15 | Eutactic structure types of ionic compounds

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We may use simple elemental (unary) structures as a starting point for describing the structures of many simple ionic compounds composed of two (binary) or three (ternary) elements. In order to construct an ionic compound it is first important to appreciate that (1) different elements have different radii and (2) their structures are not so much governed by a densest packing of hard spheres but rather by the electrostatic interactions between the ions.

Ionic radii

Ionic radius is a key metric of ionc solids that provide predictive power to a particular structures likely hood of formation and relative stability. Our ability to assign an ionic radius is premised upon a singular emiprical fact.

The distance between two elements is empirically constant (< 10% s.u.)[1] in all know structures provided:

1. The oxidation states are constant

2. The coordination number is constant

3. The spin state is constant

Based on these observations, using X-ray crystallography, Shannon and Prewitt[2][3] compiled a table of standard ionic radii that remains in use today as the most common reference. Generally ionic radius decreases as:

1. the charge of the ion becomes more positive

2. as the the principle quantum number (n) decreases

3. as the effective nuclear charge increases (moving right in the same row of the periodic table)

4. as coordination number decreases

5. For first row transition metals all else being equal low spin states have a smaller ionic radius than high spin states

The most most prominent trend in ionic radii it that anions are much larger than cations. For this reason the structures of ionic compounds are often best described as an elemental arrangement of anions with cations occupying the voids between anions for charge balance. Such a image is not physically meaningful considering that electrostatic repulsion makes such an arrangement of anions unstable without the presence of charge balancing cations. None the less the simplest model for predicting the structures simple ionic compounds is based on this idea of cations filling voids using Pauling’s radius ratio and “no rattling” rules.

Voids within close packed structures

Fig. 29 shows the “voids” within the close packed arrangement of spheres that cations may fill within pauling ion packing model. There are three unique 3D void spaces: an octahedral hole, one tetrahedral hole oriented upwards, and, one tetrahedral hole oriented downwards. Necessarily the octahedral hole must be larger than the tetrahedral holes in a close packing of hard spheres.

Fig. 29 Octahedral and tetrahedral voids within a close packed arrangement of spheres. The same voids are present in both the ccp and hpc structure types.

Pauling’s no rattling rule

By geometric analysis of each void within a packing of anions, Pauling’s no rattling rule attemps to predict which cations will occupy which voids based on the relative radii of the cations and the anions. Each can be dirived by inscribing the largest possible sphere (the cation) that can fit in the void. If the size of the the cation is smaller than this incrsibed sphere then Paulings rule[4] predicts that it would “rattle” within its anion cage and result in an unstable configuration. These ratios are summarized for common voids found in real crystal structures in Fig. 30. As coordination number increases the minimum allow radius of the cation also increases. Note for the dodecahedral void (CN = 12) that the stucture described is one where both anion and cation occupy the same close packed arrangement of atoms and thus ideally their raddii should be equal (\(\frac{r_+}{r_-}=1\)).

Fig. 30 Pauling’s radius ration rules for predicting the stability of ionic compounds based on the principle of not rattling. Cations to anion ionic radius ratios equal to or greater than the values shown are predicted stable in each respective coordination geometry. For clarity, the cations and anions center positions are illustrated with a fixed radius.

Structures with a eutactic arrangement of anions

In truth, owing to electrostatic repulsion the anions in an ionic lattice will never close pack as though they are hard spheres as we imagined for the elements. A better model for their formation is that proposed by O’Keeffe[5] where an ionic lattice can be imagined as anions arranged to yield the highest possible volume structure allowed while keeping their bond distances with cations constant. Such structures are referred to as maximum volume structures and O’Keeffe found that in many cases the anions will ideally arrange themselves in a geometry isostructural to the elemental close packed structures but with a larger than expected unit cell. Such an ideal arrangement of anions in an ionic solid is referred to as a eutactic structure. Within this paradigm many of the common ionic structures can be summarized using eutactic ccp and hcp arrangements of anions as a foundation.

Anion Arrangement	Interstitial Sites			Examples
	T+	T–	O
ccp	-	-	1	NaCl (rocksalt)
	1	-	-	ZnS (blende/sphalerite)
	1/8	1/8	1/2	MgAl2O4 (spinel)
	-	-	1/2	CdCl2
	-	-	1/3	CrCl3
	1	1	-	K2O (antifluorite)
hcp	-	-	1	NiAs
	1	-	-	ZnS (wurtzite)
	-	-	1/2	CdI2
	-	-	1/2	TiO2 (rutile)*
	-	-	2/3	Al2O3 (corundum)
	1/8	1/8	1/2	Mg2SiO4 (olivine)

Rocksalt (NaCl)

The rock salt structure, Fig. 31 is composed of a eutactic ccp lattice of anions with all of the octahedral holes filled by cations. The coordination number of every cation is six and the coordination number of every anion is six. The cations and anions are in a one-to-one stoichiometric ratio and therefore much have equal and opposite oxidation states to from this structure (e.g. \(FOS_{Na} =+1\) and \(FOS_{Cl} = –1\))Sodium chloride (rock salt) is just one of many compounds that has this structural arrangement of cations and anions. Other compounds include KCl, MgO, MnO, FeO, NiO, AgCl, PbSe, PbTe.

Formula: NaCl
Space group: \(Fm\bar{3}m\)
Lattice: Cubic-F
Cell: \(a = 5.657\,\text{Å}\)
Z: 4   V: \(181.1\,\text{Å}^3\)

Fig. 31 Crystal structure of rocksalt (NaCl). Na is shown in yellow and Cl in green.

Sphalerite (cubic-ZnS)

Also known as zinc blende, the sphalerite structure (Fig. 32) is composed of a eutactic ccp lattice of anions with half all of the tetrahedral holes filled by cations. The structure is related to that of diamond where in ZnS, sites alternate between Zn⁺² and S^–2. All of the Zn and S site both have a coordination number of four. As in rock salt there are just as many Zn as S sites within this structure type and therefore the cations and anions much have equal and opposite charges. The compounds HgTe, ZnSe, ZnTe, CuCl as well as the common semiconductors CdS, CdSe, CdTe, GaN, GaAs, InP, and InAs have the sphalerite structure type.

Formula: ZnS
Space group: \(F\bar{4}3m\)
Lattice: Cubic-F
Cell: \(a = 5.409\,\text{Å}\)
Z: 4   V: \(158.3\,\text{Å}^3\)

Fig. 32 Crystal structure of Sphalerite (ZnS, aka zinc blende). Zn is shown in grey and S in yellow.

Spinel (MgAl₂O₄)

The spinel structure (Fig. 33) is composed of a eutactic ccp lattice of anions with one quarter of the tetrahedral holes filled by 2+ and half of the octahedral holes filled by 3+ cations. The structure can be envisioned as being composed of vertex sharing Al₄O₄ cubane clusters that form a diamondoid structure with tetrahedral Mg²⁺ cations in the interstitial sites. The compounds Fe₃O₄, Mn₃O₄, Co₃O₄, CuFe₂O₄, and NiFe₂O₄, ZnCr₂O₄, CdIn₂S₄, CdIn₂S₄, CdIn₂S₄, and CuCo₂Se₄ among others adopt the spinel structure type.

Formula: Al₂MgO₄
Space group: \(Fd\bar{3}m\)
Lattice: Cubic-F
Cell: \(a = 8.084\,\text{Å}\)
Z: 8   V: \(528.2\,\text{Å}^3\)

Fig. 33 Crystal structure of Sphalerite (ZnS, aka zinc blende). Zn is shown in grey and S in yellow.

CdCl₂

The CdCl₂ structure (Fig. 34) is composed of a eutactic ccp lattice of anions with only half of the octhedral holes filled by cations to yield a compound with only 2D bond network connectivity. In this layered compound the three distinct layers for the anion ABCA… stacking motif are aligned normal to the c-axis of th unit cell; this is the every 4 layer is perfectly superimposed. In order for the cations to fill every other octahedral hole 6 close packed layers must becontained within the unit cell if yield the repeat pattern A|B CA B*C A|*B where * denotes the cation locations. Structures that adopt the CdCl₂ structure type include MgCl₂, MnCl₂, FeCl₂, and CoCl₂.

Formula: CdCl₂
Space group: \(R\bar{3}m\)
Lattice: Trigonal-R
Cell: \(a = 3.85\,\text{Å},\, c = 17.46\,\text{Å},\, \gamma=120º\)
Z: 3   V: \(224.1\,\text{Å}^3\)

Fig. 34 Crystal structure of CdCl₂. Cd is shown in yellow and Cl in greens.

CrCl₃

The CrCl₃ structure (Fig. 35) is composed of a eutactic ccp lattice of anions with only one third of the octhedral holes filled by cations to yield a layered compound very similar to CdCl₂ with ordered octahedral site vacancies. Structures that adopt the CrCl₃ structure type include FeCl₃, VCl₃,

Formula: CrCl₃
Space group: \(P3_2 12\)
Lattice: Trigonal-P
Cell: \(a = 6.017\,\text{Å},\, c = 17.3\,\text{Å},\, \gamma=120º\)
Z: 6   V: \(542.4\,\text{Å}^3\)

Fig. 35 Crystal structure of CrCl₃. Cr is shown in black and Cl in green.

Antifluorite (K₂O)

The antifluorite structure (Fig. 36) is composed of a eutactic ccp lattice of anions with all of the tetrahedral holes filled by cations. Antifluorite is an example of an antistructure, where the parent compound fluorite (CaF₂) has the same exact structural arrangment of atoms except the cations and anions have swapped places. Note the relationship in stoichiometry between K₂O and CaF₂. Within the antifluorite structure the cations have a coordination number of four and the anions have a coordination number of eight. Compounds that adopt the antifluorite structure type are numerous: Li₂O, Na₂O, Na₂S, Rb₂O, Rb₂S, K₂S, K₂Se, and K₂Te. Many more compounds adopt the fluorite structure type including CaF₂, SrF₂, SrCl₂, BaF₂, \(\beta\)-PbF₂, PbO₂, CeO₂, ThO₂, and UO₂.

Formula: Na₂O
Space group: \(Fm\bar{3}m\)
Lattice: Cubic-F
Cell: \(a = 5.55\,\text{Å}\;\)
Z: 4   V: \(171.0\,\text{Å}^3\)

Fig. 36 Crystal structure of Na₂O which adopts the antifluorite structure type. Na is shown in yellow and O in red.

NiAs

The NiAs structure type (Fig. 37) is composed of a eutactic hcp lattice of anions with all of the octahedral holes filled by cations. Both the Ni and As sites have a coordination number of six. Many of the NiAs structures are intermetallic phases. The compounds NiS, NiSb, NiSe, NiSe, NiTe, PtSn, PtBi, MnBi, CrSe, and CrSb adopt the NiAs structure type.

Formula: NiAs
Space group: \(P6_3mc\)
Lattice: Hexagonal-P
Cell: \(a = 3.602\,\text{Å},\, c = 5.009\,\text{Å},\, \gamma=120º\)
Z: 2   V: \(56.28\,\text{Å}^3\)

Fig. 37 Crystal structure of NiAs. Ni is shown in green and As in purple.

Wurtzite (ZnS)

The Wurtzite (ZnS) structure (Fig. 38) is a polymorph of the sphalerite (also ZnS) structure where the sulfide anions have an hcp arrangement instead of ccp. While sphalerite is cubic and isotropic, wurtzite is hexagonal and therefore uniaxial. As in sphalerite all of the the Zn sites are tetrahedral with a coordination number of 4. The sulfur atoms also have a coordination number of 4. Structures that adopt the wurtzite structure type include ZnO, ZnSe, ZnTe, BeO, MnS, AgI, AlN, GaN, InN, and SiC. As with sphalerite, many of these compounds are important semiconductors. Silicon carbide has a large number of uses as a coating, abrasive, structural material, semiprecious stone, foundry material, and semiconductor.

Formula: ZnS
Space group: \(P6_3mc\)
Lattice: Hexagonal-P
Cell: \(a = 3.811\,\text{Å},\, c = 6.234\,\text{Å},\, \gamma=120º\)
Z: 2   V: \(78.41\,\text{Å}^3\)

Fig. 38 Crystal structure of ZnS (wurtzite). Zn is shown in grey and S in yellow.

CdI₂

The CdI₂ structure (Fig. 39) is very similar to the layered CdCl₂ structure except that the iodide anions are in a eutactic hcp arrangement. Because of this the repeat unit along the c-axis is smaller than that of CdCl₂. Again half of the octahedral holes are filled by cations in a pattern A|*B A|*B where * indicated the locations of the cations. The compounds CaI₂, CoI₂, MgI₂, PbI₂, VCl₂, TiCl₂, Mg(OH)₂, Ca(OH)₂, Co(OH)₂, Ni(OH)₂, and Cd(OH)₂ adopt the CdI₂ structure type.

Formula: CdI₂
Space group: \(P6_3mc\)
Lattice: Hexagonal-P
Cell: \(a = 4.24\,\text{Å},\, c = 13.67\,\text{Å},\, \gamma=120º\)
Z: 2   V: \(212.8\,\text{Å}^3\)

Fig. 39 Crystal structure of CdI₂. Cd is shown in yellow and I in purple.

Rutile (TiO₂)

The rutile structure (Fig. 40) is also composed of a distorted hcp lattice of oxide anions with half of the octahedral holes filled by cations with a different ordering of the cations to yield a 3D structure in contrast to the 2D structure of CdI₂. The Ti sites have a coordination number of six and the O sites have a coordination number of three. The rutile structure is one of multiple polymorphs of TiO₂ including anatase and brookite. Compounds that adopt the rutile structure type include CrO₂, GeO₂, IrO₂, SnO₂, WO₂, PbO₂, CoF₂, MgF₂, FeF₂, and PdF₂.

Formula: TiO₂
Space group: \(P\frac{4_2}{m}nm\)
Lattice: Tetragonal-P
Cell: \(a = 4.594\,\text{Å},\, c = 2.958\,\text{Å}\)
Z: 2   V: \(62.42\,\text{Å}^3\)

Fig. 40 Crystal structure of rutile (TiO₂). Ti is shown in white and O in red.

Corundum (Al₂O₃)

The corundum (Al₂O₃) structure type (Fig. 41) is composed of oxides anions arranged in a eutactic hcp configuration with Al³⁺ ions occupying 2/3 of the octahedral holes to yield a structure with 3D bond network connectivity. Ternary phases with mixed cations adopt the similarly structured Ilmenite structure type. Compounds with this structure type include Al₂O₃ and it doped minerals ruby (Cr-doped) and sapphire (Ti-doped), along with Fe₂O₃ (hematite), Ti₂O₃, V₂O₃, and Ga₂O₃. The compounds MgTiO₃, MnTiO₃, FeTiO₃, and CoTiO₃, NiTiO₃, CdTiO₃, ZnTiO₃, MgSnO₃, NiMnO₃, and NaSbO₃ adopt the ternary ilmenite structure type.

Formula: Al₂O₃
Space group: \(R\bar{3}c\)
Lattice: Trigonal-R
Cell: \(a = 5.12\,\text{Å},\,\alpha=55.28º\)
Z: 3   V: \(224.13\,\text{Å}^3\)

Formula: TiFeO₃
Space group: \(R\bar{3}\)
Lattice: Trigonal-R
Cell: \(a = 5.087\,\text{Å},\,c = 13.75\,\text{Å},\,\gamma=120º\)
Z: 6   V: \(308.0\,\text{Å}^3\)

Fig. 41 Crystal structures of corundum (Al₂O₃) and ilmenite (FeTiO₃). Al is shown in blue, O in red, Fe in orange, and Ti in white.

Olivine (Mg₂SiO₄)

The olivine structure type (Fig. 42) is composed of oxides anions arranged in a eutactic hcp configuration with Si⁴⁺ ions occupying 1/8 of the tetrahedral holes and Mg occupying 1/2 of the octahedral holes to yield a structure with 3D bond network connectivity. Like spinels there is some flexibility in the valence distribution between the two cation sites that yields a diversity of mixed cation materials. Compounds that adopts olivine like structures include Fe₂SiO₄, Al₂BeO₄, LiFePO₄, LiMnPO₄, NaSmGeO₄, Mn₂SiS₄ , Mg₂SiS₄, Ca₂GeS₄, and Na₂BeF₄.

Formula: Mg₂SiO₄
Space group: \(Pbnm\)
Lattice: Orthorhombic-P
Cell: \(a = 4.756\,\text{Å},\, b = 10.21\,\text{Å},\, c = 5.980\,\text{Å}\)
Z: 4   V: \(290.3\,\text{Å}^3\)

Fig. 42 Crystal structure of the olivine Mg₂SiO₄. Mg is shown in green, Si in teal, and O in red.

References

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[1]

Bergerhoff, Brandenberg International Tables of Crystallography C 2006, 9, 778.

[2]

Shannon, Prewitt Acta. Cryst. B 1969, 25, 925.

[3]

Shannon Acta. Cryst. A 1976, 32, 751.

[4]

Pauling The Nature of the Chemical Bond 1960, 544.

[5]

O’Keeffe Acta Cryst A 1977, 33, 924.