Cation-anion radius ratio: Difference between revisions

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*Critical Radius Ratio*. This diagram is for coordination number six: 4 anions in the plane shown, 1 above the plane and 1 below. The stability limit is at r_C/r_A = 0.414

In condensed matter physics and inorganic chemistry the cation-anion radius ratio is the ratio of the ionic radius of the cation to the ionic radius of the anion in a cation-anion compound. This is simply given by $r_{C}/r_{A}$ .

According to Pauling's rules for crystal structures, the allowed size of the cation for a given structure is determined by the critical radius ratio.^[1] If the cation is too small, then it will attract the anions into each other and they will collide hence the compound will be unstable due to anion-anion repulsion; this occurs when the radius ratio drops below 0.155.

At the stability limit the cation is touching all the anions and the anions are just touching at their edges (radius ratio = 0.155). Beyond this stability limit (radius ratio < 0.155) the compound may be stable.

The table below gives the relation between radius ratio and coordination number, which may be obtained from a simple geometrical proof.^[2]

Radius Ratio	Coordination number	Type of void	Example
< 0.155	2	Linear
0.155 - 0.225	3	Triangular Planar	B₂CO₃^{[citation needed]}
0.225 - 0.414	4	Tetrahedral	ZnS, CuCl
0.414 - 0.732	6	Octahedral	NaCl, MgO
0.732 - 1.000	8	Cubic	CsCl, NH₄Br

The radius ratio rule was first proposed by Gustav F. Hüttig in 1920. ^[3] ^[4] In 1926 Victor Goldschmidt ^[3] extended the use to ionic lattices. ^[5] ^[6] ^[7]

References

^ Linus Pauling (1960) Nature of the Chemical Bond, p. 544, at Google Books
^ Toofan J. (1994) Journal of Chemical Education 71(9): 147 (and Erratum p.749) A Simple Expression between Critical Radius Ratio and Coordination Numbers
^ ^a ^b The Origin of the Ionic-Radius Ratio Rules William B. Jensen and William B. Jensen Journal of Chemical Education 2010 87 (6), 587-588 doi:10.1021/ed100258f
^ Hüttig, G. F. (1920), Notiz zur Geometrie der Koordinationszahl. Z. Anorg. Allg. Chem., 114: 24–26. doi:10.1002/zaac.19201140103
^ V. Goldschmidt, T. Barth, G. Lunde, W. Zachariasen, Geochemische Verteilungsgesetze der Elemente. VII. Die Gesetze der Krystallochemie, Dybwad: Oslo, 1926, pp. 112-117.
^ V. Goldschmidt, Geochemische Verteilungsgesetze der Elemente. VIII. Untersuchungen über Bau und Eigenschaften von Krystallen, Dybwad: Oslo, 1927, pp. 14-17
^ V. Goldschmidt, “Crystal Structure and Chemical Constitution,” Trans. Faraday Soc. 1929, 25, 253-283. doi:10.1039/TF9292500253

[1] Linus Pauling (1960) Nature of the Chemical Bond, p. 544, at Google Books

[2] Toofan J. (1994) Journal of Chemical Education 71(9): 147 (and Erratum p.749) A Simple Expression between Critical Radius Ratio and Coordination Numbers

[jensen-3] The Origin of the Ionic-Radius Ratio Rules William B. Jensen and William B. Jensen Journal of Chemical Education 2010 87 (6), 587-588 doi:10.1021/ed100258f

[4] Hüttig, G. F. (1920), Notiz zur Geometrie der Koordinationszahl. Z. Anorg. Allg. Chem., 114: 24–26. doi:10.1002/zaac.19201140103

[5] V. Goldschmidt, T. Barth, G. Lunde, W. Zachariasen, Geochemische Verteilungsgesetze der Elemente. VII. Die Gesetze der Krystallochemie, Dybwad: Oslo, 1926, pp. 112-117.

[6] V. Goldschmidt, Geochemische Verteilungsgesetze der Elemente. VIII. Untersuchungen über Bau und Eigenschaften von Krystallen, Dybwad: Oslo, 1927, pp. 14-17

[7] V. Goldschmidt, “Crystal Structure and Chemical Constitution,” Trans. Faraday Soc. 1929, 25, 253-283. doi:10.1039/TF9292500253

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 |0.732 - 1.000||8||[[Cubic crystal system|Cubic]]||CsCl, NH<sub>4</sub>Br
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+The radius ratio rule was first proposed by Gustav F. Hüttig in 1920. <ref name=jensen>''The Origin of the Ionic-Radius Ratio Rules'' William B. Jensen and William B. Jensen Journal of Chemical Education 2010 87 (6), 587-588 {{DOI|10.1021/ed100258f}} </ref> <ref>Hüttig, G. F. (1920), Notiz zur Geometrie der Koordinationszahl. Z. Anorg. Allg. Chem., 114: 24–26. {{doi|10.1002/zaac.19201140103}}</ref> In 1926 [[Victor Goldschmidt]] <ref name=jensen /> extended the use to ionic lattices. <ref>V. Goldschmidt, T. Barth, G. Lunde, W. Zachariasen, Geochemische Verteilungsgesetze der Elemente. VII. Die Gesetze der Krystallochemie, Dybwad: Oslo, 1926, pp. 112-117.</ref> <ref>V. Goldschmidt, Geochemische Verteilungsgesetze der Elemente. VIII. Untersuchungen über Bau und Eigenschaften von Krystallen, Dybwad: Oslo, 1927, pp. 14-17</ref> <ref>V. Goldschmidt, “Crystal Structure and Chemical Constitution,” Trans. Faraday Soc. 1929, 25, 253-283. {{doi|10.1039/TF9292500253}}</ref>
 ==See also==