INO-05

Goldschmidt tolerance factor

t = (rA + rX) / (√2 (rB + rX)). Perovskite packing.

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Crystal chemistryGoldschmidt

Governing equation

t=\dfrac{r_A+r_X}{\sqrt{2}(r_B+r_X)}

where

r_A: Radius A (Å)
r_B: Radius B (Å)
r_X: Radius X (Å)
t: Tolerance factor (—)

Lecture brief

Historical brief

Pauling and Allred–Rochow electronegativity, Kapustinskii lattice energy, CFSE and Goldschmidt radii organise the periodic solid. The sheets predict bond character and crystal packing. This sheet (INO-05 — Goldschmidt tolerance factor) is the form associated with Goldschmidt. Working symbols: $r_A$ , $r_B$ , $r_X$ $\rightarrow$ $t$ . t ≈ 1 cubic, 0.71–0.9 octahedral tilt, t > 1 hexagonal stacking.

Purpose

Purpose: compute

t

from

r_A

r_B

r_X

in Inorganic chemistry via

t=\dfrac{r_A+r_X}{\sqrt{2}(r_B+r_X)}

t = (rA + rX) / (√2 (rB + rX)). Perovskite packing. Use it when a real inorganic chemistry question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

r_A = 1.440\,\mathrm{Å}

r_B = 0.610\,\mathrm{Å}

r_X = 1.400\,\mathrm{Å}

, the governing relation

t=\dfrac{r_A+r_X}{\sqrt{2}(r_B+r_X)}

yields

t = 0.999\,\mathrm{—}

. Ionic radii in Å for ABX3. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Radius A r_A1.44 ÅRadius B r_B0.61 ÅRadius X r_X1.4 Å

Outputs

Tolerance factor t0.999 —

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Narration of this film

Ionic radii in Å for ABX3.

t ≈ 1 cubic, 0.71–0.9 octahedral tilt, t > 1 hexagonal stacking.

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