 2.20 Å.

01Numerator: 1.88 + 2.20 = 4.08 Å.
02Denominator: 1.414 × (1.19 + 2.20) = 1.414 × 3.39 = 4.79 Å.
03t = 4.08 / 4.79 ≈ 0.85 — inside the usual perovskite window.
04Yet the black perovskite phase of CsPbI₃ is metastable at room temperature and tends to convert to a yellow non-perovskite (δ) phase.

RESULTCsPbI₃ passes the tolerance-factor screen but is not the stable room-temperature phase.

Halide perovskites are especially sensitive to the limits of t. Use τ, hull distance and experimental evidence before relying on a pass.

Typical interpretation of the tolerance factor

t	Typical structure	Example
> 1.0	A cation too large: hexagonal perovskite polytypes or B-site off-centring	BaTiO₃ (≈ 1.06, tetragonal)
≈ 0.9–1.0	Cubic or nearly cubic perovskite (mild tilting is common below ~1)	SrTiO₃ (≈ 1.00); CaTiO₃ (≈ 0.97, orthorhombic)
≈ 0.71–0.9	Tilted perovskite (orthorhombic or rhombohedral)	GdFeO₃-type structures
< ~0.71	Perovskite unlikely; ilmenite or other structures	—

When to use it — and when not to

Use it when

Pre-screening A/B/X combinations for perovskite oxides, fluorides and halides.
Predicting whether a substitution will cause tilting or off-centring.
Choosing A-site or B-site dopants that keep the structure stable.
Explaining structural trends across a perovskite family.

Don’t rely on it when

Non-ABX₃ structures (use Pauling-type reasoning or hull distance instead).
Molecular A-site cations without a well-defined effective radius — results depend heavily on the chosen radius.
As the only screen for halide perovskites, where t has well-known false positives.

Common mistakes

Using radii for the wrong coordination number.

Use CN 12 for A and CN 6 for B and X where available, and state any substitutions.

Ignoring the octahedral factor.

Check μ = r_B / r_X as well; a good t with too small a B cation still fails.

Treating t within the window as proof of stability.

Confirm with Bartel’s τ, hull distance across sources and, ideally, an experimental structure.

Mixing oxidation-state assumptions.

Fix charges first; Mn³⁺ and Mn⁴⁺, for example, have very different radii.

Applying it in Lattice Graph

LatticeGraph lets you take perovskite candidates that pass a tolerance-factor screen and check them against computed stability and experimental structures in one place.

01Search the ABX₃ composition (or a family of compositions) across computed databases.
02Check hull distance and cross-source confidence for the perovskite structure.
03Look for an experimental structure in COD to see which polymorph has actually been made.

DATASETS

Materials ProjectOQMDAFLOWJARVIS-DFTCrystallography Open Database (COD)

Frequently asked questions

Why √2?