Nets as representations of sheets of tetrahedra

Planar 3-connected nets may be used as compact representations of the connectivity of silicate sheets that have the stoichiometry [T_2nO_5n] where n = 1–24; an example is shown in Fig. 1. In the sheet of corner-linked tetrahedra (Fig. 1a), all tetrahedra link to three other tetrahedra, i.e. they are 3-connected. In Fig. 1b, the vertices of the net represent the tetrahedra and the edges of the net represent the linkage between the tetrahedra. All tetrahedra in Fig. 1a have their apical vertices pointing in the same direction (up in terms of the viewer), and the tetrahedra of the six-membered rings are designated as being in the u⁶ arrangement (Hawthorne, Reference Hawthorne2015a); note that such an arrangement is not inherent in Fig. 1b unless we specifically colour the vertices to indicate u and d behaviour of the analogous tetrahedra. Also emphasised in Fig. 1b is the three-connected nature of the net vertices and the unit cell of the net (which contains two vertices and five edges: [Si₂O₅]). Common 3-connected nets are listed in Table 2 and illustrated in Fig. 2. Nearly all of these nets correspond to known structures of silicate minerals. There are an infinite number of other 3-connected plane nets, but these need not be considered until structural analogues are discovered or suspected. There are certain geometrical variations in single-layer sheets of tetrahedra that do not change the linkage of the corresponding net, and hence stoichiometry is conserved by these variations (Hawthorne, Reference Hawthorne2015a). However, these variations play an important role in the formation of more complicated sheets, and also are key features in linkage between the sheet and the interstitial complex.

Fig. 1. (a) The mica sheet of tetrahedra; (b) the 6³ net with its unit cell shown by dotted black lines; and (c) the mica sheet showing that the tetrahedra all point in the same direction, and that the O(br) anions (shown as yellow circles) are planar. Net vertices: red circles; net edges: green lines. Modified from Hawthorne (Reference Hawthorne2015a).

Fig. 2. The simpler 3-connected plane nets: (a) the 6³ net; (b) the 4.8² net; (c) the 3.12² net; (d) the 4.6.12 net; (e) the (4.6.8)₂(6.8²)₁ net; (f) the (5².8)₁(5.8²)₁ net; (g) the (4.6.10)₄(6².10)₁ net; (h) the (3.8²)₁(6.8²)₁ net; (i) (5².8)₁(5.6²)₁(5.6.8)₂(6².8)₁ net; and (j) the (5.6.7)₄(5.7²)₁(6².7)₁ net; the unit cell of each net is shown by dotted lines.

Table 2. Simple 3-connected plane nets.

Planar and folded sheets of tetrahedra

In Figs 1 and 3, all O(bridging) [ = O(br)] anions are shown as yellow circles; note that in Fig. 1, the O(br)] anions lie in the plane of the net. Figure 3a shows the sheet in sanbornite with tetrahedra at the vertices of a 6³ net; the view from one direction shows that the O(br) anions are very non-planar whereas the view from the orthogonal direction shows that the O(br) anions are quasi-planar. The arrangement of tetrahedra in cuprorivaite (Fig. 3b) shows a sheet with tetrahedra at the vertices of a 4.8² net. The view of the sheet from both horizontal directions shows that the O(br) anions are very non-planar in each direction, and the sheet in Fig. 3b is repetitively folded about (fold) axes parallel to both viewing directions orthogonal to the sheet. Here, we will not consider linkage between sheets and extra-sheet species, but note that geometrically-planar sheets tend to link to units involving edge-sharing octahedra coordinating medium-sized di- and trivalent cations whereas folded sheets tend to link to polymerisations of more highly coordinated cation polyhedra.

Fig. 3. Folded sheets of tetrahedra; (a) sanbornite: the 6³ silicate sheet and views in the plane of the sheet, showing folding in one direction; and (b) cuprorivaite: 4.8² silicate sheet and views in the plane of the sheet, showing folding in two directions. Yellow circles represent O(br) anions.

The relative orientation of tetrahedra in sheets of tetrahedra

In the net of Fig. 1b, each vertex represents a tetrahedron. However, a vertex has no orientation relative to the plane of the net, which is not the case for a tetrahedron. The apical ([1]-coordinated) anion of a 3-connected tetrahedron may lie on one side of the sheet or the other. Thus tetrahedra in a sheet may all point in the same direction or the tetrahedra may point in different directions. In Fig. 1, the tetrahedra all point in one direction (see Fig. 1c) which we designate as u (up towards the reader). Figures 4a,b show tetrahedra at the vertices of the 6³ net, and both Figs 4a and 4b show that the tetrahedra point in both directions, u and d (down), relative to the plane of the sheet. There are two distinct six-membered rings in Fig. 4a; in one ring, all tetrahedra point in the same direction: (u⁶); in the other ring, tetrahedra point in different directions: four point up (in the same direction as the tetrahedra in the first ring) and two point down (in the opposite direction to the tetrahedra of the first ring), and the sequence around the ring gives the symbol (u²du²d) (Fig. 4a). Thus the attitude of the tetrahedra in a sheet may be represented by these (u–d) strings. For a single-layer sheet, the designation of a specific tetrahedron as u (or d) is arbitrary. In single-layer sheets, we adopt the convention whereby the direction of the larger number of tetrahedra is defined as u. For a double-layer sheet, tetrahedra of the upper parent-layer sheet that point away from the plane containing the oikodoméic operation are defined as u.

Fig. 4. The occurrence of u–d tetrahedra in the structure of gyrolite; (a) shows the tetrahedra at the vertices of a 6³ net with u and d tetrahedra indicated; (b) shows a cross-sectional view of the sheet, showing the tetrahedra pointing both ways and the planar nature of the O(br) anions (shown as yellow circles); and (c) the net of vertices in which red circles represent u tetrahedra and the yellow circles represent d tetrahedra; the unit cell is shown by heavy broken lines and the unit cell of the parent 63 net (cf. Fig. 2a) is shown by dotted red lines.

The apical anions of the u and d tetrahedra of Fig. 4 lie out of the plane of the T–O–T linkages. However, this is not necessarily the case. Figure 5a shows the sheet of tetrahedra in ferronordite-(Ce). As is apparent in the cross-section of a narrow slice of the sheet (Fig. 5b), the presence of tetrahedra with their edges in the upper and lower surfaces of the sheet allows the u and d tetrahedra not to project above or below the sheet itself, and produces a new type of tetrahedron which we will denote as o. Although this type of arrangement is more common in sheets involving 4-connected tetrahedra, as in ferronordite-(Ce), it does occur in sheets with only 3-connected tetrahedra (as in the minerals of the gadolinite supergroup (Bačík, Reference Bačík, Miyawaki, Atencio, Cámara and Fridrichová2017), see below).

Fig. 5. The sheet of 3-connected (orange) and 4-connected (violet) tetrahedra in ferronordite-(Ce): (a) plan view of the sheet; and (b) view of a thin ribbon (between the dashed lines of Fig. 5a) of the sheet in the plane of the sheet.

We also need to define the directions u and d relative to the rest of the structure. As noted below, silicate sheets may have more than one layer of tetrahedra (Liebau, Reference Liebau1985; Hawthorne, Reference Hawthorne2015a). In double-layer silicate structures, we will define d tetrahedra of the upper layer as linking to tetrahedra in the lower layer, and hence u tetrahedra link to the rest of the structure. In single-layer silicates, we do not have this internal definition of direction. In this case, we generally define the majority of tetrahedra as u and the minority as d, and hence the designation of u and d in these cases is more arbitrary.

In order to represent the information of the u and d directions of tetrahedra, it is necessary to use a slightly more complicated net nomenclature. The nets in Fig. 2 show the unit cells in dotted lines. Consider the net 6³ (Fig. 2a) and its associated silicate sheet (Fig. 1a); these have a unit cell that contains two vertices/tetrahedra. Consider the silicate sheet in Fig. 4a; the topology of this sheet is based on the 6³ net but the unit cell has to be larger in order to represent the u and d nature of the tetrahedra; this is equivalent to colouring the vertices of the net different colours according to the u or d nature of the tetrahedron corresponding to that vertex. It is obvious from Fig. 4a that the unit cell of the 6³ net in Fig. 2 is not adequate to do this. The net corresponding to the sheet in Fig. 4a is shown in Fig. 4c with the u tetrahedra shown as red vertices and the d tetrahedra shown as yellow vertices. The corresponding unit-cell in Fig. 4c is shown as heavy broken lines, and part of the unit cell of the parent 6³ net is shown as dotted red lines. It is apparent that the true unit-cell is four times the size of the parent unit-cell. We wish to retain the number of constituent vertices in the net symbol, and hence this number will need to be contained in the net. We may do this by enclosing the reduced net symbol. i.e. the set of vertices with any common factor removed from the stoichiometric coefficients and placed outside a pair of square brackets. Thus the nets in Figs 2a–d are expressed as [6³]₂, [4.8²]₄, [3.12²]₆ and [4.6.12]₁₂. For the net in Fig. 2e, there are six vertices in the unit cell and there are two distinct vertices, (4.6.8) and (6.8²) in the ratio 2:1; hence the net symbol is [(4.6.8)₂(6.8²)₁]₂ such that the product of the sum of the stoichiometric coefficients within the square brackets and the subscript outside the square brackets is equal to the number of vertices in the unit cell of the net: (2 + 1) x 2 = 6 (Fig. 2e). We note that this approach is only a notation for the recording the u and d directions of tetrahedra. A more rigorous method of describing u and d tetrahedra based on orientational matrices was developed by Krivovichev and Burns (Reference Krivovichev and Burns2003) and amplified by Krivovichev (Reference Krivovichev2009). The former has the advantage of simplicity whereas the latter has potential for combining with structure-generating functions (Hawthorne, Reference Hawthorne2015a) to rigorously derive all possible sheet arrangements.

Multi-layer tetrahedron-sheets and oikodoméic operations

Liebau (Reference Liebau1985) divided sheet silicates into two types: single-layer sheets and double-layer sheets. In Fig. 6a, tetrahedra lie at the vertices of a 6³ net, and all tetrahedra have their apical vertices concealed below the plane of the figure. However, viewing perpendicular to the sheet (Figs. 6b,c) shows that there is another single-layer of tetrahedra directly underlying the upper net, and the lower tetrahedra have their apical vertices pointing upward. The sheet of Fig. 6 is a double-layer sheet with a mirror (or pseudo-mirror) plane relating the upper and lower layers of the sheet.

Fig. 6. A double-layer silicate sheet; (a) d tetrahedra at the vertices of a 6³ net; (b) and (c) views of the sheet parallel to the plane of the sheet, showing that the double sheet has a lower-layer component in which u tetrahedra occur at the vertices of a 6³ net; the upper and lower payers are related by a mirror (or pseudo-mirror) plane shown by the red line and labelled m.

Hawthorne (Reference Hawthorne2015a) introduced a series of topological operations that change the bond topology of a parent net. Stoichiometry is not conserved but changes systematically in accord with the particular operation, giving rise to more complicated nets that can represent more complicated sheet structures. These operations are designated as oikodoméic operations as they involve the act of building new structural arrangements (from the Greek word oikodomé: the act of building). There are three classes of oikodoméic operations that can affect nets or sheets of connected tetrahedra: [1] insertion, whereby vertices of different connectedness are inserted into the edges of a parent net; [2,3] replication operations, whereby a single-layer sheet is replicated, reoriented and linked to the original single-layer sheet to produce a double-layer sheet of tetrahedra. Class-2 oikodoméic operations replicate the parent layer about apical anions of d tetrahedra (Figs 7a,b), whereas class-3 operations replicate the parent layer about the central cations of d tetrahedra (Figs 7c,d). How do these oikodoméic operations differ from symmetry operators? A symmetry element is part of the symmetry of an already existing arrangement, and the corresponding symmetry operation describes the transformation of part of the arrangement to geometrical congruence with another part of the arrangement, whereas oikodoméic operations generate arrangements with the corresponding (topological) symmetry from a simpler parent arrangement of tetrahedra.

Fig. 7. Oikodoméic operations replicating and reorienting the upper single-layer tetrahedra from above the plane of the operation to below the plane of the operation; (a) the mirror operation acting though apical anions of the upper single-layer parent sheet; (b) the two-fold rotation operation acting though apical anions of the upper single-layer parent sheet; (c) the mirror operation though the central T cations of tetrahedra shared between the upper and lower single-layer sheets; and (d) the two-fold rotation operation acting though the central T cations of tetrahedra shared between the upper and lower single-layer sheets.

Recently, three silicate minerals have been described as triple-layer: günterblassite, (K,Ca)_3–xFe[(Si,Al)₁₃O₂₅(OH,O)₄](H₂O)₇ (Chukanov et al., Reference Chukanov, Rastsvetaeva, Aksenov, Pekov, Zubkova, Britvin, Belakovskiy, Schüller and Ternes2012a); umbrianite, K₇Na₂Ca₂[Al₃Si₁₀O₂₉]F₂Cl₂ (Sharygin et al., Reference Sharygin, Pekov, Zubkova, Khomyakov, Stoppa and Pushcharovsky2013); and hillesheimite, (K,Ca,□)₂(Mg,Fe,Ca,□)₂[(Si,Al)₁₃O₂₃(OH)₆](OH)(H₂O)₈ (Chukanov et al., Reference Chukanov, Rastsvetaeva, Aksenov, Pekov, Zubkova, Britvin, Belakovskiy, Schüller and Ternes2013). These also may be generated from parent sheets by oikodoméic replication operations.

Interstitial constituents

The idea of Binary Structure Representation (e.g. Hawthorne and Schindler, Reference Hawthorne and Schindler2008) considers structures as being partitioned into two parts, a strongly bonded (usually anionic) Structural Unit and a (usually cationic) Interstitial Complex that binds the structural units into a continuous structure. The development of structure hierarchies focuses on the structural units, but the interstitial complex is also of great interest as the Principle of Correspondence of Lewis-acidity – Lewis-basicity (Hawthorne, Reference Hawthorne2012a, Reference Hawthorne2015b) allows analysis of the factors that control the chemical compositions and aspects of the structural arrangements of both the structural unit and the interstitial complex (e.g. Hawthorne and Schindler, Reference Hawthorne and Schindler2008, Schindler and Hawthorne, Reference Schindler and Hawthorne2001a,Reference Schindler and Hawthorneb,Reference Schindler and Hawthornec, Reference Schindler and Hawthorne2004, Reference Schindler and Hawthorne2008; Schindler et al., Reference Schindler, Hawthorne and Baur2000, Reference Schindler, Huminicki and Hawthorne2006). As we plan to do this as part of our future work on sheet-silicate minerals, we shall describe the stereochemistry of the interstitial complex, i.e. the coordination of the cation constituents and (H₂O), in particular the role of (H₂O) as Transformer (H₂O)^t, Non-Transformer (H₂O)ⁿ, Inverse-Transformer (H₂O)ⁱ and solely hydrogen-bonded (H₂O)^z groups (Hawthorne, Reference Hawthorne1992; Hawthorne and Schindler, Reference Hawthorne and Schindler2008; Hawthorne and Sokolova, Reference Hawthorne and Sokolova2012) unless the details are obscured by positional disorder.

Single 2-connected vertices

Figure 8 shows three inserted plane nets derived from the 6³ net. In Fig. 8a, 2-connected vertices have been inserted into all edges of the 6³ net (a class-1 oikodoméic operation), maintaining the original translational symmetry. The original six-membered rings have the u–d sequence (ududud) and become twelve-membered rings on insertion of the 2-connected tetrahedra; the resulting net is (12²)₃(12³)₂. Figure 8b shows the net of the silicate sheet in zeophyllite (Table 3) and Fig. 8c shows the corresponding silicate sheet in zeophyllite. The geometrical symmetry of the archetype 6³ net is preserved and the unit cell is the same in both nets (cf. Figs 1b and 8b). The interstitial complex consists of three distinct Ca²⁺ ions with coordination numbers [6] (= six O^2–), [8] (= four O^2– and four F^–) and [8] (= four O^2–, three F^– and one (H₂O)^t group). The O(1) anion is bonded to Si(1) and accepts three hydrogen bonds from the (H₂O)^t group, and one F^– ion accepts a hydrogen bond from the (H₂O) group.

Fig. 8. Nets and corresponding structures derived from the 3-connected plane net 6³ by insertion of 2-connected vertices between 3-connected vertices; (a) the (12²)₃(12³)₂ net; (b) the [(12²)₃(12³)₂]₁ net in the structure of zeophyllite; (c) the sheet of tetrahedra in zeophyllite; (d) the (10²)₄(10³)₄ net; (e) the [(10²)₄(10³)₄]₁ net in the structure of tumchaite; (f) the sheet of tetrahedra in tumchaite; (g) the (8²)₄(8³)₈ net; (h) the [(8²)₄(8³)₈]₁ net in the structure of kvanefjeldite; and (i) the sheet of tetrahedra in kvanefjeldite. Yellow circles: two-connected vertices.

Table 3. Single-layer sheet-silicates based on 3-connected nets with inserted 2-connected vertices.

References: (1) Merlino (Reference Merlino1972); (2) Chukanov et al. (Reference Chukanov, Yakubovich, Pekov, Belakovsky and Massa2008), Yakubovich et al. (Reference Yakubovich, Massa and Chukanov2008); (3) Kolitsch et al. (Reference Kolitsch, Merlino and Holstam2012); (4) Subbotin et al. (Reference Subbotin, Merlino, Pushcharovsky, Pakhomovsky, Ferro, Bodanova, Vloshin, Sorokhtina and Zubkova2000); (5) Johnsen et al. (Reference Johnsen, Leonardsen, Fälth and Annehed1983), Petersen et al. (Reference Petersen, Johnsen, Leonardsen and Rønsbo1984); and (6) Grew et al. (Reference Grew, Peacor, Rouse, Yates, Su and Marquez1996).

*The u–d arrangement refers to the parent 6³ net.

**P = planar; 1F = folded in one direction.

The sheets in britvinite and molybdophyllite (Table 3) are also based on the (12²)₃(12³)₂ net, but the u–d sequence in the parent 6³ sheet is (u⁶) for both minerals. The interstitial complex in britvinite consists of twenty distinct Pb²⁺ ions with coordination numbers from [6] to [10] and coordinating anions O^2–, (OH)^–, F^– and Cl^–. The interstitial complex in molybdophyllite consists of four distinct Pb²⁺ ions with coordination numbers from [6] to [9] and six distinct Mg²⁺ ions each of which is coordinated by six O^2– ions.

In Fig. 8d, 2-connected vertices have been inserted into two-thirds of the edges of the 6³ net, again maintaining the original translational symmetry. The original six-membered rings become ten-membered rings and the resulting net is (10²)₄(10³)₄. Figure 8e shows the net of the silicate sheet in tumchaite (Table 3) and Fig. 8f shows the corresponding silicate sheet. The net corresponding to the sheet of tetrahedra is strongly geometrically distorted (Fig. 8e), but is [(10²)₄(10³)₄]₁ and topologically identical to the ideal inserted net in Fig. 8d. This geometrical distortion causes a doubling in the size of the unit cell (Fig. 8e) relative to that of the parent net (Fig. 1b). The interstitial complex in tumchaite consists of one distinct Zr⁴⁺ ion with coordination number [6] and one distinct Na⁺ ion which is coordinated by five O^2– ions and two (H₂O) groups.

In Fig. 8g, 2-connected vertices are inserted into one-third of the edges of the 6³ net, maintaining the original translational symmetry. As noted by Hawthorne (Reference Hawthorne2015a), the resulting arrangement is very different from those of the nets in Figs 8a and 8d. The six-membered ring in Fig. 8g does not have trans symmetry and hence the unit cell must span more than one single six-membered ring. As a result, the unit cell must be larger than that of the parent net, and a further doubling is caused by the geometrical distortion of the sheet. Figure 8h shows the net of the silicate sheet in kvanefjeldite (Table 3) and Fig. 8i shows the corresponding silicate sheet. The interstitial complex in kvanefjeldite consists of one distinct Ca²⁺ ion with coordination number [6] and two distinct Na⁺ ions both of which are coordinated by seven O^2– ions.

Pairs of 2-connected vertices

In Fig. 9, pairs of 2-connected vertices have been inserted into one-third of the edges of the 6³ net, maintaining the original translational symmetry. The resultant net is (14²)₆(14³)₂, and in the corresponding sheet of tetrahedra in hyttsjöite (Fig. 9a; Table 3), the 2-connected tetrahedra are coloured yellow. The sheet is quite corrugated and also shows considerable geometrical distortion from a geometrically holosymmetric 6³ net. The structure of hyttsjöite also contains a discontinuous layer of tetrahedra (Fig. 9b) which is linked into a continuous layer by pairs of short bonds involving lone-pair-stereoactive Pb²⁺, and this arrangement accounts for the very high Si:O ratio: 1:3.00 (Table 3). The interstitial complex in hyttsjöite is quite complicated. It consists of three distinct Pb²⁺ ions coordinated solely by O^2– with coordination numbers [8], [8] and [6], respectively, one Ba²⁺ coordinated by twelve O^2– ions, three distinct Ca²⁺ ions with coordination numbers [6] (= six O^2–) and [9] (= eight O^2– ions and one (H₂O) group, and nine O^2– ions, respectively), one Fe³⁺ and one Mn²⁺ each coordinated by six O^2– ions.

Fig. 9. Net and corresponding structure derived from the 3-connected plane net 6³ by insertion of pairs of 2-connected vertices between all 3-connected vertices; (a) the (14²)₆(14³)₂ net and the sheet of tetrahedra in hyttsjöite; and (b) the partly disconnected layer of tetrahedra in hyttsjöite, showing short Pb²⁺–O bonds that link it into a sheet. Yellow circles: two-connected vertices; yellow tetrahedra: two-connected tetrahedra; large red circles: lone-pair-stereoactive Pb²⁺; and blue tetrahedra: one-connected tetrahedra.

The 6³ net

The most common single-layer sheet-silicate minerals are based on this net (Fig. 1b; Table 4); note that all tetrahedra are in the u configuration (hence the nets have the (u⁶) arrangement) and are planar (i.e. not folded). Table 4 also lists the single-layer sheet-silicate minerals based on the 6³ net which have tetrahedra of their six-membered rings in arrangements other than (u⁶). These minerals are dominated by planar (P) sheets of tetrahedra. We will not discuss the common silicate-mineral groups here as their crystal structures and crystal chemistry have been dealt with in detail elsewhere (e.g. Brigatti and Guggenheim, Reference Brigatti, Guggenheim, Mottana, Sassi, Thompson and Guggenheim2002; Brigatti et al., Reference Brigatti, Guggenheim and Poppi2003). The structures of the smectites are not well-known; some are T–O–T structures and others are double-layer structures that we will deal with later in the paper. Hanjiangite has a large interlayer component and is probably related to the surite–ferrisurite series (Hayase et al., Reference Hayase, Dristas, Tsutsumi, Otsuka, Tanabe, Sudo and Nishiyama1978; Uehara et al., Reference Uehara, Yamazaki and Tsutsumi1997; Kampf et al., Reference Kampf, Jackson, Sidder, Foord and Adams1992) and niksergievite (Saburov et al., Reference Saburov, Britvin, Bekenova, Sergieva, Kotelnikov, Chukanov and Yagovkina2005), with complex polytypism (Merlino, Reference Merlino2014) possibly giving rise to several structural variants.

Table 4. Single-layer sheet-silicates based on the 6³ net with mixed u–d arrangements.

References: (1) Bailey (Reference Bailey1988); (2) Brigatti and Poppi (Reference Brigatti and Poppi1993), Brigatti et al. (Reference Brigatti, Guggenheim and Poppi2003), Brigatti and Guggenheim (Reference Brigatti, Guggenheim, Mottana, Sassi, Thompson and Guggenheim2002); (3) Liu et al. (Reference Liu, Li, Mao, Wu, Liu, Su, Xiong and Yu2012), Merlino (Reference Merlino2014); (4) Merlino (Reference Merlino1988a); (5) McDonald and Chao (Reference McDonald and Chao2007); (6) Giester et al. (Reference Giester, Lengauer, Pristacz, Rieck, Topa and von Bezing2016); (7) Yakovenchuk et al. (Reference Yakovenchuk, Ivanyuk, Pakhomovsky, Selivanova and Mikhailova2011); (8) Pant (Reference Pant1968); (9) Garvie et al. (Reference Garvie, Devouard, Groy, Cámara and Buseck1999), Vortmann et al. (Reference Vortmann, Rius, Marler and Gies1999); (10) Hesse and Liebau (Reference Hesse and Liebau1980); (11) Annehed et al. (Reference Annehed, Fälth and Lincoln1982); (12) Evans (Reference Evans1973); (13) Grice (Reference Grice1991); (14) Kampf et al. (Reference Kampf, Rossman and Housley2009); (15) Artioli and Galli (Reference Artioli and Galli1994), Chiari et al. (Reference Chiari, Giustetto and Ricchiardi2003), Giustetto and Chiari (Reference Giustetto and Chiari2004), Post and Heaney (Reference Post and Heaney2008); (16) Cámara et al. (Reference Cámara, Garvie, Devouard, Groy and Buseck2002); (17) Chukanov et al. (Reference Chukanov, Britvin, Blass, Belakovskiy and Van2012b); (18) Hawthorne et al. (Reference Hawthorne, Abdu, Tait and Back2013); (19) Pluth et al. (Reference Pluth, Smith, Pushcharovsky, Semenov, Bram, Riekel, Weber and Broach1997); (20) Ferraris et al. (Reference Ferraris, Khomyakov, Belluso and Soboleva1998); (21) Post et al. (Reference Post, Bish and Heaney2007); (22) Fahey et al. (Reference Fahey, Ross and Axelrod1960), Biedl and Preisinger (Reference Biedl and Preisinger1962); and (23) Capitani and Mellini (Reference Capitani and Mellini2006), Dódony et al. (Reference Dódony, Pósfai and Buseck2002).

*P = planar; 1F = folded in one direction; M = octahedrally coordinated cations, T = tetrahedrally coordinated cations, A = interstitial cations.

Let us examine the sheet in gyrolite (Fig. 10a). Some of the six-membered rings are reasonably close to showing 6-fold rotational symmetry, whereas others are strongly distorted from this arrangement. If we examine the configurations of the apical vertices of the tetrahedra in the different rings (Fig. 10a), we see that the less-distorted rings have a (u⁶) arrangement of their apical vertices, whereas the more-distorted rings have a (u²du²d) arrangement where some apical vertices lie above the plane of the sheet and others lie below the plane of the sheet. Despite vertices pointing in different directions, the T–O–T linkages within the sheet are planar (or nearly so).

Fig. 10. The 6³ sheets of tetrahedra in (a) gyrolite and (b) silinaite. In gyrolite, the (u⁶) ring links only to d tetrahedra, and hence ribbons of like-pointing tetrahedra cannot form; in silinaite, (u³d³) rings link such that chains of tetrahedra form in one direction, allowing modulation of the sheet.

Compare the sheet in gyrolite (Fig. 10a) with the sheet in silinaite (Fig. 10b). In silinaite, there is only one type of six-membered ring: (u³d³). Inspection of the cross-sections of gyrolite (Fig. 10a) and silinaite (Fig. 10b) shows that the bridging anions within the sheets behave somewhat differently; although they are topologically equivalent, there are significant geometrical differences. In gyrolite, the bridging anions are planar (or their deviations from planarity are not spatially modulated) whereas in silinaite, the bridging anions are topologically planar but their deviations from planarity are spatially modulated in one direction (Fig. 10b). In silinaite, the u tetrahedra are arranged in ribbons, interspersed with ribbons of d tetrahedra, and the parallel arrangement of these ribbons allows the sheet to modulate in a direction orthogonal to these ribbons. In gyrolite, the (u⁶) ring links only to d tetrahedra (Fig. 10a) and hence ribbons of u tetrahedra do not occur (as the definition of the direction of u is arbitrary, similarly ribbons of d tetrahedra are not present). Thus in gyrolite, the lack of ribbons of u (and d) tetrahedra inhibits modulation. Hence we expect two types of single-sheet arrangements: (1) planar arrangements, and (2) modulated (or potentially modulated) arrangements.

Planar sheets of u–d tetrahedra

Gyrolite (Fig. 11a), ellingsenite (Fig. 11b), martinite (Fig. 11c) and natrosilite (Fig. 11d) contain planar sheets. Gyrolite, ellingsenite and martinite contain topologically identical sheets: there are two types of six-membered rings, a (u⁶) ring and a (u²du²d) ring, and the (u⁶) ring is completely surrounded by (u²du²d) rings, whereas each (u²du²d) ring is surrounded by two (u⁶) rings and four (u²du²d) rings, and the planar unit-cell contains one (u⁶) ring and three (u²du²d) rings. In martinite, one third of the d tetrahedra are occupied by B³⁺. The interstitial complex in gyrolite consists of eight distinct Ca²⁺ ions and one distinct Na⁺ ion. There are three [7]-coordinated Ca²⁺ ions, each of which is bonded to 5 O^2– ions and two (OH)^– groups, and five [6]-coordinated Ca²⁺ ions, with coordinations ${\rm O}_5^{{\rm 2\ndash}} $OH)^– (×3), ${\rm O}_3^{{\rm 2\ndash}} $(OH$)_3^\ndash $ (×1) and ${\rm O}_2^{{\rm 2\ndash}} $(H₂O$)_4^{\rm t} $ (×1). There is also one [6]-coordinated Na⁺ ion bonded to 6 (H₂O)^t groups. The interstitial complex in ellingsenite consists of two distinct Ca²⁺ ions, two distinct Na⁺ ions, and one site that is occupied by 50% Ca²⁺ and 50% Na⁺. The Ca²⁺ ions are [6]-coordinated, one by six O^2– ions and the other by five O^2– ions and one (OH)^– group. Both Na⁺ ions are [8]-coordinated, one by six O^2– ions and two (OH)^– groups, and the other by five O^2– ions, two (OH)^– groups and one (H₂O)^t group. The site containing Ca_0.50 + Na_0.50 is [6]-coordinated by five O^2– ions and one (OH)^– group. The formula given for ellingsenite is not compatible with the refined structure; this issue is discussed in the Appendix and a revised formula is suggested. The interstitial complex in martinite consists of two sites occupied by Ca²⁺ ions and seven sites occupied by Na⁺ ions. There are two Ca²⁺ ions bonded to ${\rm O}_6^{{\rm 2\ndash}} $ F and O₆^2– ions, respectively, and seven distinct Na⁺ ions, four of which are [8]-coordinated by ${\rm O}_6^{{\rm 2\ndash}} {\rm F}_2^\ndash $, ${\rm O}_5^{{\rm 2\ndash}} $(OH$)_2^\ndash $(H₂O)ⁿ, ${\rm O}_6^{{\rm 2\ndash}} $(H₂O$)_2^{\rm n} $ and ${\rm O}_6^\ndash $ F^–(H₂O)ⁿ, one of which is [7]-coordinated by ${\rm O}_6^{{\rm 2\ndash}} $ F^–, and two of which are positionally disordered such that both sites cannot be locally occupied, with coordinations ${\rm O}_5^{{\rm 2\ndash}} $(OH)^–(H₂O)ⁿ and ${\rm O}_3^{{\rm 2\ndash}} $(OH)^–(H₂O)ⁿ, respectively.

Fig. 11. Planar 6³ sheets of u–d tetrahedra in (a) gyrolite, (b) ellingsenite, (c) martinite and cairncrossite, and (d) natrosilite.

In natrosilite, there is only one type of six-membered ring, a (ududud) ring, and thus each (ududud) ring is surrounded by other (ududud) rings. It should be noted that neither (ududud) rings nor combinations of (u⁶) and (u²du²d) rings can produce modulated sheets. The interstitial complex in natrosilite consists of two sites occupied by Na⁺ ions with coordinations ${\rm O}_5^{{\rm 2\ndash}} $ and ${\rm O}_6^{{\rm 2\ndash}} $, respectively.

Folded sheets of u–d tetrahedra

Sanbornite and kanemite (Fig. 12a) and makatite (Fig. 12b) contain folded sheets (cf. Fig. 3) of six-membered rings of tetrahedra. The sheets differ in their arrangements of u–d tetrahedra; in sanbornite and kanemite, there are two types of rings, (ud²ud²) and (u²d¹u²d¹), whereas in makatite there is only one type of ring, (u⁴d²). As is apparent from Fig. 12, the sheets are folded strongly: in one direction, cf. Fig. 3, leading to a very non-planar arrangement of O(br) anions; in the other direction, adjacent tetrahedra point u and d and the O(br) anions are only slightly non-planar. In sanbornite and kanemite, the folding is in phase in that the topological repeat in the sheet along the fold axis is one pair of tetrahedra, whereas in makatite, the topological repeat along the fold axis is two pairs of tetrahedra, giving a different appearance to the sheet in the direction of the fold axes (Fig. 12). In kanemite, there is one interstitial Na⁺ ion coordinated by (H₂O$)_6^{\rm n} $. In sanbornite, there is one interstitial Ba²⁺ ion coordinated by ${\rm O}_9^{{\rm 2\ndash}} $. In makatite, there are three interstitial Na⁺ ions coordinated by (H₂O$)_6^{\rm n} $ (×2) and ${\rm O}_3^{{\rm 2\ndash}} $(H₂O$)_2^{\rm n} $, respectively.

Fig. 12. Folded 6³ sheets of u–d tetrahedra in (a) sanbornite and kanemite and (b) makatite; cross-sections of each sheet show the folding in one direction, and the u–d symbols indicate the different u–d arrangements in each type of sheet.

Modulated sheets of u–d tetrahedra

Above, we saw that in order for sheets to be modulated in a particular direction, the sheets must contain parallel ribbons of u tetrahedra and ribbons of d tetrahedra (Fig. 10b). A convenient way of classifying such modulated sheets is by the width (i.e. the number of tetrahedra) across the ribbon. The minimum width of these ribbons is two tetrahedra, and those structures thus formed are shown in Fig. 13. In pentagonite (Fig. 13a), there are two types of six-membered rings, (u²d⁴) and (u⁴d²), that share two adjacent similarly pointing tetrahedra to form ribbons of similarly pointing tetrahedra. Inspection of Fig. 13a shows no perceptible sign of modulation of the sheet, but the possibility of modulation exists in terms of the linkage of tetrahedra. In pentagonite, there are two interstitial cations, one V⁴⁺ coordinated by ${\rm O}_5^{{\rm 2\ndash}} $ and one Ca²⁺ coordinated by ${\rm O}_6^{{\rm 2\ndash}} $(H₂O)^t, plus an (H₂O)^z group. In silinaite, there is only one type of six-membered ring, (u³d³), that shares vertices with adjacent rings such that like-pointing tetrahedra form fairly extended ribbons (Fig. 13b), and there is a pronounced modulation with a wavelength of four tetrahedra orthogonal to these ribbons. There are two interstitial cations, Li⁺ coordinated by ${\rm O}_4^{{\rm 2\ndash}} $ and Na⁺ coordinated by ${\rm O}_2^{{\rm 2\ndash}} $(H₂O$)_4^{\rm n} $.

Fig. 13. Modulated 6³ sheets (ribbon width = 2u2d tetrahedra) of u–d tetrahedra in (a) pentagonite, (b) silinaite and (c) plumbophyllite.

In plumbophyllite, there are two types of six-membered rings, (u⁴d²) and (u²d⁴), that share vertices with adjacent rings such that like-pointing tetrahedra form fairly extended ribbons (Fig. 13c), and there is a pronounced modulation with a wavelength of four tetrahedra orthogonal to these ribbons; however, note that the sheet is more folded than is the case in silinaite (Fig. 13b), and hence the modulation tends to appear blurred where viewed parallel to the ribbons (Fig. 13c). There is one interstitial lone-pair stereoactive Pb²⁺ cation coordinated by ${\rm O}_6^{{\rm 2\ndash}} $(H₂O)^t where the site containing (H₂O) is only half-occupied.

Sheets with a ribbon width of four tetrahedra are shown in Fig. 14. In palygorskite (Fig. 14a) and raite (Fig. 14b), there are two types of six-membered rings, (u⁶) and (u³d³), that share three adjacent similarly pointing tetrahedra to form ribbons of similarly pointing tetrahedra four tetrahedra wide. There is only a very slight modulation of this sheet in palygorskite and a more prominent modulation in raite; the magnitude of the modulation is more a function of the strip of octahedra to which the sheet is attached than a characteristic of the sheet itself (Guggenheim and Eggleton, Reference Guggenheim and Eggleton1998). In palygorskite, the interstitial cations are all [6]-coordinated; there are two sites each half-occupied by Mg²⁺ and Al³⁺ (Giustetto and Chiari, Reference Giustetto and Chiari2004), plus interstitial (H₂O) groups. In raite, there are two interstitial Mn²⁺ cations coordinated by ${\rm O}_4^{{\rm 2\ndash}} $(OH)₂, two Na⁺ cations coordinated by ${\rm O}_6^{{\rm 2\ndash}} $ and (H₂O)₆, respectively, and one quarter-occupied site containing Ti⁴⁺ coordinated by ${\rm O}_2^{{\rm 2\ndash}} $(H₂O)₄.

In kalifersite (Fig. 15a), there are two types of six-membered rings, (u⁶) and (d⁶), in the ratio 2:1. The (u⁶) rings link to form an upward-pointing ribbon six-tetrahedra wide, and these ribbons are linked laterally by a ribbon of (d⁶) rings that form a downward-pointing ribbon four-tetrahedra wide, forming a modulation in which the widths of the ribbons with like-pointing tetrahedra are different. The interstitial cations are five [6]-coordinated Fe²⁺ ions with coordinations ${\rm O}_6^{{\rm 2\ndash}} $ (×2) and ${\rm O}_4^{{\rm 2\ndash}} $(OH)₂ (×3), and three K⁺ ions (+ minor Na⁺) with coordinations ${\rm O}_2^{{\rm 2\ndash}} $(H₂O)₄ (×2) and (H₂O)₆. In sepiolite (Fig. 15b), there are two types of six-membered rings, (u⁶) and (d⁶), in the ratio 1:1. The (u⁶) rings link to form an upward-pointing ribbon six-tetrahedra wide, and the (d⁶) rings link to form a downward-pointing ribbon six-tetrahedra wide. Interstitial cations are four distinct Mg²⁺ ions with coordinations ${\rm O}_4^{{\rm 2\ndash}} $(OH$)_2^\ndash $ (×3) and ${\rm O}_4^{{\rm 2\ndash}} $(H₂O$)_2^{\rm t $ plus four interstitial (H₂O)^z sites that may show partial occupancy and positional disorder. In antigorite (Fig. 15c), there are four types of six-membered rings, (u⁶), (u⁴d²), (u²d⁴) and (d⁶), in the ratio 2:1:1:2. The (u⁶) and (u⁴d²) rings link to form an upward-pointing ribbon seven-tetrahedra wide, the (u²d4²) and (d⁶) rings link to form a downward-pointing ribbon seven-tetrahedra wide, and antigorite shows the most prominent modulation of these modulated sheet-silicates.

Fig. 14. Modulated 6³ sheets (ribbon width = 4u4d tetrahedra) of u–d tetrahedra in (a) palygorskite and (b) raite.

Fig. 15. Modulated 6³ sheets of u–d tetrahedra in (a) kalifersite (ribbon width = 6u4d), (b) sepiolite (ribbon width = 6u6d) and (c) antigorite (ribbon width = 7u7d).

Inspection of Figs 13–15 indicates that (u⁶) rings are generally much less distorted away from planar hexagonal symmetry than six-membered rings containing both u and d tetrahedra. It is obvious that commensurate modulation and folding are connected to linkage requirements between the sheets and the non-tetrahedrally coordinated parts of the structures, but the diversity of the latter, involving variations in stoichiometry, coordination number and ligancy of non-tetrahedrally coordinated cations, differing amounts of H and the ensuing hydrogen bond networks, are beyond the scope of the present work.

The 4.8² net

Details of the 4.8² net are shown in Fig. 2b. There are both four-membered and eight-membered rings but only one type of vertex: one four-membered ring and two eight-membered rings meet at each vertex, and the unit cell contains [Si₄O₁₀]. Table 5 lists the single-layer sheet-silicate minerals based on this net.

Table 5. Single-layer sheet-silicates based on the 4.8², 4.6.12, (4.6.8)₂(6.8²)₁, (5².8)₂(5.8²)₁, (4.6.12), (4.6.10)₄(6².10)₁, (5.6.7)₄(5.7²)₁(6².7)₁ and (4².14)₁₂(4.6.14)₈(6.14²)₄ nets with mixed u–d arrangements.

References: (1) Foit et al. (Reference Foit, Phillips and Gibbs1973), Rinaldi et al. (Reference Rinaldi, Gatta and Angel2010); (2) Segalstad and Larsen (Reference Segalstad and Larsen1978), Demartin et al. (Reference Demartin, Pilati, Diella, Gentile and Gramaccioli1993); (3) Cámara et al. (Reference Cámara, Oberti, Ottolini, Della Ventura and Bellatreccia2008); (4) Ximen and Peng (Reference Ximen and Peng1985), Miyawaki et al. (Reference Miyawaki, Matsubara, Yokoyama and Okamoto2007); (5) Demartin et al. (Reference Demartin, Minaglia and Gramaccioli2001), Miyawaki et al. (Reference Miyawaki, Matsubara, Yokoyama and Okamoto2007); (6) Yakubovich et al. (Reference Yakubovich, Matvienko, Voloshin and Simonov1983); (7) Rastsvetaeva et al. (Reference Rastsvetaeva, Pushcharovskii, Pekov and Voloshin1996); (8) Miyawaki et al. (Reference Miyawaki, Nakai and Nagashima1985); (9) Foord et al. (Reference Foord, Gaines, Crock, Simmons and Barbosa1986); Cooper and Hawthorne (Reference Cooper and Hawthorne2017); (10) Bartl and Pfeifer (Reference Bartl and Pfeifer1976), Ståhl (Reference Ståhl1993); (11) Dunn et al. (Reference Dunn, Rouse, Norberg and Peacor1978); (12) Matsueda et al. (Reference Matsueda, Miura, Rucklidge and Kato1981); (13) Evans (Reference Evans1973); (14) Zubkova et al. (Reference Zubkova, Filinchuk, Pekov, Pushcharovsky and Gobechiya2010); (15) Zubkova et al. (Reference Zubkova, Filinchuk, Pekov, Pushcharovsky and Gobechiya2010); (16) Zubkova et al. (Reference Zubkova, Pekov, Pushcharovsky and Chukanov2009); (17) Chakoumakos et al. (Reference Chakoumakos, Fernandez-Baca and Boatner1993), Bensch and Schur (Reference Bensch and Schur1995); (18) Lin et al. (Reference Lin, Liao and Wang1992), Giester and Rieck (Reference Giester and Rieck1994), Knight et al. (Reference Knight, Henderson and Clark2010); (19) Pabst (Reference Pabst1943), Hazen and Finger (Reference Hazen and Finger1983); (20) Giester and Rieck (Reference Giester and Rieck1996); (21) Uvarova et al. (Reference Uvarova, Sokolova, Hawthorne, Agakhanov and Pautov2004a); (22) Szymański et al. (Reference Szymański, Owens, Roberts, Ansell and Chao1982); (23) Livingstone et al. (Reference Livingstone, Atkin and Hutchison1976); (24) Perrault and Szymański (Reference Perrault and Szymański1982); (25) Kabalov et al. (Reference Kabalov, Sokolova, Pautov and Schneider1998); (26) Yang et al. (Reference Yang, Downs, Yang and Allen2011); (27) Takeuchi et al. (Reference Takéuchi, Kawada, Irimaziri and Sadanaga1969), Kato and Takéuchi (Reference Kato and Takéuchi1983); (28) Kato and Watanabe (Reference Kato and Watanabe1992); (29) Ozawa et al. (Reference Ozawa, Takéuchi, Takahata, Donnay and Donnay1983); (30) Ozawa et al. (Reference Ozawa, Takéuchi, Takahata, Donnay and Donnay1983); (31) Dunn and Peacor (Reference Dunn and Peacor1984); (32) Kasahev and Sapozhnikov (Reference Kasahev and Sapozhnikov1978); (33) Fleet (Reference Fleet1965); (34) Gebert et al. (Reference Gebert, Medenbach and Flörke1983), Lazebnik et al. (Reference Lazebnik, Lazebnik and Makhotko1984); (35) Es'kova et al. (Reference Es'kova, Semenov, Khomyakov, Kazakova and Shumyatskaya1974), Shumyatsaya et al. (Reference Shumyatsaya, Voronkov and Pyatenko1980); (36) Cámara et al. (Reference Cámara, Ottolini, Devouard, Garvie and Hawthorne2006); (37) Alberti and Galli (Reference Alberti and Galli1983); (38) Merlino (Reference Merlino1983); (39) Uvarova et al. (Reference Uvarova, Sokolova, Hawthorne, Pautov and Agakhanov2004b); (40) Grice and Gault (Reference Grice and Gault1995); (41) Heinrich et al. (Reference Heinrich, Eggleton and Guggenheim1994); (42) Yamnova et al. (Reference Yamnova, Egorov-Tismenko and Khomyakov1996); and (43) Krivovichev et al. (Reference Krivovichev, Pakhomovsky, Ivanyuk, Mikhailova, Men'shikov, Armbruster, Selivanova and Meisser2007).

*P = planar; 1F = folded in one direction.

The minerals of the gadolinite supergroup (Table 5) show the simplest type of 4.8² net (Fig. 16a). The B³⁺ tetrahedra point both up and down in datolite, but do not project above or below the plane of the sheet. Instead, they link to Si⁴⁺ tetrahedra which have edges in the top and bottom surfaces of the sheet (Fig. 16b) and hence occur entirely within the body of the sheet. In general, the ordering of cations in this structure type is very strong, with Si⁴⁺ and (Be²⁺, B³⁺ and Al³⁺) occupying discrete tetrahedra (Fig. 16). Bačík et al. (Reference Bačik, Fridrichová, Uher, Pršek and Ondrejka2014) wrote the general formula of the minerals of this group as W₂XZ₂T₂O₈V₂ where W = Ca²⁺, REE ³⁺ (Y³⁺ + lanthanoids), Bi³⁺; X = Fe²⁺, □ (vacancy), Mg²⁺, Mn²⁺, Zn²⁺, Cu²⁺, Al³⁺, Fe³⁺; Z = B³⁺, Be²⁺, Li⁺; T = Si⁴⁺, B³⁺, Be²⁺, S⁶⁺, P⁵⁺; V = O^2–, OH^–, F^–, and divided the minerals into two subgroups on the basis of the Z-site occupancy: the datolite subgroup where Z = B³⁺, and the gadolinite subgroup where Z = Be²⁺. The interstitial cations occupy the W, X and Z sites. In the datolite-subgroup minerals, the W site is [8]-coordinated and is occupied by Ca²⁺ coordinated by ${\rm O}_6^{{\rm 2\ndash}} $(OH$)_2^\ndash $; the X site is vacant in datolite, and is occupied by Fe²⁺ in homilite (Table 5) where it is [6]-coordinated by ${\rm O}_4^{{\rm 2\ndash}} $(OH$)_2^\ndash $. In the gadolinite-subgroup minerals, the W site is [8]-coordinated and is occupied by REE ³⁺ coordinated by ${\rm O}_6^{{\rm 2\ndash}} $(OH$)_2^\ndash $; the X site is vacant in hingganite minerals, and is occupied by Fe²⁺ in gadolinite minerals (Table 5) where it is [6]-coordinated by ${\rm O}_4^{{\rm 2\ndash}} $(OH$)_2^\ndash $. Minasgeraisite-(Y) is more complicated as it has triclinic P1 (rather than monoclinic P2₁/a) symmetry; there is prominent order of Ca²⁺, Bi³⁺ and REE ³⁺ over four symmetrically distinct W sites and the two X sites are vacant (occupied by minor Mn²⁺). Moreover, preliminary results suggest that (at least) some hingganite minerals have similar symmetry and cation order as minasgeraisite-(Y) (Cooper and Hawthorne, Reference Cooper and Hawthorne2017).

Fig. 16. The 4.8² net and sheet in datolite; (a) the geometrically distorted 4.8² net; and (b) the sheet of tetrahedra.

The apophyllite-group minerals (Fig. 17a) and cavansite (Fig. 17b) have planar sheets. Both minerals have u and d tetrahedra in both four-membered and eight-membered rings. However, the patterns of u and d tetrahedra are different in both types of ring. In the apophyllite structure, there are (u⁴) and (d⁴) four-membered rings that combine to give (u²d²u²d²) eight-membered rings (Fig. 17a), whereas in cavansite, there is only one type of four-membered ring, (u²d²), that combines to give (u⁴d⁴) eight-membered rings (Fig. 17b). The interstitial complex in the apophyllite structure consists of one distinct Ca²⁺ ion that is [7]-coordinated by ${\rm O}_4^{{\rm 2\ndash}} $(F^–,OH^–)(H₂O)₂, and one K⁺ ion that is [8]-coordinated by (H₂O)₈. The interstitial complex in cavansite consists of one distinct Ca²⁺ ion that is [8]-coordinated by ${\rm O}_4^{{\rm 2\ndash}} $(H₂O)₄, and one V⁴⁺ ion that is [5]-coordinated by ${\rm O}_4^{{\rm 2\ndash}} $(OH)^–.

Fig. 17. Kinked planar 4.8² sheets of u–d tetrahedra in (a) apophyllite and (b) cavansite.

The 4.8² sheets in cryptophyllite, mountainite and shlykovite are shown in Fig. 18, and give us some insight into the coupling of u and d tetrahedra in the different rings. In all three structures, the four-membered rings have the configuration (u³d¹). However, in cryptophyllite (Fig. 18a) and shlykovite (Fig. 18b), the eight-membered ring has the configuration (u⁴d¹u²d¹) whereas in mountainite (Fig. 18c), the eight-membered ring has the configuration (u³d¹u³d¹). Inspection of Fig. 18 shows how this difference arises. Each four-membered ring links to four other four-membered rings. In cryptophyllite and shlykovite, all adjacent four-membered rings have the same orientation, whereas in mountainite this is not the case. In mountainite (Fig. 18c), two adjacent four-membered rings have the same orientation and the other two four-membered rings have a different orientation (rotated by ~180°); the result is a different combination of u and d tetrahedra in the eight-membered rings formed by linkage of the four-membered rings. It is apparent that these differences in u–d arrangements of tetrahedra play a major role in the diversity of linkage to the non-tetrahedrally coordinated constituents of these structures, and this will be examined more rigorously in a later paper. The interstitial complex in cryptophyllite consists of one Ca²⁺ ion that is [6]-coordinated by O₅(H₂O), and two K⁺ ions [8]-coordinated by O₆(H₂O)₂ and O₃(H₂O)₅, respectively. The interstitial complex in shlykovite consists of one Ca²⁺ ion that is [6]-coordinated by ${\rm O}_5^{{\rm 2\ndash}} $(H₂O), and one K⁺ ion that is [8]-coordinated by ${\rm O}_6^{{\rm 2\ndash}} $(H₂O)₂. The interstitial complex in mountainite consists of one Ca²⁺ ion that is [6]-coordinated by O₅(H₂O), one K⁺ ion [8]-coordinated by O₆(H₂O)₂, and one Na⁺ ion coordinated by ${\rm O}_2^{{\rm 2\ndash}} $(OH)^–(H₂O)₅.

Fig. 18. Planar 4.8² sheets of u–d tetrahedra in (a) cryptophyllite, (b) shlykovite and (c) mountainite.

Cuprorivaite (Fig. 19a) and ekanite (Fig. 19b) have doubly folded sheets with u and d tetrahedra in both four-membered and eight-membered rings. Unlike apophyllite (Fig. 17a) and cavansite (Fig. 17b), cuprorivaite and ekanite have the same pattern of u and d tetrahedra in their four-membered and eight-membered rings: (u⁴), (d⁴) and (u²d²u²d²). Thus the sheets in apophyllite and cavansite (Fig. 17), although based on the same type of 3-connected net, are significantly different, whereas the sheets in cuprorivaite and ekanite (Fig. 19) are topologically identical. There are four minerals with the cuprorivaite structure in the so-called ‘gillespite group’ (Table 5). The interstitial complex in these minerals consists of one distinct M²⁺ ion (M²⁺ = Ca²⁺, Ba²⁺ or Sr²⁺) that is [8]-coordinated by ${\rm O}_8^{{\rm 2\ndash}} $, and one Cu²⁺ (or Fe²⁺) ion that is [4]-coordinated by ${\rm O}_4^{{\rm 2\ndash}} $. There are five minerals in the ekanite group (Table 5). The interstitial complex in ekanite consists of one distinct Ca²⁺ ion that is [8]-coordinated by ${\rm O}_8^{{\rm 2\ndash}} $, and one Th⁴⁺ ion that is [8]-coordinated by ${\rm O}_8^{{\rm 2\ndash}} $.

Fig. 19. Folded 4.8² sheets of u–d tetrahedra in (a) cuprorivaite and (b) ekanite.

The 3.12² net

As far as we are aware, there are no sheet-silicate minerals based on this 3-connected net, despite its simplicity. Three-membered rings are unusual in silicates but do occur, e.g. the benitoite-group minerals (Hawthorne, Reference Hawthorne1987). Possibly this is connected with the association of large and small rings in a sheet (Fig. 2c), although there are structures based on the 4.6.12 net. Perhaps the occurrence of an intermediate-sized ring relieves strain in the sheet.

The 4.6.12 net

This net contains three four-membered rings, two six-membered rings and one twelve-membered ring in the unit cell (Fig. 2d) in the ratio 3:2:1, and there is only one type of vertex: (4.6.12) with 12 vertices per unit cell. There are six structures with sheets based on this net: pyrosmalite-(Fe), pyrosmalite-(Mn), schallerite, friedelite, mcgillite and nelenite (Table 5). The sheets are topologically identical in these minerals, but link to sheets of octahedra in different ways to form polytypes (Takéuchi et al., Reference Takéuchi, Ozawa and Takahata1983). In the sheets of schallerite and nelenite, an oxygen anion bridging two (SiO₄) tetrahedra also links to an As³⁺ cation, thereby violating the valence-sum rule, and hence there must be significant disorder involving vacancies, Si⁴⁺ and As³⁺ that is not resolved in the current refined structure. The four-membered rings in the As³⁺-free minerals have the sequence (u²d²), there are two six-membered rings with the sequences (u⁶) and (d⁶), and one twelve-membered ring with the sequence (u²d²u²d²u²d²) (Fig. 20). Note that there is significant modulation of this sheet. The interstitial complex in the pyrosmalite-group minerals consists of four distinct M²⁺ ions (M²⁺ = Fe²⁺ or Mn²⁺) that are [6]-coordinated by ${\rm Cl}_6^\ndash $, ${\rm O}_4^{{\rm 2\ndash}} {\rm Cl}_2^\ndash $, ${\rm O}_6^{{\rm 2\ndash}} $ and ${\rm O}_5^{{\rm 2\ndash}} $Cl^–, respectively.

Fig. 20. The 4.6.12 sheet in the structures of pyrosmalite-(Fe), pyrosmalite-(Mn) and schallerite. Note the slight modulation in the sheet.

The (4.6.8)₂(6.8²)₁ net

Details of the (4.6.8)₂(6.8²)₁ net are shown in Fig. 2e. This net contains four-, six- and eight-membered rings and there are two types of vertex: (4.6.8) and (6.8²) in the ratio 2:1; the unit cell contains [Si₆O₁₅]. Table 5 lists the single-layer sheet-silicate minerals based on this net, and their structures are shown in Fig. 21. Despite the differing appearance of these sheets, they do show the linkage of this net, but differ in their patterns of u and d tetrahedron vertices (Table 5). In armstrongite (Fig. 21a), the four-membered and six-membered rings have the sequences (u²d²) and (ud³ud), and they link such that the eight-membered rings have the sequence (u⁴d⁴). In dalyite (Fig. 21b), all rings show the same u–d sequence as in armstrongite but the sheet is strongly sheared relative to that of armstrongite. The interstitial complex in armstrongite consists of one distinct Ca²⁺ ion that is [7]-coordinated by ${\rm O}_5^{{\rm 2\ndash}} $(H₂O)₂, and one Zr⁴⁺ that is coordinated by ${\rm O}_6^{{\rm 2\ndash}} $; there are two distinct interstitial transformer (H₂O) groups. The interstitial complex in dalyite consists of one Na⁺ ion that is [8]-coordinated by ${\rm O}_8^{{\rm 2\ndash}} $, and one Zr⁴⁺ ion that is [6]-coordinated by ${\rm O}_6^{{\rm 2\ndash}} $.

Fig. 21. Folded 4.6.8 sheets in the structures of (a) armstrongite, (b) dalyite and davanite, and (c) sazhinite-(Ce) and sazhinite-(La).

In sazhinite-(Ce) and sazhinite-(La) (Fig. 21c), four-membered rings have the sequences (u⁴) and (d⁴), six-membered rings have the sequences (u²du²d) and (ud²ud²), and they link such that the eight-membered rings have the sequence (u²dud²ud). These structures show an interesting contrast with the structures of the 4.8² net (Figs 4b, 10; Table 5); they are each folded, as is apparent in Fig. 21, but unlike the 4.8² structure of Fig. 5b, they are folded only in one direction. The interstitial complex in sazhinite-(Ce) and sazhinite-(La) consists of one distinct M³⁺ ion that is [7]-coordinated by ${\rm O}_7^{{\rm 2\ndash}} $, and two Na⁺ ions, the coordination of which is somewhat ambiguous: there is one Na⁺ ion with a coordination of either ${\rm O}_5^{{\rm 2\ndash}} $ or ${\rm O}_5^{{\rm 2\ndash}} $(H₂O), and one Na⁺ ion with a coordination of either ${\rm O}_3^{{\rm 2\ndash}} $(H₂O)₂ or ${\rm O}_3^{{\rm 2\ndash}} $(H₂O)₅; one (H₂O) group is either a hydrogen-bonded group or an inverse transformer group, depending on the coordinations chosen for the Na⁺ cations, and the other (H₂O) group is a hydrogen-bonded group.

The (5².8)₂(5.8²)₁ net

This net contains 5- and 8-membered rings (Fig. 2f) and there are two types of vertex: (5².8) and (5.8²) in the ratio 2:1; the unit cell contains [Si₆O₁₅], and this net topology occurs in the structures of nekoite, okenite and zeravshanite (Table 5). Nekoite (Fig. 22a), has two distinct five-membered rings with u–d arrangements (u³d²) and (u²dud) and one eight-membered ring with the arrangement (u⁶d²), and the sheet is planar. Okenite (Fig. 22b) has one distinct five-membered ring with the u–d arrangement (u⁴d) and one eight-membered ring with the arrangement (u⁷d), and the sheet is also planar. Thus the sheets in nekoite and okenite are geometrical isomers, they have the same topology (connectivity of chemical bonds) but the geometrical details are distinct in that the u–d arrangements of tetrahedra are different in each mineral. In okenite, the sheets alternate with layers of [Si₆O₁₆] chains (Fig. 22c) which has the stoichiometric effect of decreasing the connectivity of the silicate part of the structure (Table 5).

Fig. 22. Planar (5².8)₂(5.8²)₁ sheets in the structures of (a) nekoite; (b) okenite; (c) chains of tetrahedra in okenite; and (d) zeravshanite; note the one-dimensional folding of the silicate sheet in zeravshanite. Legend as in Fig. 1.

Zeravshanite (Fig. 22d) is significantly different. It has two distinct five-membered rings with u–d arrangements (u³d²) and (u²d³) and one eight-membered ring with the arrangement (u⁴d⁴). Moreover, unlike nekoite and okenite, the sheet in zeravshanite is folded in one direction. The interstitial complex in nekoite consists of three distinct Ca²⁺ ions that are all [6]-coordinated by ${\rm O}_4^{{\rm 2\ndash}} $(H₂O)₂; there are seven distinct (H₂O) groups, four are transformer, one is non-transformer and two are hydrogen bonded only. The interstitial complex in okenite consists of six distinct sites occupied by Ca²⁺ ions; four of these sites are fully occupied and are [6]-coordinated by ${\rm O}_5^{{\rm 2\ndash}} $(H₂O) (×3) and ${\rm O}_6^{{\rm 2\ndash}} $, and two are half-occupied and coordinated by ${\rm O}_2^{{\rm 2\ndash}} $(H₂O)₅ and (H₂O)₆, respectively. There are thirteen distinct (H₂O) sites, not all of which are fully occupied; four (H₂O) groups are transformer, seven are non-transformer and two are hydrogen-bonded only. The interstitial complex in zeravshanite consists of two Cs⁺ ions coordinated by ${\rm O}_9^{{\rm 2\ndash}} $ and ${\rm O}_8^{{\rm 2\ndash}} $, respectively, one Na⁺ ion coordinated by ${\rm O}_6^{{\rm 2\ndash}} $(H₂O)₂, and two Zr⁴⁺ ions each of which are coordinated by ${\rm O}_6^{{\rm 2\ndash}} $. There is one interstitial transformer (H₂O) group.

Miscellaneous nets

There are several nets (Table 5) that just have just one example each of a sheet-silicate unit.

The (4.6.10)₄(6².10)₁ net has four-membered, six-membered and ten-membered rings in the ratio 2:2:1 (Fig. 23a). In the structure of varennesite (Table 5), all four-membered rings have the u–d arrangement (u²d²), the six-membered rings have the arrangements (u⁶) and (d⁶), and the two types of ten-membered ring have the arrangements (u³d²u³d²) and (d³u²d³u²). The sheet is planar but with a strong modulation in both directions (Fig. 23a). The interstitial complex in varennesite consists of three Na⁺ ions coordinated by ${\rm O}_4^{{\rm 2\ndash}} $(OH)^–(H₂O), ${\rm O}_3^{{\rm 2\ndash}} $(H₂O)₃, and ${\rm O}_2^{{\rm 2\ndash}} $(H₂O)₄, and one Mn²⁺ ion coordinated by ${\rm O}_4^{{\rm 2\ndash}} $(OH$)_2^\ndash $; there are six interstitial (H₂O) groups, five of which are non-transformer and one of which is hydrogen-bonded only.

Fig. 23. Miscellaneous 3-connected plane nets and their corresponding structures; (a) the (4.6.10)₄(6².10)₁ net and sheet in varennesite; (b) the (5.6.7)₄(5.7²)₁(6².7)₁ net and sheet in bementite; and (c) the (5².8)₁(5.6²)₁(5.6.8)₂(6².8)₁ net and sheet in intersilite.

The (5.6.7)₄(5.7²)₁(6².7)₁ net has five-membered, six-membered and seven-membered rings in the ratio 1:1:1 (Fig. 23b). In the structure of bementite (Table 5), there are two types of five-membered rings with u–d arrangements (u³d²) and (u²d³), two types of six-membered rings with the arrangements (u⁶) and (d⁶), and the two types of seven-membered rings with the arrangements (u⁵d) and (u²d⁵). The sheet is planar but with a modulation in one direction (Fig. 23b). The interstitial complex in bementite consists of fifteen distinct Mn²⁺ ions all of which are [6]-coordinated by ${\rm O}_2^{{\rm 2\ndash}} $(OH$)_4^\ndash $ (×6), ${\rm O}_3^{{\rm 2\ndash}} $(OH$)_3^\ndash $ (×7) and ${\rm O}_4^{{\rm 2\ndash}} $(OH$)_2^\ndash $ (×2).

The (5².8)₁(5.6²)₁(5.6.8)₂(6².8)₁ net has five-membered, six-membered and eight-membered rings in the ratio 2:2:1 (Fig. 23c). In the structure of intersilite (Table 5), there are two types of five-membered rings with u–d arrangements (u²dud) and (udud²), three types of six-membered rings with the arrangements (u⁶), (u³d³) and (d⁶), and two types of eight-membered rings with the arrangements (u⁴du²d) and (d⁴ud²u). The sheet is planar (Fig. 23c) with no significant modulation. The interstitial cations in intersilite consists of Mn²⁺, Ti⁴⁺ and Na⁺, but the interatomic distances and coordinations are not compatible with these cations, suggesting considerable positional disorder between the sheets.

The (4².14)₁₂(4.6.14)₈(6.14²)₄ net has four-membered, six-membered and fourteen-membered rings in the ratio 3:2:1 (Fig. 24a). In the structure of yakovenchukite-(Y) (Fig. 24b; Table 5), there is one type of four-membered ring with the arrangement (u²d²), two types of six-membered ring with the arrangements (u²du²d) and (ud²ud²), and one type of fourteen-membered ring with the arrangement (ud²ud²udu²du²d). The sheet is folded in one direction (Fig. 24b). The interstitial complex in yakovenchukite-(Y) consists of two K⁺ (+ some Na⁺) ions coordinated by ${\rm O}_7^{{\rm 2\ndash}} $(H₂O)₂ and ${\rm O}_7^{{\rm 2\ndash}} $(H₂O), two Ca²⁺ ions coordinated by ${\rm O}_6^{{\rm 2\ndash}} $(H₂O)₂ and ${\rm O}_5^{{\rm 2\ndash}} $(H₂O), and two Y³⁺ ions each of which is coordinated by ${\rm O}_6^{{\rm 2\ndash}} $; there are three interstitial (H₂O) groups, two of which are non-transformer and one of which is hydrogen-bonded only.

Fig. 24. Miscellaneous 3-connected plane nets and their corresponding structures; (a) the (4².14)₁₂(4.6.14)₈(6.14²)₄ net, and (b) the sheet in yakovenchukite-(Y).

The 6³ net

The most common double-layer sheet-silicate minerals are based on this net (Table 7). Hexacelsian (Fig. 36a), cymrite (Fig. 36a), dmisteinbergite (Fig. 36b) and burckhardtite (Fig. 36b) have a simple (d⁶) arrangement that is repeated by the oikodoméic operation m (class-2) to give a sheet of the form [TO₂]_n, and the sheet in burckhardtite shows slight geometrical distortion relative to the sheets in hexacelsian and cymrite. This double-layer sheet is not restricted to silicate minerals; for example, it occurs in minjiangite, Ba[Be₂(PO₄)₂] (Rao et al., Reference Rao, Hatert, Wang, Gu, Dal Bo and Dong2015). In kampfite (Fig. 36c), the six-membered rings are all quite distorted from geometrical hexagonal symmetry, a feature that is presumably imposed on the sheet by the interstitial linkage between the double-layer sheets. Note that the formula given for kampfite is not compatible with the crystal structure: the structure shows that all anions bonded to tetrahedrally coordinated Si⁴⁺ and Al³⁺ are each linked to two T cations and hence the formula must contain [T₁₆O₃₂], not [T₁₆O₃₁]; this issue is discussed in the Appendix and a revised formula is suggested.

Fig. 36. Double-layer sheets of tetrahedra based on the 3-connected plane net 6³ plus an oikodoméic operation: (a) the (d⁶) arrangement in hexacelsian and cymrite; (b) the (d⁶) arrangement in dmisteinbergite and burckhardtite; and (c) the (d⁶) arrangement in kampfite.

The interstitial complex in hexacelsian consists of one Ba²⁺ ion coordinated by ${\rm O}_{12}^{{\rm 2\ndash}} $ anions. The interstitial complex in cymrite consists of four Ba²⁺ ions [11]-coordinated by ${\rm O}_{10}^{{\rm 2\ndash}} $(H₂O) (×3) and ${\rm O}_{11}^{{\rm 2\ndash}} $ [see Gagné and Hawthorne (Reference Gagné and Hawthorne2016) for minimum and maximum bondlengths observed in alkaline-earth–O^2– bonds]. The interstitial complex in dmisteinbergite consists of one Ca²⁺ ion coordinated by ${\rm O}_6^{{\rm 2\ndash}} $ anions. The interstitial complex in burckhardtite consists of one Pb²⁺ ion coordinated by ${\rm O}_9^{{\rm 2\ndash}} $ anions and showing lone-pair-stereoactive behaviour, and one site occupied by Fe²⁺ and Fe³⁺ ions and coordinated by ${\rm O}_6^{{\rm 2\ndash}} $ anions. The interstitial complex in kampfite consists of three Ba²⁺ ions coordinated by ${\rm O}_{10}^{{\rm 2\ndash}} $Cl^–, ${\rm O}_{10}^{{\rm 2\ndash}} {\rm Cl}_4^\ndash $ and ${\rm O}_6^{{\rm 2\ndash}} {\rm Cl}_6^\ndash $, together with (CO₃) groups.

In fedorite and lalondeite (Fig. 37a), the upper 6³ sheet has the u–d arrangements (u⁶) and (u²du²d) in the ratio 1:3. In naujakasite (Fig. 37b, the 6³ nature of the double sheet is not immediately apparent (centre figure) as the class-2 oikodoméic operation does not correspond to rigorous reflection symmetry; however, when the upper sheet only is illustrated (top figure), the 6³ nature of the parent single sheet, with its (u²d⁴) arrangement, becomes apparent. The interstitial complexes in fedorite and lalondeite consist of four octahedrally coordinated sites containing different amounts of Ca²⁺ and Na⁺ with coordinations ${\rm O}_6^{{\rm 2\ndash}} $, ${\rm O}_5^{{\rm 2\ndash}} $F^– (×2) and ${\rm O}_4^{{\rm 2\ndash}} {\rm F}_2^\ndash $, and three sites with strong positional disorder occupied by Na⁺ and K⁺ and coordinated by partly disordered O^2– and (H₂O). The interstitial complexes in naujakasite and manganonaujakasite consist of one octahedrally coordinated Fe²⁺ (Mn²⁺) ion coordinated by ${\rm O}_6^{{\rm 2\ndash}} $, and by three Na⁺ ions the coordinations of which are very irregular, but are ${\rm O}_5^{{\rm 2\ndash}} $ or ${\rm O}_7^{{\rm 2\ndash}} $, ${\rm O}_4^{{\rm 2\ndash}} $ or ${\rm O}_5^{{\rm 2\ndash}} $ and ${\rm O}_5^{{\rm 2\ndash}} $ or ${\rm O}_8^{{\rm 2\ndash}} $, respectively.

Fig. 37. (a) The (u⁶) and (u²du²d) arrangement in fedorite and lalondeite; and (b) the (u²d⁴) and (ud²ud²) arrangement in naujakasite.

The 4.8² net

The double-layer sheet-silicate minerals based on this net are listed in Table 7. Macdonaldite, monteregianite-(Y) and rhodesite (Fig. 38a) and delhayelite and fivegite (Fig. 38b) have u–d arrangements (u³d)₁(u⁴du²d)₁ but differ in the occupancy of the vertices. In macdonaldite, monteregianite-(Y) and rhodesite, the sheets are silicate, whereas in delhayelite and fivegite, the sheet is an aluminosilicate and Al³⁺ occupies the tetrahedra that link the single sheets into a double sheet (i.e. all the d tetrahedra in the upper sheet). Note that there is an acid silicate-group in macdonaldite. Carletonite (Fig. 38c) has a more highly connected double-sheet with the u–d arrangement (u²d²)₄(u²d²u²d²)₂(udududud)₂ in a completely silicate sheet. All double-layer arrangements are generated from the parent single-layer sheet by the class-2 oikodoméic operation m. The interstitial complex in macdonaldite consists of one Ba²⁺ ion coordinated by ${\rm O}_4^{{\rm 2\ndash}} $(H₂O)₆, three Ca²⁺ ions coordinated by ${\rm O}_4^{{\rm 2\ndash}} $(H₂O)₂ (× 2) and ${\rm O}_4^{{\rm 2\ndash}} $(OH$)_2^\ndash $, and seven distinct (H₂O) groups, five of which are transformer groups and two of which are non-bonded groups. The interstitial complex in monteregianite-(Y) consists of two K⁺ ions coordinated by ${\rm O}_6^{{\rm 2\ndash}} $(H₂O)₄ (x 2), three Na⁺ ions coordinated by ${\rm O}_5^{{\rm 2\ndash}} $(H₂O)₂ ${\rm O}_4^{{\rm 2\ndash}} $(H₂O)₂ (x 2), and one Y³⁺ ion coordinated by ${\rm O}_6^{{\rm 2\ndash}} $; in addition, there are six distinct non-transformer (H₂O) groups. The interstitial complex in rhodesite consists of one K⁺ ion coordinated by ${\rm O}_6^{{\rm 2\ndash}} $(H₂O)₄, two Ca²⁺ ions coordinated by ${\rm O}_4^{{\rm 2\ndash}} $(H₂O)₂ and ${\rm O}_6^{{\rm 2\ndash}} $, respectively, with three distinct transformer (H₂O) groups. The interstitial complex in delhayelite consists of one Na⁺ ion coordinated by ${\rm O}_8^{{\rm 2\ndash}} $(H₂O)₂, one K⁺ ion coordinated by ${\rm O}_6^{{\rm 2\ndash}} $(H₂O)₅, and two Ca²⁺ ions coordinated by ${\rm O}_5^{{\rm 2\ndash}} $(H₂O) (x 2) and ${\rm O}_6^{{\rm 2\ndash}} $, respectively, with four distinct (H₂O) groups. The interstitial complex in fivegite consists of three K⁺ ions coordinated by ${\rm O}_7^{{\rm 2\ndash}} $Cl^– (x 2) and ${\rm O}_6^{{\rm 2\ndash}} $Cl^–₂, and two Ca²⁺ ions, one coordinated by ${\rm O}_5^{{\rm 2\ndash}} $(OH)^– and the other coordinated by ${\rm O}_3^{{\rm 2\ndash}} $(OH$)_2^\ndash $ (although one of the Ca²⁺–O^2– distances is unrealistically short, cf. Gagné and Hawthorne, Reference Gagné and Hawthorne2016). The interstitial complex in carletonite consists of three Na⁺ ions coordinated by ${\rm O}_4^{{\rm 2\ndash}} $F^–(H₂O), ${\rm O}_5^{{\rm 2\ndash}} $(H₂O) and ${\rm O}_8^{{\rm 2\ndash}} $, respectively, one K⁺ ion coordinated by ${\rm O}_9^{{\rm 2\ndash}} $, and one Ca²⁺ ion coordinated by ${\rm O}_7^{{\rm 2\ndash}} $F^–; there are one transformer and one non-transformer (H₂O) groups.

Fig. 38. Double-sheets of tetrahedra based on the 3-connected plane net 4.8² plus an oikodoméic operation: (a) the (u³d)₁(u⁴du²d)₁ arrangement in macdonaldite, monteregianite-(Y) and rhodesite; (b) the (u³d)₁(u⁴du²d)₁ arrangement in delhayelite and fivegite, note how all d tetrahedra are occupied by Al³⁺; and (c) the (u²d²)₄(u²d²u²d²)₂(udududud)₂ arrangement in carletonite.

The crystal structure of seidite-(Ce) was not determined by solution of the structure from single-crystal diffraction data; it is a proposed structure-model based on chemical composition, electron and X-ray powder-diffraction data, and proposed similarities with the structure of miserite (Scott, Reference Scott1976). The parent upper-layer sheet in seidite-(Ce) (Fig. 39a) is a 4.8² net with one type of four-membered ring with the u–d arrangement (udud) and one type of eight-membered ring with the arrangement (u²d²u²d²). The topological unit-cell contains four tetrahedra but the crystallographic unit-cell is larger as is apparent from the fact that all nearest-neighbour four-membered rings are not geometrically equivalent. A fragment of the upper-layer sheet is shown in Fig. 39b (left) with the u tetrahedra shaded in yellow and the d tetrahedra shaded in brown to better illustrate the class-2 oikodoméic operation generating the lower-layer sheet from the parent upper-layer sheet. There is a 2₁ screw-rotation operation that generates the corresponding fragment from the lower-layer sheet (Fig. 39b, centre), producing the double-layer-sheet fragment in Fig. 39b (right). The resulting double-layer sheet is shown in Fig. 39c and its cross-section is shown in Fig. 39d. The interstitial complex in seidite-(Ce) consists of two Na⁺ ions coordinated by ${\rm O}_4^{{\rm 2\ndash}} $(H₂O)₃ and ${\rm O}_3^{{\rm 2\ndash}} $(H₂O)₄, one Ce³⁺ ion coordinated by ${\rm O}_3^{{\rm 2\ndash}} $(H₂O)₃, and one Ti⁴⁺ ion coordinated by ${\rm O}_4^{{\rm 2\ndash}} $(OH$)_2^\ndash $; there are five non-transformer (H₂O) groups.

Fig. 39. Double-sheets of tetrahedra based on the 3-connected plane net 4.8² plus an oikodoméic operation in seidite-(Ce): (a) the upper-layer parent sheet; (b) sketch of the linkage of four-membered rings in the upper- and lower-layer sheets; and (c,d) the double-layer sheet in plan and cross-section.

Miscellaneous complex 3-connected nets

The simple nets are the basis of the more common groups of minerals, and also a single net can be the basis of several significantly different structures. This is not the case with the more complicated nets of Table 8. The parent single-layer of chiappinoite-(Y) is based on the 4.6.8 net (Fig. 40a); the four-membered ring has the arrangement (d⁴), the six-membered ring has the arrangement (u²d⁴), and the eight-membered ring has the arrangement (d²u²d²u²) (Fig. 40b). The sheet is folded (Fig. 40c) and a class-3 oikodoméic operation m gives rise to the double-layer sheet (Fig. 40d). The interstitial complex in chiappinoite-(Y) consists of one Y³⁺ ion coordinated by ${\rm O}_8^{{\rm 2\ndash}} $ and one Mn²⁺ ion coordinated by ${\rm O}_8^{{\rm 2\ndash}} $.

Fig. 40. Double-sheet of tetrahedra based on the 3-connected plane net 4.6.8 plus a class-3 oikodoméic m operation in chiappinoite-(Y): (a) the upper-layer parent net; (b) the upper-layer parent sheet in plan and (c) in cross-section; and (d) the double-layer sheet in plan showing the class-3 oikodoméic operation m.

The structure of ajoite has a parent (5.6²)₂(5.6.7)₄(5.7²)₂(6².7)₆ net (Fig. 41a) with 5-, 6- and 7-membered rings as the basis of its single-layer parent sheet. The upper-layer parent sheet has the following u–d structure: (u⁶)₃(u⁴d³)₂(u³d²)₂(u²du²d)₁ (Fig. 41b) and the oikodoméic operation generating the double-layer structure is a class-2 m operation (Fig. 41c). The interstitial complex in ajoite consists of one positionally disordered K⁺ ion coordinated by ${\rm O}_8^{{\rm 2\ndash}} $, and ten Cu²⁺ ions coordinated by ${\rm O}_2^{{\rm 2\ndash}} $(OH$)_4^\ndash $, O₃(OH)₃ (×3), O₄(OH)₂ (×4) and O₅(OH) (×2); there are two non-bonded interstitial (H₂O) groups.

Fig. 41. Double-layer sheets of tetrahedra based on miscellaneous complex 3-connected plane nets plus an oikodoméic operation: (a) the parent (5.6²)₂(5.6.7)₄(5.7²)₂(6².7)₆ net in ajoite; (b) the parent upper single-layer sheet in ajoite; (c) the double-layer sheet in ajoite; (d) the parent (5².7)₈(5.6.7)₈(6.7²)₄(5.6.8)₈(5.7.8)₈ net in armbrusterite; (e) the parent upper single-layer sheet in armbrusterite; and (f) the double-layer sheet in armbrusterite.

The (5.6.7)₄(5.6.8)₄(5.7.8)₄(5.7²)₄(6².7)₂ net (Fig. 41d) is the basis of the silicate sheet in armbrusterite (Fig. 41e) and has five-, six-, seven- and eight-membered rings. The u–d arrangements in the parent upper-layer sheet are as follows: (u⁵)₄(u⁴d)₄(u₆)₄(u⁵d²)₄(u⁴du²d)₂, and the oikodoméic operation generating the double-layer structure is a class-2 m operation (Fig. 41f). The interstitial complex in armbrusterite consists of two Na⁺ ions coordinated by ${\rm O}_6^{{\rm 2\ndash}} $ and ${\rm O}_5^{{\rm 2\ndash}} $(OH)^–, one Mn³⁺ ion coordinated by ${\rm O}_6^{{\rm 2\ndash}} $, five Mn²⁺ ions coordinated by ${\rm O}_4^{{\rm 2\ndash}} $(OH$)_2^\ndash $ (×5), and eight disordered K⁺ ions and disordered interstitial (H₂O) groups.

The (5.6²)₆(6³)₆(5.6.8)₂₄ net that is the basis of the silicate sheet in stilpnomelane (Fig. 42a) (and presumably its Mg- and Mn-analogues lennilenapeite and franklinphilite) has five-, six- and eight-membered rings. In stilpnomelane, the u–d arrangements are as follows: (u³d²), (u⁶), (d⁶) and (u²d²u²d²), and the oikodoméic operation is a class-2 m operation. The (5.6²)₈(5.7²)₄(5.6.7)₈(6².7)₁₂ net is the basis of the silicate sheet in bannisterite (Fig. 42b) and has five-, six- and seven-membered rings. In bannisterite, the u–d arrangements are as follows: (u³d²), (u⁶), (u²du²d), (u³d⁴) and (u⁵d²), and the oikodoméic operation is m. The (4.5.12)₂₄(5.6²)₁₂(6³)₁₂(5.6.12)₂₄ net is the basis of the silicate sheet in parsettensite (Fig. 42c) and has four-, five-, six- and twelve-membered rings. In parsettensite, the u–d arrangements are as follows: (d⁴), (u³d²), (u⁶) and (u²d²u²d²u²d²), and the oikodoméic operation is a class-2 m operation. The interstitial complexes in stilpnomelane, bannisterite and parsettensite consist of sheets of octahedrally coordinated Fe²⁺ or Mn²⁺ with additional alkali- and alkaline-earth cations and (H₂O) groups that either have not been located or are strongly positionally disordered.

Fig. 42. Double-layer sheets of tetrahedra based on miscellaneous complex 3-connected plane nets plus an oikodoméic operation: (a) the (5.6²)₆(6³)₆(5.6.8)₂₄ net and sheet in stilpnomelane; (b) the (5.6²)₈(5.7²)₄(5.6.7)₈(6².7)₁₂ net and sheet in bannisterite; and (c) the (4.5.12)₂₄(5.6²)₁₂(6³)₁₂ (5.6.12)₂₄ net and sheet in parsettensite.

The structure of ganophyllite (Table 8) has been the object of much work since its description by Hamberg (Reference Hamberg1890). Noe and Veblen (Reference Noe and Veblen1999) refined the ganophyllite structure in a subcell (a _sub x 3 = a) and presented high-resolution transmission electron microscopy evidence of incommensurate behaviour in some crystals (or parts of crystals). Eggletonite (Table 8) was described as the Na analogue of ganophyllite, and the structure was refined on a subcell (Peacor et al., Reference Peacor, Dunn and Simmons1984). Hughes et al. (Reference Hughes, Rakovan, Bracco and Gunter2003) solved the structure of an orthorhombic dimorph of tamaite (Table 8) and showed that the parent layer of the silicate double-layer sheet is based on the (5.6²)₆(5.6.7)₄(6².7)₁₀ net (Fig. 43). This net has five-, six- and seven-membered rings with the following u–d arrangements in the parent single-layer: (u³d²), (u⁶), (u²du²d) and (u⁶d²), and the oikodoméic operation is m. The interstitial complexes in ganophyllite, eggletonite and tamaite consist of sheets of octahedrally coordinated Fe²⁺ or Mn²⁺ with additional alkali- and alkaline-earth cations and (H₂O) groups that either have not been located or are strongly positionally disordered.

Fig. 43. Double-layer sheets of tetrahedra based on miscellaneous complex 3-connected plane nets plus an oikodoméic operation: the (5.6²)₆(5.6.7)₄(6².7)₁₀ net and sheet in tamaite.

The (3.8²)(6.8²) net (Fig. 44a) consists of three-membered, six-membered and eight-membered rings in the ratio 2:1:3, and corresponds to the parent-layer silicate sheet in zussmanite (Fig. 44b) and presumably to the isostructural coombsite (Table 8). The u–d arrangements are (d³), (u⁶) and (u²d²u²d²), and the oikodoméic operation is a class-3 m operation. The interstitial complex consists of two Fe²⁺ ions coordinated by ${\rm O}_3^{{\rm 2\ndash}} $(OH$)_3^\ndash $ (×2), one site occupied by Fe²⁺ and smaller ions and coordinated by ${\rm O}_6^{{\rm 2\ndash}} $, and one K⁺ ion coordinated by ${\rm O}_{{\rm 12}}^{{\rm 2\ndash}} $.

Fig. 44. Double-layer sheets derived by class-3 oikodoméic operations: (a) the (3.8²)(6.8²) net; and (b) the corresponding arrangements of tetrahedra in zussmanite, showing the class-3 oikodoméic operation m.