Introduction

An idea concept for a description of a crystal structure using the “fundamental building blocks” approach was developed in the twentieth century. Currently, it has become a useful tool to describe a number of physical properties of crystalline matter as well as for the modular design of a crystal structure. An example of such a property that is mostly dependent on the arrangement of the building blocks or modules is ionic conductivity, where the ion migration ability often directly depends on the size of channels or interatomic distances. The modules can be selected based on a crystal chemical analysis of similar structure types; however, this can be difficult if no modular series for similar compounds are known.

According to (Hawthorne 1994), the fundamental building block (FBB) is a tightly bonded unit within the structure that can be considered an inorganic analog of a molecule in an organic structure. The FBB is usually a homo- or heteropolyhedral cluster of coordination polyhedra with the strongest bond-valence linkages in the structure (Hawthorne 1994). In some cases, FBBs can be considered modules that build up crystal structures (Taylor and West 1928; Donnay and Donnay 1953; Dornberger-Schiff and Grell-Nirmann 1961; Megaw 1973; Moore and Araki 1976; Thompson 1978; Sandomirskiy and Belov 1984; Hawthorne 1985; 1994; Veblen 1991; Merlino 1997; Krivovichev 1999a, b; Ferraris et al. 2004; Makovicky 2018).

In this article, microblock descriptions of the crystal structures of Na2SO4–K2SO4 sulfates, Ca2SiO4 polymorphic modifications (P63/mmc, Pnma, Pn21a, Pn21/n), natural bubnovaite K2Na8Ca(SO4)6, and dobrovolskyite Na4Ca(SO4)3 were proposed for the first time. These structures could be considered to consist of microblocks, which were inherited from their disordered high-temperature parent microblocks. In accordance with Megaw 1973, a relationship between the structures described in this article is the “aristotype (basic structure)–hettotypes (derivative structures)” one.

Crystal structures of Na2SO4–K2SO4 sulfates

Six mineral species are known in the Na2SO4–K2SO4 system: thénardite Na2SO4 (Rasmussen et al. 1996), metathénardite Na2SO4 (Pekov et al. 2019), natroaphthitalite Na3K(SO4)2 (Shchipalkina et al. 2020), belomarinaite KNaSO4 (Filatov et al. 2019), aphthitalite K3Na(SO4)2 (Okada and Ossaka 1980), and arcanite K2SO4 (McGinnety 1972). Metathénardite is a high-temperature modification of α-Na2SO4 (P63/mmc) stabilized at room temperature by a bivalent cation admixture, and thénardite transforms into α-Na2SO4 (P63/mmc) at approximately 420 °C (Rasmussen et al. 1996). All other minerals of this series undergo a reversible phase transition to high-temperature modifications isotypical with α-K2SO4 and α-Na2SO4 (both—P63/mmc) (Eysel 1973). The crystal structures of α-Na2SO4 and α-K2SO4 were refined using single-crystal X-ray diffraction data collected at 420 and 640 °C, respectively (Arnold et al. 1981; Naruse et al. 1987). The geometrical parameters of these minerals are listed in Table 1. Both coexistence and coherent conjugation of domains of crystal structures of the compounds of the Na2SO4–K2SO4 series are described in Shchipalkina et al. 2021 and references therein.

Table 1 Unit cell parameters of some aphthitalite-like minerals
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The crystal structures of Na3K(SO4)2, KNaSO4, and K3Na(SO4)2 under ambient conditions have practically the same unit cell parameters as their high-temperature modifications (Table 1). These structures consist of complex columns of the KO12, NaO6, MO10 (M = K, Na) and SO4 polyhedra (Moore 1973, 1976, 1981; Hawthorne et al. 2000). Crystal structures of aphthitalite-like minerals can be described as pseudo-close-packed cationic arrays with hexagonal layers parallel to (001) (O’Keeffe and Hyde 1985; Krivovichev and Filatov 1999; Vegas 2000; Gorelova et al. 2016; Krivovichev 2017; Shablinskii et al. 2021) as well as mixed frameworks consisting of columns formed by octahedral–tetrahedral clusters of MO6 octahedra that share six vertices with six adjacent SO4 tetrahedra (Shablinskii et al. 2021). These clusters represent microblocks derived by Voronkov et al. 1975, which are the “elementary bricks” of the frameworks (Krivovichev 2017). According to the cited paper, these microblocks are the modules or fundamental building blocks (FBBs).

The crystal structures of the α-Na2SO4 and α-K2SO4 high-temperature modifications consist of a disordered SO4 tetrahedron and two symmetrically independent sites for alkali cations. Notably, dynamic disorder exists in the structures, which leads to the formation of inherited microblocks. One of the alkali cations occupying the octahedral MO6 (M = Na, K) site is surrounded by six disordered tetrahedra forming the fundamental building block (FBB) (Fig. 1) similar to the microblocks found in the mixed (heteropolyhedral) frameworks (Voronkov et al. 1975) (see Fig. 4).

Fig. 1
figure 1

Parent FBB of α-Na2SO4 (edge model) in a ac projection, b ab projection. White/red circles show oxygen atoms with partial occupancies of 1/3 and 2/3

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According to (Miyake et al. 1980; Arnold et al. 1981; Eysel et al. 1985; Naruse et al. 1987; Rasmussen et al. 1996; Pekov et al. 2019 and references terrain), the disordered tetrahedron can be represented as the SO4 tetrahedra of different orientations, and only one is possible at a time. Only five orientations of the SO4 tetrahedra were established by an analysis of single-crystal neutron and X-ray diffraction data, which were designated as “up” (apex model) (Fig. 2a), “down” (apex model) (Fig. 2b), “left” (edge model) (Fig. 2c), “tilting” (edge model) (Fig. 2d) and “right” (edge model) (Fig. 2e). In the “apex model,” one of the vertices of the SO4 tetrahedron is oriented up or down randomly in the [001] direction. In the case of the “edge model,” one of the edges of the tetrahedron is parallel to the L3 axis, and the remaining two vertices lie on the xy¼ mirror plane in the α–K2SO4 structure (Arnold et al. 1981). In the “edge model,” three different orientations can be obtained by a thermally induced rotation of the SO4 tetrahedron around the z-axis by 120°. Thus, in the trigonal crystal system, it is possible to derive 25 microblocks from the MO6 octahedron (M = Na, K) that shares six corners with six adjacent disordered tetrahedra. This number can be reduced to 15 if the microblocks that can be obtained from others are removed using symmetry operations (see Fig. 3).

Fig. 2
figure 2

The 5 types of the different orientations of the SO4 tetrahedron, which can be derived from the parent FBB: a “up” (apex model), b “down” (apex model), c “left” (edge model), d “tilting” (edge model), e “right” (edge model). Yellow circles—S, red circles—O

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Fig. 3
figure 3

Octahedral–tetrahedral clusters of trigonal symmetry, which can be formed after cooling of parent FBB

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After the analysis of the crystal structures of the minerals of the Na2SO4–K2SO4 series that are similar to the high-temperature α-Na2SO4 (see Table 1), all structures (excluding Na2SO4-Fddd) clearly contain these microblocks with the SO4 tetrahedra in one of the described orientations (see Figs. 5, 6, 7), as if the tetrahedron remains in this orientation after a cooling of the high-temperature α-Na2SO4 and K2SO4 modifications. Based on this, the fundamental building block (FBB) of the high-temperature modifications can be deduced as the “parent” block of the octahedral–tetrahedral microblocks that compose the mixed frameworks of the aphthitalite-like minerals investigated in this work. A transition from the parent block to the inherited block occurs upon cooling of the high-temperature modification that is structurally similar to α-Na2SO4. The parent FBB consists of a NaO6 octahedron surrounded by six disordered SO4 tetrahedra (a tetrahedron with disordered oxygen atoms) (Fig. 1). This disordered tetrahedron has five (“apex model”) or nine (“edge model”) vertices. As mentioned above, the disordered tetrahedra can be represented as five different orientations of the ordered SO4 tetrahedra.

Description of the [M(TO4)6] microblocks

The numbers 1, 2, 31, 3, and 32 are assigned to denote the tetrahedron in the “up” (apex model) (Fig. 2a), “down” (apex model) (Fig. 2b), “left” (edge model) (Fig. 2c), “tilting” (edge model) (Fig. 2d), and “right” (edge model) (Fig. 2d) orientations, respectively; The numbers 31, 3, and 32 are assigned because they can be obtained from each other by a 120° rotation around the c axis. In these minerals and structurally similar compounds, only the tetrahedron orientations can be used to define the microblocks since the octahedron maintains the same orientation. Then, a microblock of any symmetry can be designated as a set of six digits, where the first three digits correspond to the three upper orientations of the tetrahedra, while the second three digits correspond to the three lower orientations of the tetrahedra; for example, (1)(31)(2)(1)(31)(2) is a possible notation. If the microblock designation is combined with its structural formula, there will be (SO4-2)(SO4-2)(SO4-2)Na(SO4-2)(SO4-2)(SO4-2) for aphthitalite and (SO4-31)(SO4-31)(SO4-31)Na(SO4-31)(SO4-31)(SO4-31) for Na2SO4 (Cmcm) if the sodium atom is in the octahedral site. All microblocks derived by Voronkov et al. 1975 have –3m or 3m symmetry. As mentioned above, if the symmetry of the microblock is 3m, then 25 types of microblocks can be combinatorically derived, which can be reduced to 15 (Fig. 3).

With a decrease in the symmetry of the microblock, the number of combinatorically derived microblocks significantly increases. If the microblock has trigonal symmetry, then it can be defined by two-digit numbers: the first number corresponds to the upper part of the microblock (one octahedron and three tetrahedra), and the second number corresponds to the lower part of the microblock (one octahedron and three tetrahedra) (Fig. 4).

Fig. 4
figure 4

Basic microblocks proposed by Voronkov et al. as FBB’s in the mixed anionic octahedral–tetrahedral frameworks (redrawn picture of Voronkov et al. 1975)

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Furthermore, the configurations of the tetrahedra in the microblocks could also be of transitional forms. Therefore, the parameters that describe the ideal configurations that can be inherited from the parent (high-temperature) microblock needs to be introduced. The M–O–T angles and M–O–T–O torsion angles were selected as geometric parameters and calculated using experimental data for the parent microblock. Ideal parameters for each configuration are listed in Table 2.

Table 2 Geometric parameters for different configurations of the SO4 tetrahedra in the microblocks inherited from the parent one
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Notably, these microblocks can be considered as defect closest packings of anions, which was shown by Voronkov et al. 1975.

Temperature-dependent structural transformations in the Na2SO4–K2SO4 system, Ca2SiO4, bubnovaite, and dobrovolskyite

The internal energy of crystals increases with temperature growth. As a rule, this is shown by a decreasing density of the crystal structure with a decrease in the coordination number and an increase in the bond lengths, the size of polyhedra and its rearrangement (Filatov 1990; Filatov et al. 2017). With a decrease in temperature, the thermal motion of atoms gradually decreases, and the degree of their difference increases. Thus, atoms tend to become ordered through different crystallographic sites of a structure.

To show the process of structural transformations from high-temperature modification to low-temperature modification, a number of examples are considered to determine the effect of a decrease in temperature on the microblocks in the crystal structures. For this consideration, the compounds of the Na2SO4–K2SO4 system are convenient to use as examples, since all framework crystal structures are composed of these microblocks.

Transformations of the compounds in the Na2SO4–K2SO4 system

According to (Eysel 1973; Belousova et al. 2021), the K3Na(SO4)2 and KNaSO4 compounds transform into P–3m1 and P3m1 at 400 and 480 °C, respectively, upon cooling from the isotypical Na2SO4 (P63/mmc) high-temperature modification. A similar situation likely occurs during the cooling of the high-temperature Na3K(SO4)2 phase (P3m1), but this phase exists only as an impurity-stabilized (Cu) modification. The structures of low-temperature modifications consist of microblocks 11 and 22. As mentioned above, these microblocks can be inherited from the parent microblocks upon cooling (Fig. 5). A combination of the ionic radii of the [VI]K and [VI]Na atoms is suitable for maintaining the trigonal symmetry of the microblock, which is not the case for the end-members of the Na2SO4–K2SO4 series.

Fig. 5
figure 5

Parent FBB (left) and microblocks (right) in the K3Na(SO4)2 crystal structure (K atoms—violet color, Na—blue color)

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According to high-temperature X-ray diffraction data, the Na2SO4 (P63/mmc) high-temperature modification transforms upon cooling at 240 and 230 °C into Pbnm and Cmcm phases, respectively (Rasmussen et al. 1996). At room temperature, the Cmcm modification transforms into a modification with the Fddd space group. A structural relationship is observed during cooling. In this case, a microblock is formed that can be inherited from the high-temperature modification, but the symmetry of the microblock is lower than that of the trigonal microblock (Fig. 6). The decrease in the symmetry of the microblock occurs because the octahedra of the upper and lower microblocks share only edges; they do not share faces and not connected through three SO4 tetrahedra. This interconnection of the polyhedra occur due to a smaller ionic radius of the [VI]Na atom that causes the edges to be shared between the octahedra; however, the radius is large enough to prevent the formation of interconnections with the three SO4 tetrahedra. The microblock in the Pbnm polymorphic modification can be denoted as (SO4-32)(SO4-31)(SO4-3)Na(SO4-3)(SO4-31)(SO4-32), and that in the Cmcm modification can be denoted as (SO4-31)(SO4-31)(SO4-3)Na(SO4-3)(SO4-31)(SO4-31). The difference between the microblocks is a rotation of one of the tetrahedra by 120° because the modification with the Pbnm space group is the intermediate one and exists only in a limited temperature range (230–240 °C).

Fig. 6
figure 6

Changes in microblocks and crystal structures of Na2SO4 upon cooling. The matrix–column pair corresponding to the change in the coordinated system, and the splitting of the Wyckoff positions under the change of symmetry are given in Tables S1 and S2

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The high-temperature modification of K2SO4 (P63/mmc) transforms into K2SO4 (Pmcn) upon cooling to 587 °C (Arnold et al. 1981). These structures are similar, and the structure of the low-temperature phase consists of microblocks with configurations of tetrahedrons 1 and 2, as in K3Na(SO4)2, but the symmetry of microblocks is lower than trigonal (Fig. 7). The decrease in symmetry is associated with the large ionic radius of the [VI]K atom, which leads to a distortion of the octahedral site and microblock at room temperature. The low-temperature microblock can be denoted as (SO4-1)(SO4-2)(SO4-2)K(SO4-1)(SO4-2)(SO4-2).

Fig. 7
figure 7

Changes in microblocks and crystal structures of K2SO4 upon cooling

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Ca2SiO4 transformations

The polymorphic modifications of Ca2SiO4 are structurally similar to the Na2SO4 modifications (Moore 1981), while the high-temperature modifications are isotypical (Udagawa et al. 1977; Mumme et al. 1996). However, P–3m1 is sometimes considered an alternative space group for the high-temperature phase (Mumme et al. 1996). Upon cooling, the following sequence of polymorphic transformations occurs: P63/mmc (1545–1250 °C) → Pnma (1250–1060 °C) → Pna21 (1060–680 °C) → P21/n (680–630 °C) → Pcmn (630 °C) (Mumme et al. 1996; Yamnova et al. 2011). In the modification with the Pcmn space group, microblocks no longer exist. All microblocks in the modifications P63/mmc → Pnma → Pna21 → P21/n contain tetrahedrons in orientations 1 and 2. The distortion of the microblocks and reduction in their trigonal symmetry are associated with an increase in the coordination number of the Ca atom from 6 to 8 with decreasing temperature; thus, it can no longer be located in the octahedral site. The microblocks in Ca2SiO4 (Pnma), Ca2SiO4 (Pna21), and Ca2SiO4 (P21/n) can be denoted as (SiO4-2)(SiO4-1)(SiO4-2)Ca(SiO4-2)(SiO4-1)(SiO4-2), (SiO4-2)(SiO4-1)(SiO4-2)Ca(SiO4-1)(SiO4-2)(SiO4-1), and (SiO4-1)(SiO4-2)(SiO4-1)Ca(SiO4-2)(SiO4-1)(SiO4-2), respectively. Previously, the modular description of Ca2SiO4 polymorphic transformations was described by Zvyagin and Pushcharovsky (1993), but the description provided in this paper does not use the microblocks discussed in our study (Fig. 8).

Fig. 8
figure 8

Changes in microblocks and crystal structures of Ca2SiO4 upon cooling

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Description of the microblock in bubnovaite and dobrovolskyite

Mixed frameworks in the crystal structures of the new minerals of bubnovaite K2Na8Ca(SO4)6 (Gorelova et al. 2016) and dobrovolskyite Na4Ca(SO4)3 (Shablinskii et al. 2021) are similar to the structure of aphthitalite consisting of microblocks that can be derived from the parent microblock. Petrovite was found in association with dobrovolskyite (Filatov et al. 2020). However, there are Ca atoms in the chemical composition of the minerals. Both minerals (bubnovaite and dobrovolskyite) are superstructures related to aphthitalite with a ratio of unit cell parameters of 2 × 2 × 3 and 3 × 3 × 3, respectively.

The [M(SO4)6] microblocks in bubnovaite share corners with the SO4 tetrahedra, forming a mixed framework consisting of two symmetrically independent columns elongated through the c axis and located on the L3 axis. The SO4 tetrahedra are presented in the crystal structure in two orientations: 1 “up” and 2 “down”. Therefore, the crystal structure is composed of microblocks 11 and 22. A two-digit designation is used for their microblocks since their symmetries are trigonal. The first column of microblocks need to be considered as a pseudo-column since one of the Na sites has a partial occupation. The sequences of the microblocks in the pseudo-column and in a continuous column are (21)(22)(11)(21)(22)(11) and (11)(22)(21)(11)(22)(21), respectively; these are associated with a disordering of the tetrahedra in the orientation 1 “up” and 2 “down” in one site. The disordering is associated with the partial occupation of the sodium atoms in one of the sites. Microblock 11 is formed when the sodium atoms are in the site. When the site is vacant, the tetrahedron takes the 2 “down” orientation (Fig. 9a).

Fig. 9
figure 9

Microblocks in the crystal structure of bubnovaite: a the first and b the second symmetrically independent columns. Blue circles show Na, violet circles—K, gray circles—Ca, yellow circles—S, red circles—O

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The second column contains the same disordered SO4 tetrahedron leading to the disordering of the microblocks. The sequence of the microblocks in the column can be denoted as (21)(22)(11)(21)(22)(11) with a high probability or (11)(22)(21)(11)(22)(21) with a low probability (Fig. 9b).

The crystal structure of dobrovolskyite can be described as a three-dimensional framework consisting of Na–O and Ca–O polyhedra and SO4 tetrahedra. The framework can be described as three symmetrically independent rods elongated through the c axis and composed of octahedral–tetrahedral clusters formed by one central NaO6 or CaO6 octahedron sharing six corners with six adjacent SO4 tetrahedra (Shablinskii et al. 2021). The first type of rod is located on the L3 axis (x = 0; y = 0), and the second (x ≈2/3; y ≈ 2/3) and third (x ≈2/3; y ≈ 0) types of rods are located on general sites.

In the structure of dobrovolskyite, these microblocks sharing corners with the tetrahedra and faces with the octahedra form three symmetrically independent rods elongated through the c axis. These rods are interconnected through the SO4 tetrahedra and form a three-dimensional mixed framework. The Na3O7, Na4O10, Na9O8, and Ca1O6 polyhedra are located in the cavities of the framework. Microblocks in all three symmetrically independent rods include the SO4 tetrahedra in all five orientations that can be inherited from the parent block: 1, 2, 3, 31 and 32 (Fig. 2). The first rod is located on the L3 axis; therefore, the designation of two digits for the microblocks with trigonal symmetry is used. The sequence of microblocks in the rod from top to bottom is (23)(33)(332)(3231)(3131)(311) (Fig. 10). Microblocks 23, 33, 332, 3231, 3131, and 311 have formulas (S5O4)3Na12(S6O4)3, (S1O4)3Na1(S5O4)3, (S2O4)3Na5(S1O4)3, (S2O4)3Na2(S3O4)3, (S4O4)3Na8(S3O4)3 and (S4O4)3Na10(S6O4)3, respectively.

Fig. 10
figure 10

Three types of rods in the crystal structure of dobrovolskyite

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Rods 2 and 3 as well as the microblocks are highly distorted since they are not located on the L3 axis (Fig. 10).

All orientations of the tetrahedra that can be inherited from the parent microblock of the high-temperature phase in the crystal structure of dobrovolskyite Na4Ca(SO4)3 are possible. Almost all tetrahedra in the crystal structure are disordered. Apparently, dobrovolskyite is an intermediate quenched high-temperature phase similar to α-Na2SO4 and stabilized by the Cu, Mg, and K admixtures.

The different numbers and types of microblocks in these rods make it possible to explain the formation of the superstructures of dobrovolskyite and bubnovaite in contrast to aphthitalite and α-Na2SO4. The number of the microblocks in one rod in dobrovolskyite and bubnovaite is 6, and that in aphthitalite is 2, which perfectly correlates with the ratios of the c parameter (3:1) in these minerals. The difference in these microblocks can be used to explain the increase in the parameter of the unit cell.

Role of changing the coordination number of the cation in the M(TO4)6 microblock

An increase in the coordination number upon cooling is a well-known mechanism of densification of a crystal structure. However, some exceptions are known, such as several REEBO3 borates (Biryukov et al. 2020). If the coordination number of the cation in the octahedral site in the microblock increases upon cooling, then the microblock is often reduced to its trigonal symmetry. However, with some distortion of the microblock (11), the coordination number of the cation can be increased to 12 due to the apical vertices of all six SO4 tetrahedra without reducing the symmetry. Such an example is observed in bubnovaite, where the K atom is such a cation (Fig. 11). In the (12) microblock, the coordination number of the cation can be increased to 9 without reducing the symmetry of the microblock due to the 3 apical vertices of the SO4 tetrahedra. If the coordination number of the cation is not equal to 12 or 9 upon cooling, then the symmetry of the microblock decreases. A good example is the Ca atoms in Ca2SiO4 and the K atoms in β-K2SO4; here, the coordination number increases to 8, which is accompanied by an arrangement of tetrahedra where four apical vertices are directed toward the cation and two are directed away from the cation (Fig. 12). Thus, the inheritance of the various microblocks upon cooling is also determined by the type of cation occupying the octahedral site in the parent block.

Fig. 11
figure 11

Various microblocks inherited upon cooling from the parent K(SO4)6 due to a change in the coordination number of the cation

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Fig. 12
figure 12

Change in the Ca(SiO4)6 microblock due to a change in the coordination number of the cation upon cooling

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Conclusions

The minerals and compounds structurally related to α-Na2SO4 and the derived superstructures were formed as a result of the cooling of the high-temperature phases containing the disordered parent microblock in their structures. Here, the inheritance driving force was the tendency of the structure to become ordered upon cooling. Only microblocks 1, 2, 3, 5, and 6 (in our designation 11, 22, 33, 31, 32) could be formed upon cooling of the parent microblock. The other microblocks could be formed in a different way. The reasons for the formation of a microblock from the parent microblock were mainly determined by the ionic radius and type of cation occupying the octahedral site.

If these sites are occupied only by large K cations (for example, in K2SO4), microblocks with tetrahedron configurations 1 “down” and 2 “up” will be formed upon cooling, but the symmetry of the microblock and structure will be reduced due to an increase in the coordination number of the K atoms. If the trigonal symmetry of the microblocks and structure is preserved, the K cation can be surrounded by 12 oxygen atoms in the microblock (11). The K cation can be surrounded by only 6 oxygen atoms in the alternating microblock (22). A mismatch between the coordination number of the K atom and that of the cation in the microblock (22) with trigonal symmetry leads to a reduction in the symmetry of the microblock and crystal structure. The situation is similar in the Ca2SiO4 compound (except Ca2SiO4-Pcmn), where microblocks are also inherited from the same parent unit. The coordination number of the Ca atom increases from 6 to 8 upon cooling, which is accompanied by a reduction in the symmetry of the microblock and crystal structure.

The combination of the K and Na cations (aphthitalite K3Na(SO4)2 and belomarinaite KNaSO4) is favorable for maintaining the trigonal symmetry of the crystal structure and microblocks upon cooling. In the microblocks alternating in the rods elongated through the c axis, the K atom occupies a 12-vertex site of the microblock (11), while the Na atom occupies an octahedral site of the microblock (22). If these sites contain only Na cations (Na2SO4), the microblocks with the tetrahedra in orientation 3 are inherited instead of the microblocks (11) and (22) to compensate for the insufficient size of the cation and the values of the coordination number. In this case, the NaO6 octahedra will share edges, not faces as in other compounds.

The crystal structures of bubnovaite K2Na8Ca(SO4)6 and dobrovolskyite Na4Ca(SO4)3 have an definite chemical and structural relationship with the structures of the Na2SO4–K2SO4 system. In dobrovolskyite, microblocks with all possible configurations of the SO4 tetrahedra can be derived from the parent microblock.

Octahedra of the M(TO4)3 microblock in the columns can be connected through three tetrahedra or through shared corners, edges and faces. The MO6 octahedra can be connected through three tetrahedra in orientation 3, maintaining the trigonal symmetry of the column consisting of microblocks. The MO6 octahedra can share faces when they contain tetrahedra in orientations 1, 2, 31, and 32, maintaining the trigonal symmetry. Additionally, the MO6 octahedra can share corners and edges if they contain all orientations of the tetrahedra but without preserving the trigonal symmetry of the microblock.

The formation of minerals with these described structural features facilitates great prospects for the synthesis of compounds with useful properties. Initial reagents consisting of similar microblocks can be selected for successful synthesis. Then, during the synthesis, compounds with these crystal structures are formed from these microblocks and can potentially exhibit useful properties.