The Na2-nHn[Zr(Si2O7)]mH2O Minerals and Related Compounds (n = 0–0.5; m = 0.1): Structure Refinement, Framework Topology, and Possible Na+-Ion Migration Paths

The Na2−nHn[Zr(Si2O7)]·mH2O family of minerals and related compounds (n = 0–0.5; m = 0.1) consist of keldyshite, Na3H[Zr2(Si2O7)2], and parakeldyshite, Na2[Zr(Si2O7)], and synthetic Na2[Zr(Si2O7)]·H2O. The crystal structures of these materials are based upon microporous heteropolyhedral frameworks formed by linkage of Si2O7 groups and ZrO6 octahedra with internal channels occupied by Na+ cations and H2O molecules. The members of the family have been studied by the combination of theoretical (geometrical–topological analysis, Voronoi migration map calculation, structural complexity calculation), and empirical methods (single-crystal X-ray diffraction, microprobe analysis, and Raman spectroscopy for parakeldyshite). It was found that keldyshite and parakeldyshite have the same fsh topology, while Na2ZrSi2O7·H2O is different and has the xat topology. The microporous heteropolyhedral frameworks in these materials have a 2-D system of channels suitable for the Na+-ion migration. The crystal structure of keldyshite can be derived from that of parakeldyshite by the Na+ + O2− ↔ OH− + substitution mechanism, widespread in the postcrystallization processes in hyperagpaitic rocks.


Single-Crystal X-Ray Diffraction
The crystal-structure studies of parakeldyshite were carried out at the X-ray Diffraction Resource Centre of St. Petersburg State University on an Agilent Technologies Xcalibur EOS diffractometer equipped with the CCD detector using monochromatic MoKα radiation (λ = 0.71069 Å) at room temperature. More than a hemisphere of diffraction data was collected (scanning step 1°, exposure time 10 s). The absorption correction was done empirically using spherical harmonics implemented in the SCALE ABSPACK calibration algorithm in the CrysAlysPro software package [35]. The unit-cell parameters were determined and refined by the least squares method using 1364 independent reflections. The structure was refined using the SHELXL software package [36]. The data are deposited in CCDC under Entry No. 22040710. The coordination number of Na determined by number bonds with maximal length constrained 3.10 Å. Crystal data, data collection information, and refinement details are given in Table 1. Atom coordinates and isotropic parameters of atomic displacements are given in Table S1, interatomic distances in Table S2, and the anisotropic parameters of atomic displacements are given in Table S3.

Composition
The chemical composition of parakeldyshite was determined at the 'Geomodel' resource center of St. Petersburg State University using the scanning electron microscope Hitachi S-3400N equipped by INCA 500 WDS detector operating at 20-30 nA and 20 kV. The analyses were performed with the beam size of 5 µm and the counting time of 10-20/10 s on peaks/background for each chemical element. Quartz (Si), corundum (Al), calcite (Ca), halite (Na), zircon (Zr), rutile (Ti), hematite (Fe), celestine (Sr), and rhodonite (Mn) were used as standards. An average chemical composition based on 5 analyzes (in wt.%): ZrO 2

Single-Crystal X-ray Diffraction
The crystal-structure studies of parakeldyshite were carried out at the X-ray Diffraction Resource Centre of St. Petersburg State University on an Agilent Technologies Xcalibur EOS diffractometer equipped with the CCD detector using monochromatic MoKα radiation (λ = 0.71069 Å) at room temperature. More than a hemisphere of diffraction data was collected (scanning step 1 • , exposure time 10 s). The absorption correction was done empirically using spherical harmonics implemented in the SCALE ABSPACK calibration algorithm in the CrysAlysPro software package [35]. The unit-cell parameters were determined and refined by the least squares method using 1364 independent reflections. The structure was refined using the SHELXL software package [36]. The data are deposited in CCDC under Entry No. 22040710. The coordination number of Na determined by number bonds with maximal length constrained 3.10 Å. Crystal data, data collection information, and refinement details are given in Table 1. Atom coordinates and isotropic parameters of atomic displacements are given in Table S1, interatomic distances in Table S2, and the anisotropic parameters of atomic displacements are given in Table S3.

Geometrical-Topological Analysis
Geometrical-topological analysis of the crystal structures of keldyshite-related compounds was carried out using algorithms implemented in the ToposPro software package (https://topospro.com/) [33]. Maps of the migration of Na + -ions were constructed by the Voronoi method, which was shown to be efficient for various types of ionic conductors [34,37,38]. The radius of an elementary channel (R chan ) suitable for Na + -ion migration was chosen as 2.0 Å, similar to that reported previously [38,39].
Topological analysis of the crystal structures of keldyshite-related compounds also included the determination of the type basic grid, the construction of tiling, and the search for topologically similar inorganic compounds. The base grid is a graph whose vertices are the centers of gravity of the structural units, i.e., SiO 4 tetrahedra and ZrO 6 octahedra [40]. After contraction of doubly connected nodes ("bridging" oxygen atoms), a 4,6-coordinated grid was obtained ( Figure 2).
The topological classification of the atomic nets in crystal structures was carried out in accordance with the following basic principle [41]: atomic nets with the same set of topological indices (coordination sequence, vertex symbols) belong to the same topological type [42]. In the case of the presence of stable polyhedral units in the crystal structure, the classification was carried out according to the basic grid [40]. Determination of the topological mesh type was performed using the ToposPro complex of the TopCryst web service (http://topcryst.com), which contains data on about 190,000 topological types of the nets.
The tiling theory, which is actively used to study and analyze the crystal structures of zeolites [43] and zeolite-related materials with heteropolyhedral frameworks [44][45][46], was introduced and developed by M. O'Keeffe [47]. This approach allows study of the smallest cavities in inorganic frameworks that can be used to fill the entire crystal space [47]. Since the grids in the crystal structures of keldyshite and parakeldyshite have the same topology, the set of tilings in these structures is the same. The tiling theory, which is actively used to study and analyze the crystal structures of zeolites [43] and zeolite-related materials with heteropolyhedral frameworks [44][45][46], was introduced and developed by M. O'Keeffe [47]. This approach allows study of the smallest cavities in inorganic frameworks that can be used to fill the entire crystal space [47]. Since the grids in the crystal structures of keldyshite and parakeldyshite have the same topology, the set of tilings in these structures is the same.

Raman Spectroscopy
The Raman spectrum (RS) was obtained using a Horiba Jobin-Yvon LabRam HR 800 spectrometer (Geomodel Resource Center, St. Petersburg State University) from the surface of a parakeldyshite crystal at room temperature and a wavelength of 514 nm in the range from 4000 to 80 cm −1 . The baseline correction was carried out using the algorithms implemented in the OriginPro 8.1 software package.

Raman Spectroscopy
The Raman spectrum (RS) was obtained using a Horiba Jobin-Yvon LabRam HR 800 spectrometer (Geomodel Resource Center, St. Petersburg State University) from the surface of a parakeldyshite crystal at room temperature and a wavelength of 514 nm in the range from 4000 to 80 cm −1 . The baseline correction was carried out using the algorithms implemented in the OriginPro 8.1 software package.

Single-Crystal X-ray Diffraction
The crystal structures of microporous zirconium silicates are based upon frameworks consisting of ZrO 6 octahedra and SiO 4 tetrahedra linked via common O atoms. According to the structural classification proposed by Ilyushin and Blatov [40], the crystal structures of keldyshite, NaH  The tiling theory, which is actively used to study and analyze the crystal structures of zeolites [43] and zeolite-related materials with heteropolyhedral frameworks [44][45][46], was introduced and developed by M. O'Keeffe [47]. This approach allows study of the smallest cavities in inorganic frameworks that can be used to fill the entire crystal space [47]. Since the grids in the crystal structures of keldyshite and parakeldyshite have the same topology, the set of tilings in these structures is the same.

Raman Spectroscopy
The Raman spectrum (RS) was obtained using a Horiba Jobin-Yvon LabRam HR 800 spectrometer (Geomodel Resource Center, St. Petersburg State University) from the surface of a parakeldyshite crystal at room temperature and a wavelength of 514 nm in the range from 4000 to 80 cm −1 . The baseline correction was carried out using the algorithms implemented in the OriginPro 8.1 software package.

Single-Crystal X-Ray Diffraction
The crystal structures of microporous zirconium silicates are based upon frameworks consisting of ZrO6 octahedra and SiO4 tetrahedra linked via common O atoms. According to the structural classification proposed by Ilyushin and Blatov [40], the crystal structures of keldyshite, NaH[Zr(Si2O7)], parakeldyshite, Na2[Zr(Si2O7)], and Na2[Zr(Si2O7)]•H2O contain polyhedral microensembles (PME) MT6 of the A-1 type ( Figure 3).  In the terms proposed in [5], the crystal structure of parakeldyshite can be described as a framework consisting of the M 2 T 4 -type secondary building units (SBUs) with Na atoms in adjacent cavities ( Figure 4a). Each ZrO 6 octahedron is linked to six SiO 4 tetrahedra, which in turn are linked to two Zr octahedra each. In the terms proposed in [5], the crystal structure of parakeldyshite can be described as a framework consisting of the M2T4-type secondary building units (SBUs) with Na atoms in adjacent cavities ( Figure 4a). Each ZrO6 octahedron is linked to six SiO4 tetrahedra, which in turn are linked to two Zr octahedra each. The crystal structure of parakeldyshite from albitized pegmatite of Takhtarvumvorr Mt., Khibiny, Russia was solved in the space group P1 and refined to final R1 = 0.024 [for 1364 F 2 > 4σ(F 2 )]. In general, the structure model proposed in [17] was confirmed. The bond lengths in polyhedra vary significantly and are equal to 2.047-2.139 (2) (3) Å for the ZrO6, Si1O4, Si2O4, Na1O8, and Na2O7 polyhedra, respectively. The degrees of distortion of polyhedra (based on bond lengths) calculated according to Baur [48] for the Si1O4, Si2O4, ZrO6, Na1O8, and Na2O7 polyhedra are equal to 0.01369, 0.01171, 0.01626, 0.04376, and 0.07001, respectively. According to our data, there are no additional Na sites described in [27]. In contrast to keldyshite, there are two independent positions of Na1 and Na2 with a coordination number (CN) of 8, respectively, in the crystal structure of parakeldyshite (Figure 5a). The crystal structure of keldyshite ( Figure 5b) differs from that of parakeldyshite and contains only one independent Na site with sevenfold coordination. The uneven distribution of Na in keldyshite results in the slight framework deformation manifested by the change of the Si-O-Si angle of 126.7(9)° compared to 127.7(8)° in parakeldyshite. At the same time, the shape of the channels changes significantly as can be clearly seen in the projection of the MT layer (Figure 6b).
The crystal structure of Na2[Zr(Si2O7)]•H2O is based on the M2T6 type of SBUs ( Figure 4c). As in the structures of keldyshite and parakeldyshite, the ZrO6 octahedron is linked through common vertices to six SiO4 tetrahedra, each connected to three Zr-centered octahedra. The topological difference of the MT-framework (Figure 5c) from those observed in keldyshite and parakeldyshite is The crystal structure of parakeldyshite from albitized pegmatite of Takhtarvumvorr Mt., Khibiny, Russia was solved in the space group P1 and refined to final R 1 = 0.024 [for 1364 F 2 > 4σ(F 2 )]. In general, the structure model proposed in [17] was confirmed. The bond lengths in polyhedra vary significantly and are equal to 2.047-2.139(2), 1.601-1.672(2), 1.600-1.669(2), 2.443-2.913(3), and 2.384-2.913(3) Å for the ZrO 6 , Si1O 4 , Si2O 4 , Na1O 8 , and Na2O 7 polyhedra, respectively. The degrees of distortion of polyhedra (based on bond lengths) calculated according to Baur [48] for the Si1O 4 , Si2O 4 , ZrO 6 , Na1O 8 , and Na2O 7 polyhedra are equal to 0.01369, 0.01171, 0.01626, 0.04376, and 0.07001, respectively. According to our data, there are no additional Na sites described in [27]. In contrast to keldyshite, there are two independent positions of Na1 and Na2 with a coordination number (CN) of 8, respectively, in the crystal structure of parakeldyshite (Figure 5a). In the terms proposed in [5], the crystal structure of parakeldyshite can be described as a framework consisting of the M2T4-type secondary building units (SBUs) with Na atoms in adjacent cavities (Figure 4a). Each ZrO6 octahedron is linked to six SiO4 tetrahedra, which in turn are linked to two Zr octahedra each.  The crystal structure of parakeldyshite from albitized pegmatite of Takhtarvumvorr Mt., Khibiny, Russia was solved in the space group P1 and refined to final R1 = 0.024 [for 1364 F 2 > 4σ(F 2 )]. In general, the structure model proposed in [17] was confirmed. The bond lengths in polyhedra vary significantly and are equal to 2.047-2.139(2), 1.601-1.672(2), 1.600-1.669(2), 2.443-2.913(3), and 2.384-2.913(3) Å for the ZrO6, Si1O4, Si2O4, Na1O8, and Na2O7 polyhedra, respectively. The degrees of distortion of polyhedra (based on bond lengths) calculated according to Baur [48] for the Si1O4, Si2O4, ZrO6, Na1O8, and Na2O7 polyhedra are equal to 0.01369, 0.01171, 0.01626, 0.04376, and 0.07001, respectively. According to our data, there are no additional Na sites described in [27]. In contrast to keldyshite, there are two independent positions of Na1 and Na2 with a coordination number (CN) of 8, respectively, in the crystal structure of parakeldyshite (Figure 5a). The crystal structure of keldyshite ( Figure 5b) differs from that of parakeldyshite and contains only one independent Na site with sevenfold coordination. The uneven distribution of Na in keldyshite results in the slight framework deformation manifested by the change of the Si-O-Si angle of 126.7(9)° compared to 127.7(8)° in parakeldyshite. At the same time, the shape of the channels changes significantly as can be clearly seen in the projection of the MT layer (Figure 6b).
The crystal structure of Na2[Zr(Si2O7)]•H2O is based on the M2T6 type of SBUs ( Figure 4c). As in the structures of keldyshite and parakeldyshite, the ZrO6 octahedron is linked through common vertices to six SiO4 tetrahedra, each connected to three Zr-centered octahedra. The topological difference of the MT-framework (Figure 5c) from those observed in keldyshite and parakeldyshite is

Topological Analysis
The topological type of the base grid in the crystal structures of keldyshite and parakeldyshite is fsh, while that in Na2[Zr(Si2O7)]•H2O is xat (Figure 7). Table 2 shows data on related inorganic compounds of the fsh and xat topological types. Keldyshite and parakeldyshite consists of one type of the [4 3 .6 3 ] tiles formed by three four-membered rings and three six-membered rings (Figure 8). In the case of Na2[Zr(Si2O7)]•H2O, there are two types of tiles: [6 3 ] t-kah and [4 6 .6 3 ] t-afo ( Figure 8). Na atoms are located inside all [4 3 .6 3 ] tiles in parakeldyshite and only half of these tiles are filled in keldyshite. In the crystal structure of Na2[Zr(Si2O7)]•H2O, one Na site is located in the t-kah tile, whereas another one is within the t-afo tile.

Topological Analysis
The topological type of the base grid in the crystal structures of keldyshite and parakeldyshite is fsh, while that in Na 2 [Zr(Si 2 O 7 )]·H 2 O is xat (Figure 7). Table 2 shows data on related inorganic compounds of the fsh and xat topological types. Keldyshite and parakeldyshite consists of one type of the [4 3 .6 3 ] tiles formed by three four-membered rings and three six-membered rings (Figure 8). In the case of Na 2 [Zr(Si 2 O 7 )]·H 2 O, there are two types of tiles: [6 3 ] t-kah and [4 6 .6 3 ] t-afo ( Figure 8). Na atoms are located inside all [4 3 .6 3 ] tiles in parakeldyshite and only half of these tiles are filled in keldyshite. In the crystal structure of Na 2 [Zr(Si 2 O 7 )]·H 2 O, one Na site is located in the t-kah tile, whereas another one is within the t-afo tile.

Raman Spectroscopy
The Raman spectrum of parakeldyshite from the albitites of Takhtarvumchorr Mt. is shown in (Figure 9). The most intense spectral lines are similar to those for parakeldyshite from the Alluaiv Mt., Lovozero alkaline massif [52], RRUF, 120048 [59]. The bands in the range 850-1020 cm −1 correspond to stretching vibrations of Si-O bonds. Two intense absorption bands at 968 and 1017 cm −1 are attributed to asymmetric stretching vibrations of Si-O-Si bonds, while three bands at 850, 905, and 941 cm −1 are attributed to symmetric vibration modes of similar bonds [60][61][62]. The non-typical band at 718 cm −1 can be associated with symmetric stretching vibrations of the Si-O-Si bridging oxygen in sorosilicate groups [63]. The bands in the range 450-600 cm −1 correspond to asymmetric bending vibrations of Si-O bonds in tetrahedra [62]. Bands of different intensities in the region 350-450 cm −1 belong to symmetric deformation vibration modes in SiO 4 tetrahedra [63]. The most intense absorption band at 331 cm −1 is attributed to bending vibration modes of the Zr-O bonds in octahedra, and the bands in the range 90-300 cm −1 correspond to symmetric bending vibrations of bonds in octahedra or translational Crystals 2020, 10, 1016 9 of 14 vibrations [64]. The absence of bands in the region 3000-3800 cm −1 ( Figure S1) indicates the absence of OH groups in the structure of parakeldyshite, confirming its unchanged nature.
band at 718 cm can be associated with symmetric stretching vibrations of the Si-O-Si bridging oxygen in sorosilicate groups [63]. The bands in the range 450-600 cm −1 correspond to asymmetric bending vibrations of Si-O bonds in tetrahedra [62]. Bands of different intensities in the region 350-450 cm −1 belong to symmetric deformation vibration modes in SiO4 tetrahedra [63]. The most intense absorption band at 331 cm −1 is attributed to bending vibration modes of the Zr-O bonds in octahedra, and the bands in the range 90-300 cm −1 correspond to symmetric bending vibrations of bonds in octahedra or translational vibrations [64]. The absence of bands in the region 3000-3800 cm −1 ( Figure  S1) indicates the absence of OH groups in the structure of parakeldyshite, confirming its unchanged nature.

Discussion
According to the approach of matrix (self)assembly of the crystal structures from SBUs proposed by Ilyushin for sodium zirconium silicates, all possible SBUs variants are defined as М2Тn (n = 2, 4, 6) [5]. The crystal structure of keldyshite/parakeldyshite is based upon the M2T4 blocks, while the structure of Na2[Zr(Si2O7)]•H2O is based upon the M2T6 blocks. This fact indicates different formation conditions and the impossibility of the transformation of one structure type into another through the rearrangement of Na + -ions. Indeed, the conditions of the formation of phases in the Na2CO3-ZrO2-SiO2-H2O hydrothermal system are different: parakeldyshite crystallizes at 450 °C [5], while the Na2[Zr(Si2O7)]•H2O phase appears at a temperature of about 200 °C [28]. According to [65], keldyshite is a product of the sequential transformation of parakeldyshite under hypergenic conditions with the preservation of the overall framework topology. According to our data on the migration of Na + -ions, such transition is possible.

Discussion
According to the approach of matrix (self)assembly of the crystal structures from SBUs proposed by Ilyushin for sodium zirconium silicates, all possible SBUs variants are defined as М 2 Т n (n = 2, 4, 6) [5]. The crystal structure of keldyshite/parakeldyshite is based upon the M 2 T 4 blocks, while the structure of Na 2 [Zr(Si 2 O 7 )]·H 2 O is based upon the M 2 T 6 blocks. This fact indicates different formation conditions and the impossibility of the transformation of one structure type into another through the rearrangement of Na + -ions. Indeed, the conditions of the formation of phases in the Na 2 CO 3 -ZrO 2 -SiO 2 -H 2 O hydrothermal system are different: parakeldyshite crystallizes at 450 • C [5], while the Na 2 [Zr(Si 2 O 7 )]·H 2 O phase appears at a temperature of about 200 • C [28]. According to [65], keldyshite is a product of the sequential transformation of parakeldyshite under hypergenic conditions with the preservation of the overall framework topology. According to our data on the migration of Na + -ions, such transition is possible.
The unit-cell parameters (  The paths of the Na + -ion migration obtained using the Voronoi method are two-dimensional, running through all the crystallographic positions of Na ( Figure 10). Thus, the migration of Na + -ions in the keldyshite-related zirconium silicates occurs along a two-periodic network of channels. Diffusion is possible when Na + -ions move from cavity to cavity (from one tile to another) through four-membered and six-membered rings. Calculating the radius of these rings (Table 4) and comparing it with the threshold value of 2.0 Å, we can conclude that free migration of Na + in these structures will occur only through six-membered rings. Moreover, in the case of parakeldyshite, there is one six-membered ring that is too narrow (marked in red in Table 4) for sodium to move along. However, the migration paths through the remaining six-membered rings form a 2-D migration map, which is consistent with the result obtained by the Voronoi method.
running through all the crystallographic positions of Na ( Figure 10). Thus, the migration of Na + -ions in the keldyshite-related zirconium silicates occurs along a two-periodic network of channels. Diffusion is possible when Na + -ions move from cavity to cavity (from one tile to another) through four-membered and six-membered rings. Calculating the radius of these rings (Table 4) and comparing it with the threshold value of 2.0 Å, we can conclude that free migration of Na + in these structures will occur only through six-membered rings. Moreover, in the case of parakeldyshite, there is one six-membered ring that is too narrow (marked in red in Table 4) for sodium to move along. However, the migration paths through the remaining six-membered rings form a 2-D migration map, which is consistent with the result obtained by the Voronoi method.
The refinement of the crystal structure of parakeldyshite from the Takhtarvumchorr pegmatite demonstrates the absence of splitting of the Na sites. According to the chemical data and Raman spectroscopy, the studied sample of parakeldyshite is the extreme Na-member of the possible keldyshite-parakeldyshite series. The migration paths analysis by the Voronoi method showed that all three studied phases have a 2-D system of channels (Figures 10 and 11), within which the migration of Na + cations is possible. These data confirm the possibility of transition from parakeldyshite to keldyshite by the Na + + O 2-↔ OH -+ □ substitution scheme, which is widespread in postcrystallization processes in peralkaline rocks.   The refinement of the crystal structure of parakeldyshite from the Takhtarvumchorr pegmatite demonstrates the absence of splitting of the Na sites. According to the chemical data and Raman spectroscopy, the studied sample of parakeldyshite is the extreme Na-member of the possible keldyshite-parakeldyshite series. The migration paths analysis by the Voronoi method showed that all three studied phases have a 2-D system of channels (Figures 10 and 11), within which the migration of Na + cations is possible. These data confirm the possibility of transition from parakeldyshite to keldyshite by the Na + + O 2− ↔ OH − + substitution scheme, which is widespread in postcrystallization processes in peralkaline rocks.

Parakeldyshite Na2ZrSi2O7
Keldyshite NaZr(Si2O6OH) 6 2. The structural complexity IG,total was calculated according to the method proposed in [66]. The calculated values for keldyshite/parakeldyshite and Na2ZrSi2O7•H2O are 76.107 and 86.606 (bits/u.c.), respectively. The identity of the structural complexity values for keldyshite and parakeldyshite emphasizes their structural similarity. The increase in structural complexity with the decreasing crystallization temperature is in agreement with the general tendency observed for hydrothermal systems [66,67].
Supplementary Materials: The following materials are available online at www.mdpi.com/xxx/s1, Figure S1: title, Table S1: Fractional atomic coordinates (×10 4 ) and equivalent isotropic displacement parameters (Å 2 × 10 3 ) for parakeldyshite. Table S2: selected interatomic distances in parakeldyshite. Table S3: anisotropic parameters of atomic displacements in parakeldyshite.  The structural complexity I G,total was calculated according to the method proposed in [66]. The calculated values for keldyshite/parakeldyshite and Na 2 ZrSi 2 O 7 ·H 2 O are 76.107 and 86.606 (bits/u.c.), respectively. The identity of the structural complexity values for keldyshite and parakeldyshite emphasizes their structural similarity. The increase in structural complexity with the decreasing crystallization temperature is in agreement with the general tendency observed for hydrothermal systems [66,67].