Silicate

Where we refer to a silicate unit (e.g. silicate chain, ribbon and/or tube), let it be understood that the unit must contain Si⁴⁺ but may also contain other tetrahedrally coordinated cations: e.g. T = P⁵⁺, V⁵⁺, As⁵⁺, Al³⁺, Fe³⁺, B³⁺, Be²⁺, Zn²⁺ and Mg²⁺. For simplicity of expression, we refer to such compositions as silicates, whether or not the dominant tetrahedrally coordinated cation is Si⁴⁺, as we require them to contain Si⁴⁺ as an essential constituent. Chains and ribbons that contain no Si⁴⁺-tetrahedra will be discussed if such minerals provide insight into the relation between composition and structure; examples include the aluminate minerals addibischoffite and warkite. Synthetic compounds will be discussed if they contain chains, ribbons or tubes of Si⁴⁺-tetrahedra that are topologically unique or intermediate between other chains in the hierarchy.

We will refer to a tetrahedron or any other higher coordination polyhedron by its central cation if the anions coordinating that cation are solely O²⁻ or if the identity of such anions is unclear. It follows that the expression Si⁴⁺-tetrahedron (tetrahedrally coordinated Si⁴⁺) or Na⁺-octahedron (octahedrally coordinated Na⁺) represents a (Si⁴⁺O₄)⁴⁻-tetrahedron or a (Na⁺O₆)¹¹⁻-octahedron, respectively and a ‘T-tetrahedron’ represents a (TO⁴)ⁿ⁻ tetrahedron, where T is one or more unspecified tetrahedrally coordinated cation. If the anions that coordinate any given cation are anything other than O²⁻ (i.e. (OH)⁻, (H₂O), F⁻), the expanded notation for that cation-polyhedron will be given, e.g. (Na⁺O₄(OH)₂)⁹⁻-octahedron or (SiO₃(OH))³⁻-tetrahedron. Where coordination number is not expressed in this notation, it will be appended to the central cation of the respective polyhedron or ion (e.g. ^[7]Na⁺-polyhedron or ^[8]Ba⁺ ion).

We define chains, ribbons, and tubes of T-tetrahedra as follows:

• Chain: a silicate unit of (TO₄)ⁿ⁻ tetrahedra that link infinitely in a single direction and that can be broken by eliminating a single linkage between adjacent (TO₄)ⁿ⁻ tetrahedra (Fig. 1a).

• Ribbon: a silicate unit of (TO₄)ⁿ⁻ tetrahedra that link infinitely in a single direction and that cannot be broken by eliminating a single linkage between adjacent (TO₄)ⁿ⁻ tetrahedra (Fig. 1b).

• Tube: a silicate unit of (TO₄)ⁿ⁻ tetrahedra that link infinitely in a single direction, and also link orthogonal to the direction of polymerisation to form a hollow cylinder. A silicate tube cannot be broken by eliminating a single linkage between adjacent (TO₄)ⁿ⁻ tetrahedra (Fig. 1c).

For simplicity, we will refer to all chain-, ribbon- and tube-silicate minerals and structures as chain silicates and chains, respectively. A layer is a single planar or semi-planar array of ions. A sheet is a single planar or semi-planar array of linked polyhedra.

Fig. 1. Definitions of a silicate chain, ribbon and tube: (a) a chain: if the linkage between two tetrahedra is broken, the one-dimensional polymerisation is lost; (b) a ribbon: if the linkage between two tetrahedra is broken, the one-dimensional polymerisation is not lost; (c) a tube: if the linkage between two tetrahedra is broken, the one-dimensional polymerisation is not lost, and also the tetrahedra fold round on themselves perpendicular to the direction of polymerisation to form a hollow cylinder. Dashed black lines show the geometrical repeat unit of the chain, ribbon and tube.

Low-acidity ^[4]T cations: Alkali metals

There is some ambiguity about the [4]-coordinated ions Li⁺, Na⁺, K⁺, Rb⁺ and Tl⁺. These ions all have Lewis acidities (Gagné and Hawthorne, Reference Gagné and Hawthorne2017) lower than the cut-off of 0.30 vu used by Hawthorne and Schindler (Reference Hawthorne and Schindler2008) to exclude ions from the structural unit, and using this criterion, we would exclude such ions from the chain. Conversely, from a topological perspective, one might wish to include all tetrahedrally coordinated cations. In some cases, the same structure types may incorporate different cations with a range of valence states at a tetrahedrally coordinated site; for example, in the milarite structure (Gagné and Hawthorne, Reference Gagné and Hawthorne2016), the T2 site in the framework may be occupied by Li⁺ (e.g. berezanskite, Hawthorne et al., Reference Hawthorne, Sokolova, Pautov, Agakhanov and Karpenko2016; sogdianite; Sokolova et al., Reference Sokolova, Hawthorne and Pautov2000), and various divalent (Be²⁺, Mg²⁺, Fe²⁺ and Zn²⁺) and trivalent (B³⁺, Al³⁺ or Fe³⁺) cations. It seems undesirable to separate isostructural minerals on the basis of Lewis acidity of the T cation and we include such minerals in the same group here.

The geometrical repeat unit and the ^cT_r expression

In the tetrahedron and ball-and-stick representations, we assign a geometrical repeat unit in which the geometry of the chain (lengths and angles of linkages between tetrahedra) is preserved. The geometrical repeat unit contains the minimum number of tetrahedra (n_g) required to generate the chain through translation operations. It is necessary to specify the numbers of 1-, 2-, 3- and 4-connected tetrahedra that comprise n_g to describe the geometrical repeat unit of a chain. To do this, we denote a tetrahedron by T, its connectivity by the superscript c (c = 1–4) and the number of such tetrahedra in the geometrical repeat unit by the subscript r. The expression ^cT_r = ¹T_r²T_r³T_r⁴T_r represents the possible connectivities of tetrahedra in the repeat unit, and the number of terms with r ≠ 0 in the ^cT_r expression is defined as its rank. The majority of chains contain only 2- and 3-connected vertices (i.e. ^cT_r has a rank of 2 as r = 0 for ¹T_r and⁴T_r), but some chains also contain 1- and/or 4-connected vertices. As an example, consider the tetrahedron (Fig. 2a) and ball-and-stick (Fig. 2b) representations of the [Si₆O₁₈]¹²⁻ chain in sapphirine-supergroup minerals. The ball-and-stick representation shows three types of vertices: 3-connected (red circles), 2-connected (blue circles) and 1-connected (yellow circles) (Fig. 2b). The repeat unit contains two of each of these types of vertex (n_g = 6), and the ^cT _r expression for the aenigmatite-type chain is written as ^cT_r = ¹T₂²T₂³T₂⁴T₀ = ¹T₂²T₂³T₂ (rank = 3).

We can generate all possible ^cT _r expressions for chains with tetrahedron connectivities, c, of 1, 2, 3 and 4 where r = 1 to ∞ by sequentially increasing the values of c and r. There are various ways in which this may be done. However, the most useful way is to order these in terms of increasing rank of ^cT_r (i.e. the number of individual ^cT_r vaues). For a given rank, we sequentially increase the value of c for r = 1 to ∞; Table 2 shows the ^cT_r expressions produced in this way. For a rank of 1, where c = 1, a chain is not possible: ¹T₁ corresponds to single tetrahedron; ¹T₂ corresponds to a [T₂O₇] dimer, and no further linkages are possible without changing the value of c (Table 2). Thus, ²T₁ is the simplest possible chain arrangement, followed by ²T₂, ²T₃, ²T₄ etc. For higher ranks, we order ^cT_r first in terms of c and then in terms of r, hence for a rank of 2: ²T_r³T_r, we have ²T₁³T_r, ²T₂³T_r, ²T₃³T_r……. etc. Thus we have a rationale for both generating all theoretical chains and ordering observed chains into a preliminary hierarchy.

Table 2. Hierarchical ordering of ^cT_r values where r = 1–∞ and c = 1–4.

The topological repeat unit

Graphical representations of chains have a topological repeat unit in which only the topological properties are preserved. The topological repeat unit contains the minimum number of vertices (n_t) required to generate the chain through infinite linkage in a single direction. By analogy with the geometrical repeat unit, we may describe the topological repeat unit using the expression ^cV_r = ¹V_r²V_r³V_r⁴V_r where ^cV_r denotes the connectivity of vertices (V) rather than tetrahedra (T).

In many chains, tetrahedra are topologically identical but geometrically distinct. This often results in chains with geometrical and topological repeat units that contain different numbers of tetrahedra and vertices. Figures 3a,b show the tetrahedron and ball-and-stick representations of the chain in astrophyllite-supergroup minerals, where ^cT_r = ³T₂¹T₂ is the connectivity of the tetrahedra in the geometrical repeat unit. Figure 3c shows the graphical representation of the same chain with a topological repeat unit that contains two vertices rather than four as shown in the ball-and-stick representation (Fig. 3b); this is because the direction of branching of 1-connected vertices (or tetrahedra) does not affect the topology of the linkage. It follows that we may describe the topological repeat unit in astrophyllite-supergroup minerals as ^cV_r = ³V₁¹V₁. Hence the ^cV_r expression for a topological repeat unit may be derived by multiplying the values of r in the respective ^cT_r expression by n_t/n_g. For some chains, ribbons and tubes, a graphical representation will not be given if such a representation further obscures the connectivity of vertices and/or if ^cT_r = ^cV_r (n_g = n_t).

Fig. 3. (a) Tetrahedral, (b) ball-and-stick and (c) graphical representations of the chain in astrophyllite-supergroup minerals where ^cT _r ≠ ^cV _r.

Group and class

In Mineralogy, a group is defined as two or more minerals with the same or similar structure and chemical composition, a group may belong to a single supergroup and/or have multiple subgroups, and supergroups may belong to a single class or subclass (Mills et al., Reference Mills, Hatert, Nickel and Ferraris2009). Here, we also use the words group and class in their general sense: [1] group indicates a collection of minerals with same ^cT_r and/or ^cV_r expression; [2] class organises minerals with ^cT_r expressions in which the tetrahedron connectivity is the same (c) but not necessarily the number of tetrahedra with that connectivity (r). For example, the hierarchy class ³T_r contains the groups ³T₄, ³T₆, ³T₈, ³T₁₂, ³T₁₆, ³T₁₇, ³T₃₂ and ³T₅₆.

²T₁ chains

The metagermanate chain, [GeO₃] (Krivovichev et al., Reference Krivovichev, Filatov and Semenova1998, fig. 4.C1), is the only known structure with ²T₁ chains (Figs 4a,b) in which each 2-connected (GeO₄)⁴⁻-tetrahedron is geometrically and topologically identical (Fig. 4c). ²T₁ chains are the simplest possible chain-type and constitute the first group of the ²T_r class where r = 1 (Tables 2 and 3).

Fig. 4. (a, b) Tetrahedral representation of the ²T₁ [GeO₃] chain viewed orthogonal to the length of the chain; (c) a graphical representation of the chain where red points (vertices) represent Si⁴⁺-tetrahedra and black lines (edges) represent linkages between adjacent Si⁴⁺-tetrahedron. Dashed black lines show the geometrical and topological repeat unit of the chain.

²T₂ chains

²T₂ chains are extremely common and comprise one of the largest groups of minerals in the structure hierarchy (Table 3). The geometrical repeat unit of ²T₂ chains contains two (TO₄)ⁿ⁻-tetrahedra (n_g = 2) that are linked to form [T₂O₆]ⁿ⁻ groups that polymerise in a single direction to form the ²T₂ chain in which both tetrahedra in the repeat unit may point in the same or opposing directions (Figs 5a–d). The topological repeat unit contains a single vertex that is topologically identical to all other vertices: n_t = 1, as is the case for all ²T_r chains. The pyroxene supergroup are by far the most abundant minerals with this chain type; they have been described in considerable detail elsewhere (e.g. Papike et al., Reference Papike, Prewitt, Sueno and Cameron1973; Ohashi and Finger, Reference Ohashi and Finger1974; Cameron and Papike, Reference Cameron and Papike1981; Bruno et al., Reference Bruno, Carbonin and Molin1982; Rossi et al., Reference Rossi, Smith, Ungaretti and Domeneghetti1983; Tribaudino et al., Reference Tribaudino, Benna and Bruno1989; Redhammer et al., Reference Redhammer, Amthauer, Roth, Tippelt and Lottermoser2006; Nestola et al., Reference Nestola, Tribaudino, Boffa Ballaran, Liebske and Bruno2007; Abdu and Hawthorne, Reference Abdu and Hawthorne2013), and we will consider them here only briefly. Typically, ²T₂ chains extend along the c-axis and link to ribbons of edge-sharing octahedra; in diopside, there are alternating layers of ²T₂ chains and sheets of Mg²⁺-octahedra and ^[7]Ca²⁺-polyhedra (Figs 5e,f). This general stacking sequence of chains of tetrahedra and sheets of octahedra (or higher-coordination polyhedra) is common in chain silicates that contain ²T₂ chains.

Fig. 5. (a, b, c) Tetrahedral representations of the ²T₂ chain in pyroxenes where both tetrahedra in the geometrical repeat unit point in the same direction and (d) a ball-and-stick representation of the chain. The structure of diopside projected (e) onto (100) and (f) along the c-axis. Dashed black lines show the geometrical repeat unit of the chain.

Lintisite-group minerals include lintisite, punkaruaivite, eliseevite, kukisvumite and manganokukisvumite (Table 3) all of which contain ²T₂ chains that occur in two distinct layers. Lintisite and punkaruaivite contain ^[4]Li⁺ and are of particular interest as Li⁺-tetrahedra form chains of edge-sharing tetrahedra, and we will describe the structure here. Other minerals that contain silicate ribbons in addition to ²T₂ pyroxene-like chains, such as vinogradovite and paravinogradovite (see below), will be discussed in the ^cT_r section that corresponds to the ribbon rather than the ²T₂ chain. For all known chain-, ribbon- and tube-silicate minerals, tetrahedra link via corners through a common bridging anion. Such anions can link to a maximum of two Si⁴⁺-tetrahedra, allowing a maximum Si⁴⁺-connectivity of four. For a given (SiO₄)⁴⁻-tetrahedron the average mean bond-valence is 1.0 vu and therefore the valence sum rule is satisfied at the O²⁻ bridging anion (Si–O–Si). However, if substitution of Si⁴⁺ by a cation with a smaller formal valence and a lower Lewis acidity (Gagné and Hawthorne, Reference Gagné and Hawthorne2017) occurs, the mean bond-valence contribution to the bridging anion decreases which allows additional T–O linkages. This T-cation may have a connectivity of greater than four and form arrangements not observed in units composed predominately of Si⁴⁺-tetrahedra. This situation occurs in lintisite (and punkaruaivite), in which Li⁺-tetrahedra (mean bond-valence of 0.25 vu) have a connectivity of six and form ⁴T₄⁶T₂ [Li₂Si₄O₁₂]⁶⁻ ([Li₂Si₄O₁₀(OH)₂]⁴⁻ in punkaruaivite) ribbons, the only example of this arrangement in chain-silicate minerals; the calculated bond-valence sum for ^[4]Li⁺ (X1 cation) in lintisite is 0.92 vu.

In lintisite, there are two distinct ways in which ²T₂ chains link to the rest of the structure (Fig. 6a): [1] ²T₂ chains link two adjacent sheets of Ti⁴⁺-octahedra and ^[8]Na⁺-polyhedra (Na1) along the a-axis (Figs 6b); and [2] ²T₂ chains link sheets of Ti⁴⁺-octahedra and ^[8]Na⁺-polyhedra (Na1) to chains of edge-sharing (LiO₄)⁷⁻-tetrahedra (X1) (Figs 6c). This linkage of chains of (SiO₄)⁴⁻- and (LiO₄)⁷⁻-tetrahedra (X1) form ⁴T₄⁶T₂ [Li₂Si₄O₁₂]⁶⁻ ribbons (Figs 6d,e) that extend along the c-axis. In lintisite, channels extend along the c-axis and are occupied by Na⁺ that form (NaO₂(H₂O)₄)³⁻-octahedra (Na2 and W1) (Fig. 6a). In punkaruaivite, ²T₂ chains and ⁴T₄⁶T₂ [Li₂Si₄O₁₀(OH)₂]⁴⁻ ribbons link to ribbons of edge-sharing (TiO₄(OH)₂)⁶⁻-octahedra instead of sheets as the Na1 site is vacant. Channels are occupied solely by (H₂O) groups (W1) rather than Na2 cations as in lintisite, and the bond-valence sum at the X1-cation is 1.02 vu. Eliseevite contains only ²T₂ chains and no ⁴T₄⁶T₂ ribbon as the X1 cation forms (Li(H₂O)₄(OH)₂)⁻-octahedra rather than Li⁺-tetrahedra. Here edge-sharing ribbons of (Li(H₂O)₄(OH)₂)⁻-octahedra are linked along the a-axis to sheets of Ti⁴⁺-octahedra and ^[8]Na⁺-polyhedra via ²T₂ chains. Although the bond-valence sums at the X1-cation in lintisite and punkaruaivite suggest Li⁺ is [4]-coordinated in this structure type, the bond-valence sums at the X1-cation in eliseevite for [4]- and [6]-coordinated Li⁺ are 0.75 and 0.90 vu, respectively. In kukisvumite and manganokukisvumite, Zn²⁺ and Mn²⁺ occupy the X1 site, respectively, and have both been described as tetrahedrally coordinated cations. The incident bond-valence sum at the X1-cation for Zn²⁺ in kukisvumite is 1.16 vu, but there are no other anions close to Zn, and Zn must be tetrahedrally coordinated. This apparent bond-valence deficiency at the X1-cation is presumably the result of the half-occupancy of the site by Zn²⁺ with the real Zn²⁺–O distances much shorter and the □–O distances much longer than the observed distances.

Fig. 6. The structure of lintisite projected (a) along the c-axis, (b, c) onto (100), and the [Li₂Si₄O₁₂]⁶⁻ ribbon of [SiO₄]⁴⁻ and [LiO₄]⁷⁻ tetrahedra projected (d) onto (100) and (e) along the c-axis. Fine dashed black lines outline the unit cell which is halved along the a-axis in (a). The H atoms associated (H₂O) groups have been omitted for clarity.

Although most chain silicates with ²T₂ chains contain alternating layers of chains of tetrahedra and sheets of octahedra (or higher-coordination polyhedra), there are different sequences of layer stacking that involve such chains. The carpholite-group minerals (Basso and Carbone, Reference Basso and Carbone2010) include carpholite, balipholite, ferrocarpholite, magnesiocarpholite, vanadiocarpholite and potassiccarpholite with the general formula A ₂BM ₂Al₄[Si₄O₁₂]V ₄W ₄ (Tait et al., Reference Tait, Hawthorne, Grice, Jambor and Pinch2004) (Table 3). In all group members, the A site is dominated by vacancy with minor Na⁺ and the B site is also vacant in all members except balipholite and potassiccarpholite where it is occupied by Ba²⁺ and K⁺, respectively. In carpholite, both M-octahedra are occupied by Mn²⁺ and share edges with Al³⁺-octahedra, forming ribbons that extend along [001]. These ribbons are linked to each other along [100] by chains of Al³⁺-octahedra, forming channels that are occupied by ²T₂ [Si₂O₆]⁴⁻ chains (Figs 7a,b). In magnesiocarpholite and ferrocarpholite, Mg²⁺ and Fe²⁺ substitute for Mn²⁺ at one or both of the M sites. In balipholite and potassiccarpholite, Mg²⁺/Li⁺ and Mn²⁺/Li⁺ occupy the M sites, respectively and in vanadiocarpholite V³⁺ substitutes for Al³⁺. In all carpholite-group minerals, the V and W sites are occupied by (OH)⁻ groups except for potassiccarpholite where F⁻ occupies the W site.

Fig. 7. The structure of carpholite projected (a) onto (010) and (b) along the c-axis. The structure of nchwaningite projected (c) onto (100) and (d) along the c-axis. Fine dashed black lines outline the unit cell and H atoms associated with (OH)⁻ and (H₂O) groups have been omitted for clarity.

The ²T ₂ chains in nchwaningite link to sheets of (MnO₂(OH)₃(H₂O))⁵⁻-octahedra along the a-axis and the b-axis (Fig. 7c). Each Si⁴⁺–Mn²⁺ layer, shown in Fig. 7d, is linked to an identical layer along the b-axis via hydrogen bonding associated with (H₂O) and (OH)⁻ groups that occupy interlayer space and coordinate both Si⁴⁺ and Mn²⁺ ions. In lorenzenite, layers of ²T₂ chains link to a double-layer sheet of Ti⁴⁺-octahedra and $^{[7]}\hbox{Na}^+$-polyhedra rather than a single-layer sheet (Figs 8a,b). In shattuckite, ²T₂ chains link to continuous, modulated single-layer sheets of (CuO₄(OH)₂)⁸⁻-octahedra (Cu1 and Cu2) (Fig. 8c) and chains of Cu²⁺-octahedra (Cu3) (Fig. 8d) that occur in layers parallel to the modulated sheets (001) (Fig. 8e). In yegorovite, both tetrahedra in the repeat unit are acid silicate (silanol) groups; (SiO₃(OH))³⁻ and point in oblique directions with respect to each another (Figs 9a–d), a notable geometric difference from the more common ²T₂ minerals. Here, ²T₂ chains link to modulated single-layer sheets of (NaO(H₂O)₃(OH)₂)³⁻- (Na1 and Na3) and (NaO(H₂O)₄(OH))²⁻-octahedra (Na2 and Na4) that are parallel to (100) (Figs 9e,f). In aerinite, there are three distinct modes of linkage between ²T₂ chains and the rest of the structure. All ²T₂ chains link to chains of Fe³⁺-octahedra that extend along and inside and outside of tubes of Al³⁺-octahedra and ^[7]Ca²⁺-polyhedra and link such tubes to each other (Fig. 10a). Each chain of Fe³⁺-octahedra links to three different ²T₂ chains (Fig. 10b). The way in which ²T₂ chains and chains of Fe³⁺-octahedra link within tubes and between tubes of Al³⁺-octahedra and ^[7]]Ca²⁺-polyhedra is shown in Figs 10c,d, respectively. Aerinite also contains (H₂O), (OH)⁻ and (CO₃)²⁻ groups and we refer readers to Rius et al. (Reference Rius, Crespi, Roig and Melgarejo2009) for a more detailed structural description. Many synthetic compounds that contain ²T₂ [Si₂O₆]⁴⁻ chains have been also been described, examples of which are listed in Table 3.

Fig. 8. The structure of lorenzenite projected (a) onto (100) and (b) along the c-axis. The structure of shattuckite projected (c, d) onto (010) and (e) along the c-axis. Fine dashed black lines outline the unit cell and H atoms associated with (OH)⁻ groups have been omitted for clarity.

Fig. 9. (a, b, c) Tetrahedral representations of the ²T ₂ chain in yegorovite where both tetrahedra in the geometrical repeat unit point in oblique directions, and (d) a ball-and-stick representation of the chain. The structure of yegorovite projected (e) onto (001) and (f) along the a-axis. Fine dashed black lines outline the unit cell and H atoms associated with (OH)⁻ and (H₂O) groups have been omitted for clarity.

Fig. 10. The structure of aerinite projected (a) along the c-axis and (b, c, d) the structural modules that contain ²T₂ chains viewed orthogonal to the c-axis. Fine dashed black lines outline the unit cell and H atoms associated with (OH)⁻ and (H₂O) groups and C atoms associated with (CO₃) groups have been omitted for clarity.

²T₃ chains

In ²T₃ chains there are three geometrically distinct tetrahedra in the geometrical repeat unit, and all minerals that contain ²T₃ chains are listed in Table 4. There has been considerable work done on the effect of M-site substitutions on chain geometry, compositional limits and hydrogen bonding. In these minerals, M-site substitutions affect the periodicity, the geometrical aspects of the chain, and the stacking patterns of the structural elements. They are also affected by temperature (e.g. Ohashi and Finger, Reference Ohashi and Finger1976, Reference Ohashi and Finger1978; Liebau, Reference Liebau1980; Nagashima et al., Reference Nagashima, Imaoka, Fukuda and Pettke2018; Prewitt and Peacor, Reference Prewitt and Peacor1964) but these effects will not be discussed in detail here.

Of the minerals that contain ²T₃ chains, wollastonite-group minerals are the most abundant and may be divided into two categories, anhydrous and hydrous, based on the absence or presence of (OH)⁻ groups (Liebau, Reference Liebau1980) (Table 4). Wollastonite (A and M polytypes), dalnegorskite, bustamite, mendigite and ferrobustamite do not contain (OH)⁻, whereas pectolite, schizolite, murakamiite, tanohataite, serandite, berrydawsonite-(Y) and vistepite contain (OH)⁻ groups (Table 4). The geometry of the chain in wollastonite-group minerals is shown in Figs 11a–d. In general, these chains contain three distinct tetrahedra and consist of c-shaped trimers that link along the b-axis, parallel to ribbons of octahedra and/or other higher coordination polyhedra (Fig. 11e), These ribbons occur in layers that alternate with layers of ²T₃ chains (Fig. 11f). Thompson et al. (Reference Thompson, Yang and Downs2016) examined the relations between pyroxenoids and pyroxenes in detail.

Fig. 11. (a, b, c) Tetrahedral representations of the ²T ₃ chain in wollastonite-group minerals and (d) a ball-and-stick representation of the chain. The structure of wollastonite-2M viewed (e) orthogonal to the b-axis and (f) along the b-axis. Dashed black lines show the geometrical repeat unit of the chain and fine dashed black lines outline the unit cell.

The most common wollastonite polytypes are 1A and 2M, with 3A, 4A, 5A and 7A polytypes being less common, resulting from different stacking sequences (Henmi et al., Reference Henmi, Kawahara, Henmi, Kusachi and Takéuchi1983). Wollastonite contains three octahedrally coordinated sites that are fully occupied by Ca²⁺ (M1–M3) and form ribbons three octahedra wide (Figs 11e,f). The other anhydrous wollastonite-group minerals have the general formula M1₂M2₂M3M4[Si₃O₉]₂ and include dalnegorskite, bustamite, ferrobustamite and mendigite (Table 4). These minerals contain four sites M1–M4 occupied by Ca²⁺, Mn²⁺ and Fe²⁺ where the M4 site is always occupied by Ca²⁺. In bustamite (Figs 12a,b) and ferrobustamite (Figs 12c,d), the M1 site is occupied by Ca²⁺ > Fe²⁺ and Ca²⁺ > Mn²⁺, respectively and the M3 site is occupied by Mn²⁺ in bustamite and Fe²⁺ in ferrobustamite. In the structures of bustamite, ferrobustamite and dalnegorskite, the M2 sites are occupied dominantly by Ca²⁺. In mendigite, the M1, M2 and M3 sites are dominated by Mn²⁺ (Figs 12e,f). Dalnegorskite represents the compositional limit of Ca²⁺ in the bustamite-type structure where M1, M2 and M4 are fully occupied by Ca²⁺ and M3 is occupied by Mn²⁺.

Fig. 12. The structure of (a, b) bustamite, (c, d) ferrobustamite and (e, f) mendigite viewed (a, c, e) orthogonal to the b-axis and (b, d, f) along the b-axis. M-site labels in (a) are also applicable to (c) and (e). Fine dashed black lines outline the unit cell.

The general formula of the hydrous wollastonite-group minerals can be written as [M3(M1,M2)₂(Si₃O₈(OH)], where the M3 site is occupied either by Na⁺ or Li⁺, and M1 and M2 are occupied by Ca²⁺, Mn²⁺, Y³⁺, Sn²⁺ and Sc³⁺ (Takéuchi et al., Reference Takéuchi, Koto and Yamanaka1976a,Reference Takéuchi, Kudoh and Yamanakab; Mellini et al., Reference Mellini, Merlino, Orlandi and Rinaldi1982; Hybler et al., Reference Hybler, Petříček, Jurek, Skála and Císařová1997; Nagashima et al., Reference Nagashima, Imaoka, Fukuda and Pettke2018). The M1 and M2 octahedra are typically occupied by Ca²⁺ and Mn²⁺, and form ribbons two polyhedra wide. Unlike the anhydrous varieties, ribbons of octahedra are linked to each other by M3 cations, forming sheets parallel to (101). In pectolite, M1 and M2 are occupied by Ca²⁺ and M3 is occupied by Na⁺ (Figs 13a,b). In barrydawsonite-(Y), M1 is occupied by Ca²⁺, M2 is occupied by Na⁺ and Y³⁺ (+REE), and M3 is occupied by Na⁺. In schizolite, M1 and M2 are occupied by Ca²⁺ and Mn²⁺, respectively, and M3 is occupied by Na⁺ (Figs 13c,d). In murakamiite (Figs 13e,f) and tanohataite (Figs 13g,h) M1 and M2 are occupied by Ca²⁺ and Mn²⁺, respectively, and M3 is occupied by Li⁺. In serandite, M1 and M2 are occupied by Mn²⁺ and M3 is occupied by Na⁺ (Figs 13i,j). In vistepite, the T3 site is occupied by B³⁺, and ribbons are three octahedra wide. Unlike most hydrous wollastonite-group minerals, adjacent ribbons of octahedra are not linked by Na⁺ or Li⁺ to form sheets. It follows that the vistepite structure more closely resembles that of bustamite and mendigite despite containing (OH)⁻ groups. Here, M1 and M2 are occupied by Mn²⁺, M3 is occupied by Sn²⁺ and M4 is vacant (Figs 14a,b). In cascandite, M1 is occupied by Ca²⁺, M2 is occupied by Sc³⁺ and M3 is vacant (Figs 14c,d). In general, hydrous wollastonite-group minerals contain one (OH)⁻ group that acts as a bridging anion between (SiO₃(OH)⁻)³⁻ tetrahedra (T1) and (CaO₅(OH⁻))⁹⁻ octahedra (M1); a more detailed discussion of hydrogen bonding in these minerals is given by Nagashima et al. (Reference Nagashima, Imaoka, Fukuda and Pettke2018).

Fig. 13. The structure of (a, b) pectolite, (c, d) schizolite, (e, f) murakamiite, (g, h) tanohataite and (i, j) serandite viewed (a, c, e, g, i) orthogonal to the b-axis and (b, d, f, h, j) along the b-axis. M-site labels in (a) are also applicable to (c), (e), (g) and (i) and fine dashed black lines outline the unit cell. The H atoms associated with (OH)⁻ groups have been omitted for clarity.

Fig. 14. The structure of vistepite viewed (a) orthogonal to the a-axis and (b) along the a-axis, ^{T 3}[BO₄] tetrahedra are shown in dark blue. The structure of cascandite projected (c) onto (100) and (d) along the c-axis. Fine dashed black lines outline the unit cell and H atoms associated with (OH)^– groups have been omitted for clarity.

In hilairite, ²T₃ chains extend along the c-axis and link to isolated Zr⁴⁺-octahedra to form an open framework (Figs 15a,b). Zr⁴⁺-octahedra occur in rows that extend along the c-axis, forming channels that are partly occupied by Na⁺ (Na1 and Na2) and (H₂O) groups; these channels are labelled 1 in Fig. 15a. Figure 15b shows the position of Na1 atoms and channels labelled 2 that contain Na2 atoms and (H₂O) groups. Other members of the hilairite group include calciohilairite, komkovite, sazykinaite-(Y), pyatenkoite-(Y) and synthetic K⁺-, Rb⁺-, Pb²⁺-, Sr²⁺-, Ba²⁺-, Ca²⁺- and Cs⁺-exchanged analogues of hilairite (Pekov et al., Reference Pekov, Chukanov, Kononkova and Pushcharovsky2003, Reference Pekov, Grigorieva, Zubkova, Turchkova and Pushcharovsky2010a; Zubkova et al., Reference Zubkova, Pekov, Turchkova, Pushcharovskii, Merlino, Pasero and Chukanov2007, Reference Zubkova, Kolitsch, Pekov, Turchkova, Vigasina, Pushcharovsky and Tillmans2009a) (Table 4).

Fig. 15. The structure of hilairite projected (a) orthogonal to the c-axis and (b) along the c-axis. Channel 1 in (a) is occupied by Na1, Na2 and (H₂O) groups and channel 2 in (b) is occupied by Na2 and (H₂O) groups. The (H₂O) groups, Na1 and Na2 in (a) and (H₂O) groups and Na2 in (b) have been omitted for clarity. Fine dashed black lines outline the unit cell.

The ²T₃ chain in umbite is geometrically distinct from the ²T₃ chain in wollastonite- and the hilairite-group minerals. In umbite, chains extend along the c-axis and link to isolated Zr⁴⁺-octahedra (Fig. 16a), forming an open framework that contains channels that also extend along the c-axis, resembling hilairite-group minerals (Figs 15a–b). In umbite, K2 atoms and (H₂O) groups occur in channels labelled 1 and K1 atoms occur within channels labelled 2 in Fig. 16b. In paraumbite, the H⁺-exchanged variety of umbite, the K1 site is partly occupied by H⁺ and in synthetic Cs⁺-exchanged umbite, Cs⁺ partly occupies the K1 site (Khomyakov et al., Reference Khomyakov, Voronkov, Kobyashev and Polezhaeva1983a; Fewox et al., Reference Fewox, Clearfield and Celestian2011). Kamenevite is the Ti⁴⁺-analogue of umbite and kostylevite is a monoclinic polymorph of umbite.

Fig. 16. The structure of umbite projected (a) onto (010) and (b) along the c-axis. In (b), channel 1 is occupied by K2 and (H₂O) groups and channel 2 is occupied by K1. The (H₂O) groups, K1 and K2 atoms have been omitted for clarity. Fine dashed black lines outline the unit cell.

Foshagite, hillebrandite, jennite, ‘metajennite’, plombièrite (tobermorite-14Å), riversideite (tobermorite-9.3Å) and whelanite can be chemically classified as calcium–silicate–hydrates (C–S–H minerals) (Table 4) and have received considerable attention due to their structural and compositional similarities to the tobermorite-group minerals and their role in the hydration of Portland cement (Vigfusson, Reference Vigfusson1931; Gard and Taylor, Reference Gard and Taylor1976; Taylor, Reference Taylor1992; Cong and Kirkpatrick, Reference Cong and Kirkpatrick1996; Richardson, Reference Richardson2008; Meller et al., Reference Meller, Kyritsis and Hall2009). Synthetic C–S–H phases produced by cement chemists are often poor quality and not suitable for X-ray diffraction analysis. Consequently, natural analogues of such phases are studied instead which correspond to many of the ²T₃, C–S–H minerals described here. In foshagite, ²T₃ chains link to ribbons of (CaO₄(OH)₂)⁸⁻- and (CaO₅(OH))⁹⁻-octahedra (Ca1–Ca4) (Fig. 17a). Ribbons four octahedra wide occur in layers parallel to (101) that alternate with layers of ²T₃ chains (Fig. 17b), an arrangement that is similar to that of the wollastonite-group minerals (Figs 11e–f). In hillebrandite, ²T₃ chains occupy and extend along tunnels in a framework of (Ca(O,OH)₆)-octahedra and (Ca(O,(OH))₇)-polyhedra.

Fig. 17. The structure of (a, b) foshagite and (c, d) jennite viewed (a, c) orthogonal to the b-axis and (b, d) along the b-axis. The structure of plombièrite (tobermorite-14Å) projected (e) onto (100) and (f) along the b-axis. In (f), layers that contain ²T ₃ chains are labelled T, layers that contain sheets of Ca²⁺-polyhedra are labelled O and layers that contain interstitial Ca²⁺-polyhedra and (H₂O) groups are labelled I. Fine dashed black lines outline the unit cell which is halved along the c-axis in (f). The H atoms associated with (OH)⁻ and (H₂O) groups have been omitted for clarity.

In jennite, ²T₃ chains extend along the b-axis and link to a framework of Ca²⁺-octahedra and ^[7]Ca²⁺-polyhedra (Ca1–Ca5). This framework consists of ribbons of (CaO₄(OH)₂)⁸⁻-octahedra that extend along the b-axis (Ca1 and Ca3) and are cross-linked to adjacent ribbons by isolated (CaO₂(H₂O)₄)²⁻-octahedra (Ca5), forming sheets parallel to (001) (Fig. 17c). These sheets link along the a-axis via ribbons of (CaO₃(OH)₂(H₂O))⁶⁻- and (CaO₄(OH)₂(H₂O))⁸⁻-polyhedra (Ca2 and Ca4, respectively) (Fig. 17d). Jennite contains four transformer (H₂O)^t groups and three (OH)⁻ groups.

The ²T₃ [Si₃O₈(OH)]⁵⁻ chains in plombièrite (tobermorite-14Å) link to sheets of (CaO₆(OH))¹¹⁻- (Ca1) and (CaO₆(H₂O))¹⁰⁻-polyhedra (Ca2) that are parallel to (100), forming layers with the composition [Ca₄Si₆O₁₆(OH)₂(H₂O)₂]²⁻ (Fig. 17e). These layers link along the c-axis via chains of (CaO₂(H₂O)₄)-octahedra (Ca3) that extend along the b-axis, parallel to ²T₃ chains (Fig 17f). Plombièrite contains one (OH)⁻ group that bridges (SiO₃(OH)⁻)³⁻-tetrahedra and (CaO₆(OH))¹¹⁻-polyhedra (Ca1). Plombièrite also contains three (H₂O)^t and two (H₂O)^z groups that bond to interlayer Ca3 atoms, a layer that separates adjacent ²T₃ chains and prevents them from polymerising and forming ²T₂³T₄ [Si₆O₁₆]⁸⁻ ribbons as in tobermorite-11Å (see below). In riversideite (tobermorite-9.3Å),²T₃ [Si₃O₈(OH)]⁵⁻ chains link to similar sheets of (CaO₆(OH))¹¹⁻-polyhedra, Ca²⁺-octahedra and ^[7]Ca²⁺-polyhedra but occur within a structure that is much more condensed due to a lower H₂O content. For a more detailed description of hydrated C–S–H minerals, refer to the section on tobermorite-11Å. In both plombièrite and riversideite (tobermorite-group minerals), the following stacking sequence is observed; OTITOTIT, where ‘O’ represents a layer of Ca²⁺-polyhedra, ‘T’ a layer containing ²T₃ chains and ‘I’, an interstitial layer of (H₂O) groups and Ca²⁺-polyhedra that may or may not be present depending on the hydration state (Fig. 17f). In whelanite, the stacking sequence is OTCTOTCT (Fig. 18a), where ‘O’ represents a layer of (CaO₆(OH,H₂O))-polyhedra, ‘T’ is a layer containing ²T₃ chains, and ‘C’ is a layer of (Cu(O,(OH))₆)- and (Ca(O,(OH))₆)-octahedra. In whelanite, the TCT block has OD character with two overlapping, half-occupied ²T₃ chains in which all T sites (Si1a, Si1b and Si2) and anions involved in Si–O–Si linkages are half occupied; anions associated with Si⁴⁺-tetrahedra but not involved in Si–O–Si linkages are half occupied by O²⁻ and half-occupied either by (OH)⁻ or by (H₂O). Despite chemical, spectroscopic and structural evidence for (CO₃)²⁻ groups in whelanite, its position in the structure has yet to be determined due to problems of disorder. Kampf et al. (Reference Kampf, Mills, Merlino, Pasero, McDonald, Wray and Hindman2012) provide a detailed description of both MDO polytypes and their OD characteristics. Figure 18b shows the structure of whelanite in which both overlapping, ²T₃ chains are shown, one in red and the other in orange.

Fig. 18. The structure of whelanite projected (a) along the b-axis and (b) onto (100). In (a), the TOTCTOT stacking sequence is labelled and in (b) one of the overlapping, half-occupied ²T₃ chains is shown as orange tetrahedra on the other chain is shown as red tetrahedra. Fine dashed black lines outline the unit cell and H atoms associated with (OH)⁻ and (H₂O) groups and C atoms associated with (CO₃) groups have been omitted for clarity.

²T₄ chains

The ²T₄ [Si₄O₁₂]⁸⁻ chain in batisite-group minerals (Table 5) contains four distinct Si⁴⁺-tetrahedra that form c-shaped tetramers (Figs 19a–d). In batisite-group minerals, ²T₄ chains extend along the a-axis and link to chains of corner-sharing Ti⁴⁺-octahedra (Fig. 19e). Each Ti⁴⁺-octahedra links to four distinct ²T₄ chains, and chains of octahedra and tetrahedra occur in layers that alternate along the a-axis. Batisite-group minerals also contain three sites A1, A2 and A3. In batisite, these sites are occupied by Ba²⁺, Na⁺ and Na⁺, respectively (Fig. 19f). In scherbakovite, A1 and A2 are occupied by K⁺ and A3 is occupied by Na⁺. In noonkanbahite, A1, A2 and A3 are occupied by Ba²⁺, K⁺ and Na⁺, respectively. Although the presence of (OH)⁻ is only included in the ideal formula of scherbakovite (Table 5), there is evidence for partial occupancy of one O²⁻ site by (OH)⁻ in batisite and noonkanbahite (Uvarova et al., Reference Uvarova, Sokolova, Hawthorne, Liferovich, Mitchell, Pekov and Zadov2010; Zolotarev et al., Reference Zolotarev, Zhitova, Gabdrakhmanova, Krzhizhanovskaya, Zolotarev and Krivovichev2017).

Fig. 19. (a, b, c) Tetrahedral representations of the ²T ₄ chain in batisite-group minerals and (d) a ball-and-stick representation of the chain. The structure of batisite projected (e) onto (100) and (f) along the c-axis. Dashed black lines outline the geometrical repeat unit of the chain and fine dashed black lines outline the unit cell.

The ²T₄ [Si₄O₁₂]⁸⁻ chains in haradaite extend along the c-axis and are linked to each other along the b-axis by sheets of Sr²⁺-polyhedra (Sr1) (Fig. 20a). Chains are also linked along the c-axis by ^[5]V⁴⁺-polyhedra. Sheets of ²T₄ chains and ^[5]V⁴⁺-polyhedra and sheets of Sr²⁺-polyhedra are parallel to (001) and alternate along the b-axis (Fig. 20b). In suzukiite, the Ba²⁺-analogue of haradaite, ²T₄ chains are linked to each other by sheets of ^[11]Ba²⁺-polyhedra. In ohmilite, ²T₄ chains extend along the b-axis and link to a complex layer of Ti⁴⁺-octahedra (M1) and Sr²⁺-polyhedra (Fig. 20c). Corner-sharing Ti⁴⁺-octahedra form chains that extend along the b-axis and link to two ²T₄ chains, forming [Si₈O₂₄(Ti₂O₂)]¹²⁻ ribbons. These ribbons link to each other along the a-axis by (SrO₈(H₂O))¹⁴⁻- (Sr1), (SrO₆(H₂O)₂)¹⁰⁻- (Sr2) and (SrO₇(H₂O))¹²⁻-polyhedra (Sr3) (Fig. 20d). Ohmilite contains one (H₂O)ⁿ and two (H₂O)^t groups. Mizota et al. (Reference Mizota, Komatsu and Chihara1983) have suggested that the substitution Ti⁴⁺ + O²⁻ ↔ Fe³⁺ + OH⁻ may be responsible for the partial occupancy of the M1 site by Fe³⁺ and the OH-stretch in the IR spectrum of ohmilite.

Fig. 20. The structure of haradaite projected (a) onto (001) and (b) along the a-axis (b). The structure of ohmilite projected (c) onto (100) and (d) along the b-axis (c) and into the b-axis. Fine dashed black lines outline the unit cell and H atoms associated with (OH)⁻ and (H₂O) groups have been omitted for clarity.

In the OD structure of fukalite, six MDO polytypes are known; here we describe the fukalite structure based on the MDO1 polytype. Here, ²T₄ chains are strongly modulated, resembling chains in batisite, haradaite and ohmilite. In fukalite, there are two types of layers that are parallel to (100) and alternate along the b-axis (Fig. 21a). Layer 1 consists of planar sheets of (CaO₆(OH))¹¹⁻-polyhedra (Ca1–Ca4) and layer 2 consists of ribbons of (CaO₄(OH)₂)⁸⁻- and (CaO₅(OH)₂)¹⁰⁻-polyhedra (Ca5–Ca8) that link along the c-axis to form a modulated sheet. Layers are linked along the b-axis by (CO₃)²⁻ groups and ²T₄ chains that extend along the a-axis (Fig. 21b). There are four (OH)⁻ groups that bridge Ca²⁺-polyhedra of layer 1 to Ca²⁺-polyhedra of layer 2.

Fig. 21. The structure of fukalite projected (a) along the a-axis and (b) onto (001). In (a), (CO₃) groups are shown in dark grey and H atoms associated with (OH)⁻ groups are omitted for clarity. Fine dashed black lines outline the unit cell.

The ²T₄ chain in taikanite (Figs 22a–d) is geometrically distinct from the ²T₄ chain in the batisite-group minerals. In taikanite, Si⁴⁺-tetrahedra form chains that extend along the b-axis and link to chains of edge-sharing ^[8]Sr²⁺-polyhedra that also extend along the b-axis (Fig. 22e). These chains link along the a-axis via chains of edge-sharing Mn²⁺-octahedra that extend along the c-axis and via ^[8]Ba²⁺-polyhedra. In taikanite, ^[8]Sr²⁺- and ^[8]Ba²⁺-polyhedra and ²T₄ chains occur in layers that alternate along the c-axis (Fig. 22f). The ²T₄ [Si₄O₁₀(OH)₂]⁶⁻ chain in krauskopfite contains (Si(O,OH)₄)-tetrahedra but does not consist of c-shaped tetramers like the chains in batisite-group minerals (Fig. 19b); instead chains are more extended (linear) (Figs 23a–d). In krauskopfite, chains link to chains of edge-sharing ^[8]Ba²⁺-polyhedra (Fig. 23e) that occur in layers parallel to (001) and alternate with layers of ²T₄ chains (Fig. 23f). Krauskopfite contains six H⁺ ions, two of which are associated with (OH)⁻ groups and form acid silicate groups and four are associated with (H₂O) groups bonded to ^[8]Ba²⁺-polyhedra.

Fig. 22. (a, b, c) Tetrahedral representations of the ²T ₄ chain in taikanite and (d) a ball-and-stick representation of the chain. The structure of taikanite projected (e) onto (001) and (f) along the b-axis. Dashed black lines outline the geometrical repeat unit of the chain and fine dashed black lines outline the unit cell.

Fig. 23. (a, b, c) Tetrahedral representations of the ²T ₄ chain in krauskopfite and (d) a ball-and-stick representation of the chain. The structure of krauskopfite projected (e) onto (100) and (f) along the c-axis. Dashed black lines outline the geometrical repeat unit of the chain and fine dashed black lines outline the unit cell. The H atoms associated with (OH)⁻ and (H₂O) groups have been omitted for clarity.

Balangeroite and gageite are asbestiform minerals that have monoclinic (2M) and triclinic (1A) polytypes due to their OD character (Bonaccorsi et al., Reference Bonaccorsi, Ferraris and Merlino2012) (Table 5). Both minerals contain ²T₄ chains that show a higher degree of chain extension than other ²T₄ chains (Figs 24a–d). Chains extend along the b-axis and link to a framework of predominately Mg²⁺-, Mn²⁺- and Fe²⁺-octahedra. Balangeroite and gageite contain twenty-four distinct octahedrally coordinated M-sites that polymerise to form [1] a ribbon of edge-sharing octahedra, and [2] a column of edge-sharing octahedra. These units extend along the b-axis and each occurs in two crystallographically distinct orientations: sites M1–M12 comprise the ribbon and sites M13–M24 comprise the column (Fig. 24e). Here, there are eight distinct Si⁴⁺-tetrahedra that form ²T₄ chains that occupy channels in the framework of octahedra. Each ²T₄ chain links to two crystallographically distinct columns (Fig. 24f) and to one ribbon (Fig. 24g). Balangeroite and gageite contain 20 (OH)⁻ groups that each link to three M-site cations. The Ni–Fe analogue of balangeroite occurs as intergrowths in antigorite serpentinite (Evans and Kuehner, Reference Evans and Kuehner2011).

Fig. 24. (a, b, c) Tetrahedral representations of the ²T ₄ chain in balangeroite and (d) a ball-and-stick representation of the chain. The structure of balangeroite projected (e) along the b-axis and (f, g) the two modes of linkage between ²T ₄ chains and the interstitial structure projected onto (001). Dashed black lines outline the geometrical repeat unit of the chain and fine dashed black lines outline the unit cell. The H atoms associated with (OH)⁻ groups have been omitted for clarity.

Many novel, synthetic compounds contain ²T₄ chains such as BaUO₂[Si₂O₆] (Plaisier et al., Reference Plaisier, Ijdo, de Mello Donego and Blasse1995), NaY[Si₂O₆] (Toebbens et al., Reference Toebbens, Kahlenberg and Kaindi2005), Cu₃Na₂[Si₄O₁₂] (Kawamura and Kawahara, Reference Kawamura and Kawahara1976), Ca₃Mn₂O₂[Si₄O₁₂] (Moore and Araki, Reference Moore and Araki1979) and NaGd[P₄O₁₂] (Amami et al., Reference Amami, Fend and Trabelsi-Ayedi2005).

²T₅ chains

The rhodonite-group minerals rhodonite, ferrorhodonite and vittinkiite (Table 5) contain ²T₅ chains in which Si⁴⁺-tetrahedra link to form c-shaped trimers that are linked by [Si₂O₇]⁶⁻ dimers (Figs 25a–d). Here, ²T₅ chains and ribbons of higher coordination polyhedra (M1–M5) occur in alternating layers that are parallel to (100).

Fig. 25. (a, b, c) Tetrahedral representations of the ²T ₅ chain in rhodonite-group minerals and (d) a ball-and-stick representation of the chain. The structure of rhodonite viewed (e) orthogonal to [110] and (f) along [110]. Dashed black lines outline the geometrical repeat unit of the chain.

The general formula for rhodonite-group minerals may be written as M ₅M _1–3M ₄[T ₅O₁₅], and a new nomenclature scheme was recently provided by Shchipalkina et al. (Reference Shchipalkina, Pekov, Chukanov, Biagioni and Pasero2019b). In these structures, ²T₅ chains extend along [110] and link adjacent ribbons of higher-coordination polyhedra. In vittinkiite, ideally Mn₅[Si₅O₁₅], all M sites are occupied by Mn²⁺. However, such compositions are uncommon, as most rhodonite-group minerals show considerable chemical variability over the M sites (Mason, Reference Mason1975), incorporating Ca²⁺, Fe²⁺, Mg²⁺ and Zn²⁺ in addition to Mn²⁺. The [7]-coordinated M5-site in rhodonite is dominated by Ca²⁺ (Figs 25e,f) and as a result, the formula for rhodonite is ideally CaMn₄[Si₅O₁₅]. In ferrorhodonite, M4 is occupied by Fe²⁺, M1–M3 are occupied by Mn²⁺, and M5 is occupied by Ca²⁺. Figures 26a–f highlights M-site substitution in these three minerals. Rhodonite-group minerals with significant Zn²⁺ and Mg²⁺ have been reported and details on the ordering of M-site cations and the effects of composition on rhodonite structures are given by Peacor and Niizeki (Reference Peacor and Niizeki1963), Peacor et al. (Reference Peacor, Essene, Brown and Winter1978a), Ohashi and Finger (Reference Ohashi and Finger1975), Nelson and Griffen (Reference Nelson and Griffen2005), Leverett et al. (Reference Leverett, Williams and Hibbs2008) and Shchipalkina et al. (Reference Shchipalkina, Chukanov, Pekov, Aksenov, McCammon, Belakovskiy, Britvin, Koshlyakova, Schäfer, Scholz and Rastsvetaeva2017). The compositional range of the rhodonite-group minerals overlaps strongly with that of the bustamite-group minerals and in a minor way with that of pyroxmangite (Shchipalkina et al., Reference Shchipalkina, Pekov, Chukanov, Biagioni and Pasero2019b).

Fig. 26. The structure of (a, b) vittinkiite, (c, d) rhodonite and (e, f) ferrorhodonite viewed (a, c, e) orthogonal to [110] and (b, d, f) along [110]. M-site labels in (a) are also applicable to (c) and (e).

The general formula of lithiomarsturite, marsturite, nambulite and natronambulite can be written as M ₅M ₄M _1–3[T ₅O₁₄(O,OH)] (Table 5). The T1 tetrahedra in lithiomarsturite and marsturite and the T4 tetrahedra in natronambulite and nambulite are acid silicate groups; (SiO₃(OH))³⁻, and ²T₅ chains extend along [110]. The M1–M3 sites in marsturite and the M1 and M3 sites in lithiomarsturite are occupied by Mn²⁺, the M5 site is occupied by Na⁺ and Li⁺, respectively, and the M4 site is occupied by Ca²⁺ in both minerals. In marsturite, ^[7]Ca²⁺-polyhedra (M4) and (NaO₇(OH))¹⁴⁻-polyhedra (M5) link adjacent ribbons of M1–M3-octahedra to form sheets that alternate with layers of ²T₅ chains (Figs 27a,b). In lithiomarsturite, the M5 site is occupied by Li⁺ that forms (LiO₄(OH))⁸⁻-polyhedra and the M2 site is occupied by Ca²⁺ that forms (CaO₅(OH))⁹⁻-octahedra (Figs 27c,d). Although the coordination of Li⁺ in lithiomarsturite, nambulite and natronambulite is not yet agreed upon, Nagashima et al. (Reference Nagashima, Armbruster, Kolitsch and Pettke2014a) used bond-valence arguments to show that Li⁺ is probably [5]-coordinated; here we show Li⁺ as a [5]-coordinated cation (Fig. 27c). In nambulite and natronambulite, the octahedrally coordinated M1–M3 sites are occupied by Mn²⁺, the M4 site is occupied by ^[7]Mn²⁺ and the M5 site is occupied by Li⁺ that forms (LiO₄(OH))⁸⁻-polyhedra in nambulite (Figs 27e,f), and by Na⁺ that forms (NaO₇(OH))¹⁴⁻-polyhedra in natronambulite (Figs 27g,h). In nambulite and natronambulite, the M4 and M5 polyhedra link adjacent ribbons of M1–M3 octahedra to form a sheet similar to that in marsturite and lithiomarsturite. In santaclaraite, T1 is an (SiO₃(OH))³⁻-tetrahedron and ²T₅ chains link to rhodonite-type ribbons of (Mn²⁺(O,(OH),H₂O)₆)-octahedra and Ca²⁺-polyhedra.

Fig. 27. The structure of (a, b) marsturite, (c, d) lithiomarsturite, (e, f) nambulite and (g, h) natronambulite viewed (a, c, e, g) orthogonal to [110] and (b, d, f, h) along [110]. Here, M-site labels in (a) are also applicable to (c), (e) and (g) and H atoms associated with (OH)⁻ groups have been omitted for clarity.

The general formula for the babingtonite-group minerals can be written as A ₂M ₁M ₂[T ₅O₁₄(OH)] (Table 5). The geometrical repeat unit of the ²T₅ [Si₅O₁₄(OH)]⁹⁻ chain in babingtonite contains four Si⁴⁺-tetrahedra and one (SiO₃(OH))³⁻-tetrahedron. These ²T₅ chains link to sheets of ^[8]Ca²⁺-polyhedra (Ca1 and Ca2) and Fe²⁺-octahedra (M1 and M2) that are parallel to (110) (Figs 28a,b). In manganbabingtonite, the M1 site is occupied by Mn²⁺ and in scandiobabingtonite M2 is occupied by Sc³⁺. Various synthetic compounds containing ²T₅ chains have been described including Mn²⁺₅[Si₅O₁₅] (synthetic vittinkiite) and LiMn²⁺₄[Si₅O₁₄(OH)] (synthetic nambulite) (Ito, Reference Ito1972).

Fig. 28. The structure of babingtonite projected (a) onto (001) and (b) orthogonal to the c-axis. Fine dashed black lines outline the unit cell and H atoms associated with (OH)^– groups have been omitted for clarity.

²T₆ chains

The ²T₆ [Si₆O₁₈]¹²⁻ chains in stokesite, georgechaoite and gaidonnayite are modulated in two directions and have been described geometrically as spiral chains (Vorma, Reference Vorma1963; Chao, Reference Chao1985; Yuan et al., Reference Yuan, Guowu and Guangming2017). These ²T₆ chains extend along the b-axis and are modulated along both the c-axis and the a-axis (Figs 29a–d). Chains of tetrahedra link to chains of Sn²⁺-octahedra and (CaO₄(H₂O)^T₂)⁶⁻-octahedra (Fig. 29e), and adjacent chains of octahedra are linked along the c-axis by ²T₆ chains (Fig. 29f). In georgechaoite and gaidonnayite, ²T₆ chains extend along [101] and are modulated along the b-axis (Figs 30a–d). Chains of tetrahedra link to chains of (NaO₄(H₂O)^T₂)⁷⁻- and (ZrO₆)⁸⁻-octahedra that are cross-linked by (KO₄(H₂O)₂)⁷⁻-octahedra in georgechaoite (Figs 30e,f) and by (NaO₄(H₂O)^T₂)⁷⁻-polyhedra in gaidonnayite. In gaidonnayite, Na sites contain minor amounts of K⁺ that may be substituted by Cs⁺ in synthetic Cs-exchanged varieties such as Cs₄Zr₂[Si₆O₁₈](H₂O)₄ (Celestian et al., Reference Celestian, Lively and Xu2019). Synthetic K₈Sr₂[Si₆O₁₈] also contains geometrically similar ²T₆ chains (Kahlenberg et al., Reference Kahlenberg, Kaindl and Sartory2007).

Fig. 29. (a, b, c) Tetrahedral representations of the ²T ₆ chain in stokesite and (d) a ball-and-stick representation of the chain. The structure of stokesite projected (e) onto (001) and (f) along the a-axis. The H atoms of both (H₂O) groups are shown as red circles. Dashed black lines outline the geometrical repeat unit of the chain and fine dashed black lines outline the unit cell.

Fig. 30. (a, b, c) Tetrahedral representations of the ²T ₆ chain in georgechaoite and (d) a ball-and-stick representation of the chain. The structure of georgechaoite projected (e) onto (001) and (f) along the a-axis. Dashed black lines outline the geometrical repeat unit of the chain and fine dashed black lines outline the unit cell. The H atoms associated (H₂O) groups have been omitted for clarity.

²T₇ chains

Pyroxferroite and pyroxmangite are of particular interest as they are the only minerals that contain ²T₇ chains (Figs 31a–d). Chains of tetrahedra link to sheets of octahedra and [7]-coordinated polyhedra (M1–M7) that occur in layers that alternate with layers of ²T₇ chains along [011] (Figs 31e,f). Compositions close to either end-member are rare; there is extensive Fe²⁺–Mn²⁺ solid-solution that also can incorporate significant amounts of Mg²⁺, Ca²⁺, Na⁺ and minor Al³⁺ and Cr³⁺. In pyroxmangite and pyroxferroite, M1–M4 are octahedrally coordinated, M6 is [5]- or [6]-coordinated, and M5 and M7 are [7]-coordinated. In pyroxmangite, M1–M4 are typically occupied predominantly by Mn²⁺ with subordinate Mg²⁺ and Fe²⁺; however, Mg²⁺ dominates at M2–M4 in synthetic Mg-rich pyroxmangite (Finger and Hazen, Reference Finger and Hazen1978). The M6 site is preferentially occupied by Mg²⁺ but may also be occupied by Mn²⁺ and/or Fe²⁺, and M5 and M7 are occupied by Mn²⁺ with minor Mg²⁺ and Fe²⁺. In pyroxferroite, Ca²⁺ preferentially occupies the [7]-coordinated M5 and M7 sites, the M1–M4 sites are occupied by Fe²⁺ with subordinate Mn²⁺ and Ca²⁺, and the M6 site is preferentially occupied by Mg²⁺ but may also contain Mn²⁺ and/or Fe²⁺.

Fig. 31. (a, b, c) Tetrahedral representations of the ²T ₇ chain in pyroxmangite and (d) a ball-and-stick representation of the chain. The structure of pyroxmangite viewed (e) orthogonal to the c-axis and (f) along the c-axis. Dashed black lines outline the geometrical repeat unit of the chain and fine dashed black lines outline the unit cell.

²T₉ chains

The ²T₉ chain occurs only in synthetic ferrosilite III (Table 5) where it extends along the c-axis (Figs 32a–d). There are nine octahedrally coordinated sites (M1–M9) that are occupied by Fe²⁺ and form ribbons that extend parallel to [001]. The ²T₉ chains and ribbons of Fe²⁺-octahedra occur in alternating layers (Figs 32e,f), comparable to the structures of pyroxmangite (Figs 31e,f) pyroxferroite and the rhodonite-group minerals (Figs 25e,f and 26a–f).

Fig. 32. (a, b, c) Tetrahedral representations of the ²T ₉ chain in synthetic ferrosilite III and (d) a ball-and-stick representation of the chain. The structure of synthetic ferrosilite III viewed (e) orthogonal to the c-axis and (f) along the c-axis. Dashed black lines outline the geometrical repeat unit of the chain and fine dashed black lines outline the unit cell.

²T₁₂ chains

The geometrical repeat unit of the chain in alamosite contains twelve Si⁴⁺-tetrahedra that polymerise to form ²T₁₂ chains that extend parallel to [101]. The ²T₁₂ chains are modulated in two directions, resulting in a spiral chain (Figs 33a–d) resembling the ²T₆ chains in stokesite (Figs 29a–d). In alamosite, ²T₁₂ chains link to spiral ribbons of [5]-, [7]- and [6]-coordinated Pb²⁺-polyhedra that occupy three sites: Pb1, Pb2 and Pb3 (Figs 33e,f).

Fig. 33. (a, b, c) Tetrahedral representations of the ²T ₁₂ chain in alamosite and (d) a ball-and-stick representation of the chain. The structure of alamosite projected (e) onto (010) and (f) along the c-axis. Pb1 and Pb2 atoms have been omitted for clarity. Dashed black lines outline the geometrical repeat unit of the chain and fine dashed black lines outline the unit cell.

²T₂₄ chains

The ²T ₂₄ chain occurs only in synthetic Na₂₄Y₈(Si₂₄O₇₂). The geometrical repeat unit of this chain contains twenty-four Si⁴⁺-tetrahedra that polymerise to form a spiral chain modulated along the a- and c-axes (Figs 34a–d). A complete structure description is given by Maksimov et al. (Reference Maksimov, Kalinin, Merinov, Ilyukhin and Belov1980).

Fig. 34. (a, b, c) Tetrahedral representations of the ²T ₂₄ chain in synthetic Na₂₄Y₈[Si₂₄O₇₂] projected (a) onto (100), (b) onto (001), (c) along the b-axis and (d) a ball-and-stick representation of the chain. Dashed black lines outline the geometrical repeat unit of the chain.

³T₄ ribbons

The geometrical repeat unit of the ribbons in vinogradovite, paravinogradovite and bigcreekite (Table 6) contains four distinct tetrahedra that polymerise to form ³T₄ ribbons (Figs 35a–c). Topologically, these ³V₂ ribbons (Fig. 35d) are identical to many of the ³T_r ribbons described in the following sections. Vinogradovite and paravinogradovite contain ²T₂ pyroxene-like chains in addition to ³T₄ ribbons. Vinogradovite contains two distinct tetrahedra: T1 and T2; the T1 tetrahedra form the ²T₂ chains and the T2 tetrahedra form the ³T₄ ribbons. Paravinogradovite contains seven distinct Si⁴⁺-tetrahedra (T1–T7) and one Al³⁺-tetrahedra (T8): T1–T4 tetrahedra form the ²T₂ chains and T5–T8 tetrahedra form the ³T₄ ribbons. In vinogradovite, ²T₂ chains and ³T₄ ribbons extend along the c-axis and link to sheets of Ti⁴⁺-octahedra (M1) and ^[8]Na⁺-polyhedra (X1) that extend parallel to [001] (Fig. 36a). In Fig. 36b, channel 1 is occupied by Na⁺(and K⁺) ions (A1) and (H₂O) groups (W1 and W2). In paravinogradovite, chains and ribbons of tetrahedra extend along the a-axis and link to sheets of [TiO₄(OH)₂]⁶⁻- (M1–M4) and [NaO₅(OH)]¹⁰⁻-octahedra (X1–X2) that extend parallel to (100) (Fig. 36c). Due to its OD character, there are several partly occupied sites in paravinogradovite. As shown in Fig. 36d, channel 1 is partly occupied by Na⁺ (X3), resulting in a discontinuous sheet. Multiple partly occupied sites occur within channel 2, including W1–W3, A1–A4 (Na⁺) and A5 (K⁺). Paravinogradovite contains four H⁺ sites associated with (OH)⁻ groups (Fig. 36d).

Fig. 35. (a, b) Tetrahedral representation of the ³T₄ ribbon in paravinogradovite projected (a) orthogonal to the a-axis (b) onto (010), (c) a ball-and-stick and (d) a graphical representation of the ribbon. The T8 site is occupied by Al³⁺. Dashed black lines outline the geometrical and topological repeat unit of the ribbon.

Fig. 36. The structure of vinogradovite projected (a) onto (100) and (b) along the c-axis. In (b), channel 1 is occupied by Na(K)⁺-polyhedra and H atoms associated with (H₂O) groups have been omitted for clarity. The structure of paravinogradovite projected (c) onto (001) and (d) along the a-axis. In (d), channel 1 is partly occupied by Na⁺-polyhedra and channel 2 is occupied by Na⁺-polyhedra and (H₂O) groups which have been omitted for clarity. Paravinogradovite contains four H sites (H1–H4) associated with (OH)⁻ that are shown as red circles. Fine dashed black lines outline the unit cell which is halved along the a-axis in (b).

Table 6. Minerals with ³T_r ribbons and tubes.

References: (1) Kalsbeek and Rønsbo (Reference Kalsbeek and Rønsbo1992), Rastsvetaeva et al. (Reference Rastsvetaeva, Simonov and Belov1968), Rastsvetaeva and Andrianov (Reference Rastsvetaeva and Andrianov1984); (2) Khomyakov et al. (Reference Khomyakov, Kulikova, Sokolova, Hawthorne and Kartashov2003); (3) Basciano et al. (Reference Basciano, Groat, Roberts, Gault, Dunning and Walstrom2001); (4) Völlenkle et al. (Reference Völlenkle, Wittmann and Nowotny1968); (5) Dörsam et al. (Reference Dörsam, Kahlenberg and Fischer2003); (6) Robinson and Fang (Reference Robinson and Fang1970); (7) Gatta et al. (Reference Gatta, Rotiroti, McIntyre, Guastoni and Nestola2008), Fang et al. (Reference Fang, Robinson and Ohya1972); (8) Cannillo et al. (Reference Cannillo, Rossi and Ungaretti1973), Zubkova et al. (Reference Zubkova, Ksenofontov, Kabalov, Chukanov, Nedelko, Pekov and Pushcharovsky2011); (9) Agakhanov et al. (Reference Agakhanov, Pautov, Karpenko, Sokolova, Abdu, Hawthorne, Pekov and Siidra2015); (10) Grigor'eva et al. (Reference Grigor'eva, Zubkova, Pekov, Kolitsch, Pushcharovsky, Vigasina, Giester, Dordević, Tillmanns and Chukanov2011); (11) Grigor'eva et al. (Reference Grigor'eva, Zubkova, Pekov, Kolitsch, Pushcharovsky, Vigasina, Giester, Dordević, Tillmanns and Chukanov2011); (12) Mellini and Merlino (Reference Mellini and Merlino1978), Hogarth et al. (Reference Hogarth, Chao, Plant and Steacy1974); (13) Brandão et al. (Reference Brandão, Rocha, Reis, dos Santos and Jin2009), Pozas et al. (Reference Pozas, Rossi and Tazzoli1975); (14) Golovachev et al. (Reference Golovachev, Drozdov, Kuz'min and Belov1971), Rozhdestvenskaya et al. (Reference Rozhdestvenskaya, Bannova, Nikishova and Soboleva2004); (15) Khomyakov et al. (Reference Khomyakov, Kurova and Nechelyustov1992), Karimova and Burns (Reference Karimova and Burns2007); (16) Aksenov et al. (Reference Aksenov, Rastsvetaeva, Chukanov and Kolitsch2014), Chukanov et al. (Reference Chukanov, Aksenov, Rastsvetaeva, Blass, Varlamov, Pekov, Belakovskiy and Gurzhiy2015b); (17) Rozhdestvenskaya and Nikishova (Reference Rozhdestvenskaya and Nikishova1998), Ghose and Wan (Reference Ghose and Wan1979); (18) Schmidmair et al. (Reference Schmidmair, Kahlenberg and Grießer2018); (19) Brandão et al. (Reference Brandão, Rocha, Reis, dos Santos and Jin2009); (20) Kornev et al. (Reference Kornev, Maksimov, Lider, Ilyukhin and Belov1972), Kawamura and Kawahara (Reference Kawamura and Kawahara1977), Durand et al. Reference Durand, Vilminot, Richard-Plouet, Derory, Lambour and Drillon1997, Cadoni and Ferraris (Reference Cadoni and Ferraris2011); (21) Peacor and Buerger (Reference Peacor and Buerger1962b), Pyatenko and Pudovkina (Reference Pyatenko and Pudovkina1960), Wagner et al. (Reference Wagner, Parodi, Semet, Robert, Berrahma and Velde1991), Kolitsch and Tillmanns (Reference Kolitsch and Tillmanns2004), Schingaro et al. (Reference Schingaro, Mesto, Lacalamita, Scordari, Kaneva and Vladykin2017); (22) Kolitsch and Tillmanns (Reference Kolitsch and Tillmanns2004); (23) Siidra et al. (Reference Siidra, Krivovichev and Depmeier2009); (24) Hung et al. (Reference Hung, Wang, Kao and Lii2003); (25) Merlino (Reference Merlino1969), Bagiński et al. (Reference Bagiński, Macdonald, White and Jeżak2018), Merlino and Biagioni (Reference Merlino and Biagioni2018); (26) Johnsen et al. (Reference Johnsen, Nielsen and Sotofte1978), Upton et al. (Reference Upton, Hill, Johnsen and Petersen1978); (27) Dunn et al. (Reference Dunn, Rouse, Cannon and Nelen1977), Ghose and Wan (Reference Ghose and Wan1978); (28) Rastsvetaeva et al. (Reference Rastsvetaeva, Rozenberg, Khomyakov and Rozhdestvenskaya2003), Dorfman et al. (Reference Dorfman, Rogachev, Goroshchenko and Uspenskaya1959), Rozhdestvenskaya et al. (Reference Rozhdestvenskaya, Nikishova, Bannova and Lasebnik1987); (29) Rastsvetaeva et al. (Reference Rastsvetaeva, Rozenberg, Khomyakov and Rozhdestvenskaya2003), Khomyakov et al. (Reference Khomyakov, Nechelyustov, Krivokoneva, Rastsvetaeva, Rozenberg and Rozhdestvenskaya2009); (30) Rozhdestvenskaya et al. (Reference Rozhdestvenskaya, Nikishova and Lazebnik1996), Nikishova et al. (Reference Nikishova, Lazebnik, Rozhdestvenskaya, Emelyanova and Lazebnik1996); (31) Rozhdestvenskaya and Evdokimov (Reference Rozhdestvenskaya and Evdokimov2006), Scott (Reference Scott1976), Rozhdestvenskaya et al. (Reference Rozhdestvenskaya, Nikishova, Bannova and Lasebnik1987), Kaneva et al. (Reference Kaneva, Lacalamita, Mesto, Schingara, Scordari and Vladykin2014); (32) Men'shikov (Reference Men'shikov1984), Konev et al. (Reference Konev, Vorobiev, Paradina and Sapozhnikov1987), Rozhdestvenskaya et al. (Reference Rozhdestvenskaya, Mugnaioli, Schowalter, Schmidt, Czank, Depmeier and Rosenauer2017); (33) Schafer and Schleid (Reference Schäfer and Schleid2007); (34) Hung et al. (Reference Hung, Wang, Kao and Lii2003); (35) Kahlenberg and Manninger (Reference Kahlenberg and Manninger2014); (36) Rozhdestvenskaya et al. (Reference Rozhdestvenskaya, Mugnaioli, Czank, Depmeier, Kolb and Merlino2011), Chiragov and Shirinova (Reference Chiragov and Shirinova2004), Matesanz et al. (Reference Matesanz, Garcia-Guinea, Crespo-Feo, Lopez-Arce, Valle-Fuentes and Correcher2008), Rozhdestvenskaya et al. (Reference Rozhdestvenskaya, Mugnaioli, Czank, Depmeier, Kolb, Reinholdt and Weirich2010), Rozhdestvenskaya et al. (Reference Rozhdestvenskaya, Kogure, Abe and Drits2009); (37) Sassi et al. (Reference Sassi, Gramlich, Miéhé-Brendié, Josien, Paillaud, Benggedach and Patarin2003); (38) Moore et al. (Reference Moore, Sen Gupta, Schlemper and Merlino1987).

* Indicates the ^cT_r expression of an additional structural unit including a chain, ribbon, tube, cluster or sheet of [TO₄]^n– tetrahedra in the respective mineral.

The ³T ₄ ribbon in bigcreekite extends along the a-axis and links to modulated sheets of (BaO₂(H₂O)₇)²⁻-polyhedra that are parallel to (100) (Fig. 37a). ³T₄ ribbons and sheets of Ba²⁺-polyhedra alternate along the b-axis (Fig. 37b). This type of ³T₄ ribbon occurs in several synthetic compounds including Li₄(SiGe₃O₁₀) and Cs₂H₂(Si₄O₁₀) (Völlenkle et al., Reference Völlenkle, Wittmann and Nowotny1968; Dörsam et al., Reference Dörsam, Kahlenberg and Fischer2003).

Fig. 37. The structure of bigcreekite projected (a) onto (001) and (b) along the a-axis. Fine dashed black lines outline the unit cell and H atoms associated with (H₂O) groups have been omitted for clarity.

³T₆ ribbons

The epididymite group includes epididymite, eudidymite, elpidite and yusupovite (Table 6), all of which contain ³T₆ [Si₆O₁₅]⁶⁻ ribbons with the vertex degree ³V₂ (Figs 38a–d). In the two dimorphs, epididymite and eudidymite, ribbons extend along [001] and are linked to adjacent chains along [100] and [010] by pairs of edge-sharing Be²⁺-tetrahedra, forming an open-framework of tetrahedra. Cavities within this framework are occupied by (H₂O) and Na⁺ that forms (NaO₆(H₂O))¹¹⁻- and (NaO₅(H₂O)₂)⁹⁻-polyhedra in epididymite and eudidymite, respectively. Although both minerals contain frameworks of tetrahedra rather than ribbons of tetrahedra, they have been added to Table 6 for purposes of comparison. In elpidite, ³T₆ ribbons extend along [100] and are linked to each other along [010] and [001] by Zr⁴⁺- and (NaO₄(H₂O)₂)⁷⁻-octahedra rather than Be²⁺-tetrahedra, forming an open-framework that contains ³T₆ ribbons. Cavities within this framework are occupied by (H₂O) and Na⁺ that forms (NaO₆(H₂O))¹¹⁻-polyhedra (Figs 39a,b). In yusupovite, the monoclinic dimorph of elpidite, ³T₆ ribbons extend along [010] and there are six T sites rather that three as in the other epididymite-group minerals. Various synthetic, K⁺- and Rb⁺-exchanged analogues of elpidite (K₂Zr[Si₆O₁₅](H₂O) and Rb₂Zr[Si₆O₁₅](H₂O)) have been described by Grigor'eva et al. (Reference Grigor'eva, Zubkova, Pekov, Kolitsch, Pushcharovsky, Vigasina, Giester, Dordević, Tillmanns and Chukanov2011).

Fig. 38. (a, b) Tetrahedral representation of the ³T₆ ribbon in epididymite group projected (a) onto (010), (b) onto (100), (c) a ball-and-stick and (d) a graphical representation of the ribbon. Dashed black lines outline the geometrical and topological repeat unit of the ribbon.

Fig. 39. The structure of elpidite projected (a) onto (010) and (b) onto (100). The [NaO₄(H₂O)₂]^7–-octahedra (Na2) are associated with the W1 site and [NaO₆(H₂O)]^11–-polyhedra (Na1) are shown as green circles and are associated with the W2 site. The Zr⁴⁺-octahedra in (b) have been omitted for clarity. Fine dashed black lines outline the unit cell.

³T₈ ribbons and tubes

The ³T₈ [Si₈O₂₀]⁸⁻ ribbon in caysichite-(Y) has a geometrical repeat unit that contains a pair of four-membered rings that are geometrically distinct from each another (Figs 40a–c). This ³V₂ ribbon (Fig. 40d) is topologically identical to the ribbon in epididymite-group minerals (Fig. 38d) paravinogradovite, vinogradovite and bigcreekite (Fig. 35d). In caysichite-(Y), [Si₈O₂₀]⁸⁻ ribbons extend along the c-axis and are linked to ribbons of (Ca,REE)- and Y-polyhedra (Ca1 and Y1) which are coordinated by oxygen atoms from (CO₃)²⁻ groups, (H₂O) groups and (OH)⁻ groups (Fig. 41a). Each [Y₄(Ca,REE)₄(CO₃)₆(OH)(H₂O)₇] ribbon extends along the c-axis and is linked to four other ribbons along the a-axis and b-axis, forming an open framework. The ³T₈ ribbons extend along the tunnels of this framework and each links to four of these ribbons (Fig. 41b).

Fig. 40. (a, b) Tetrahedral representation of the ³T₈ ribbon in caysichite-(Y) projected (a) onto (010), (b) onto (100), (c) a ball-and-stick and (d) a graphical representation of the ribbon. Dashed black lines outline the geometrical and topological repeat unit of the ribbon.

Fig. 41. The structure of caysichite-(Y) projected (a) onto (100) and (b) along the c-axis. In (a), the C and O atoms of the (CO₃) groups are shown as dark grey and red circles, respectively. In (a) and (b), Y³⁺ ions are shown as teal circles. In (a), H atoms associated with (H₂O) and (OH)⁻ groups are omitted and in (b) C atoms associated with (CO₃) groups are also omitted for clarity. Fine dashed black lines outline the unit cell.

The ³T₈ [Si₈O₂₀]⁸⁻ tube in litidionite-group minerals extends along the a-axis and consists of two linked chains of four-membered rings, each topologically identical to the ³T₂²T₂ chain in revdite (see below) and the ³T₄²T₄ chain in vlasovite (see below). In litidionite, adjacent four-membered rings of opposing chains link to each other across the tube via two tetrahedra, forming eight-membered rings that link along the a-axis (Figs 42a,b), and a six-membered ring that can be viewed into the a-axis (Fig. 42c). In this tube n_g = n_t as shown in Fig. 42d.

Fig. 42. (a, b, c) Tetrahedral representation of the ³T₈ tube in litidionite projected (a, b) orthogonal to the a-axis and (c) along the a-axis, (d) a ball-and-stick (graphical) representation of the tube. Dashed black lines outline the geometrical and topological repeat unit of the tube.

Members of the litidionite group include manaksite, fenaksite and calcinaksite, and there are isostructural synthetic structures such as K₂Ca[Si₄O₁₀] (Schmidmair et al., Reference Schmidmair, Kahlenberg and Grießer2018), KNaM[Si₄O₁₀] (M = Cu²⁺, Mn²⁺ and Fe²⁺) (Brandão et al., Reference Brandão, Rocha, Reis, dos Santos and Jin2009) and Na₂M[Si₄O₁₀] (M = Co²⁺, Ni²⁺, Cu²⁺ and Mn²⁺) (Kornev et al., Reference Kornev, Maksimov, Lider, Ilyukhin and Belov1972, Kawamura and Kawahara, Reference Kawamura and Kawahara1977, Durand et al., Reference Durand, Vilminot, Richard-Plouet, Derory, Lambour and Drillon1997, Cadoni and Ferraris, Reference Cadoni and Ferraris2011). Agrellite (Table 6) is not isostructural with the litidionite-group minerals but contains a tube that is topologically identical. In litidionite, ³T₈ tubes link along the c-axis via ribbons of ^[5]Cu²⁺- and ^[7]Na⁺-polyhedra and ^[8]K⁺ ions occupy cavities within the tube (Figs 43a,b). In agrellite, tubes of tetrahedra extend along the c-axis and link along the b-axis via sheets of (CaO₅F₂)¹⁰⁻-polyhedra (Ca1 and Ca4) and (CaO₅F)⁹⁻-octahedra (Ca2 and Ca3). The Ca1 site may be partly occupied by REEs and Sr²⁺, and cavities within the tube of tetrahedra are occupied by ^[8]Na⁺ ions (Figs 43c,d). In both structures, ³T₈ tubes and ribbons of higher coordination polyhedra occur in layers that alternate along [001] in litidionite-group minerals and along [010] in agrellite. Fenaksite and manaksite are the Fe²⁺- and Mn²⁺-analogues of litidionite, and calcinaksite is the Ca²⁺-analogue of litidionite that also contains (H₂O).

Fig. 43. The structure of litidionite viewed (a) orthogonal to the a-axis and (b) along the a-axis. The structure of agrellite viewed (c) orthogonal to the c-axis and (d) along the c-axis. In (d), F1 and F2 anions are shown as green circles. Fine dashed black lines outline the unit cell.

The tube in narsarsukite extends along the c-axis and consists of four-membered rings of corner-sharing tetrahedra that link to adjacent tetrahedra, along [001], through two tetrahedra of each ring (Figs 44a–c). Topologically, this ³V₈ tube (Fig. 44d) is similar to the tube in litidionite but not identical (Fig. 42d). This tube extends parallel to chains of corner-sharing (TiO₅(O,OH,F)-octahedra that extend along the c-axis and each link to four ³T₈ tubes to form an open framework; channels in this framework are occupied by ^[7]Na⁺ ions (Figs 45a,b). Schingaro et al. (Reference Schingaro, Mesto, Lacalamita, Scordari, Kaneva and Vladykin2017) report the partial substitution Ti⁴⁺ + O²⁻ ↔ Fe³⁺ + F⁻, (OH)⁻ at octahedrally coordinated sites. There are various synthetic compounds with ³T₈ ribbons and tubes: K₂Sc[Si₄O₁₀]F (Kolitsch and Tillmanns, Reference Kolitsch and Tillmanns2004), Pb₆O[Si₆Al₂]O₂₀ (Siidra et al., Reference Siidra, Krivovichev and Depmeier2009) and K₂In[Si₄O₁₀](OH) (Hung et al., Reference Hung, Wang, Kao and Lii2003) (Table 6).

Fig. 44. (a, b, c) Tetrahedral representation of the ³T₈ tube in narsarsukite projected (a) orthogonal to the c-axis, (b) onto (010), (c) along the c-axis and (d) a ball-and-stick (graphical) representation of the tube. Dashed black lines outline the geometrical and topological repeat unit of the tube.

Fig. 45. The structure of narsarsukite projected (a) onto (010) and (b) along the c-axis. Fine dashed black lines outline the unit cell and H atoms associated with (OH)^– groups have been omitted for clarity

³T₁₂ ribbons and tubes

In tuhualite, ³T₁₂ ribbons extends along the c-axis and are modulated along the b-axis (Figs 46a–d). This ³V₂ ribbon (Fig. 46e) is topologically identical to the ribbons in paravinogradovite, vinogradovite, bigcreekite, (Fig. 35d) the epididymite-group minerals (Fig. 38d) and caysichite-(Y) (Fig. 40d). Tuhualite contains three octahedrally coordinated sites occupied by Fe²⁺, Fe³⁺ (Fe1 and Fe2) and Na⁺ (Na1); these octahedra polymerise to form sheets that link to ³T₁₂ ribbons. This linkage forms channels that extend along the a-axis and are occupied by octahedrally coordinated Na⁺ and (H₂O) groups with subordinate K⁺ (Fig. 47a). Sheets of octahedra and ³T₁₂ ribbons occur in layers that alternate along the a-axis (Fig. 47b). In zektzerite and emeleusite, the Li⁺-analogues of tuhualite, ³T₁₂ ribbons link to sheets of Fe³⁺-octahedra (Zr⁴⁺-octahedra in zektzerite), Na⁺-polyhedra and Li⁺-tetrahedra. In both minerals, adjacent chains are linked to each other by Li⁺-tetrahedra, forming a framework rather than a ribbon of tetrahedra. However, both minerals are included in Table 6 as they belong to the tuhualite group.

Fig. 46. (a, b, c) Tetrahedral representation of the ³T₁₂ ribbon in tuhualite projected (a) onto (010), (b) orthogonal to the c-axis, (c) along the c-axis, (d) a ball-and-stick and (e) a graphical representation of the ribbon. Dashed black lines outline the geometrical and topological repeat unit of the ribbon.

Fig. 47. The structure of tuhualite projected (a) onto (100) and (b) along the c-axis. Fine dashed black lines outline the unit cell and H atoms associated with (H₂O) groups have been omitted for clarity.

The ³T₁₂ [Si₁₂O₃₀]¹²⁻ tubes in canasite and fluorcanasite extend along [010] and consist of two linked ribbons of six-membered rings (Figs 48a,b) that form a tube (Fig. 48c). Topologically, this ³V₁₂ tube (Fig. 48d) is similar to the [Si₁₂O₃₀]¹²⁻ tube in charoite (see below). In both minerals, ³T₁₂ tubes link to corrugated sheets of Ca²⁺- and Na⁺-octahedra that are parallel to (100). Channels within each ³T₁₂ tube are occupied by K⁺ ions (K1–K4) and by an additional (H₂O) group in fluorcanasite. In canasite, there are four (OH)⁻ sites whereas in fluorcanasite, there are two F sites (F2 and F3), an (OH) site and a split site that typically contains F > (OH) (F1), all of which bond to Na⁺- and Ca²⁺-octahedra. In fluorcanasite, ³T₁₂ tubes and sheets of (CaO₄(OH)F)⁸⁻-octahedra (Ca1–Ca3) and (NaO₄F₂)⁹⁻-octahedra (Na1–Na3) occur in layers that alternate along the c-axis (Figs 49a,b). Frankamenite is the triclinic polymorph of fluorcanasite. In miserite, ³T₁₂ tubes extend along [001] and link to a more complicated slab that occurs in layers parallel (100). These slabs are composed of (CaO_7–xF_x)- and (CaO_6–xF_x)-polyhedra (where x = 1–2) and [Si₂O₇]⁶⁻ dimers. Cavities are occupied by K⁺ ions (K1–K2) and (H₂O) groups (Figs 49c,d).

Fig. 48. (a, b, c) Tetrahedral representation of the ³T₁₂ tube in canasite projected (a) onto (100), (b) orthogonal to the b-axis, (c) along the b-axis and (d) a ball-and-stick (graphical) representation of the tube. Dashed black lines outline the geometrical and topological repeat unit of the tube.

Fig. 49. The structure of fluorcanasite projected (a) onto (001) and (b) along the b-axis. The structure of miserite projected (c) onto (010) and (d) along the c-axis. In both structures, (OH)^– and (H₂O) groups are shown as red circles and F anions are shown as green circles. Fine dashed black lines outline the unit cell.

Other ³T_n tubes

There are several synthetic compounds that contain ³T₁₆ tubes and ribbons that are topologically distinct, with the vertex connectivity ³T₈ or ³T₁₆. Figures 50a–d show the ³T₁₆ ribbon in synthetic Cs₄Y₂[Si₈O₂₀]F₄ that consists of four- and eight-membered rings of Si⁴⁺-tetrahedra that link along [010]; this ribbon has a vertex degree ³V₈ (Fig. 50e). Although this ribbon somewhat resembles that of a tube (Fig. 50c), it does not form a contiguous hollow cylinder (Figs 50b,d) and is therefore a ribbon. One can also more rigorously differentiate ribbons and tubes on the basis of topology. All ribbon-graphs can be represented in 2-dimensions in which no edges cross and no vertices overlap (Fig. 50e); this is not possible for tube-graphs. This distinction represents an important topological divide in 1-dimensional graphs and will be discussed in detail in a subsequent paper. In synthetic Cs₄Y₂[Si₈O₂₀]F₄, ³T₁₆ ribbons link to each other via chains of (YO₄F₂)⁷⁻-octahedra to form an open framework in which cavities are occupied by Cs⁺ ions (Schäfer and Schleid, Reference Schäfer and Schleid2007). Synthetic K₄In₂[Si₈O₂₀](OH)₂ contains ³T₁₆ tubes that consist of four-, six- and eight-membered rings of Si⁴⁺-tetrahedra that link along [010] (Figs 51a,b); this tube has a vertex degree ³V₁₆ (Fig. 51c). The structure of synthetic K₄In₂[Si₈O₂₀](OH)₂ is similar to that of synthetic Cs₄Y₂[Si₈O₂₀]F₄ where (YO₄F₂)⁷⁻-octahedra are replaced by (InO₄(OH)₂)⁷⁻-octahedra and Cs⁺ is replaced by K⁺ (Hung et al., Reference Hung, Wang, Kao and Lii2003). The other synthetic compounds isostructural with K₄In₂[Si₈O₂₀](OH)₂ are K₄Lu₂[Si₈O₂₀](OH)₂ and Ru₄Lu₂[Si₈O₂₀]F₂ (Kahlenberg and Manninger, Reference Kahlenberg and Manninger2014).

Fig. 50. (a, b, c) Tetrahedral representation of the ³T₁₆ ribbon in synthetic Cs₄Y₂[Si₈O₂₀]F₄ projected (a) orthogonal to the b-axis, (b) onto (001), (c) along the b-axis, (d) a ball-and-stick and (e) a graphical representation of the ribbon. Dashed black lines outline the geometrical and topological repeat unit of the ribbon.

Fig. 51. (a, b, c) Tetrahedral representation of the ³T₁₆ tube in synthetic K₄In₂[Si₈O₂₀](OH)₄ projected (a) orthogonal to the b-axis, (b) along the b-axis and (c) a ball-and-stick (graphical) representation of the tube. Dashed black lines outline the geometrical and topological repeat unit of the tube-ribbon.

Charoite has several polytypes; for simplicity we briefly describe only one: charoite-96; for more detailed descriptions, see Rozhdestvenskaya et al. (Reference Rozhdestvenskaya, Mugnaioli, Czank, Depmeier, Kolb, Reinholdt and Weirich2010, Reference Rozhdestvenskaya, Mugnaioli, Czank, Depmeier, Kolb and Merlino2011). Charoite contains a ³T₁₇ tube (Figs 52a–c), a ³T₁₂ tube (Figs 52d–f) and a ²T₄³T₂ ribbon (Figs 52g–i), all of which extend along [001]. Each tube and ribbon links to an open framework of (NaO₅(OH))¹⁰⁻- and Ca²⁺-octahedra. Charoite contains eleven sites occupied by alkali-metal and alkaline-earth-metal ions: K1–K7 (K7 is mostly vacant), Sr1 ions and W1–W3 groups occupy cavities within the ³T₁₇ and ³T₁₂ tubes and within the framework of octahedra (Fig. 53). The (OH)⁻ groups are assumed to form (Si(O,OH)₄), acid silicate groups and link to Na⁺ ions to form (NaO₅OH)¹⁰⁻-octahedra. Denisovite has a similar structure but contains only ³T₁₂ and ²T₄³T₂ ribbons that extend along [001] and link to an open framework of Ca²⁺- and Na⁺-octahedra that are coordinated predominately by O²⁻ and subordinate F⁻ and (OH)⁻. In denisovite, cavities within the ³T₁₂ tubes and within the framework of octahedra are occupied by K⁺ ions and (H₂O) groups.

Fig. 52. (a, b, c) Tetrahedral representation of the ³T₁₇ tube in charoite projected (a) onto (100), (b) along the c-axis and (c) a ball-and-stick representation of this tube. (d, e, f) the ³T₁₂ tube in charoite projected (d) orthogonal to the c-axis, (e) along the c-axis and (f) a ball-and-stick representation of this tube. (g, h, i) the ²T₄³T₂ ribbon in charoite projected (g) onto (100), (h) along the c-axis and (i) a ball-and-stick representation of this ribbon. Dashed black lines outline the geometrical and topological repeat unit of the tube or ribbon.

Fig. 53. The structure of charoite projected onto (001). Fine dashed black lines outline the unit cell and H atoms associated with (H₂O) and (OH)⁻ groups have been omitted for clarity.

Synthetic Na₁₆[Si₃₂O₆₄(OH)₁₆] contains ³T₃₂ ribbons that are topologically identical to the other ³T_n, ³V₂ ribbons described above. These ³T₃₂ ribbons consist of four-membered rings of Si⁴⁺- and (SiO₃(OH))³⁻-tetrahedra that link along [102] to form planar ribbons. The ³T₃₂ ribbons occur in layers perpendicular to (010) and link to sheets of Na⁺-polyhedra that are also perpendicular to (010) (Sassi et al., Reference Sassi, Gramlich, Miéhé-Brendié, Josien, Paillaud, Benggedach and Patarin2003).

Ashcroftine-(Y) is the largest and most complicated inosilicate that is currently known. It contains ³T₅₆ tubes that consist of complex cages of Si⁴⁺-tetrahedra that link to each other along [001] by four ³T₁–³T₁ linkages. Each cage consists of four-, seven- and eight-membered rings (Figs 54a–e). Due to the complicated nature of the ashcroftine-(Y) structure, we do not provide a total structure description or figure here and instead, refer readers to Moore et al. (Reference Moore, Sen Gupta, Schlemper and Merlino1987).

Fig. 54. (a, b, c) Tetrahedral representation of the ³T₅₆ tube in ashcroftine projected (a) onto (010), (b) orthogonal to the c-axis, (c) along the c-axis, and (d, e) a ball-and-stick (graphical) representation of the tube. Dashed black lines outline the geometrical and topological repeat unit of the tube.

¹T₂³T₂ chains

The astrophyllite supergroup includes minerals from the astrophyllite, kupletskite and devitoite groups (Table 7) (Sokolova et al., Reference Sokolova, Cámara, Hawthorne and Ciriotti2017a), all of which contain ¹T₂³T₂ [Si₄O₁₂]⁸⁻ chains that consist of a ²T₂ chain in which each tetrahedron is decorated by a 1-connected tetrahedron (Figs 55a–d). The geometrical repeat unit contains four tetrahedra whereas the topological repeat unit contains only two vertices (¹V₁³V₁) (Fig. 55e). The ¹T ₂³T ₂ chains extend along the a-axis and link to adjacent chains along the b-axis via octahedrally or [5]-coordinated Ti⁴⁺, Nb⁵⁺ or Fe³⁺ (plus minor other ions) occupying the D site to form the H-sheet (Fig. 56a). There are four octahedrally coordinated M-sites (M1–M4) that are occupied dominantly by Fe²⁺ (astrophyllite group) and Mn²⁺ (kupletskite group) although minor amounts of other cations (e.g. Zn²⁺; Piilonen et al., Reference Piilonen, LaLonde, McDonald, Gault and Larsen2003a,Reference Piilonen, McDonald and LaLondeb, Reference Piilonen, Pekov, Back, Steede and Gault2006) can be incorporated. The resulting octahedra share edges to form a continuous sheet, called the O-sheet (Fig. 56b). The O-sheet is enclosed between two H-sheets to form an HOH block (Fig. 56c) (Sokolova, Reference Sokolova2012; Sokolova et al., Reference Sokolova, Cámara, Hawthorne and Ciriotti2017a), similar to the HOH block in TS-block minerals (Sokolova, Reference Sokolova2006; Sokolova and Cámara, Reference Sokolova and Cámara2017). Species that occur between adjacent TS-blocks form the I-block. Astrophyllite-supergroup minerals that contain HOH blocks that link to one another directly (along the c-axis) through common vertices of the D-polyhedron are designated as Type-1 structures and include the astrophyllite- and kupletskite-group minerals. Astrophyllite-supergroup minerals that contain HOH blocks that do not link directly are designated as type-2 structures and are members of the devitoite group (Table 7).

Fig. 55. (a, b, c) Tetrahedral representations of the ¹T₂³T₂ chain in astrophyllite projected (a) onto (010), (b) orthogonal to the a-axis, (c) along the a-axis, (d) a ball-and-stick and (e) a graphical representation of the chain. Dashed black lines outline the geometrical and topological repeat unit of the chain.

Fig. 56. The structure of astrophyllite highlighting the (a) H-sheet (and I-block cations) and (b) O-sheet projected orthogonal to the a-axis and (c) the HOH-block and I-block viewed along the a-axis. The A and B, I-block cations are labelled, and the H atoms associated with (OH)⁻ groups are shown as red circles. Fine dashed black lines outline the unit cell.

Table 7. Minerals with ¹T_r³T_r and ²T₂³T_r ribbons.

References: (1) Cámara et al. (Reference Cámara, Sokolova, Abdu and Hawthorne2010), Zhitova et al. (Reference Zhitova, Krivovichev, Hawthorne, Krzhizhanovskaya, Zolotarev, Abdu, Yakovenchuk, Pakhomovsky and Goncharov2017), Woodrow (Reference Woodrow1967), Shi et al. (Reference Shi, Ma, Li, Yamnova and Pushcharovskii1998), Piilonen et al. (Reference Piilonen, McDonald and LaLonde2003b); (2) Agakhanov et al. (Reference Agakhanov, Pautov, Sokolova, Abdu and Karpenko2016); (3) Uvarova et al. (Reference Uvarova, Sokolova, Hawthorne, Agakhanov and Pautov2008), Agakhanov et al. (Reference Agakhanov, Pautov, Uvarova, Sokolova, Hawthorne and Karpenko2008, Reference Agakhanov, Pautov, Sokolova, Abdu and Karpenko2016); (4) Nickel et al. (Reference Nickel, Rowland and Charette1964), Cámara et al. (Reference Cámara, Sokolova, Abdu and Hawthorne2010); (5) Stepanov et al. (Reference Stepanov, Bekenova, Levin, Sokolova, Hawthorne and Dobrovol'skaya2012); (6) Kapustin (Reference Kapustin1973), Sokolova and Hawthorne (Reference Sokolova and Hawthorne2016), Sokolova et al. (Reference Sokolova, Cámara, Hawthorne and Ciriotti2018a); (7) Christiansen et al. (Reference Christiansen, Johnsen and Stahl1998), Piilonen et al. (Reference Piilonen, McDonald and Lalonde2001, Reference Piilonen, Pekov, Back, Steede and Gault2006); (8) Cámara et al. (Reference Cámara, Sokolova, Abdu and Hawthorne2010); (9) Cámara et al. (Reference Cámara, Sokolova, Abdu and Hawthorne2010), Piilonen et al. (Reference Piilonen, Lalonde, Mcdonald and Gault2000); (10) Sokolova et al. (Reference Sokolova, Day, Hawthorne, Kasatkin, Downs, Horváth and Pfenninger-Horváth2019); (11) Kampf et al. (Reference Kampf, Rossman, Steele, Pluth, Dunning and Walstrom2010); (12) Sokolova et al. (Reference Sokolova, Cámara, Hawthorne, Semenov and Ciriotti2017b), Sokolova and Cámara (Reference Sokolova and Cámara2008), Shi et al. (Reference Shi, Ma, Li, Yamnova and Pushcharovskii1998); (13) Khomyakov et al. (Reference Khomyakov, Cámara, Sokolova, Abdu and Hawthorne2011); (14) Sokolova et al. (Reference Sokolova, Day, Hawthorne and Kristiansen2018b); (15) Cámara et al. (Reference Cámara, Sokolova, Abdu and Hawthorne2014), Ferraris et al. (Reference Ferraris, Ivaldi, Khomyakov, Sobolova, Belluso and Pavese1996); (16) Cámara et al. (Reference Cámara, Sokolova, Hawthorne, Rowe, Grice and Tait2013); (17) Oberti et al. (Reference Oberti, Hawthorne, Ungaretti and Cannillo1993); (18) Evans and Mrose (Reference Evans and Mrose1966), Evans and Mrose (Reference Evans and Mrose1977); (19) Yang et al. (Reference Yang, Downs, Evans and Pinch2014), Kolitsch et al. (Reference Kolitsch, Merlino, Belmonte, Carbone, Cabella, Lucchetti and Ciriotti2018); (20) Czank and Bissert (Reference Czank and Bissert1993); (21) Bakakin and Soloveva (Reference Bakakin and Soloveva1970); (22) Veblen and Burnham (Reference Veblen and Burnham1978b); (23) Konishi et al. (Reference Konishi, Akai and Kurokawa1993); (24) Tateyama et al. (Reference Tateyama, Shimoda and Sudo1978), Ams et al. (Reference Ams, Jenkins, Boerio-Goates, Morcos, Navrotsky and Bozhilov2009); (25) Hesse and Liebau (Reference Hesse and Liebau1980), Filipenko et al. (Reference Filipenko, Pobedimskaya, Ponomarev and Belov1971); (26) Merlino (Reference Merlino1983); (27) Downs et al. (Reference Downs, Pinch, Thompson, Evans and Megaw2016); (28) Rastsvetaeva and Aksenov (Reference Rastsvetaeva and Aksenov2011); (29) Hesse and Liebau (Reference Hesse and Liebau1980), Wang et al. (Reference Wang, Xu, Zhou, Yu and Qiu2015); (30) Hesse and Liebau (Reference Hesse and Liebau1980).

* Indicates the ^cT_r expression of an additional structural unit including a chain, ribbon, tube, cluster or sheet of [TO₄]^n– tetrahedra in the respective mineral.

In astrophyllite-group minerals, the A and B sites are occupied by K⁺ and Na⁺, respectively (Figs 56a), except in nalivkinite and bulgakite (in which the A site is occupied by Li⁺ with associated (H₂O) to complete its coordination), and in tarbagatite and bulgakite (in which the B site may contain dominant Ca²⁺). In kupletskite-group minerals, the A and B sites are occupied by K⁺ and Na⁺, respectively, except in kupletskite-(Cs) in which the A site is occupied by Cs⁺. Laverovite is a kupletskite-group mineral but differs from the others in having Zr⁴⁺ occupying the D site. The devitoite-group minerals have much more diverse I-blocks, in accord with the absence of direct linkage between adjacent HOH blocks, and also more ordered O-sheets. Thus, lobanovite has (most importantly) Na⁺ at the M1 site and Mg²⁺ at the M4 site. Sveinbergeite has (H₂O) at the A site, Ca²⁺ at the B site, and 1 apfu Fe³⁺ at the M sites. Devitoite (Figs 57a,b) has Ba²⁺ at both the A and B sites, [5]-coordinated Fe³⁺ at the D site, and (PO₄)³⁻₂ and (CO₃)²⁻ groups in the I-block; in heyerdahlite, Mn²⁺ occupies the M sites.

Fig. 57. The structure of devitoite highlighting the (a) H- and O-sheet projected onto (001) and the (b) HOH-block and I-block viewed along the a-axis. In (b), the C and O atoms of the (CO₃) groups are shown as dark grey and red circles, respectively. The I-block, [PO₄] groups are shown as yellow polyhedra and ^[5]Fe³⁺-polyhedra are shown in dark pink to differentiate them from Fe²⁺-octahedra. Fine dashed black lines outline the unit cell and H atoms associated with (OH)^– groups are omitted for clarity.

¹T₂³T₄ ribbons

The ¹T₂³T₄ [Si₆O₁₇]¹⁰⁻ ribbon in nafertisite consists of ribbons of edge-sharing hexagons, where each six-membered ring is decorated by two 1-connected tetrahedra (Figs 58a–e). The ¹T₂³T₄ ribbons extends along [100] and are linked along [001] to sheets of (Fe²⁺O₄(OH)₂)⁸⁻-octahedra (M1–M3) that are parallel to the a–b plane (110) (O-sheet) (Fig. 59a). Here, ¹T₂³T₄ ribbons occur in layers parallel to (110) and are linked to adjacent ribbons along [010] via (ZrO₅F)⁷⁻-octahedra (D site), forming the H-sheet. The linkage of O- and H-sheets forms HOH blocks that are linked to adjacent blocks along [001] through common F⁻ anions of the D-site octahedra. Similar structure modules occur in astrophyllite-supergroup minerals (Sokolova et al., Reference Sokolova, Cámara, Hawthorne and Ciriotti2017a) and TS-block minerals (Sokolova, Reference Sokolova2006). In nafertisite, HOH blocks are separated by the A, W1, B and C sites that form the I-block (Fig. 59b). The A site is split into two disordered sites, A1 and A2, where K⁺ at A1 forms partly occupied (KO₈F)¹⁶⁻-polyhedra, and Na⁺ at A2 forms (NaO₄(H₂O)F)⁸⁻-octahedra. Due to the short distance between the A1 and W1 sites, they cannot both be locally occupied. Na⁺ fully occupies the B site forming (NaO₈F₂)¹⁷⁻-polyhedra and the C site is occupied by Na⁺ with minor Rb⁺ and Cs⁺. Nafertisite contains two (OH)⁻ groups that are associated with the M1–M3 octahedra of the O-sheet.

Fig. 58. (a, b, c) Tetrahedral representations of the ¹T₂³T₄ ribbon in nafertisite projected (a) onto (010), (b) onto (001), (c) along the a-axis, (d) a ball-and-stick and (e) a graphical representation of the ribbon. Dashed black lines outline the geometrical and topological repeat unit of the ribbon.

Fig. 59. The structure of nafertisite highlighting the (a) H- and O-sheets projected onto (001) and the (b) HOH-block and I-block viewed along the a-axis. The H atoms associated with (OH)⁻ groups linked to octahedra of the O-sheet are shown as small red circles, (H₂O) groups (W1) are shown as larger red circles and I-block cations are shown as green circles. Fine dashed black lines outline the unit cell which is halved along the c- and b-axes in (b).

¹T₂³T₆ ribbons

The ¹T₂³T₆ [Si₈O₂₂]¹²⁻ ribbon in veblenite consists of ribbons of edge-sharing hexagons of tetrahedra decorated by single, 1-connected tetrahedra along its periphery (Figs 60a–d). The ribbons link through dominantly Nb⁵⁺-octahedra (D site) to form a discontinuous H-sheet, and adjacent D-octahedra are linked along [010] by [Si₂O₇]⁶⁻ dimers. The (Fe^2+/3+(O,OH)₆)- and (Mn²⁺(O,OH)₆)-octahedra share common edges to form a modulated O-sheet parallel to (001) (Fig. 61a). The H- and O-sheets link to form an HOH block (cf. astrophyllite-supergroup minerals, (Fig. 56). The HOH blocks are linked along [001] by an I-block that contains the A1 and A2 sites occupied by K⁺, the B site which is dominantly vacant (with minor Na⁺), and (H₂O) which occupies five sites (W1–W5) (Fig. 61b).

Fig. 60. (a, b) Tetrahedral representations of the ¹T₂³T₆ ribbon in veblenite projected (a) orthogonal to the a-axis, (b) along the a-axis, (c) a ball-and-stick and (d) a graphical representation of the ribbon. Dashed black lines outline the geometrical and topological repeat unit of the ribbon.

Fig. 61. The structure of veblenite highlighting the (a) H- and O-sheet projected onto (001) and the (b) HOH-block and I-block viewed along the a-axis. Here, I-block, (H₂O) groups are shown as red circles. Fine dashed black lines outline the unit cell and H atoms associated with (OH)⁻ groups are omitted for clarity.

²T₂³T₂ chains and ribbons

The general formula of the amphibole-supergroup minerals is AB₂C₅T₈O₂₂W₂ where A = □, Na⁺, K⁺ and Ca²⁺; B = Na⁺, Li⁺, Ca²⁺, Mn²⁺, Fe²⁺ and Mg²⁺; C = Mg²⁺, Fe²⁺, Mn²⁺, Al³⁺, Fe³⁺, Mn³⁺, Ti⁴⁺ and Li⁺; T = Si⁴⁺, Al³⁺ and Ti⁴⁺; W = (OH)⁻, F⁻, Cl⁻ and O²⁻ (Hawthorne Reference Hawthorne, Veblen and Ribbe1981, Reference Hawthorne1983b,Reference Hawthornec; Hawthorne and Oberti, Reference Hawthorne, Oberti, Hawthorne, Oberti, Della Ventura and Mottana2007; Hawthorne et al., Reference Hawthorne, Oberti, Harlow, Maresch, Martin, Schumacher and Welch2012). The ²T₂³T₂ ribbon in the amphibole-supergroup minerals has a geometrical repeat unit that contains four tetrahedra and consists of edge-sharing hexagons of tetrahedra that extend in the c-direction (Figs 62a–d). There is a strip of edge-sharing octahedra (M1–M3), occupied by the C-group cations, that links to ²T₂³T₂ ribbons in the a- and b-directions (Fig. 63a). The M4 site is situated at the periphery of the strip of octahedra and is occupied by B-group cations; it is surrounded by eight oxygen atoms not all of which necessarily bond to the central cation. The A site occurs at the centre of a large cavity between the back-to-back ²T₂³T₂ ribbons (Fig. 63b), but the A-group cations actually occupy the off-centered sites A(2) and A(m). There are (currently) six known structural variants of the amphibole arrangement. We may divide these structures into two types: (1) those that involve different stacking sequences in the a-direction (C2/m, Pnma and Pnmn), and (2) those that are derivatives of (1) and involve differences in coordination (P2₁/m) and/or topochemistry (P2/a or C1) (Hawthorne and Oberti, Reference Hawthorne, Oberti, Hawthorne, Oberti, Della Ventura and Mottana2007). Complete details of the amphibole structure and chemistry are given in Hawthorne et al. (Reference Hawthorne, Oberti, Della Ventura and Mottana2007) and Oberti et al. (Reference Oberti, Hawthorne, Cannillo, Cámara, Hawthorne, Oberti, Della Ventura and Mottana2007), and Hawthorne (Reference Hawthorne2012b) has related the bond topologies and chemical compositions of the T–O–T (pyroxene, amphibole, pyribole and mica supergroups) and HOH (astrophyllite, nafertisite, veblenite and mica supergroups) minerals via (algebraic) generating functions.

Fig. 62. (a, b) Tetrahedral representations of the ²T₂³T₂ ribbon in amphibole-supergroup minerals projected (a) orthogonal to the c-axis, (b) onto (100), (c) a ball-and-stick and (d) a graphical representation of the ribbon. Dashed black lines outline the geometrical and topological repeat unit of the ribbon.

Fig. 63. The structure of richterite projected (a) onto (100) and (b) along the c-axis (a). The structure of plancheite projected (c) onto (100) and (d) along the c-axis. Fine dashed black lines outline the unit cell and H atoms associated with (OH)⁻ groups are omitted for clarity

Lavinskyite-2O (1M) and plancheite both contain ²T₂³T₂ ribbons that link to both sides of a continuous sheet of (Cu(O,OH)₆)-octahedra (M1–M3), forming a T–O–T block. In lavinskyite, these blocks are linked to each other by K⁺ ions (A site) that occupy cavities between back-to-back ²T₂³T₂ ribbons. T–O–T blocks are also linked by ^[5]Cu²⁺- and Li⁺-polyhedra (M4 site). Plancheite is isostructural with lavinskyite-2O where ²T₂³T₂ ribbons link to sheets of (CuO₄(OH)₂)⁸⁻-octahedra (Fig. 63c), (H₂O) groups replace K⁺ at the A site, and ^[5]Cu²⁺ replaces Li⁺ at the M4 site (Fig. 63d). These minerals contain ²V₂³V₂ ribbons that are topologically identical to the ribbons in amphibole-supergroup minerals. For a more detailed description of lavinskyite and plancheite see Evans and Mrose (Reference Evans and Mrose1966, Reference Evans and Mrose1977), Yang et al. (Reference Yang, Downs, Evans and Pinch2014) and Kolitsch et al. (Reference Kolitsch, Merlino, Belmonte, Carbone, Cabella, Lucchetti and Ciriotti2018).

The ²T₂³T₂ chains in synthetic Li₂Mg₂[Si₄O₁₁] and Fe₃[BeSi₃O₉(OH)]₂ (Figs 64a–f) are topologically identical to the [Si₈O₂₂]¹²⁻ chains in vlasovite (see below) and the [Si₄O₆(OH)₅]¹⁻ chain in revdite (see below) but not with the ²V₂³V₂ ribbon in amphibole-supergroup minerals (Fig. 62d). In synthetic Li₂Mg₂[Si₄O₁₁], T1–T4 are occupied by Si⁴⁺ (Figs 64a,b), and in synthetic Fe₃[BeSi₃O₉(OH)]_2,T1–T3 are occupied by Si⁴⁺ and T4 is occupied by Be²⁺ (Figs 64c,d). In synthetic Li₂Mg₂[Si₄O₁₁], ²T₂³T₂ chains extend along the c-axis and link to ribbons of Mg²⁺-octahedra and Li⁺-polyhedra that also extend along the c-axis; chains and ribbons occur in layers that alternate along [110]. In synthetic Fe₃[BeSi₃O₉(OH)]₂, each ²T₂³T₂ chain links to several ribbons of Fe²⁺-octahedra, both of which extend along the c-axis, and chains of tetrahedra and ribbons of octahedra occur in layers that alternate along the a-axis. For a more detailed total structure description, see Czank and Bissert (Reference Czank and Bissert1993) and Bakakin and Soloveva (Reference Bakakin and Soloveva1970).

Fig. 64. Tetrahedral representations of the ²T₂³T₂ chains in (a, b) synthetic Li₂Mg₂[Si₄O₁₁] and (c, d) Fe₃[BeSi₃O₉OH]₂ projected (a) orthogonal to the c-axis, (b) along the c-axis, (c) onto (100), (d) along the c-axis, (e) a ball-and-stick and (f) a graphical representation of the both chains. The T4 site in synthetic Fe₃[BeSi₃O₉OH]₂ is occupied by Be²⁺ and is shown as a dark green tetrahedron. Dashed black lines outline the geometrical and topological repeat unit of the chain.

²T₂³T₄ ribbons

The ²T₂³T₄ ribbon in jimthompsonite and clinojimthompsonite has a geometrical repeat unit that contains three distinct tetrahedra. Each ribbon is planar and consists of three polymerised ²T₂ chains that form a ribbon that extends along the c-axis and consists of staggered edge-sharing hexagons (Figs 65a–c). The ribbon in clinojimthompsonite and jimthompsonite has the vertex degree ²V₂³V₄ (Fig. 65d). The geometrical relations of the triple-chain in jimthompsonite and other biopyriboles such as amphiboles have been discussed in detail (e.g. Thompson, Reference Thompson1978; Papike and Ross, Reference Papike and Ross1970; Law and Whittaker, Reference Law and Whittaker1980; Veblen and Burnham, Reference Veblen and Burnham1978a,Reference Veblen and Burnhamb; Cameron and Papike, Reference Cameron and Papike1979; Chisholm, Reference Chisholm1981) and M-site substitutions and compositional limits are considered by Maresch et al. (Reference Maresch, Welch, Gottschalk, Ruthmann, Czank and Ashbrook2009) and Jenkins et al. (Reference Jenkins, Gilleaudeau, Kawa, Dibiase and Fokin2012). In clinojimthompsonite and jimthompsonite, the M1–M5 sites are occupied by Mg²⁺ (and subordinate Fe²⁺), these octahedra polymerise to form planar ribbons that extend along the c-axis. Mg²⁺ ions form (MgO₄(OH)₂)⁸⁻-octahedra at the M1–M3 sites and Mg²⁺-octahedra at the M4 and M5 sites. Each ²T₂³T₄ ribbon links to three distinct ribbons of Mg²⁺-octahedra and links such ribbons to each other along the a- and b-axes (Figs 66a,b).

Fig. 65. (a, b) Tetrahedral representations of the ²T₂³T₄ ribbon in jimthompsonite projected (a) orthogonal to the c-axis, (b) onto (100), (c) a ball-and-stick and (d) a graphical representation of the ribbon. Dashed black lines outline the geometrical and topological repeat unit of the ribbon.

Fig. 66. The structure of jimthompsonite projected (a) onto (100) and (b) along the c-axis. Fine dashed black lines outline the unit cell and H atoms associated with (OH)^– groups are omitted for clarity.

Various synthetic compounds isostructural with clinojimthompsonite have also been reported: Na₂Mg₄[Si₆O₁₆](OH)₂ and NaMg₄[Si₆O₁₅(OH)](OH)₂ in which Na⁺ occupies the A site (vacant in clinojimthompsonite and jimthompsonite) and the M5 site (Tateyama et al.,Reference Tateyama, Shimoda and Sudo1978; Ams et al., Reference Ams, Jenkins, Boerio-Goates, Morcos, Navrotsky and Bozhilov2009) (Table 7). This ²T₂³T₄ ribbon also occurs in synthetic Ba₄[Si₆O₁₆] and links to sheets of Ba²⁺-polyhedra.

The ²T₂³T₄ [Si₆O₁₆]⁸⁻ ribbons in okenite and yangite have a geometrical repeat unit that contain three distinct tetrahedra that polymerise to form planar ribbons of alternating four- and six-membered rings that extend along the b-axis (Figs 67a–c). Geometrically, this ribbon is related to ribbons in tobermorite-11Å (see below), epididymite-group minerals (Figs 38a–c) and related synthetic compounds such as β-Na₃Y[Si₆O₁₅] (Haile and Wuensch, Reference Haile and Wuensch1997) as they all contain ribbons of two ²T₃ (wollastonite-like) chains with varying degrees of polymerisation between the tetrahedra of each chain. Topologically, this ²V₂³V₄ ribbon is similar to the ²V₂³V₂ ribbon in amphibole-supergroup minerals (Fig. 62d) but is not identical (Fig. 67d).

Fig. 67. (a, b) Tetrahedral representations of the ²T₂³T₄ ribbon in okenite projected (a) onto (100), (b) onto (001), (c) a ball-and-stick and (d) a graphical representation of the ribbon. Dashed black lines outline the geometrical and topological repeat unit of the ribbon.

In addition to ribbons, okenite also contains planar [Si₆O₁₅]⁶⁻ sheets that consist of five- and eight-membered rings of tetrahedra and correspond to a (5².8)₂(5.8²)₁ net (Hawthorne et al., Reference Hawthorne, Uvarova and Sokolova2019). Okenite contains six Ca sites (Ca1–Ca6), four of which are occupied by Ca²⁺ (Ca1–Ca4) that form planar ribbons of edge-sharing octahedra that extend along the b-axis and are linked to each other along the a-axis by ²T₂³T₄ chains (Fig. 68a). A complicated [Ca₂(H₂O)₁₂]⁴⁺ layer involving (CaO₂(H₂O)₆)²⁻- (Ca5) and (CaO₂(H₂O)₅)²⁻-polyhedra (Ca6) links adjacent sheets of tetrahedra along the c-axis. The stacking sequence, involving ribbons and sheets of tetrahedra, ribbons of Ca²⁺-octahedra and the [Ca₂(H₂O)₁₂]⁴⁺ layer, is shown in Fig. 68b. The ribbon in yangite is topologically identical to that in okenite. However, the interstitial structure of yangite is relatively simple, containing one [5]- and one [6]-coordinated sites occupied by Pb²⁺ and Mn²⁺, respectively, that form ribbons of edge-sharing octahedra. These ribbons and ²T₂³T₄ chains occur in layers that alternate along the c-axis (Fig. 68b).

Fig. 68. The structure of okenite projected (a) onto (001) and (b) along the b-axis. Here, layers that contain [Si₆O₁₆]⁸⁻ ribbons, [Si₆O₁₅]⁶⁻ sheets and [Ca₂(H₂O)₁₂]⁴⁺ dimers are labelled. Fine dashed black lines outline the unit cell and H atoms associated with (H₂O) groups are omitted for clarity.

²T₂ ³T₆ ribbons

Two topologically distinct variants of the ²T₂³T₆ ribbon occur in synthetic K₆Eu³⁺₂[Si₈O₁₉(OH)₂](OH)₂(H₂O)₁₁ and Ba₅[Si₈O₂₁]. The geometrical repeat unit of the ²T₂³T₆ ribbon in the former contains three distinct tetrahedra that polymerise to form ribbons that extend along the c-axis (Figs 69a–d). These ribbons consist of two linked ²T₂³T₂ chains similar to those in synthetic Li₂Mg₂[Si₄O₁₁] (Figs 64a,b) and vlasovite (see below). The ²T₂³T₆ ribbons link to each other via (EuO₄(OH)₂)⁷⁻-octahedra, forming an open framework in which channels are occupied by ^[8–9]K⁺ ions and (H₂O) groups. The ²T₂³T₆ ribbon in synthetic Ba₅[Si₈O₂₁] consists of two linked ²T₂³T₂ amphibole-like ribbons (Figs 69e–h) and extends along the b-axis. These ribbons link to sheets of ^[8]Ba²⁺-polyhedra that are parallel to (010).

Fig. 69. (a, b, c) Tetrahedral representations of the ²T₂³T₆ ribbon in synthetic K₆Eu³⁺₂[Si₈O₁₉(OH)₂](OH)₂(H₂O)₉ projected (a) onto (010), (b) onto (100), (c) along the c-axis and (d) a ball-and-stick (graphical) representation of the ribbon. (e, f, g) Tetrahedral representations of the ²T₂³T₆ ribbon in synthetic Ba₅[Si₈O₂₁] projected (e) onto (100), (f) onto (001), (g) along the b-axis and (h) a ball-and-stick (graphical) representation of the ribbon. Dashed black lines outline the geometrical and topological repeat unit of the ribbon.

²T₂³T₈ ribbons

The geometrical repeat unit of the ribbon in synthetic Ba₆[Si₁₀O₂₆] contains ten distinct Si⁴⁺-tetrahedra that polymerise to form ²T₂³T₈ ribbons that extend parallel to [010] (Figs 70a–d). The structure of synthetic Ba₆[Si₁₀O₂₆] is closely related to those of synthetic Ba₅[Si₈O₂₁] (Figs 69e–g) and Ba₄[Si₆O₁₆] (Hesse and Liebau, Reference Hesse and Liebau1980) in which ribbons of tetrahedra link to sheets of ^[8]Ba²⁺-polyhedra that are parallel to (010).

Fig. 70. (a, b, c) Tetrahedral representations of the ²T₂³T₈ ribbon in synthetic Ba₆[Si₁₀O₂₆] projected (a) onto (100), (b) onto (001), (c) along the b-axis and (d) a ball-and-stick (graphical) representation of the ribbon. Dashed black lines outline the geometrical and topological repeat unit of the ribbon.

²T₃³T₄ ribbons

The ²T₃³T₄ [Si₇O₁₈(OH)]⁹⁻ ribbons in tokkoite, senkevichite and tinaksite (Table 8) have a geometrical repeat unit that contains seven distinct Si⁴⁺-tetrahedra where T7 is an acid silicate group: (SiO₃OH)³⁻. Each ²T₃³T₄ ribbon consists of alternating four- and eight-membered rings (Figs 71a–e) and in tokkoite, extend along [001] and link to sheets of Ca²⁺-octahedra (M1, M3–M4) and ^[7]Ca²⁺-polyhedra (M2) that extend parallel to the b–c plane (Figs 72a,b). These sheets are linked along [100] by ²T₃³T₄ ribbons, forming large tunnels that are occupied by K⁺ ions (A1–A2) (Fig. 72c). In tinaksite and senkevichite, Ti⁴⁺ and ^[7]Na⁺ occupy the M1 and M2 sites, respectively, and Cs⁺ occupies the A1 site in senkevichite. The tokkoite → tinaksite (senkevichite) substitution: 2Ca²⁺_(M1+M2) + (F,OH)⁻ ↔ Ti⁴⁺_(M1) + Na⁺_(M2) + O⁻ was suggested by Rozhdestvenskaya and Nikishova (Reference Rozhdestvenskaya and Nikishova2002).

Fig. 71. (a, b, c) Tetrahedral representations of the ²T₃³T₄ ribbon in tokkoite projected (a) onto (100), (b) orthogonal to the c-axis, (c) along the c-axis, (d) a ball-and-stick and (e) a graphical representation of the ribbon. Dashed black lines outline the geometrical and topological repeat unit of the ribbon.

Fig. 72. The structure of tokkoite projected (a) onto (010) showing the linkage of ²T₃³T₄ ribbons to sheets of Ca²⁺-polyhedra, (b) sheets of Ca²⁺-polyhedra projected onto (100) and (c) the alternating layers of ²T ₃³T ₄ ribbons and sheets of Ca²⁺-polyhedra projected along the c-axis. Fine dashed blacked lines outline the unit cell and H atoms associated with (OH)⁻ groups are omitted for clarity

Table 8. Minerals with ²T_r³T_r ribbons.

References: (1) Rozhdestvenskaya et al. (Reference Rozhdestvenskaya, Nikishova, Lazebnik and Lazebnik1989), Lacalamita et al. (Reference Lacalamita, Mesto, Kaneva, Scordari, Pedrazzi, Vladykin and Schingaro2017); (2) Agakhanov et al. (Reference Agakhanov, Pautov, Uvarova, Sokolova, Hawthorne and Karpenko2005), Uvarova et al. (Reference Uvarova, Sokolova, Hawthorne, Agakhanov, Pautov and Karpenko2006); (3) Rogov et al. (Reference Rogov, Rogova, Voronkov and Moleva1965), Bissert (Reference Bissert1980); (4) Wenk (Reference Wenk1974), Wenk (Reference Wenk1973), Ghent et al. (Reference Ghent, Stout and Erdmer1990); (5) Wenk (Reference Wenk1974), Matsubara (Reference Matsubara1981), Aoki (Reference Aoki, Akasako and Ishida1981); (6) Fleet (Reference Fleet1977), Agrell et al. (Reference Agrell, Brown and McKie1965), Wenk et al. Reference Wenk, Biagioni, Hsiao, Lee, Tanzella, Yeh, Hodgson and Nissen1976, Worthing (Reference Worthing1987); (7) Grice and Dunn (Reference Grice and Dunn1994), Dunn et al. (Reference Dunn, Peacor, Su, Nelen and von Knorring1986), Brugger and Berlepsch (Reference Brugger and Berlepsch1997); (8) Merlino et al. (Reference Merlino, Bonaccorsi and Armbruster2001); (9) Merlino et al. (Reference Merlino, Bonaccorsi and Armbruster2001), Hamid (Reference Hamid1981); (10) Merlino et al. (Reference Merlino, Bonaccorsi and Armbruster2001), Churakov (Reference Churakov2009); (11) Merlino et al. (Reference Merlino, Bonaccorsi and Armbruster2000a); (12) Merlino et al. (Reference Merlino, Bonaccorsi and Armbruster2000a), Henmi and Kusachi (Reference Henmi and Kusachi1992), Hoffmann and Armbruster (Reference Hoffmann and Armbruster1997); (13) Biagioni et al. (Reference Biagioni, Merlino and Bonaccorsi2015); (14) Biagioni et al. (Reference Biagioni, Bonaccorsi, Merlino, Bersani and Forte2012b); (15) Biagioni et al. (Reference Biagioni, Merlino and Bonaccorsi2015); (16) Garbev (Reference Garbev2004), Kudoh and Takéuchi (Reference Kudoh and Takéuchi1979), Hejny and Armbruster (Reference Hejny and Armbruster2001), Merlino and Bonaccorsi (Reference Merlino and Bonaccorsi2008), Churakov and Mandaliev (Reference Churakov and Mandaliev2008); (17) McDonald and Chao (Reference McDonald and Chao2004); (18) Sandomirskii and Belov (Reference Sandomirskii and Belov1979), Belokoneva (Reference Belokoneva2005), Zubkova et al. (Reference Zubkova, Pushcharovsky, Giester, Pekov, Turchkova, Chukanov and Tillmanns2005), Zubkova et al. (Reference Zubkova, Pushcharovsky, Giester, Pekov, Turchkova, Tillmanns and Chukanov2006), Cruciani et al. Reference Cruciani, De Luca, Nastro and Pattison1998; (19) Krivovichev et al. (Reference Krivovichev, Yakovenchuk, Armbruster, Döbelin, Pattison, Weber and Depmeier2004a); (20) Zubkova et al. (Reference Zubkova, Pushcharovsky, Giester, Pekov, Turchkova, Tillmanns and Chukanov2006); (21) Zubkova et al. (Reference Zubkova, Pushcharovsky, Giester, Pekov, Turchkova, Chukanov and Tillmanns2005); (22) Cesbron and Williams (Reference Cesbron and Williams1980), Lópes et al. (Reference Lópes, Frost, Scholz, Xi and Amaral2014); (23) Sokolova et al. (Reference Sokolova, Hawthorne, Ball, Mitchell and Della Ventura2006), Voronkov and Pyatenko (Reference Voronkov and Pyatenko1962), Gittins et al. (Reference Gittins, Gasparrini and Fleet1973), Gobechiya et al. (Reference Gobechiya, Pekov, Pushcharovsky, Ferraris, Gula, Zubkova and Chukanov2003), Kaneva et al. (Reference Kaneva, Vladykin, Mesto, Lacalamita, Scordari and Schingaro2018); (24) Salvadó et al. (Reference Salvadó, Pertierra, Garcia-Granda, Khainakov, Garcia, Bortun and Clearfield2001); (25) Compagnoni et al. (Reference Compagnoni, Ferraris and Mellini1985), Mellini et al. (Reference Mellini, Ferraris and Compagnoni1985), Deriu et al. (Reference Deriu, Ferraris and Belluso1994); (26) Wan and Ghose (Reference Wan and Ghose1978), Ryall and Threadgold (Reference Ryall and Threadgold1966), Richmond (Reference Richmond1942); (27) Zhao et al. (Reference Zhao, Li, Chen, Li, Chu, Liu, Yu and Xu2010); (28) Plášil et al. (Reference Plášil, Fejfarová, Čejka, Dušek, Škoda and Sejkora2013), McBurney and Murdoch (Reference McBurney and Murdoch1959), Rastsvetaeva et al. (Reference Rastsvetaeva, Arakcheeva, Pushcharovsky, Atencio and Menezes Filho1997b), Burns (Reference Burns2001); (29) Zöller et al. (Reference Zöller, Tillmanns and Hentschel1992).

* Indicates the ^cT_r expression of an additional structural unit including a chain, ribbon, tube, cluster or sheet of (TO₄)^n– tetrahedra in the respective mineral.

²T₄³T₂ chains

The howieite group includes deerite, johninnesite, taneyamalite and howieite, all of which contain ²T₄³T₂ [Si₆(O,OH)₁₇] chains (Table 8). The geometrical repeat unit of each chain contains six-membered rings that link to one another through a single corner-sharing tetrahedra (Figs 73a–c). The structures of howieite (Figs 73d,e) and taneyamalite (Figs 73f,g) both contain six T sites fully occupied by Si⁴⁺ and twelve octahedrally coordinated M sites occupied dominantly by Fe²⁺ in howieite and Mn²⁺ in taneyamalite. In both minerals, octahedrally coordinated cations form [M ₁₂O₃₀] ribbons that are four octahedra wide (Figs 73d,f) and extend along the c-axis (Figs 73e,g), parallel to the ²T₄³T₂ chains. These chains and ribbons of octahedra occur in layers that alternate along the b-axis, and adjacent ribbons of octahedra are linked along [1$\bar{2}$0] by the 2-connected tetrahedra of each six-membered ring. Both minerals also contain 13 (OH)⁻ sites, three of which are associated with the [Si₆(O,OH)₁₇] chain and the rest with the [M ₁₂O₃₀] ribbon. These minerals also have a single large-cation site occupied by Na⁺ in howieite and Na⁺ with minor Ca²⁺ in taneyamalite. In both minerals, ^[8]Na⁺-polyhedra occur in layers with ribbons of octahedra and link adjacent ²T₄³T₂ chains along the b-axis (Figs 73d–g).

Fig. 73. (a, b, c) Tetrahedral representation of the ²T₄³T₂ chain in howieite-group minerals_, (b) a ball-and-stick and (c) a graphical representation of this chain. The structure of howieite projected (d) along the c-axis and (e) onto (010). The structure of taneyamalite projected (f) along the c-axis and (g) onto (010). Dashed black lines outline the geometrical and topological repeat unit of the ribbon and fine dashed black lines outline the unit cell. The H atoms associated with (OH)^– groups are omitted for clarity

The interstitial structure of deerite is closely related to that of howieite and taneyamalite, and deerite and howieite have been observed as intergrowths in the metasedimentary rocks of the Franciscan Formation, California (Fleet, Reference Fleet1977). The major difference in deerite is that ribbons of octahedra and ²T₄³T₂ chains have more than one orientation, somewhat resembling wallpaper-borate structures (Moore and Araki, Reference Moore and Araki1974; Cooper and Hawthorne, Reference Cooper and Hawthorne1998), and hence do not occur in alternating layers as they do in howieite and taneyamalite. Deerite contains nine octahedrally coordinated M-sites occupied by Fe³⁺ (with minor Al³⁺) and Fe²⁺ (with minor Mn²⁺). The Fe²⁺- and Fe³⁺-octahedra form ribbons that are six octahedra wide (Fig. 74a) and extend along the c-axis, parallel to the ²T₄³T₂ chains that each link to three interconnected ribbons (Figs 74a,b). Deerite contains five (OH)⁻ groups that coordinate Fe^2+/3+-octahedra of the ribbon.

Fig. 74. The structure of deerite projected (a) along the c-axis and (b) orthogonal to the c-axis. The structure of johninnesite projected (c) along the c-axis and (d) orthogonal to the c-axis. Fine dashed black lines outline the unit cell and H atoms associated with (OH)⁻ groups are omitted for clarity.

Johninnesite contains seven T sites (T1–T7), T1–T6 of are fully occupied by Si⁴⁺ forming [Si₆O₁₇]¹⁰⁻ chains, and T7 is occupied by As⁵⁺ which does not link to the ²T₄³T₂ chain. Johninnesite contains twelve octahedrally coordinated M-sites (M1–M12) occupied by Na⁺, Mn²⁺, Mg²⁺ and a vacancy. Edge-sharing (MnO₆)¹⁰⁻- (M4), (MnO₄(OH)₂)⁸⁻- (M5–M6) and (MgO₄(OH)₂)⁸⁻-octahedra (M7–M9) link to form discontinuous sheets parallel to (010) (Fig. 74c). The vacant M-sites (M11–M12) in the sheet are capped by As⁵⁺-tetrahedra on both sides of the sheet. Sheets of octahedra link to each other along the a-axis via ribbons of (MnO₆)¹⁰⁻- (M1), (MnO₅(OH))⁹⁻- (M2) and (MnO₃(OH)₃)⁷⁻-octahedra (M3), forming channels that extend along the c-axis that are occupied by Na⁺ ions(M10). The ²T₄³T₂ chains link to both the sheet and ribbon, and occur in layers parallel to the b-axis (Figs 74c,d). In johninnesite, the tetrahedra of each chain in the same layer point in the same direction, in contrast to the tetrahedra in howieite (Figs 73d,e) and taneyamalite (Figs 73f,g) where adjacent chains have tetrahedra that point in opposite directions. Johninnesite contains eight (OH)⁻ groups that are associated with the sheet and ribbon of octahedra.

²T₄³T₂ ribbons

Much work has been done developing classification schemes and nomenclature for tobermorite-group minerals as these species exhibit novel dehydration-hydration reactions, have a complex OD character, are used as cation-exchangers, and play a critical role in the hydration of Portland cement (Taylor, Reference Taylor1964, Reference Taylor1992; Gard and Taylor, Reference Gard and Taylor1976; Cong and Kirkpatrick, Reference Cong and Kirkpatrick1996; Garbev, Reference Garbev2004; Battocchio et al., Reference Battocchio, Monteiro and Wenk2012). Here, we will describe only briefly the structure for selected group-members; see Mitsuda and Taylor (Reference Mitsuda and Taylor1978), Merlino et al. (Reference Merlino, Bonaccorsi and Armbruster1999, Reference Merlino, Bonaccorsi and Armbruster2001), Merlino and Bonaccorsi (Reference Merlino and Bonaccorsi2008) and Biagioni et al. (Reference Biagioni, Merlino and Bonaccorsi2015) for more detailed descriptions. This group includes several calcium–silicate–hydrate (C–S–H) minerals such as tobermorite-11Å and clinotobermorite-11Å (Table 8) that contain ²T₄³T₂ ribbons that consist of eight-membered rings that link along the b-axis through two Si⁴⁺-tetrahedra (Figs 75a–e). These ²V₄³V₂ ribbons (Fig. 75f) are topologically similar but not identical to the ²V₄³V₂ chains in howieite-group minerals (Figs 73c, 75f).

Fig. 75. (a, b, c) Tetrahedral representations of the ²T₄³T₂ ribbon in (a, b) tobermorite-11Å and (c, d) clinotobermorite-11Å projected (a, c) onto (100) and (b, d) along the b-axis. (e) a ball-and-stick and (f) a graphical representation of the ribbon. Dashed black lines outline the geometrical and topological repeat unit of the ribbon.