I am studying $\ce{ABX3}$ perovskites, and I would like to calculate Goldschmidt tolerance factors for them. The $\ce{A}$ sites in these materials have a coordination number $12$.

The Shannon's paper (R. D. Shannon, Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides. Acta. Cryst. A 32, 751 (1976)) provides values for ionic radii for the coordination number $12$ only for a few elements, but doesn't do it for most other elements.

Are there any sources, where I can get ionic radii for $\text{C.N.} = 12$ for all elements?

  • 2
    $\begingroup$ Goldschmit has been shown to be a poor model to use with perovskites. $\endgroup$
    – A.K.
    Oct 18 '18 at 12:34
  • $\begingroup$ How many ionic compounds give coordination number 12? Close-packed structures forcing like charges together don't work well for ionic compounds. Our experimental database might be a bit limited. $\endgroup$ Oct 18 '18 at 12:36

The commonly used method of obtaining ionic radii for higher coordination numbers (C.N.) is to extrapolate values from the Shannon's scale [1] using the relationship between ionic radius and coordination number proposed by Zachariasen [2]:

... the bond lengths $D(N_1)$ and $D(N_2)$ for cation coordination numbers $N_1$ and $N_2$ were related as follows

$$D(N_2)=D(N_1)(N_2A_1/N_1A_2)^{1/n} \tag{1}$$

where $n + 1$ is the exponent in the Born repulsion term of the lattice energy and $A_1/A_2$ is the ratio of the Madelung constants. The quantity $A_1/A_2$ was taken to be $0.972$ for $N_2/N_1 = 12/9 = 8/6 = 4/3$ and $0.927$ for $N_2/N_1 = 12/8 = 9/6 = 6/4 = 3/2$.

Jia [3] used Zachariasen's formula and published effective ionic radii for all lanthanides with C.N. $6$ to $12$.

Using the same approach, selected ionic radii for the 12-coordinated $\ce{A}$-sites in oxo-perovskites $\ce{ABO3}$ [4] as well as halide perovskites $\ce{ABX3}$ [5] have been determined.


  1. Shannon, R. D. Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides. Acta Cryst. A 1976, 32 (5), 751–767. https://doi.org/10.1107/S0567739476001551.
  2. Zachariasen, W. H. Bond Lengths in Oxygen and Halogen Compounds of d and f Elements. Journal of the Less Common Metals 1978, 62, 1–7. https://doi.org/10.1016/0022-5088(78)90010-3.
  3. Jia, Y. Q. Crystal Radii and Effective Ionic Radii of the Rare Earth Ions. Journal of Solid State Chemistry 1991, 95 (1), 184–187. https://doi.org/10.1016/0022-4596(91)90388-X.
  4. Li, C.; Soh, K. C. K.; Wu, P. Formability of $\ce{ABO3}$ Perovskites. Journal of Alloys and Compounds 2004, 372 (1), 40–48. https://doi.org/10.1016/j.jallcom.2003.10.017.
  5. Li, C.; Lu, X.; Ding, W.; Feng, L.; Gao, Y.; Guo, Z. Formability of $\ce{ABX3}$ ($\ce{X = F, Cl, Br, I}$) Halide Perovskites. Acta Cryst. B 2008, 64 (6), 702–707. https://doi.org/10.1107/S0108768108032734.

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