Pyrochlore minerals and HTB-pyrochlore intgergrowths

Pyrochlore was the name originally given to a mineral of composition (Na,Ca)₂Nb₂O₆(OH,F), but is now the name of a mineral group with general composition (for stoichiometric minerals) A₂B₂X₆Y (Atencio et al., Reference Atencio, Andrade, Christy, Gieré and Kartashov2010). The aristotype has cubic symmetry, Fd $\bar{3}$m, with a ≈ 10.4 Å and atoms at the Wyckoff sites: A site = 16d (½,½,½); B = 16c (0,0,0); X = 48f (x,⅛,⅛); Y = 8b (⅜,⅜,⅜) where x = 0.3125 for undistorted BX₆ octahedra and x = 0.375 for undistorted AX₆Y₂ cubes (Subramanian et al., Reference Subramanian, Aravamudan and Subba Rao1983). The much stronger bonding of the B cations to X forces the value of x to be close to 0.3125, leading to strong distortion of the A cation-centred polyhedra. This is an inherent source of strain in pyrochlore structures.

The pyrochlore structure is usually described as an anion-deficient fluorite derivative structure with face-centred cubic (fcc) stacking of alternating A₃B and B₃A metal atom layers parallel to the {111}_fluorite planes (Subramanian et al., Reference Subramanian, Aravamudan and Subba Rao1983; Fuentes et al., Reference Fuentes, Montemayor, Maczka, Lang, Ewing and Amador2018; Trump et al., Reference Trump, Koohpayeh, Livi, Wen, Arpoino, Ramasse, Brydson, Feygenson, Takeda, Takigawa, Kimura, Nakatsuji, Broholm and McQueen2018). Both the A and B atoms form 3D kagomé arrays of anion-centred corner-connected tetrahedra, YA₄ and □B₄ (□ = vacancy) (Pannetier and Lucas, Reference Pannetier and Lucas1970) as shown in Fig. 4. The composition of the net of YA₄ tetrahedra is A₂Y and that of the □B₄ net with octahedral coordination of the B atoms by X is B₂X₆, leading to a representation of the pyrochlore formula that emphasises the independence of the two sublattices, B₂X₆.A₂Y. Defect pyrochlore minerals have vacancies at the A and Y sites and incorporation of H₂O giving A_2–mB₂X₆Y_1–n⋅pH₂O (Hogarth, Reference Hogarth1977) although minor loss of X can also occur under extreme alteration (Lumpkin and Ewing Reference Lumpkin and Ewing1995).

Fig. 4. Corner connected A₄ tetrahedra (blue) and B₄ tetrahedra (yellow) in pyrochlore

The pyrochlore B₂X₆ framework can be derived formally from BX₄ HTB layers by increasing the octahedral tilt angle until Xa–Xa decreases to match the BX₆ octahedral edge distance. With Xa–Xa = e, equation 1 is used to calculate the tilt angle, giving 15.8° for undistorted octahedra. At this tilt angle the same type of BX₆ octahedron that forms the HTB layer can exactly cap the smaller triangles of apical anions, Xa, as shown in Fig. 5. Adding successive HTB layers with capping of the smaller triangles on both sides of the layers leads directly to the B₂X₆ octahedral framework in pyrochlore. Formally it is given by 3BX₄ + B(cap) = B₄X₁₂ (no reaction mechanism implied).

Fig. 5. Capping of small triangles of Xa anions by octahedra in pyrochlore {111} HTB layer.

Since the establishment of a nomenclature scheme for the pyrochlore supergroup of minerals (Atencio et al., Reference Atencio, Andrade, Christy, Gieré and Kartashov2010; Christy and Atencio, Reference Christy and Atencio2013) there have been numerous publications on new minerals of the supergroup with high quality single-crystal refinements of the structures. We have calculated the octahedral tilt angles from the published structural data, after first converting the cubic Fd $\bar{3}$m cells and coordinates to R $\bar{3}$m hexagonal cells for ease of calculation of the tilt angles. In the hexagonal cell the tilt angle can be calculated separately from the vertical displacement of the equatorial anions, Xe, and from the horizontal displacement of the apical anions, Xa. The results are given in Table 1. The minerals hydrokenoralstonite with Al in the B site (Atencio et al., Reference Atencio, Andrade, Bastos Neto and Pereira2017) and hydrokenoelsmoreite-3C with W ⁶⁺ in the B site, (Mills et al., Reference Mills, Christy, Rumsey and Spratt2016) both have measured tilt angles that agree with the value of 15.8° calculated from equation 1. These two minerals have almost perfect octahedra with X–B–X angles of 90° and all X–X edges equal. The other minerals listed in Table 1 have tilt angles that deviate from the ideal value, with the deviation increasing from pyrochlore-group minerals (B = Nb) to microlites (B = Ta) to roméite-group minerals (B = Sb). For all these minerals the cubic cell x coordinate for the anion is higher than the ideal value of 0.3125 that corresponds to a perfect octahedron. With increasing x above this value the geometry is described as a flattened trigonal prism (Subramanian et al., Reference Subramanian, Aravamudan and Subba Rao1983). It is this distortion that results in different experimental tilt angles being calculated from the Xe and Xa anions, although the average stays close to the calculated value of 15.8° from equation 1.

Table 1. Structural data for pyrochlore-group minerals and compounds.

References: [1] Groult et al. (Reference Groult, Pannetier and Raveau1982); [2] Atencio et al. (Reference Atencio, Andrade, Bastos Neto and Pereira2017); [3] Mills et al. (Reference Mills, Christy, Rumsey and Spratt2016); [4] Ivanyuk et al. (Reference Ivanyuk, Yakovenchuk, Panikorovskii, Konoplyova, Pakhomovsky, Bazai, Bocharov and Krivovichev2019); [5] Yin et al. (Reference Yin, Li, Yang, Ge, Xu and Wang2015); [6] Bonazzi et al. (Reference Bonazzi, Bindi, Zoppi, Capitani and Olmi2006); [7] Witzke et al. (Reference Witzke, Steins, Doering, Schuckmann, Wegner and Pollmann2011); [8] Andrade et al. (Reference Andrade, Atencio, Chukanov and Ellena2013); [9] Kasatkin et al. (Reference Kasatkin, Britvin, Peretyazhko, Chukanov, Škoda and Agakhanov2020); [10] Halenius and Bosi (Reference Hålenius and Bosi2013); [11] Atencio et al. (Reference Atencio, Ciriotti and Andrade2013); [12] Biagioni et al. (Reference Biagioni, Orlandi, Nestola and Bianchin2013); [13] Matsubara et al. (Reference Matsubara, Kato, Shimizu, Sekiuchi and Suzuki1996); [14] Mumme et al. (Reference Mumme, Grey, Birch, Pring, Bougerol and Wilson2010); [15] Fu and Ijdo (Reference Fu and Ijdo2014); and [16] Courbion et al. (Reference Courbion, Jacoboni and De Pape1976).

As seen from Table 1, within each pyrochlore group the trigonal antiprism distortion does not appear to depend significantly on the A cation, with similar distortions for A cations varying from Pb²⁺ and Bi³⁺ to Ca²⁺, Na⁺ and H₂O (defect keno pyrochlores). There is however, a clear inverse correlation between the magnitude of the distortion and the B–X–B angle. The distortion is greatest for B = Sb which has the smallest B–X–B angles. Sb is the most electronegative of the B elements in Table 1, which suggests that the distortion occurs to increase the B ns–X 2p covalent σ-bonding strength, as we noted earlier for the HTB compounds.

The cubic pyrochlores in Table 1 all have a single B cation, so the tilt angle calculated from equation 1 is the same for all. In contrast, rhombohedral pyrochlores, with 1:3 ordering of the A and B cations, AA’₃BB’₃X₁₄, have widely different tilt angles, depending on the relative sizes of the B and B’ cations. In fluornatrocoulsellite, CaNa₃AlMg₃F₁₄ (Mumme et al., Reference Mumme, Grey, Birch, Pring, Bougerol and Wilson2010) the HTB cation, Mg²⁺, is considerably larger than the capping cation Al³⁺, and so larger tilt angles are required for the triangles of apical anions of the HTB layer to fit to the base of the capping octahedra. With Xa–Xa and e in equation 1 calculated for undistorted AlF₆ and MgF₆ octahedra from bond-valence sum (BVS) parameters for fluorides (Brese and O'Keeffe, Reference Brese and O'Keeffe1991) the calculated tilt angle is 19.5°. This compares with a mean value of 18.2° obtained from the refined crystal structure. The MgF₆ octahedra are relatively distorted with F–Mg–F angles in the range 84.1 to 91.4 and F–F edge distances of 2.63 to 2.91 Å. The tilted HTB layers in fluornatrocoulsellite are shown in Fig. 6a.

Fig. 6. [100] projections of the structures of rhombohedral pyrochlores (a) fluornatrocoulsellite, octahedral tilt angle = 19.5°; (b) Cs₂NaAl₃F₁₂, tilt angle = 3.5°. Tilts shown by red arrows.

In contrast to fluornatrocoulsellite, the 1:3 ordered rhombohedral pyrochlore Mn²⁺La₃Mn²⁺Sb₃O₁₄ (Fu and Ijdo, Reference Fu and Ijdo2014) has a smaller HTB cation, Sb⁵⁺, than the capping cation, Mn²⁺. In this case smaller octahedral tilts are required to match Xa–Xa to the base of the capping octahedron. For undistorted octahedra, application of equation 1 gives a tilt angle of 11.2°, compared to 10.5° measured from the refined structure. The MnO₆ polyhedron is a flattened trigonal antiprism with angles of 79.9 and 100.1°.

An extreme example of B cation size difference in 1:3 ordered rhombohedral pyrochlores is the defect pyrochlore of composition □₄NaAl₃F₁₂Cs₂ (Courbion et al., Reference Courbion, Jacoboni and De Pape1976). This is an example of the so-called inverse pyrochlores, where the very large Cs⁺ cation occupies the 8b site normally occupied by the Y anion. The B cations in the HTB layers are Al³⁺ and the capping cations are Na⁺. Application of equation 1 shows that a tilting angle of only 3.5° is required to match the capping NaF₆ to the HTB anions. The pyrochlore framework is shown in Fig. 6b. The Cs⁺ cations have 15 anion neighbours with distances in the range 3.03 to 3.55 Å, giving a nearly ideal BVS of 1.10.

When the Cs is replaced by Rb or K the P6₃/mmc-type octahedral tilting system is unable to provide suitably sized cavities for the smaller cations. Instead, a different system of tilting occurs which involves puckering of the kagomé nets of Al³⁺ cations to meet the bonding requirements of K⁺/Rb⁺ (Le Bail et al., Reference Le Bail, Gao, Fourquet and Jacobini1990). The tilting system belongs to the Immm space-group type reported by Campbell et al. (Reference Campbell, Howard, Averett, Whittle, Schmid, Yost and Stokes2018). As shown in Fig. 7a the Immm-type tilting involves only two-thirds of the octahedra, which form [010] chains in K₂NaAl₃F₁₂; the other octahedra are untilted. The tilting changes the pairs of equilateral triangles of Xa anions to isosceles triangles. As a result of the tilting the untilted octahedra are displaced above and below the plane of the tilted octahedra as shown in Fig. 7b. In K₂NaAl₃F₁₂ the displacements along c* are ± 0.5 Å. The planar kagomé net of cations becomes strongly corrugated as a result. Such tilting cannot occur in corner-connected HTB structures without the breaking of interlayer corner connections, but in pyrochlore-type structures it can occur, albeit with strong distortion of the capping BX₆ polyhedra as shown in Fig. 7b for the NaF₆ polyhedron in K₂NaAl₃F_12..

Fig. 7. (a) Immm-type tilting of octahedra in K₂NaAl₃F₁₂ as viewed along c*. (b) [010] projection of K₂NaAl₃F₁₂ structure, showing large displacement of untilted octahedra along c* (red arrows).

The 1:3 ordered rhombohedral pyrochlores have quite different in-plane B–X–B and out-of-plane B’–X–B angles with the difference between them reversing when B’ changes from being smaller than B to greater than B. For the antimony compound, the Sb–O–Sb angle of 138.6° is higher than the Sb–O–Sb angle in the four different roméite-group cubic pyrochlores in Table 1, which are in the narrow range 134.5 to 135.2°, suggesting that the bonding requirements for the A and A’ cations are important in this case.

The crystal structure results for the mineral hydrokenoelsmoreite-3C (Mills et al., Reference Mills, Christy, Rumsey and Spratt2016) require further consideration. In the refined structure the WO₆ octahedra are undistorted and the W is undisplaced from the ideal 16c site. This is in contrast to W in HTB compounds, A_xWO₃, where crystal structure studies show the W to be displaced from the centre of the octahedron by ~0.10 to 0.15 Å, consistent with a second-order Jahn Teller distortion for the d ⁰ cation (Pye and Dickens, Reference Pye and Dickens1979; Labbé et al., Reference Labbé, Goreaud, Raveau and Monier1978; Prinz, Reference Prinz1992). In addition W has a relatively high electronegativity, comparable with that for Sb according to the Pauling scale, and a similar B–X–B angle might be expected. A possible answer to this dilemma is that displacements of W and O do occur but they are short-range ordered and are manifested only indirectly by elevated atomic displacement parameters. The U _eq values are 0.023 Ų for W and 0.021 Ų for O in hydrokenoelsmoreite whereas they are much smaller at 0.007 Ų for Al and 0.011 Ų for OH in the related defect pyrochlore hydrokenoralstonite (Atencio et al., Reference Atencio, Andrade, Bastos Neto and Pereira2017). The U _eq values for hydrokenoelsmoreite correspond to root-mean-square displacements of ~0.15 Å. This is the same magnitude as the W displacements in HTB compounds and the anion displacements in the Sb pyrochlores.

The absence of long-range ordering of W displacements in hydrokenoelsmoreite is a specific example of the more general phenomenon of geometric frustration in the pyrochlore structure (McQueen et al., Reference McQueen, West, Muegge, Huang, Noble, Zandbergen and Cava2008; Trump et al., Reference Trump, Koohpayeh, Livi, Wen, Arpoino, Ramasse, Brydson, Feygenson, Takeda, Takigawa, Kimura, Nakatsuji, Broholm and McQueen2018). As shown in Fig. 4, the B cation kagomé network comprises a 3D array of corner-connected tetrahedra. Displacement of B involves ‘breathing’ or rotation of the B₄ tetrahedra and the inherent geometric frustration means that the displacements point in different directions, thus cancelling out in statistical displacements when modelled in Fd $\bar{3}$m. McQueen et al. (Reference McQueen, West, Muegge, Huang, Noble, Zandbergen and Cava2008) compared neutron powder XRD refinements of CaYNb₂O₇ and other pyrochlores in space group Fd $\bar{3}$m, with and without displacement of Nb along <111>. Without displacement the isotropic displacement parameter for Nb in CaYNb₂O₇ was 0.019(1) Ų. Refinement of the Nb (x,x,x) coordinate resulted in a displacement of 0.145 Å from the ideal position with a lowering of U _eq to 0.013(1) Ų and an improved fit to the data. A consequence of the distortion of the Nb kagomé network is that the Nb–O–Nb angle is changed from 132.4° in the ideal Fd $\bar{3}$m structure to values of 125, 132.4 and 139.2° in the displaced-Nb structure. Similar results could be expected from local displacements of the B cations in hydrokenoelsmoreite and other minerals.

Given that the pyrochlore structure can be derived from the HTB structure by capping of the apical anions, the possibility exists for intergrowth of the two structure types. In nature this is confirmed for the minerals phyllotungstite (Walenta, Reference Walenta1984; Grey et al., Reference Grey, Mumme and MacRae2013) and pittongite (Birch et al., Reference Birch, Grey, Mills, Bougerol, Pring and Ansermet2007; Grey et al., Reference Grey, Birch, Bougerol and Mills2006). Phyllotungstite, (□,Pb,Cs)₁₀(W,Fe)₁₄(O,OH)₄₂(H₂O)₄ is orthorhombic (pseudohexagonal), Cmcm with a = 7.298, b = 12.640 (= √3a) and c = 19.582 Å. The structure is shown in projection along [110] in Fig. 8. It comprises 6 Å wide blocks of pyrochlore intergrown along [001] with two HTB layers of width 3.8 Å, giving a c axis periodicity of 2 × (6 + 3.8) = 19.6 Å. An alternative description is in terms of ordered unit-cell twinning of the pyrochlore structure, where the twin planes lie between two HTB layers. The tilting of the octahedra in the HTB layers is required to match that of the octahedra in the pyrochlore blocks, and so ideally it has a magnitude of 15.8° (from equation 1). This is higher than for any reported HTB structures.

Fig. 8. [110] projection of the phyllotungstite structure.

In pittongite, (□,Na,H₂O)₃₂(W,Fe)₃₆(O,OH)₁₀₈(H₂O)₁₂ there are both 6 Å and 12 Å wide pyrochlore blocks intergrown with double HTB layers, giving a c axis periodicity of 2× (6 + 3.8 + 12 + 3.8) = 51.2 Å (actual c = 50.49 Å). Again, the structure can be described as unit-cell twinning of pyrochlore blocks. In this case the pyochlore blocks are of two different widths. At the twin planes, pairs of HTB layers are corner-connected. In the system Bi₂O₃–Fe₂O₃–Nb₂O₅, synthetic intergrowth phases have been structurally characterised that correspond to 12 Å-wide pyrochlore block intergrown with HTB (c = 2 × (12 + 3.9) = 31.8 Å) and to a rhombohedral form of the pittongite sequence with c = 3 × (6 + 3.9 + 12 + 3.9) = 77.4 Å, demonstrating that long-range ordering of quite complex intergrowth sequences of the two structure types is possible (Grey et al., Reference Grey, Mumme, Vanderah, Roth and Bougerol2007).

An intriguing example of a pyrochlore intergrowth structure has been reported by Crosnier et al. (Reference Crosnier, Pagnoux, Guyomard, Verbaere, Piffard and Tournoux1991) for the compound Cs₈Nb₁₀O₂₃(Si₃O₉)₂, with space group P6₃/mmc, a = 7.342 Å and c = 22.166 Å. In this compound HTB layers of corner-connected NbO₆ octahedra are capped on both sides by NbO₆, giving pyrochlore blocks of composition Nb₅O₁₈. The pyrochlore blocks are interconnected along [001] via corner sharing of the capping octahedra with cyclic Si₃O₉ clusters. Crosnier et al. (Reference Crosnier, Pagnoux, Guyomard, Verbaere, Piffard and Tournoux1991) describe the structure as an intergrowth of pyrochlore-type and benitoite-type (BaTiSi₃O₉)-type structures. The capping octahedra are common to both the benitoite blocks (with Nb replacing Ti) and the pyrochlore blocks. The intergrowth framework is then [Nb₃O₁₂Nb_2/2O_6/2] + [Nb_2/2Si₃O₃O_6/2], where 2/2 and 6/2 indicate sharing between the two blocks. The Cs⁺ cations occupy large cavities located between the hexagonal rings of the HTB layers and the Si₃O₉ rings.

Zirconolite minerals

Zirconolite was the name given to a monoclinic mineral of composition CaZrTi₂O₇, and it is now the group name for three different mineral polytypes with the same ideal formula and with substitutions of mainly rare earth elements (REE) for Ca and Nb, and Ta and Fe for Ti (Bayliss et al., Reference Bayliss, Mazzi, Munno and White1989). Like pyrochlore, the zirconolite structure can be described as an ordered anion-deficient fluorite derivative structure, with a fcc array of cations. However, while pyrochlore has an alternation of <111> fluorite cation layers of composition AB₃ and B₃A, the alternating layers of small and large cations in zirconolite have the compositions B₃B’ and A₂A’₂. Ti⁴⁺ cations occupy both the kagomé net sites (B) and sites in the hexagonal rings within the net (B’, displaced from the centre of the ring). The large cation layers contain parallel chains of edge-shared distorted cubes (A) and mono-capped trigonal antiprisms (A’).

In the aristotype mineral, zirconolite-2M, with space group C2/c and cell parameters a ≈ 12.4 Å, b ≈ 7.3 Å, c ≈ 11.4 Å and β ≈ 100.5° (Gatehouse et al., Reference Gatehouse, Grey, Hill and Rossell1981) the chains of large cations are oriented parallel to [110] and [1$\bar{1}$0] in successive A₂A’₂ layers. Zirconolite-3T, with trigonal space group P3₁21 and cell parameters a ≈ 7.3 Å and c ≈ 17.0 Å (Grey et al., Reference Grey, Mumme, Ness, Roth and Smith2003; Zubkova et al., Reference Zubkova, Chukanov, Pekov, Ternes, Schüller and Pushcharovsky2018), has the chains of large cations oriented parallel to [100], [010] and [110] in successive layers, so making angles of 120° with each other. Orthorhombic zirconolite-3O, with space group Cmca, a ≈ 7.3 Å, b ≈ 14.1 Å and c ≈ 10.2 Å, differs significantly from the other polytypes in that the kagomé net of small cations is not planar, but has a sawtooth shape with large (>1 Å) displacements of the cations out of the plane of the net (Chukanov et al., Reference Chukanov, Zubkova, Pekov, Vigasina, Polekhovsky, Ternes, Schüller, Britvin and Pushcharovsky2019). The corner-connected octahedra do not have the HTB layer topology of trans-corner connection. There is periodic shearing along an octahedral edge so that there is a mixture of trans- and cis-corner sharing of the octahedra. The same type of sawtooth HTB layers occur in the walentaite-group minerals (Grey et al., Reference Grey, Hochleitner, Rewitzer, Riboldi-Tunnicliffe, Kampf, MacRae, Mumme, Kaliwoda, Friis and Martin2020).

From the perspective of HTB layer octahedral tilting, the zirconolite 2M and 3T polytypes have the P6₃/mmc type of tilt system (symmetrical ‘breathing’, Fig. 2b) with capping of the smaller triangles of Xa anions by A'X₇ mono-capped trigonal antiprisms as shown in Fig. 9. Unlike the pyrochlores where the capping polyhedron links to the small triangles of anions in the HTB layers both above and below, the A'X₇ polyhedron in zirconolites has only one flat triangular face and so has only one-sided capping. Alternate A'X₇ polyhedra along the chains cap the layers alternately above and below the polyhedra.

Fig. 9. Capping of HTB layer with ZrO₇ in zirconolite-2M. Alternate ZrO₇ polyhedra have their capping triangular faces pointing to the HTB layer below and the HTB layer above (not shown).

For zirconolite-2M the triangular base of the ZrO₇ polyhedron has O–O = 2.97 Å, and the O–O edge of an undistorted TiO₆ octahedron from BVS calculations is 2.77 Å. Substituting these values in equation 1 gives an octahedral tilt angle of 12.8°. This is considerably smaller than for cubic pyrochlore, reflecting the larger size of the capping cation relative to the B cation. The tilt angle measured from the crystal structure is 10.5° based on the movement of the apical anions and 17° based on the displacement of the equatorial anions. The disparity between the two values reflects the large angular distortions of the TiO₆ polyhedra, with O–Ti–O angles in the range 78.4 to 98.4° and O–O distances from 2.48 to 2.97 Å. The shortest O–O distances correspond to edges that are shared with trigonal bipyramidal TiO₅ polyhedra in the hexagonal rings. The in-plane Ti–O–Ti and out-of-plane Ti–O–Zr angles have almost the same ranges, 133.1 to 138.3 and 131.9 to 136.7°, respectively. This contrasts with the rhombohedral pyrochlore, Mn₂La₃Sb₃O₁₄, which has a similar tilt angle but has quite different in-plane Sb–O–Sb and out-of-plane Sb–O–Mn angles of 128.7 and 147.6, respectively, (Table 1). The geometry of the octahedral tilting in the zirconolite-3T polytype (Grey et al., Reference Grey, Mumme, Ness, Roth and Smith2003) is almost identical to that reported above for the 2M polytype.

Alunite-supergroup minerals and related compounds

Alunite-supergroup minerals have the general formula AB₃(TX₄)₂X’₆ with wide ranges of large A cations, small B cations and with tetrahedral groups being mainly SO₄, PO₄, AsO₄ and X’ = O, OH, F and H₂O (Bayliss et al., Reference Bayliss, Kolitsch, Nickel and Pring2010). Jarosite minerals, AFe³⁺₃(SO₄)₂(OH)₆ are the most well-known of the alunite-supergroup minerals due to their role in acid mine drainage and metallurgical processes, their identification on Mars and their magnetic properties (Majzlan et al., Reference Majzlan, Glasnak, Fisher, White, Johnson, Woodfield and Boerio-Goates2010; Mills et al., Reference Mills, Nestola, Kahlenberg, Christy, Hejny and Redhammer2013).).

The minerals have rhombohedral symmetry, R $\bar{3}$m, with a ≈ 7 Å, c ≈ 17 Å and with atoms at the following Wyckoff sites: A site = 3a: 0,0,0; B site = 9d: ½, 0, ½; T site = 6c: 0,0,z; z ≈ 0.3; X1 = 6c: z ≈ 0.4; X2 = 18h: x,–x, z: x ≈ 0.22, z≈ –0.05; and X3 = 18h: x ≈ ⅛, z ≈ 0.13.

The structure can be described as based on HTB layers with strongly tilted octahedra giving small and large triangular groupings of apical anions as shown in Fig. 2b. The small triangles are capped by TX₄ tetrahedra and the large triangles are capped by AX₆ flattened trigonal antiprisms. The complete coordination for A comprises two interpenetrating trigonal antiprisms; A co-ordinates to 6 Xe anions (site X3) in the form of an elongated prism and to 6 Xa anions (site X2) forming a flattened prism. Goreaud and Raveau (Reference Goreaud and Raveau1980) have previously described the relationship of the alunite structure to that of pyrochlore with BX₆ sandwiched between opposing small triangles in the HTB layers in pyrochlore and AX₆ (the flattened trigonal prism) sandwiched between opposing large triangles in alunite (Fig. 10). In Table 2, are listed the results of octahedral tilting calculations for alunite supergroup members that cover a wide range of size differentials between the B cation and the capping T and A cations.

Fig. 10. Capping of large HTB triangles with AX₆ flattened trigonal antiprism in jarosite, compared with capping of small HTB triangles with BX₆ octahedron in pyrochlore (after Goreaud and Raveau, Reference Goreaud and Raveau1980).

Table 2. Structural data for alunite-supergroup minerals.

References: [1] Mills et al. (Reference Mills, Nestola, Kahlenberg, Christy, Hejny and Redhammer2013); [2] Majzlan et al. (Reference Majzlan, Stevens, Boerio-Goates, Woodfield, Navrotsky, Burns, Crawford and Amos2004); [3] Najorka et al. (Reference Najorka, Lewis, Spratt and Sephton2016); [4] Dzikowski et al. (Reference Dzikowski, Groat and Jambor2006); [5] Kato (Reference Kato1987); [6] Blount (Reference Blount1974); [7] U. Kolitsch (pers. comm.); [8] Cooper and Hawthorne (Reference Cooper and Hawthorne2012); and [9] Mills et al. (Reference Mills, Kartashov, Kampf and Raudsepp2010).

The jarosite minerals have the largest size disparity between B and T (B = Fe³⁺, T = S) and the largest calculated tilt angle of 22°. They also appear to be the least strained of the alunite minerals. The deviation of the O–Fe–O angles from 90° is of the order of only 1° and the octahedral tilt angles measured from the displacements of the Xa and Xe anions vary by ~2°. The changing of the A cation size has no significant effect on the tilting but there is a correlation between the A–Xe distance in the elongated trigonal prism and the cell parameter c. A linear correlation between the average ionic radius of the A cation and the c axis length has been reported for alunite-supergroup minerals by Sato et al. (Reference Sato, Nakai, Miyawaki and Matsubara2009). Because the TX₄ tetrahedra only cap the HTB layer on one side they cannot influence the interlayer spacing in the same way that the BX₆ double-capping octahedra can in pyrochlore structures. Instead the interlayer spacing in alunite-supergroup minerals is responsive to change in size of the A cation.

The plumbogummite group of minerals, gorceixite, goyazite and crandallite have a smaller Al³⁺ cation at the B site and a larger P⁵⁺ as a capping cation and the calculated tilt angle is correspondingly lowered to 18.5°. These minerals are more strained than the jarosites. The departure of the O–Al–O angles from 90° increases with decreasing A cation size from ± 2° for gorceixite to ± 3° for crandallite. In the latter mineral, the Ca cation is displaced from the 2a site to the more general 18f site (x,0,0, x = 0.044; Blount Reference Blount1974) to meet its bonding and valence requirements. The strain is also manifested in a larger disparity (3 to 4°) between the octahedral tilt angles measured from the Xa and Xe displacements. At the smaller tilting angle relative to the jarosites, the large triangle of Xa anions is not as expanded as in jarosites and the A–Xa and A–Xe distances are closer to one another, with the ratio (A–Xa)/(A–Xe) even reversing from greater than 1 to less than 1 in the case of gorceixite.

The dussertite-group minerals have As⁵⁺ as the capping cation, larger than P⁵⁺ in the plumbogummite-group minerals and the tilting angle is correspondingly smaller at 16.0°. The AlO₆ polyhedra are even more distorted trigonal antiprisms than for the plumbogummite minerals, with O–Al–O angles in the range 85.0 to 95.0 for arsenoflorencite-(La) and greater discrepancies between the measured tilting angles. In the examples in Table 2, the A cations change from large monovalent cations in jarosites to smaller divalent cations in plumbogummites to even smaller trivalent and divalent cations in the dussertite-group minerals, with the added complication of Pb²⁺ having a lone pair electron configuration and a displacement from the 2a site to an 18f site as for crandallite. The observed increasing distortion of the Al-centred trigonal antiprisms must be a direct consequence of the A cation bonding and valence requirements having a more dominant role in establishing the equilibrium atomic arrangement.

The compound NH₄(V⁵⁺O₂)₃(Se⁴⁺O₃)₂ (Vaughey et al., Reference Vaughey, Harrison, Dussack and Jacobson1994) is closely related to alunite-supergroup minerals, as is evident when its formula is written as NH₄V₃(SeO₃)₂O₆. In this compound, HTB layers of VO₆ octahedra are capped by SeO₃ trigonal pyramids. The compound has hexagonal symmetry, space group P6₃, with a = 7.137 Å and c = 11.462 Å. In contrast to alunite minerals, where the tetrahedral apices point to the centres of hexagonal rings in adjacent HTB layers, the selenite compound has two independent SeO₃ groups with the pyramids on one side of each HTB layer pointing towards the hexagonal ring in the adjacent layer and the SeO₃ on the opposite side of each HTB layer pointing towards the larger triangle of Xa anions in the adjacent layer. This arrangement has a 2-layer hexagonal repeat compared to the 3-layer rhombohedral repeat for alunite minerals. The base of the SeO₃ pyramids and the average edge distance for the AlO₆ octahedra are almost identical so the tilting angle is almost the same as in pyrochlore structures, 15.5 Å. The polar compounds A₂(WO₃)₃SeO₃ (A = NH₄, Rb and Cs; Harrison et al., Reference Harrison, Dussack, Vogt and Jacobson1995) and A₂(MoO₃)₃SeO₃ (A = Rb and Tl; Chang et al., Reference Chang, Kim and Halasyamani2010) are also related closely to the alunite structure. In these compounds, with space group P6₃ and similar cell dimensions to the above compound, the SeO₃ trigonal pyramids cap the smaller triangles of the HTB layers on only one side of the layers. The compound Cs₃Sb₃Ge₂O₁₃ (Pagnoux et al., Reference Pagnoux, Verbaere, Piffard and Tournoux1993), space group P6₃/mmc, a = 7.31 Å and c = 16.73 Å, has an alunite-related structure based on HTB layers of SbO₆ octahedra, capped with GeO₄ tetrahedra. In this structure the GeO₄ tetrahedra in adjacent layers share their apical anion giving Ge₂O₇ groups that link the HTB layer together. The base of the GeO₄ tetrahedron is somewhat larger than the SbO₆ edge length and the calculated octahedral tilting angle is 14.2°, smaller than for the selenites.

The alunite-type structure is amenable to intergrowth with other structure types. In the perhamite-group minerals, perhamite and krasnoite (Mills et al., Reference Mills, Mumme, Grey and Bordet2006, Reference Mills, Sejkora, Kampf, Grey, Bastow, Ball, Adams, Raudsepp and Cooper2012) pairs of alunite-type layers are intergrown with zeolitic groupings of cyclic Si₃O₉ linked into 4-member rings by corner sharing with AlO₄ tetrahedra. The latter corner-connect to the apical anions of the PO₄ tetrahedra of the alunite layers as shown in Fig. 11. The minerals are trigonal, P $\bar{3}$m1, with a ≈ 7.0 Å and c ≈ 20.2 Å. Perhamite has the composition Ca₃Al₈Si₃P₄O₂₄(OH)₁₄⋅8H₂O. The intergrowth can be written as 2[Al₃X₁₂P₂XX_½] + [Al₂X_2/2Si₃X₉], where X = O or OH. Ca²⁺ cations and H₂O are located within 8-sided [100] channels located between the two intergrowth components as shown in Mills et al. (Reference Mills, Mumme, Grey and Bordet2006). The alunite-type layers have the composition of plumbogummite-group minerals and the calculated octahedral tilting is 18.5° (Table 2). The mineral iangreyite, Ca₂Al₇(PO₄)₂(PO₃OH)₂(OH,F)₁₅⋅8H₂O with space group P321, a ≈ 7.00 Å and c ≈ 16.7 Å, has a closely related intergrowth structure, in which the zeolitic Al₂X_2/2Si₃X₉ groups are replaced by AlX₅ trigonal bipyramids that share their apices with the PO₄ tetrahedra of the alunite-type layers (Mills et al., Reference Mills, Kampf, Sejkora, Adams, Birch and Plásil2011).

Fig. 11. Intergrowth of crandallite-type layers and zeolitic clusters in perhamite.

Weberite minerals and related compounds

Weberite is the name originally given to a fluoride mineral found in southern Greenland, with composition Na₂MgAlF₇ (Bøgvard, Reference Bøgvard1938). Its crystal structure was determined approximately by Byström (Reference Byström1944) and refined by Giuseppetti and Tadini (Reference Giuseppetti and Tadini1978) in space group Imma, with a = 7.060 Å, b = 10.000 Å and c = 7.303 Å. Like pyrochlore and zirconolite, weberite can be described as an ordered anion-deficient fluorite structure with general formula A₂B₂X₇ in which alternate (111)_fluorite planes of composition AB₃ and A₃B form a fcc array (Cai and Nino, Reference Cai and Nino2009; Grey et al., Reference Grey, Mumme, Ness, Roth and Smith2003). The A₃ and B₃ arrays are both kagomé nets, parallel to (011) of the aristotype weberite cell with octahedral coordination of the small B cations to form a HTB network and 8-coordination of the large A cations. The weberite and pyrochlore structures differ in the location of the capping B cation in the A₃B layers. In pyrochlore the capping cation connects to eclipsed triangular arrays of Xa anions in the HTB layers on either side of the capping cation, whereas in weberite the capping cation coordinates to only two Xa anions on either side and completes its six-fold coordination with two unshared anions. As a result, the weberite small-cation framework has the composition B₂X₇ whereas the pyrochlore small-cation framework composition is B₂X₆.

The octahedra in the HTB layers of weberite undergo a different tilting scheme to that in pyrochlore. The octahedral tilting in weberite corresponds to the Cmcm space group type tilting reported by Campbell et al. (Reference Campbell, Howard, Averett, Whittle, Schmid, Yost and Stokes2018) and shown in Fig. 2c. In contrast to P6₃/mmc tilting, where the triangular groupings of Xa anions all move in towards a central point or all move out, the Cmcm tilting results in pairs of Xe anions in one triangle moving inwards and the third anion moving out, with the reverse occurring in the adjacent triangle, thus producing pairs of compressed and elongated isosceles triangles. The pair of Xa anions that moves in as a result of tilting provides the base for connection of the capping cation in weberite. The HTB layer and its capping octahedra are shown in Fig. 12.

Fig. 12. Capping of HTB layer in weberite (polytype 3T).

There are three different ways that a second HTB layer can join to two Xa anions of the capping octahedra in Fig. 13. One of the connections results in the unshared anions being in a trans configuration, shown in Fig. 13a while the other two connections have the unshared anions in a cis configuration, shown in Fig. 13b. The different sequences of capping configurations lead to several different weberite polytypes, distinguished by symmetry and the number of HTB layers. The aristotype mineral is weberite 2O (2-layer orthorhombic) and it has the unshared anions of the capping octahedra in trans configuration. Other weberite polytypes include 2M, 4M, 5M, 6M, 7M, 3T, 6T and 8O (Cai and Nino, Reference Cai and Nino2009). In all polytypes except the arisotype 2O, the unit cells have a hexagonal (a ≈ 7.3 Å) or orthohexagonal base and a repeat normal to the base of n × 6 Å, where 6 Å is the interlayer separation for the (001) HTB layers.

Fig. 13. Connectivity of capping octahedron to adjacent HTB layers in weberite. (a) trans-configuration; (b) cis-configuration.

The weberite 2M polytpe with space group C2/c and cell parameters a ≈ 12.3 Å, b ≈ 7.3 Å, c ≈ 12.8 Å and β ≈ 109.3° has the capping octahedra in cis configuration whereas the 4M polytype with doubled c axis has an alternation of cis and trans configurations. Similarly the 3T polytype, with space group P3₁21, a ≈ 7.3 Å and c ≈ 18 Å has each of the three capping octahedra in cis configuration while the 6T polytype with space group P3₁ and doubled c axis has an alternation of cis and trans octahedra.

Doped Ca₂Ta₂O₇ compounds have been structurally characterised as 5M, 6M and 7M polytypes (Grey et al., Reference Grey, Roth, Mumme, Bendersky and Minor1999, Reference Grey, Roth, Mumme, Planes, Bendersky, Li and Chenavas2001, Grey and Roth, Reference Grey and Roth2000). These compounds differ from other weberite polytypes in having periodic pseudo-unit-cell twinning of the cation layers (“pseudo” because B is reflected into A across the twin plane). The twin planes are on B₃A layers and correspond to elements of hexagonal stacking of the layers. For example, the 6M polytype has a stacking sequence hccccchcccch. In the h-stacked layers the kagomé net is corrugated, with cations displaced along [001] by ± 0.15 Å. The 5-layer cubic-stacked blocks correspond to weberite type. The 5M polytype exists in two structural forms, which have the same unit cell and space group, C2, and the same layer stacking sequence of hccchccccch, but Nb-doped 5M Ca₂Ta₂O₇ has weberite blocks between the h-stacked layers and Sm-doped 5M Ca₂Ta₂O₇ has weberite in the 5-layer-wide fcc block and pyrochlore in the 3-layer-wide block. The interface capping layer and adjacent HTB layers in this structure is shown in Fig. 14. The octahedral tilting changes from P6₃/mmc-type in the pyrochlore blocks to Cmcm-type in the weberite blocks, so that the capping octahedra shown in Fig. 14 share 3 anions with octahedra in the pyrochlore CaTa₃ layer on one side and only two anions with octahedra in the weberite CaTa₃ layer on the opposite side.

Fig. 14. Interface layer of capping octahedra between pyrochlore-type and weberite-type layers in (Ca,Sm)₂Ta₂O₇-5M.