Solid State Chemistry and its Applications. Anthony R. West

tetrahedral anion sites empty. Also, there is only one variable positional parameter, the x fractional coordinate of the 48 X atoms in the position (x, 1/8, 1/8), etc., Fig. 1.48. Various compounds form the pyrochlore structure and their differences depend on the value of x, which is usually in the range 0.31–0.36. In the extreme case that x = 0.375, the structure is derived from an undistorted fluorite with 8‐coordinate A cations, as in fluorite, but grossly distorted BX₆ octahedra. As x decreases, the 48 X atoms move off their regular tetrahedral sites and the cubic coordination of A becomes distorted: six X neighbours form a puckered hexagon and two X′ atoms are at a different distance, in apical positions, on either side of the puckered hexagon.

      At x = 0.3125, the B coordination becomes undistorted octahedral and the BX₆ octahedra link by sharing corners to form a 3D network. The A coordination may be described as 2X′ + 6X, with short A–X′ bonds. These A–X′ bonds form a 3D network, A₂X′, that interpenetrates the network of BX₆ octahedra, of stoichiometry B₂X₆. The A₂X′ net, with linear X′–A–X units and X′A₄ tetrahedra, is similar to that in cuprite, Cu₂O. The B₂X₆ and A₂X′ networks, that together form the pyrochlore structure, are shown separately in Fig. 1.48.

      Pyrochlore‐based oxides have a range of interesting properties and applications. La₂Zr₂O₇ is an electronic insulator whereas Bi₂Ru₂O_7‐δ is metallic and Cd₂Re₂O₇ is a low temperature superconductor. (Gd_1.9Ca_0.1) Ti₂O_6.9 is an oxide ion conductor and Y₂Mo₂O₇ is a spin glass. As indicated in some of the above examples, the anion content is not always 7 and indeed the amount of X′ in the general formula can have values between 0 and 1 in different pyrochlores.

       Figure 1.47 Some cation‐ordered fluorites showing cation positions relative to those in fcc fluorite.

       A. F. Wells, Structural Inorganic Chemistry, Oxford University Press (2012).

The rare earth oxides RE₂O₃ have structures derived from oxygen‐deficient fluorite in which the rare earth CN is reduced to 7 or 6. There are three structure types stable below 2000 °C: hexagonal A, monoclinic B and cubic C (or bixbyite) with A favoured by larger cations La to Pm, C by the smaller cations Gd to Lu and B by those of intermediate size. Some oxides are polymorphic and transform in the sequence C to B to A with increasing temperature, although most do not form all three polymorphs. Structure types and polymorphisms are summarised in Fig. 1.48(b) with part of the structure of A‐type La₂O₃ in (c) showing 7‐fold coordination of La in the form of a distorted cube with one corner missing. The B structure is a monoclinic distortion of A giving rise to a mixture of 6‐ and 7‐coordinate RE cations whereas the C structure has 6‐coordinate cations. The mineral name of the C structure is bixbyite, (Fe,Mn)₂O₃ and it is essentially cubic fluorite with ¼ of the oxygens missing.

The weberite structure, Na₂MgAlF₇, is another anion-deficient fluorite superstructure, closely related to pyrochlore that is shown by a range of fluorides and oxides of general formula A₂B₂X₇. The ideal symmetry of weberite is orthorhombic, space group Imma, but several structurally-related polymorphs occur, depending on composition. The anion-deficiency relative to fluorite leads to an overall reduction in cation coordination number with two types of distorted AX₈ polyhedra and two sets of BX₆ octahedra. It may be regarded as a 3D network of corner-sharing BX₆ octahedra, but also as a complex interpenetrating network of corner- and edge-sharing AO₈ polyhedra within which the anions occupy three sets of crystallographically-distinct tetrahedral sites. The cations in isolation form ccp layers, equivalent to those in the (111) orientation in fluorite and pyrochlore, but have two stoichiometries, A₃B and AB₃, that alternate. The A components of these layers also form a so-called Kagome network of interlinked hexagons and triangles which may be regarded as cp layers with ¼ of the packing atoms absent.

      

       Figure 1.48 The pyrochlore structure, which may be regarded as a distorted, 2 × 2 × 2 superstructure of a cation‐ordered, anion‐deficient fluorite. Anions in eightfold positions, e.g. in fluorite split into two groups in pyrochlore, with one group, X, in 48‐fold positions at e.g. (x, 1/8, 1/8). (b) Temperature‐dependent polymorphism of the rare earth sesquioxides.

      G. Adachi and N. Imanaka, Chem. Rev. 98, 1479 (1998).

       (c) part of the C-type La2O3 crystal structure showing 7-coordinate La.

      A range of complex oxides have weberite structures, in compositional families such as Ln₃NbO₇, better written more informatively as Ln₂(Ln, Nb)O₇: Ln = La, Nd, Gd, Dy; Ln₂B₂O₇: LnB = NdZr, SmTi; A₂Sb₂O₇: A = Ca, Sr, Pb and Ca₂(Ta, Nb)₂O₇ (see L Cai and JC Nino, Acta Cryst B 65, 269 (2009) for more details). Weberite oxides are of interest, showing a diverse range of electrical properties. Various weberite fluorides are known, with Na or Ag as the A-cations and different divalent, trivalent B cation combinations such as B²⁺: Mg, Mn, Fe, Co, Ni, Cu, Zn and B′³⁺: Sc, V, Cr, Fe, Al, Ga, In, Tl; most interest is in the magnetic interactions associated with the different transition metal combinations in the B sites within the Kagome A sublattice.

      

      1.17.13 Garnet

The garnets are a large family of complex oxides, including both minerals and synthetic materials, some of which are important ferromagnetic materials; yttrium aluminium garnet (YAG) is used as a synthetic gemstone and when doped with neodymium is the key component in YAG lasers; more recently, garnets with high Li⁺ ion conductivity have been synthesised; garnets have hardness 6.5–7.5 on the Mho scale and are used industrially as an abrasive on sandpaper. They have the general formula A₃B₂X₃O₁₂: A = Ca, Mg, Fe, etc.; B = Al, Cr, Fe, etc.; X = Si, Ge, As, V, etc. A is a large ion with a radius of ~ 1 Å and has a coordination number of eight in a distorted cubic environment. B and X are smaller ions which occupy octahedral and tetrahedral sites, respectively. Garnets with A = Y or a rare earth: Sm, Gd, Tb, Dy, Ho, Er, Tm, Yb or Lu and B, X = Fe³⁺ have interesting magnetic properties. One of the most important is yttrium iron garnet (YIG), Y₃Fe₅O₁₂, the structure of which is shown in Fig. 1.49. Many other A, B, X combinations are possible, such as those in Table 1.25.

       Figure 1.49 The garnet crystal structure.

      The unit cell of garnet is body centred cubic, a ≈ 12.4 Å, and contains eight formula units. The structure may be regarded as a framework built of corner‐sharing BO₆ octahedra and XO₄ tetrahedra. The larger A ions occupy eight‐coordinate cavities within this framework. In YIG and the rare earth garnets, the B and X ions are the same, Fe³⁺.