Supporting information
Crystallographic Information File (CIF) https://doi.org/10.1107/S0108270113020921/fn3142sup1.cif | |
Structure factor file (CIF format) https://doi.org/10.1107/S0108270113020921/fn3142Na2Ge4O9sup2.hkl | |
Structure factor file (CIF format) https://doi.org/10.1107/S0108270113020921/fn3142K2Ge4O9sup3.hkl | |
Structure factor file (CIF format) https://doi.org/10.1107/S0108270113020921/fn3142Rb2Ge4O9sup4.hkl | |
Chemical Markup Language (CML) file https://doi.org/10.1107/S0108270113020921/fn3142Na2Ge4O9sup5.cml | |
Chemical Markup Language (CML) file https://doi.org/10.1107/S0108270113020921/fn3142K2Ge4O9sup6.cml | |
Chemical Markup Language (CML) file https://doi.org/10.1107/S0108270113020921/fn3142Rb2Ge4O9sup7.cml |
Structures containing Ge in both tetrahedral and octahedral coordinations are rare, but are of interest as analogue structures for high-pressure silicate materials. Among them are garnet-type CaGeO3 (Nakatsuka et al., 2005), Ca2Ge7O16 (Redhammer et al., 2007a) and A2B(T3O9)-type compounds. Fleet & Muthupari (1998) noted that three structure types are known for this last class of materials: (i) wadeite-type materials with the space group P63/m, a structure type which is favoured by silicates such as wadeite K2ZrSi3O9, K2TiSi3O9, Cs2ZrSi3O9 or the high-pressure silicate K2SiSi3O9 (Swanson & Prewitt, 1983); (ii) the alkali germanates Na2Ge4O9, K2Ge4O9 and Rb2Ge4O9 with the space group P3c1 (Fleet & Muthupari, 1998; Völlenkle & Wittmann, 1971; Goreaud & Raveau, 1976); and (iii) Na2Si4O9 with the space group P21/n. All these structures show a tetrahedral–octahedral framework, in which three-membered rings of TO4 tetrahedra are interconnected by BO6 octahedra. In contrast with this, the alkali tetragermanates Li2Ge4O9 and NaLiGe4O9 show orthorhombic symmetry, with space group P21ca or Pcca depending on composition and temperature (Völlenkle et al., 1969; Redhammer & Tippelt, 2013). Their structures are based on crumbled [Ge3O9]n chains of GeO4 tetrahedra which are interconnected by GeO6 octahedra, and no ring structure is present.
The structures of A2Ge4O9-type materials are known in principle. Sodium tetragermanate Na2Ge4O9 was resolved, after long-standing uncertainties, by Fleet & Muthupari (1998), who determined the space group to be P3c1 rather than wadeite-type P63/m. The latter authors also noted close similarities between Na2Ge4O9, K2Ge4O9 and Rb2Ge4O9. The structure of K2Ge4O9 was first determined by Völlenkle & Wittmann (1971) using intensity data from visual inspection of Weissenberg photographic films, so no anisotropic refinement of the atomic displacement parameters is available and the uncertainties in the primary and secondary structural parameters are large. For Rb2Ge4O9, the structure was determined by Goreaud & Raveau (1976) using intensity data from photographic precession film exposures; again, only isotropic atomic displacement parameters are given, and the estimated standard uncertainties on e.g. the bond lengths are in the range of 0.05 Å, i.e. at least ten times larger than would be expected from a modern high-quality structure refinement based on single-crystal X-ray data. Within the isotpyic series A2Ge4O9, two further compositions exist, namely Ag2Ge4O9 and Tl2Ge4O9. Due to the similarities in their powder X-ray diffraction patterns, Wittmann & Modern (1965) concluded that they also belong to the isotypic series of P3c1 tetragermanates A2Ge4O9 (A = Na, K and Rb); only the lattice parameters are available for the Ag2Ge4O9 and Tl2Ge4O9 compounds up to now. In this study, we report a redetermination and structure refinement of three A2Ge4O9 (A = Na, K and Rb) compounds, giving full anisotropic atomic displacement parameters and much better precision in the bond lengths and angles than were possible in the older literature.
Na2Ge4O9 and K2Ge4O9 were obtained by chance when attempting to grow feldspar-type materials NaAlGe3O8 and KAlGe3O8, using the flux method with Na2MoO4 and K2MoO4 as the high-temperature solvent. A finely ground mixture of K2CO3 (Na2CO3), Al2O3 and GeO2 in the stoichiometry of the feldspars, together with the flux (flux to nutrient ratio of 10:1) were put into platinum crucibles, covered with a lid, and placed in a chamber furnace, which was then heated to 1373 K at a rate of 3 K min-1, held at this temperature for 24 h and cooled to 973 K at a rate of 0.02 K min-1. After dissolving the flux in hot water, for nominal KAlGe3O8 composition the experimental yield consisted of small colourless needles of the title compound [K2Ge4O9?], up to 2 mm in length, compact fine-grained aggregates of tetragonal KAlGe2O6 and K2Ge4O9, and finally glass spheres up to 0.5 mm. Chemical analysis of the needles using EDX analysis with scanning electron microscopy showed only oxygen, potassium and germanium were present, with no aluminium. For nominal NaAlGe3O8 composition the synthesis batch showed plates of NaAlGe3O8 feldspar and about 15% of large needles up to 1 mm in length of the title compound [Na2Ge4O9?]. Additionally, some glass was found.
Rb2Ge4O9 was first obtained by sintering a stoichiometric mixture of Rb2CO3 and GeO2 at 1123 K, followed by melting this material at 1273 K for 2 h in a platinum crucible and slow cooling to 873 K at a rate of 0.05 K min-1. By this procedure a solidified material was obtained, consisting of thin needles of the title compound [Rb2Ge4O9?], about 3 to 4 mm in length and 0.1 mm in diameter.
Crystal data, data collection and structure refinement details are summarized in Table 1. Analysis of the systematic absences yielded possible space groups P3c1 and P3c1 for all three compounds. Intensity statistics suggested strong evidence for a centrosymmetric space group. No evidence was found for higher symmetry, e.g. P63/m derivative wadeite type. Structure solution using direct methods was carried out for K2Ge4O9 in space group P3c1, yielding all the Ge-atom positions. For the O-atom positions, one turned out to be K and the remaining two O-atom positions were identified from a difference Fourier analysis. For the other two compounds, the structures were refined using the atomic coordinates obtained for K2Ge4O9. The structural model obtained is identical to that given by Völlenkle & Wittmann (1971), Goreaud & Raveau (1976) and Fleet & Muthupari (1998), but was transformed to give a full connectivity set with the Ge1 site at (0,0,1/2). The Ge1, Ge2, Ge3 and Ge4 sites of Fleet & Muthupari (1998) correspond to the Ge1, Ge4, Ge2 and Ge3 sites of this study, respectively, and the O1, O2, O3, O4 and O5 sites of Fleet & Muthupari (1998) correspond to the O5, O1, O2, O3 and O4 sites of this study, respectively.
All three A2Ge4O9 (A = Na, K and Rb) compounds exhibit the space group P3c1, thus confirming the findings of the literature. The general structure topology consists of the sevenfold coordinated A site (general position 12g, site symmetry 1), the octahedrally coordinated Ge1 and Ge4 sites (special positions 2b and 4d, site symmetries 3 and 3, respectively), and two tetrahedrally coordinated Ge2 and Ge3 sites (special and general positions 6f and 12g with site symmetries 2 and 1, respectively). There are five independent O-atom positions, four of them on general positions 12g and the fifth on special position 6f, site symmetry 2. This is in good agreement with the findings from the literature, confirming the older structure determinations of K2Ge4O9 (Völlenkle & Wittmann, 1971) and Rb2Ge4O9 (Goreaud & Raveau, 1976). The results for Na2Ge4O9 perfectly match those obtained by Fleet & Muthupari (1998). Fig. 1 shows an anisotropic displacement plot for K2Ge4O9, with the atomic nomenclature. It is evident that the A-site cations in particular, but also the O atoms, exhibit distinct anisotropic atomic displacement parameters, justifying the need for new structure refinements.
Within the A2Ge4O9 series, the Na+ compound naturally shows the smallest unit-cell dimensions and unit-cell volume, while the largest are found for Rb2Ge4O9. This variation corresponds well with the variations in ionic radii of the monovalent cations (1.18 Å for Na+, 1.51 Å for K+ and 1.61 Å for Rb+ in eightfold coordination; Shannon & Prewitt, 1969). For ease of comparison, the lattice parameters, selected bond lengths and angles, and distortion parameters, recalculated from the literature data, are given in Table 2 together with the data from this study. As can be seen, the lattice parameters for Na2Ge4O9 are in excellent agreement with those given by Fleet & Muthupari (1998). The lattice parameters of Völlenkle & Wittmann (1971) for K2Ge2O9 are confirmed in this study, albeit with a much better precision. The present data for Rb2Ge4O9 are somewhat larger than those given by Goreaud & Raveau (1976), but again the precision of the new data is better by two orders of magnitude.
The dominant building unit of the title compounds is a three-membered [Ge3O9] ring consisting of two Ge3O4 and one Ge2O4 tetrahedra, and having an intrinsic twofold symmetry (Fig. 2) with the twofold axis passing through atoms Ge2 and O5. Within the ab plane, the [Ge3O9] units are oriented in layers. Within each layer these are related to each other by the threefold axes in 3d, while from layer to layer they are related to each other by the 3-fold axes in the 2b position. The tetrahedral [Ge3O9] rings are not coplanar within such a layer. The planes through atom Ge2 and the two Ge3 atoms of an individual ring are inclined to each other by a tilt angle of 31.40 (2)–17.29 (2)°, depending on chemistry. Very similar values were calculated from the literature data (Table 2), with the best agreement being observed for the high-quality data for Na2Ge4O9 (Fleet & Muthupari, 1998) and the worst with the data for Rb2Ge4O9 (Goreaud & Raveau, 1976). The smallest tilts are found in the Rb compound; the angle between the plane through atoms Ge2 and Ge3 and the (001) plane decreases from 18.3 (1)° in Na2Ge4O9 to 10.0 (1)° in Rb2Ge4O9 (Table 2), i.e. the larger the A-site cation, the more coplanar are the tetrahedral rings aligned to each other with respect to the Ge2–Ge3–Ge2 plane. Within the [Ge3O9] rings, the Ge—O—Ge angles all increase with increasing size of the alkaline cation, causing the tetrahedral faces to be more crimped to each other within the ring. In Na2Ge4O9, for example, the angle between the O2–O4–O5 faces of the neighbouring Ge3 sites is 2.4 (1)°, while it increases to 13.17 (7)° in Rb2Ge4O9. However, this crimp brings atom Ge3 closer to the plane through bridging atoms O2–O2–O5 of the ring. While in Na2Ge4O9 atom Ge3 is displaced out of this plane by 0.704 (1) Å, it is 0.519 (1) Å away in Rb2Ge4O9; atom Ge2 is within this plane. For comparison, in the related structure of wadeite K2Zr(Si3O9), the [Si3O9] rings have ideal geometry, with the Si atoms being coplanar to the bridging O atoms of the ring.
The layers of [Ge3O9] rings pass through the unit cell at z \sim 0.25 and z ~3/4, although they are not stacked up on each other but are displaced as a consequence of the 3 [3-fold axis?]. This is different to wadeite, which has three-membered rings of SiO4 tetrahedra and where the layers of [Si3O9] rings are superimposed on each other parallel to the c axis due to the higher P63/m symmetry. The Ge2O4 tetrahedron with site symmetry 2 shows an average <Ge2—O> bond length of 1.776 (3) Å for Na2Ge4O9, in perfect agreement with the data given by Fleet & Muthupari (1972 1998?). While for K2Ge4O9 the data of this study are similar to those given in the literature, for the Rb2Ge4O9 compound in particular some distinct deviations in individual bond lengths are found. Goreaud & Raveau (1976) reported two quite small (1.69 Å) and two quite long (1.81 Å) Ge—O bonds, which are at the limit for Ge—O bonds in a tetrahedral coordination. However, these bonds are no longer valid on the basis of the new structure refinement of this study. The Ge2 tetrahedron generally shows larger <Ge—O> bond lengths and also a larger polyhedral volume compared with the Ge3O4 tetrahedron, but it shows a smaller polyhedral distortion, as expressed by the values of TAV (tetrahedral angle variance) and TQE (tetrahedral quadratic elongation; Table 2). Nevertheless, the tetrahedral distortion is large compared with e.g. Li2Ge4O9. Here, the most distorted GeO4 tetrahedron has TAV values of ~37°, and the average Ge—O distances range between 1.755 and 1.778 Å (Redhammer & Tippelt, 2013). The tetrahedral distortion of the title compounds is among the largest found in the literature and similar high values are valid only in e.g. lanthanide pyrogermanates with Ge2O7 groups (Redhammer et al., 2007b). Within the A2Ge4O9 series, a significant decrease of <Ge—O> distances, polyhedral volume and distortion with increasing size of the A-site cation is observed (Table 2). This suggests that the Rb compound represents the most stable and regular member of the A2Ge4O9 series and that the long Ge—O bonds in Na2Ge4O9 are energetically unfavourable. The bond-valence sums (BVS; Brese & O'Keeffe, 1991; Table 2) also increase from Na to Rb compounds, thereby becoming closer to the ideal value of 4.0 valence units (v.u.). But even in Rb2Ge4O9 the Ge2 site appears to be underbonded. For the Ge3O4 tetrahedra, <Ge—O> is smaller than for Ge2 and remains constant across the A2Ge4O9 series. The polyhedral volume increases slightly towards Rb2Ge4O9 but is also smaller than for the Ge2 site. For the Ge3 site, the data of this study fit well with the data from the literature, except for Rb2Ge4O9; besides the large estimated standard uncertainties there is again more spread in the individual bond lengths, with somewhat unrealistically long Ge3—O5 bond lengths in the Goreaud & Raveau (1976) data; these are mended by the present structure determination. The largest distortion of all the tetrahedral sites in the A2Ge4O9 compounds is found for the Ge3 site in Na2Ge4O9. With increasing size of the A-site cation, the Ge3 tetrahedra become more regular but are still distinctly distorted. In wadeite, the SiO4 tetrahedron shows a TAV value of only 38.8° (recalculated from Swanson & Prewitt, 1983). The BVS for the Ge3 tetrahdron are almost ideal and do not change from the Na to the Rb compound, neither in the literature nor in this study (Table 2).
Two different GeO6 octahedra (Ge1 and Ge4) connect the tetrahedral layers to each other to build up the tetrahedral-octahedral framework. Thus, three corners of a triangular octahedral face belong to an upper layer and three to a lower tetrahedral layer (see Fig. 3 for a graphical illustration). The Ge1O6 octahedron at the origin of the unit cell with site symmetry 3 has a regular O-atom coordination, with a <Ge—O> distance of 1.920 (2) Å for the sodium compound, in perfect accordance with Fleet & Muthupari (1998). This is a typical value found for Ge in an octahedral coordination. With increasing size of the A-site cation a decrease in <Ge—O> is again observed towards Rb2Ge4O9, accompanied by a decrease in the octahedral volume and, in particular, of the octahedral distortion (Table 2). This is not observable from the literature data, as Rb2Ge4O9 in particular deviates from the data of this study. Also, the large estimated standard uncertainties for both K2Ge4O9 and Rb2Ge4O9 would not allow the extraction of such small alterations with composition on a meaningful basis. While atom Ge1 is distinctly underbonded in Na2Ge4O9, the BVS approaches the ideal value of 4.00 in the Rb compound. The Ge1O6 site has common corners with six Ge2O4 tetrahedra, and additionally each O-atom corner is shared with an A site. The second octahedral site within the structure of the A2Ge4O9 series is the Ge4 site. It exhibits distinctly shorter <Ge—O> bond lengths and also a smaller polyhedral volume than the Ge1 site; both are similar across the series. From Table 2 a good agreement between the data from this study and the literature is observable, except for the longer Ge4—O bond lengths in the Goreaud & Raveau (1976) data for Rb2Ge4O9. The Ge4 site represents an almost-perfect octahedron with respect to polyhedral distortion, which does not change as a function of A-site composition; a similar situation exists for the polyhedral volume. The BVS of the Ge4 site show this Ge atom to be overbonded, with BVS of 4.29–4.19 v.u., decreasing with increasing size of the A-site cation. The O-atom corners of the Ge4O6 site are common to one Ge3O4 tetrahedron and an A-site cation each.
The cavities within the tetrahedral–octahedral framework are filled with the medium-to-large sized extra-framework alkaline cations Na+, K+ or Rb+. It is the bonding topology of the monovalent cations which marks the essential difference between the title compounds. The Na+ cation is in a 5+2 coordination, with five bonds between 2.442 (4) and 2.633 (4) Å, the two others being 2.722 (4) and 2.860 (4), showing a near square-pyramidal coordination. This distorted coordination is reflected in the large value of the bond-length distortion (deviation of individual A—O bonds from the mean value) of 4.18%. The low BVS of sodium (0.84 v.u.) suggests that the Na+ cation is nearly too small (and consequently has too long Na—O bonds) to fit into the cavities. This is supported by the large equivalent isotropic and anisotropic atomic displacement parameters in Na2Ge4O9 and by the fact that Li2Ge4O9 shows a different crystal structure, even though the chemical composition would fit the A2Ge4O9 series. K2Ge4O9 shows a more uniform K—O bond distribution which is close to a 6+1 coordination, a bond-length distortion of 2.06% and a less pronounced anisotropic atomic displacement. The BVS is 1.14 v.u., reflecting some overbonding of the A site. This is even more pronounced in Rb2Ge4O9 (1.19 v.u.), where the Rb+ shows a sevenfold coordination with bond lengths between 2.823 (1) and 3.013 (1) Å and a bond-length distortion of 1.92%; the coordination polyhedron may be described as a distorted prism with a triangular and a quadrangular covering face. The Rb+ cation itself shows the smallest equivalent and anisotropic atomic displacement parameters of all three compounds. The different size requirements of the alkali cations within the cavity are adjusted by a twist deformation of the tetrahedral rings and altered Ge—O—Ge angles between tetrahedral and octahedral sites. The latter angles increase distinctly by a mean of 11° between the Na and Rb compounds, while the increase in the twist of the Ge—O—Ge angles within the tetrahedral rings is 6.7° on the average (Table 2). The literature data show similar behaviour and the differences in bonding topology are valid, especially for the Rb2Ge4O9 compound.
In conclusion, our data for Na2Ge4O9 fit perfectly with the high-quality structure refinement of Fleet & Muthupari (1998). The redetermination of the crystal structures of K2Ge4O9 and Rb2Ge4O9 in this study using modern equipment confirms their space group symmetries and atomic positions, extracted by Völlenkle & Wittmann (1971) and Goreaud & Raveau (1976), respectively. We pay respect to these works as their data, obtained from film methods and visual estimates of intensity data, fit those determined here using automated methods, and the authors were able to report the principal structural features of the A2Ge4O9 compounds correctly. With the capabilities of modern diffractometers we have here given more precise data with much smaller estimated standard uncertainties and anisotropic refinements of atomic displacement parameters. This is especially important for the A-site cation and the O atoms, and may explain the large differences in A-site geometry in Rb2Ge4O9 between the Goreaud & Raveau (1976) data and those of this study.
Generally, among all three compounds, the literature data for Rb2Ge4O9 show the largest deviation in bond lengths, bond angles and distortion parameters from the data determined in this study (Table 2). The unrealistically small (1.69 Å) and/or large (1.81 Å) Ge—O distances for tetrahedral coordination in the Rb2Ge4O9 data of Goreaud & Raveau (1976) indicate the deficiencies in their structural model. These problems, together with the large uncertainties and the isotropic refinement of atomic displacements, are overcome in this study. The structural model for K2Ge4O9 given by Völlenkle & Wittmann (1971) is confirmed to a large extent and our new data `only' contribute much smaller uncertainties in the structural parameters and give full anisotropic refinement of atomic displacements. Direct comparison of the three A2Ge4O9 compounds allows the extraction of some systematic deviations with chemistry. Both tetrahedral sites are distinctly distorted but become more regular towards the Rb2Ge4O9 compound, which is associated with a decrease in the tetrahedral <Ge—O> distances with increasing A-site cation. Also for the octahedral sites, the distortion becomes smaller from the Na to the Rb compound, especially for the larger Ge1 site. This, together with the more regular A-site geometry (smaller bond length distortion), shows Rb2Ge4O9 to be the most regular and geometrically most stable member of the A2Ge4O9 series.
For all compounds, data collection: APEX2 (Bruker, 2007); cell refinement: APEX2 (Bruker, 2007); data reduction: APEX2 (Bruker, 2007); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: DIAMOND (Brandenburg, 1999); software used to prepare material for publication: WinGX (Farrugia, 2012).
Fig. 1. A view of the asymmetric unit and some symmetry-related atoms of
K2Ge4O9, showing 95% probability atomic displacement ellipsoids and the
atomic numbering scheme. [Symmetry codes: (i) -x + y, -x,
z; (ii) -y, x - y, z; (iii) -x,
-y, -z + 1; (iv) x - y, x, -z + 1;
(v) y, -x + y, -z + 1; (vi) y, x,
-z + 1/2; (vii) -y + 1, x - y, z; (viii)
-x + y + 1, -x + 1, z; (ix) -x + 1,
y, z + 1/2; (x) -y + 1, -x + 1, z - 1/2;
(xi) x, x - y, z - 1/2; (xii) -x + 1,
-y + 1, -z + 1; (xiii) x, x - y, z +
1/2.] Fig. 2. A polyhedral representation of a layer of tetrahedral [Ge3O9] rings in K2Ge4O9 at z ~ 0.25, in a view parallel to the c axis. Fig. 3. Polyhedral representations of the full structure of K2Ge4O9, (a) in a view parallel to the c axis and (b) in a view parallel to the b axis. |
Na2Ge4O9 | F(000) = 1332 |
Mr = 480.34 | Dx = 4.447 Mg m−3 |
Trigonal, P3c1 | Mo Kα radiation, λ = 0.71073 Å |
a = 11.3216 (10) Å | µ = 16.75 mm−1 |
c = 9.6946 (18) Å | T = 295 K |
V = 1076.2 (3) Å3 | Prism, colourless |
Z = 6 | 0.22 × 0.07 × 0.06 mm |
Bruker SMART APEX diffractometer | 924 independent reflections |
Radiation source: three-circle diffractometer | 853 reflections with I > 2σ(I) |
Graphite monochromator | Rint = 0.041 |
ω scan at four different ϕ positions | θmax = 28.7°, θmin = 2.1° |
Absorption correction: empirical (using intensity measurements) [multiscan absorption correction using APEX2 (Bruker, 2007)] | h = −15→15 |
Tmin = 0.441, Tmax = 0.748 | k = −14→14 |
12328 measured reflections | l = −13→12 |
Refinement on F2 | Primary atom site location: structure-invariant direct methods |
Least-squares matrix: full | Secondary atom site location: difference Fourier map |
R[F2 > 2σ(F2)] = 0.028 | w = 1/[σ2(Fo2) + (0.0118P)2 + 1.8181P] where P = (Fo2 + 2Fc2)/3 |
wR(F2) = 0.055 | (Δ/σ)max < 0.001 |
S = 1.21 | Δρmax = 0.78 e Å−3 |
924 reflections | Δρmin = −0.95 e Å−3 |
71 parameters | Extinction correction: SHELXL97 (Sheldrick, 2008) |
0 restraints | Extinction coefficient: 0.00065 (11) |
Na2Ge4O9 | Z = 6 |
Mr = 480.34 | Mo Kα radiation |
Trigonal, P3c1 | µ = 16.75 mm−1 |
a = 11.3216 (10) Å | T = 295 K |
c = 9.6946 (18) Å | 0.22 × 0.07 × 0.06 mm |
V = 1076.2 (3) Å3 |
Bruker SMART APEX diffractometer | 924 independent reflections |
Absorption correction: empirical (using intensity measurements) [multiscan absorption correction using APEX2 (Bruker, 2007)] | 853 reflections with I > 2σ(I) |
Tmin = 0.441, Tmax = 0.748 | Rint = 0.041 |
12328 measured reflections |
R[F2 > 2σ(F2)] = 0.028 | 71 parameters |
wR(F2) = 0.055 | 0 restraints |
S = 1.21 | Δρmax = 0.78 e Å−3 |
924 reflections | Δρmin = −0.95 e Å−3 |
Geometry. All s.u.'s (except the s.u. in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell s.u.'s are taken into account individually in the estimation of s.u.'s in distances, angles and torsion angles; correlations between s.u.'s in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell s.u.'s is used for estimating s.u.'s involving l.s. planes. |
Refinement. Refinement of F2 against ALL reflections. The weighted R-factor wR and goodness of fit S are based on F2, conventional R-factors R are based on F, with F set to zero for negative F2. The threshold expression of F2 > 2σ(F2) is used only for calculating R-factors(gt) etc. and is not relevant to the choice of reflections for refinement. R-factors based on F2 are statistically about twice as large as those based on F, and R-factors based on ALL data will be even larger. |
x | y | z | Uiso*/Ueq | ||
Na1 | 0.3355 (2) | 0.3294 (2) | 0.5763 (2) | 0.0284 (5) | |
Ge1 | 0 | 0 | 0.5 | 0.0090 (2) | |
Ge2 | 0.17623 (6) | 0.17623 (6) | 0.25 | 0.01015 (16) | |
Ge3 | 0.49387 (4) | 0.34451 (4) | 0.29968 (4) | 0.00891 (14) | |
Ge4 | 0.6667 | 0.3333 | 0.05427 (7) | 0.00812 (18) | |
O1 | 0.0592 (3) | 0.1546 (3) | 0.3803 (3) | 0.0112 (6) | |
O2 | 0.3252 (3) | 0.2245 (3) | 0.3532 (3) | 0.0140 (6) | |
O3 | 0.5126 (3) | 0.2500 (3) | 0.1700 (3) | 0.0133 (6) | |
O4 | 0.5922 (3) | 0.4152 (3) | 0.4454 (3) | 0.0130 (6) | |
O5 | 0.4939 (4) | 0.4939 (4) | 0.25 | 0.0131 (9) |
U11 | U22 | U33 | U12 | U13 | U23 | |
Na1 | 0.0364 (13) | 0.0201 (10) | 0.0200 (9) | 0.0077 (10) | 0.0071 (9) | −0.0017 (8) |
Ge1 | 0.0098 (3) | 0.0098 (3) | 0.0075 (5) | 0.00490 (16) | 0 | 0 |
Ge2 | 0.0100 (2) | 0.0100 (2) | 0.0094 (3) | 0.0042 (3) | 0.00118 (11) | −0.00118 (11) |
Ge3 | 0.0096 (2) | 0.0091 (2) | 0.0082 (2) | 0.00484 (18) | −0.00050 (15) | −0.00039 (15) |
Ge4 | 0.0091 (2) | 0.0091 (2) | 0.0061 (3) | 0.00456 (12) | 0 | 0 |
O1 | 0.0143 (15) | 0.0114 (14) | 0.0098 (13) | 0.0079 (13) | 0.0022 (11) | 0.0011 (11) |
O2 | 0.0092 (15) | 0.0163 (16) | 0.0127 (14) | 0.0035 (13) | 0.0005 (11) | 0.0010 (12) |
O3 | 0.0140 (15) | 0.0118 (15) | 0.0115 (13) | 0.0044 (13) | 0.0059 (12) | −0.0004 (11) |
O4 | 0.0175 (16) | 0.0126 (15) | 0.0096 (13) | 0.0080 (13) | −0.0045 (12) | −0.0015 (12) |
O5 | 0.0115 (15) | 0.0115 (15) | 0.018 (2) | 0.0066 (18) | −0.0006 (8) | 0.0006 (8) |
Na1—O2 | 2.442 (4) | Ge2—O1 | 1.757 (3) |
Na1—O1i | 2.443 (4) | Ge2—O2 | 1.795 (3) |
Na1—O5ii | 2.589 (4) | Ge2—O2viii | 1.795 (3) |
Na1—O4ii | 2.589 (4) | Ge3—O4 | 1.727 (3) |
Na1—O3iii | 2.633 (4) | Ge3—O3 | 1.732 (3) |
Na1—O2iv | 2.722 (4) | Ge3—O5 | 1.758 (2) |
Na1—O4 | 2.860 (4) | Ge3—O2 | 1.780 (3) |
Ge1—O1i | 1.920 (3) | Ge4—O4ix | 1.862 (3) |
Ge1—O1iv | 1.920 (3) | Ge4—O4x | 1.862 (3) |
Ge1—O1v | 1.920 (3) | Ge4—O4xi | 1.862 (3) |
Ge1—O1vi | 1.920 (3) | Ge4—O3xii | 1.883 (3) |
Ge1—O1 | 1.920 (3) | Ge4—O3xiii | 1.883 (3) |
Ge1—O1vii | 1.920 (3) | Ge4—O3 | 1.883 (3) |
Ge2—O1viii | 1.757 (3) | ||
O2—Na1—O1i | 79.30 (11) | O1iv—Ge1—O1vii | 180.00 (13) |
O2—Na1—O5ii | 141.97 (13) | O1v—Ge1—O1vii | 87.26 (12) |
O1i—Na1—O5ii | 128.56 (12) | O1vi—Ge1—O1vii | 92.74 (12) |
O2—Na1—O4ii | 112.49 (12) | O1—Ge1—O1vii | 87.26 (12) |
O1i—Na1—O4ii | 148.28 (14) | O1viii—Ge2—O1 | 126.9 (2) |
O5ii—Na1—O4ii | 59.54 (9) | O1viii—Ge2—O2 | 112.72 (14) |
O2—Na1—O3iii | 91.37 (12) | O1—Ge2—O2 | 99.76 (13) |
O1i—Na1—O3iii | 88.56 (12) | O1viii—Ge2—O2viii | 99.76 (13) |
O5ii—Na1—O3iii | 67.46 (9) | O1—Ge2—O2viii | 112.72 (14) |
O4ii—Na1—O3iii | 119.20 (13) | O2—Ge2—O2viii | 103.1 (2) |
O2—Na1—O2iv | 113.86 (14) | O4—Ge3—O3 | 129.09 (15) |
O1i—Na1—O2iv | 69.86 (11) | O4—Ge3—O5 | 95.04 (13) |
O5ii—Na1—O2iv | 101.26 (10) | O3—Ge3—O5 | 117.02 (10) |
O4ii—Na1—O2iv | 78.52 (11) | O4—Ge3—O2 | 108.21 (14) |
O3iii—Na1—O2iv | 141.79 (12) | O3—Ge3—O2 | 99.00 (14) |
O2—Na1—O4 | 64.24 (10) | O5—Ge3—O2 | 107.34 (16) |
O1i—Na1—O4 | 127.11 (13) | O4ix—Ge4—O4x | 91.02 (13) |
O5ii—Na1—O4 | 77.73 (10) | O4ix—Ge4—O4xi | 91.02 (13) |
O4ii—Na1—O4 | 83.37 (11) | O4x—Ge4—O4xi | 91.02 (13) |
O3iii—Na1—O4 | 57.33 (10) | O4ix—Ge4—O3xii | 91.06 (13) |
O2iv—Na1—O4 | 159.25 (12) | O4x—Ge4—O3xii | 89.76 (13) |
O1i—Ge1—O1iv | 87.26 (12) | O4xi—Ge4—O3xii | 177.78 (14) |
O1i—Ge1—O1v | 180.00 (16) | O4ix—Ge4—O3xiii | 177.78 (14) |
O1iv—Ge1—O1v | 92.74 (12) | O4x—Ge4—O3xiii | 91.06 (13) |
O1i—Ge1—O1vi | 87.26 (12) | O4xi—Ge4—O3xiii | 89.76 (13) |
O1iv—Ge1—O1vi | 87.26 (12) | O3xii—Ge4—O3xiii | 88.14 (13) |
O1v—Ge1—O1vi | 92.74 (12) | O4ix—Ge4—O3 | 89.76 (13) |
O1i—Ge1—O1 | 92.74 (12) | O4x—Ge4—O3 | 177.78 (14) |
O1iv—Ge1—O1 | 92.74 (12) | O4xi—Ge4—O3 | 91.06 (13) |
O1v—Ge1—O1 | 87.26 (12) | O3xii—Ge4—O3 | 88.14 (13) |
O1vi—Ge1—O1 | 180.00 (14) | O3xiii—Ge4—O3 | 88.14 (13) |
O1i—Ge1—O1vii | 92.74 (12) |
Symmetry codes: (i) y, −x+y, −z+1; (ii) −x+1, −y+1, −z+1; (iii) x, x−y, z+1/2; (iv) x−y, x, −z+1; (v) −y, x−y, z; (vi) −x, −y, −z+1; (vii) −x+y, −x, z; (viii) y, x, −z+1/2; (ix) x, x−y, z−1/2; (x) −x+y+1, y, z−1/2; (xi) −y+1, −x+1, z−1/2; (xii) −y+1, x−y, z; (xiii) −x+y+1, −x+1, z. |
K2Ge4O9 | Dx = 4.287 Mg m−3 |
Mr = 512.56 | Mo Kα radiation, λ = 0.71073 Å |
Trigonal, P3c1 | Cell parameters from 12842 reflections |
a = 11.8461 (4) Å | θ = 2.0–28.9° |
c = 9.8009 (4) Å | µ = 16.08 mm−1 |
V = 1191.10 (7) Å3 | T = 295 K |
Z = 6 | Prism, colourless |
F(000) = 1428 | 0.17 × 0.09 × 0.08 mm |
Bruker SMART APEX diffractometer | 1026 independent reflections |
Radiation source: three-circle diffractometer | 984 reflections with I > 2σ(I) |
Graphite monochromator | Rint = 0.026 |
ω scan at four different ϕ positions | θmax = 28.9°, θmin = 2.0° |
Absorption correction: empirical (using intensity measurements) [multiscan absorption correction using APEX2 (Bruker, 2007)] | h = −15→15 |
Tmin = 0.441, Tmax = 0.748 | k = −15→15 |
12842 measured reflections | l = −13→13 |
Refinement on F2 | Primary atom site location: structure-invariant direct methods |
Least-squares matrix: full | Secondary atom site location: difference Fourier map |
R[F2 > 2σ(F2)] = 0.014 | w = 1/[σ2(Fo2) + (0.0151P)2 + 1.1161P] where P = (Fo2 + 2Fc2)/3 |
wR(F2) = 0.035 | (Δ/σ)max = 0.001 |
S = 1.14 | Δρmax = 0.39 e Å−3 |
1026 reflections | Δρmin = −0.58 e Å−3 |
71 parameters | Extinction correction: SHELXL97 (Sheldrick, 2008) |
0 restraints | Extinction coefficient: 0.00549 (16) |
K2Ge4O9 | Z = 6 |
Mr = 512.56 | Mo Kα radiation |
Trigonal, P3c1 | µ = 16.08 mm−1 |
a = 11.8461 (4) Å | T = 295 K |
c = 9.8009 (4) Å | 0.17 × 0.09 × 0.08 mm |
V = 1191.10 (7) Å3 |
Bruker SMART APEX diffractometer | 1026 independent reflections |
Absorption correction: empirical (using intensity measurements) [multiscan absorption correction using APEX2 (Bruker, 2007)] | 984 reflections with I > 2σ(I) |
Tmin = 0.441, Tmax = 0.748 | Rint = 0.026 |
12842 measured reflections |
R[F2 > 2σ(F2)] = 0.014 | 71 parameters |
wR(F2) = 0.035 | 0 restraints |
S = 1.14 | Δρmax = 0.39 e Å−3 |
1026 reflections | Δρmin = −0.58 e Å−3 |
Geometry. All s.u.'s (except the s.u. in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell s.u.'s are taken into account individually in the estimation of s.u.'s in distances, angles and torsion angles; correlations between s.u.'s in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell s.u.'s is used for estimating s.u.'s involving l.s. planes. |
Refinement. Refinement of F2 against ALL reflections. The weighted R-factor wR and goodness of fit S are based on F2, conventional R-factors R are based on F, with F set to zero for negative F2. The threshold expression of F2 > 2σ(F2) is used only for calculating R-factors(gt) etc. and is not relevant to the choice of reflections for refinement. R-factors based on F2 are statistically about twice as large as those based on F, and R-factors based on ALL data will be even larger. |
x | y | z | Uiso*/Ueq | ||
K1 | 0.33657 (4) | 0.33480 (4) | 0.57196 (4) | 0.01681 (10) | |
Ge1 | 0 | 0 | 0.5 | 0.00634 (10) | |
Ge2 | 0.17619 (2) | 0.17619 (2) | 0.25 | 0.00681 (8) | |
Ge3 | 0.486079 (17) | 0.335234 (16) | 0.285931 (18) | 0.00685 (7) | |
Ge4 | 0.6667 | 0.3333 | 0.03798 (3) | 0.00632 (8) | |
O1 | 0.06988 (12) | 0.14984 (12) | 0.38487 (12) | 0.0097 (2) | |
O2 | 0.32552 (11) | 0.21187 (12) | 0.33163 (12) | 0.0103 (2) | |
O3 | 0.51615 (12) | 0.26357 (12) | 0.14830 (12) | 0.0118 (2) | |
O4 | 0.58170 (12) | 0.39769 (12) | 0.43018 (12) | 0.0104 (2) | |
O5 | 0.47618 (14) | 0.47618 (14) | 0.25 | 0.0123 (3) |
U11 | U22 | U33 | U12 | U13 | U23 | |
K1 | 0.01542 (19) | 0.0153 (2) | 0.01793 (19) | 0.00635 (15) | 0.00248 (15) | −0.00298 (15) |
Ge1 | 0.00647 (13) | 0.00647 (13) | 0.00609 (19) | 0.00323 (6) | 0 | 0 |
Ge2 | 0.00662 (10) | 0.00662 (10) | 0.00681 (13) | 0.00302 (9) | 0.00038 (4) | −0.00038 (4) |
Ge3 | 0.00664 (10) | 0.00724 (10) | 0.00699 (11) | 0.00371 (7) | −0.00026 (6) | −0.00011 (6) |
Ge4 | 0.00655 (10) | 0.00655 (10) | 0.00585 (15) | 0.00328 (5) | 0 | 0 |
O1 | 0.0113 (6) | 0.0086 (6) | 0.0094 (5) | 0.0050 (5) | 0.0036 (5) | 0.0017 (5) |
O2 | 0.0062 (5) | 0.0113 (6) | 0.0109 (6) | 0.0025 (5) | 0.0002 (4) | 0.0025 (5) |
O3 | 0.0107 (6) | 0.0115 (6) | 0.0107 (5) | 0.0037 (5) | 0.0038 (5) | −0.0019 (5) |
O4 | 0.0124 (6) | 0.0112 (6) | 0.0091 (5) | 0.0071 (5) | −0.0031 (5) | −0.0005 (4) |
O5 | 0.0093 (6) | 0.0093 (6) | 0.0194 (9) | 0.0053 (7) | −0.0019 (3) | 0.0019 (3) |
K1—O2 | 2.7377 (13) | Ge2—O1 | 1.7428 (12) |
K1—O1i | 2.7401 (13) | Ge2—O2 | 1.7889 (12) |
K1—O2ii | 2.7547 (13) | Ge2—O2viii | 1.7889 (12) |
K1—O4iii | 2.8128 (13) | Ge3—O3 | 1.7232 (12) |
K1—O5iii | 2.8305 (14) | Ge3—O4 | 1.7295 (12) |
K1—O3iv | 2.8471 (13) | Ge3—O5 | 1.7667 (8) |
K1—O4 | 2.9588 (13) | Ge3—O2 | 1.7816 (12) |
Ge1—O1i | 1.9078 (12) | Ge4—O4ix | 1.8649 (12) |
Ge1—O1v | 1.9078 (12) | Ge4—O4x | 1.8649 (12) |
Ge1—O1vi | 1.9078 (12) | Ge4—O4xi | 1.8649 (12) |
Ge1—O1ii | 1.9078 (12) | Ge4—O3xii | 1.8862 (12) |
Ge1—O1vii | 1.9078 (12) | Ge4—O3xiii | 1.8862 (12) |
Ge1—O1 | 1.9078 (12) | Ge4—O3 | 1.8862 (12) |
Ge2—O1viii | 1.7428 (12) | ||
O2—K1—O1i | 75.48 (4) | O1v—Ge1—O1vii | 88.58 (5) |
O2—K1—O2ii | 118.94 (5) | O1vi—Ge1—O1vii | 91.42 (5) |
O1i—K1—O2ii | 70.93 (4) | O1ii—Ge1—O1vii | 180 |
O2—K1—O4iii | 119.39 (4) | O1i—Ge1—O1 | 91.42 (5) |
O1i—K1—O4iii | 151.15 (4) | O1v—Ge1—O1 | 88.58 (5) |
O2ii—K1—O4iii | 80.25 (4) | O1vi—Ge1—O1 | 180.00 (6) |
O2—K1—O5iii | 139.58 (3) | O1ii—Ge1—O1 | 91.42 (5) |
O1i—K1—O5iii | 131.15 (3) | O1vii—Ge1—O1 | 88.58 (5) |
O2ii—K1—O5iii | 100.17 (3) | O1viii—Ge2—O1 | 126.40 (8) |
O4iii—K1—O5iii | 54.64 (3) | O1viii—Ge2—O2 | 107.98 (6) |
O2—K1—O3iv | 83.65 (4) | O1—Ge2—O2 | 104.11 (5) |
O1i—K1—O3iv | 84.96 (4) | O1viii—Ge2—O2viii | 104.11 (5) |
O2ii—K1—O3iv | 139.60 (4) | O1—Ge2—O2viii | 107.98 (6) |
O4iii—K1—O3iv | 119.49 (4) | O2—Ge2—O2viii | 104.45 (8) |
O5iii—K1—O3iv | 71.61 (3) | O3—Ge3—O4 | 126.74 (6) |
O2—K1—O4 | 60.64 (3) | O3—Ge3—O5 | 115.93 (4) |
O1i—K1—O4 | 120.00 (4) | O4—Ge3—O5 | 95.59 (5) |
O2ii—K1—O4 | 166.33 (4) | O3—Ge3—O2 | 101.04 (6) |
O4iii—K1—O4 | 88.35 (4) | O4—Ge3—O2 | 110.34 (6) |
O5iii—K1—O4 | 78.94 (3) | O5—Ge3—O2 | 106.09 (6) |
O3iv—K1—O4 | 53.31 (4) | O4ix—Ge4—O4x | 91.06 (5) |
O2—K1—O1 | 53.31 (3) | O4ix—Ge4—O4xi | 91.06 (5) |
O1i—K1—O1 | 52.13 (4) | O4x—Ge4—O4xi | 91.06 (5) |
O2ii—K1—O1 | 65.79 (3) | O4ix—Ge4—O3xii | 90.53 (5) |
O4iii—K1—O1 | 114.43 (3) | O4x—Ge4—O3xii | 88.02 (5) |
O5iii—K1—O1 | 164.71 (3) | O4xi—Ge4—O3xii | 178.18 (5) |
O3iv—K1—O1 | 122.80 (4) | O4ix—Ge4—O3xiii | 178.18 (5) |
O4—K1—O1 | 113.17 (3) | O4x—Ge4—O3xiii | 90.53 (5) |
O1i—Ge1—O1v | 180 | O4xi—Ge4—O3xiii | 88.02 (5) |
O1i—Ge1—O1vi | 88.58 (5) | O3xii—Ge4—O3xiii | 90.41 (5) |
O1v—Ge1—O1vi | 91.42 (5) | O4ix—Ge4—O3 | 88.02 (5) |
O1i—Ge1—O1ii | 88.58 (5) | O4x—Ge4—O3 | 178.18 (5) |
O1v—Ge1—O1ii | 91.42 (5) | O4xi—Ge4—O3 | 90.53 (5) |
O1vi—Ge1—O1ii | 88.58 (5) | O3xii—Ge4—O3 | 90.41 (5) |
O1i—Ge1—O1vii | 91.42 (5) | O3xiii—Ge4—O3 | 90.41 (5) |
Symmetry codes: (i) y, −x+y, −z+1; (ii) x−y, x, −z+1; (iii) −x+1, −y+1, −z+1; (iv) x, x−y, z+1/2; (v) −y, x−y, z; (vi) −x, −y, −z+1; (vii) −x+y, −x, z; (viii) y, x, −z+1/2; (ix) x, x−y, z−1/2; (x) −x+y+1, y, z−1/2; (xi) −y+1, −x+1, z−1/2; (xii) −y+1, x−y, z; (xiii) −x+y+1, −x+1, z. |
Rb2Ge4O9 | F(000) = 1644 |
Mr = 605.3 | Dx = 4.817 Mg m−3 |
Trigonal, P3c1 | Mo Kα radiation, λ = 0.71073 Å |
a = 12.1008 (6) Å | µ = 25.89 mm−1 |
c = 9.8722 (5) Å | T = 295 K |
V = 1251.91 (11) Å3 | Prism, colourless |
Z = 6 | 0.14 × 0.05 × 0.04 mm |
Bruker SMART APEX diffractometer | 1091 independent reflections |
Radiation source: three-circle diffractometer | 1027 reflections with I > 2σ(I) |
Graphite monochromator | Rint = 0.048 |
ω scan at four different ϕ positions | θmax = 28.9°, θmin = 1.9° |
Absorption correction: empirical (using intensity measurements) [multiscan absorption correction using APEX2 (Bruker, 2007)] | h = −16→16 |
Tmin = 0.298, Tmax = 0.746 | k = −16→15 |
13739 measured reflections | l = −13→13 |
Refinement on F2 | Primary atom site location: structure-invariant direct methods |
Least-squares matrix: full | Secondary atom site location: difference Fourier map |
R[F2 > 2σ(F2)] = 0.017 | w = 1/[σ2(Fo2) + (0.0186P)2 + 0.9217P] where P = (Fo2 + 2Fc2)/3 |
wR(F2) = 0.042 | (Δ/σ)max = 0.001 |
S = 1.20 | Δρmax = 0.62 e Å−3 |
1091 reflections | Δρmin = −0.89 e Å−3 |
71 parameters | Extinction correction: SHELXL97 (Sheldrick, 2008) |
0 restraints | Extinction coefficient: 0.00650 (19) |
Rb2Ge4O9 | Z = 6 |
Mr = 605.3 | Mo Kα radiation |
Trigonal, P3c1 | µ = 25.89 mm−1 |
a = 12.1008 (6) Å | T = 295 K |
c = 9.8722 (5) Å | 0.14 × 0.05 × 0.04 mm |
V = 1251.91 (11) Å3 |
Bruker SMART APEX diffractometer | 1091 independent reflections |
Absorption correction: empirical (using intensity measurements) [multiscan absorption correction using APEX2 (Bruker, 2007)] | 1027 reflections with I > 2σ(I) |
Tmin = 0.298, Tmax = 0.746 | Rint = 0.048 |
13739 measured reflections |
R[F2 > 2σ(F2)] = 0.017 | 71 parameters |
wR(F2) = 0.042 | 0 restraints |
S = 1.20 | Δρmax = 0.62 e Å−3 |
1091 reflections | Δρmin = −0.89 e Å−3 |
Geometry. All s.u.'s (except the s.u. in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell s.u.'s are taken into account individually in the estimation of s.u.'s in distances, angles and torsion angles; correlations between s.u.'s in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell s.u.'s is used for estimating s.u.'s involving l.s. planes. |
Refinement. Refinement of F2 against ALL reflections. The weighted R-factor wR and goodness of fit S are based on F2, conventional R-factors R are based on F, with F set to zero for negative F2. The threshold expression of F2 > 2σ(F2) is used only for calculating R-factors(gt) etc. and is not relevant to the choice of reflections for refinement. R-factors based on F2 are statistically about twice as large as those based on F, and R-factors based on ALL data will be even larger. |
x | y | z | Uiso*/Ueq | ||
Rb1 | 0.337178 (18) | 0.335293 (17) | 0.56932 (2) | 0.01206 (9) | |
Ge1 | 0 | 0 | 0.5 | 0.00325 (12) | |
Ge2 | 0.17740 (2) | 0.17740 (2) | 0.25 | 0.00375 (9) | |
Ge3 | 0.48398 (2) | 0.333022 (18) | 0.27825 (2) | 0.00399 (9) | |
Ge4 | 0.6667 | 0.3333 | 0.02886 (4) | 0.00335 (10) | |
O1 | 0.07656 (13) | 0.14755 (13) | 0.38704 (14) | 0.0069 (3) | |
O2 | 0.32683 (13) | 0.20855 (13) | 0.31913 (14) | 0.0077 (3) | |
O3 | 0.51870 (14) | 0.27242 (13) | 0.13727 (15) | 0.0090 (3) | |
O4 | 0.57704 (13) | 0.38950 (13) | 0.42159 (15) | 0.0077 (3) | |
O5 | 0.46957 (16) | 0.46957 (16) | 0.25 | 0.0101 (4) |
U11 | U22 | U33 | U12 | U13 | U23 | |
Rb1 | 0.00989 (12) | 0.01141 (13) | 0.01378 (14) | 0.00451 (8) | 0.00190 (7) | −0.00279 (7) |
Ge1 | 0.00249 (15) | 0.00249 (15) | 0.0048 (2) | 0.00124 (7) | 0 | 0 |
Ge2 | 0.00267 (12) | 0.00267 (12) | 0.00533 (17) | 0.00091 (11) | 0.00028 (5) | −0.00028 (5) |
Ge3 | 0.00293 (12) | 0.00353 (12) | 0.00574 (14) | 0.00179 (8) | −0.00012 (7) | 0.00007 (7) |
Ge4 | 0.00280 (12) | 0.00280 (12) | 0.00445 (18) | 0.00140 (6) | 0 | 0 |
O1 | 0.0076 (6) | 0.0053 (6) | 0.0082 (7) | 0.0034 (5) | 0.0045 (5) | 0.0025 (5) |
O2 | 0.0037 (6) | 0.0077 (7) | 0.0102 (7) | 0.0017 (6) | −0.0003 (5) | 0.0019 (5) |
O3 | 0.0065 (6) | 0.0101 (7) | 0.0085 (7) | 0.0026 (6) | 0.0038 (5) | −0.0019 (5) |
O4 | 0.0089 (7) | 0.0077 (7) | 0.0082 (7) | 0.0055 (6) | −0.0033 (5) | −0.0008 (5) |
O5 | 0.0061 (7) | 0.0061 (7) | 0.0185 (11) | 0.0034 (8) | −0.0021 (4) | 0.0021 (4) |
Rb1—O2i | 2.8228 (14) | Ge2—O1v | 1.7346 (14) |
Rb1—O2 | 2.8768 (14) | Ge2—O1 | 1.7346 (14) |
Rb1—O1ii | 2.8881 (13) | Ge2—O2 | 1.7878 (14) |
Rb1—O5iii | 2.9503 (16) | Ge2—O2v | 1.7878 (14) |
Rb1—O4iii | 2.9528 (14) | Ge3—O3 | 1.7203 (14) |
Rb1—O3iv | 2.9667 (14) | Ge3—O4 | 1.7228 (14) |
Rb1—O4 | 3.0130 (14) | Ge3—O5 | 1.7682 (9) |
Rb1—O1 | 3.3437 (14) | Ge3—O2 | 1.7842 (14) |
Rb1—O3v | 3.3820 (15) | Ge4—O4ix | 1.8700 (14) |
Ge1—O1vi | 1.9067 (13) | Ge4—O4x | 1.8700 (14) |
Ge1—O1ii | 1.9067 (13) | Ge4—O4xi | 1.8700 (14) |
Ge1—O1vii | 1.9067 (13) | Ge4—O3xii | 1.8908 (14) |
Ge1—O1 | 1.9067 (13) | Ge4—O3xiii | 1.8908 (14) |
Ge1—O1i | 1.9067 (13) | Ge4—O3 | 1.8908 (14) |
Ge1—O1viii | 1.9067 (13) | ||
O2i—Rb1—O2 | 121.71 (5) | O4iii—Rb1—O2iv | 132.37 (4) |
O2i—Rb1—O1ii | 71.94 (4) | O3iv—Rb1—O2iv | 48.29 (4) |
O2—Rb1—O1ii | 74.87 (4) | O4—Rb1—O2iv | 99.83 (3) |
O2i—Rb1—O5iii | 98.46 (3) | O1—Rb1—O2iv | 101.44 (3) |
O2—Rb1—O5iii | 138.73 (3) | O3v—Rb1—O2iv | 165.44 (3) |
O1ii—Rb1—O5iii | 132.00 (4) | O5—Rb1—O2iv | 142.79 (2) |
O2i—Rb1—O4iii | 79.83 (4) | O1vi—Ge1—O1ii | 180.00 (7) |
O2—Rb1—O4iii | 121.63 (4) | O1vi—Ge1—O1vii | 89.25 (6) |
O1ii—Rb1—O4iii | 151.77 (4) | O1ii—Ge1—O1vii | 90.75 (6) |
O5iii—Rb1—O4iii | 52.26 (3) | O1vi—Ge1—O1 | 89.25 (6) |
O2i—Rb1—O3iv | 138.62 (4) | O1ii—Ge1—O1 | 90.75 (6) |
O2—Rb1—O3iv | 80.99 (4) | O1vii—Ge1—O1 | 89.25 (6) |
O1ii—Rb1—O3iv | 83.50 (4) | O1vi—Ge1—O1i | 90.75 (6) |
O5iii—Rb1—O3iv | 73.71 (3) | O1ii—Ge1—O1i | 89.25 (6) |
O4iii—Rb1—O3iv | 119.74 (4) | O1vii—Ge1—O1i | 180 |
O2i—Rb1—O4 | 168.84 (4) | O1—Ge1—O1i | 90.75 (6) |
O2—Rb1—O4 | 58.73 (4) | O1vi—Ge1—O1viii | 90.75 (6) |
O1ii—Rb1—O4 | 117.31 (4) | O1ii—Ge1—O1viii | 89.25 (6) |
O5iii—Rb1—O4 | 80.02 (3) | O1vii—Ge1—O1viii | 90.75 (6) |
O4iii—Rb1—O4 | 90.69 (4) | O1—Ge1—O1viii | 180.00 (7) |
O3iv—Rb1—O4 | 51.75 (4) | O1i—Ge1—O1viii | 89.25 (6) |
O2i—Rb1—O1 | 68.66 (4) | O1v—Ge2—O1 | 125.76 (9) |
O2—Rb1—O1 | 53.19 (4) | O1v—Ge2—O2 | 106.20 (6) |
O1ii—Rb1—O1 | 51.00 (5) | O1—Ge2—O2 | 106.15 (6) |
O5iii—Rb1—O1 | 165.94 (2) | O1v—Ge2—O2v | 106.15 (6) |
O4iii—Rb1—O1 | 117.58 (4) | O1—Ge2—O2v | 106.20 (6) |
O3iv—Rb1—O1 | 119.54 (4) | O2—Ge2—O2v | 104.65 (9) |
O4—Rb1—O1 | 111.47 (4) | O3—Ge3—O4 | 125.67 (7) |
O2i—Rb1—O3v | 74.81 (4) | O3—Ge3—O5 | 115.46 (5) |
O2—Rb1—O3v | 82.50 (4) | O4—Ge3—O5 | 96.27 (6) |
O1ii—Rb1—O3v | 120.01 (4) | O3—Ge3—O2 | 101.98 (7) |
O5iii—Rb1—O3v | 100.97 (3) | O4—Ge3—O2 | 111.01 (7) |
O4iii—Rb1—O3v | 49.03 (4) | O5—Ge3—O2 | 105.25 (7) |
O3iv—Rb1—O3v | 146.22 (3) | O4ix—Ge4—O4x | 91.08 (6) |
O4—Rb1—O3v | 94.55 (4) | O4ix—Ge4—O4xi | 91.08 (6) |
O1—Rb1—O3v | 70.86 (3) | O4x—Ge4—O4xi | 91.08 (6) |
O2i—Rb1—O5 | 124.40 (3) | O4ix—Ge4—O3xii | 89.92 (6) |
O2—Rb1—O5 | 50.91 (4) | O4x—Ge4—O3xii | 87.90 (6) |
O1ii—Rb1—O5 | 124.58 (4) | O4xi—Ge4—O3xii | 178.58 (6) |
O5iii—Rb1—O5 | 100.094 (8) | O4ix—Ge4—O3xiii | 178.58 (6) |
O4iii—Rb1—O5 | 71.72 (3) | O4x—Ge4—O3xiii | 89.92 (6) |
O3iv—Rb1—O5 | 96.93 (3) | O4xi—Ge4—O3xiii | 87.90 (6) |
O4—Rb1—O5 | 45.87 (3) | O3xii—Ge4—O3xiii | 91.11 (6) |
O1—Rb1—O5 | 83.64 (4) | O4ix—Ge4—O3 | 87.90 (6) |
O3v—Rb1—O5 | 50.37 (2) | O4x—Ge4—O3 | 178.58 (6) |
O2i—Rb1—O2iv | 90.96 (4) | O4xi—Ge4—O3 | 89.92 (6) |
O2—Rb1—O2iv | 103.031 (11) | O3xii—Ge4—O3 | 91.11 (6) |
O1ii—Rb1—O2iv | 50.52 (3) | O3xiii—Ge4—O3 | 91.11 (6) |
O5iii—Rb1—O2iv | 83.86 (3) |
Symmetry codes: (i) x−y, x, −z+1; (ii) y, −x+y, −z+1; (iii) −x+1, −y+1, −z+1; (iv) x, x−y, z+1/2; (v) y, x, −z+1/2; (vi) −y, x−y, z; (vii) −x+y, −x, z; (viii) −x, −y, −z+1; (ix) x, x−y, z−1/2; (x) −x+y+1, y, z−1/2; (xi) −y+1, −x+1, z−1/2; (xii) −y+1, x−y, z; (xiii) −x+y+1, −x+1, z. |
Experimental details
(Na2Ge4O9) | (K2Ge4O9) | (Rb2Ge4O9) | |
Crystal data | |||
Chemical formula | Na2Ge4O9 | K2Ge4O9 | Rb2Ge4O9 |
Mr | 480.34 | 512.56 | 605.3 |
Crystal system, space group | Trigonal, P3c1 | Trigonal, P3c1 | Trigonal, P3c1 |
Temperature (K) | 295 | 295 | 295 |
a, c (Å) | 11.3216 (10), 9.6946 (18) | 11.8461 (4), 9.8009 (4) | 12.1008 (6), 9.8722 (5) |
V (Å3) | 1076.2 (3) | 1191.10 (7) | 1251.91 (11) |
Z | 6 | 6 | 6 |
Radiation type | Mo Kα | Mo Kα | Mo Kα |
µ (mm−1) | 16.75 | 16.08 | 25.89 |
Crystal size (mm) | 0.22 × 0.07 × 0.06 | 0.17 × 0.09 × 0.08 | 0.14 × 0.05 × 0.04 |
Data collection | |||
Diffractometer | Bruker SMART APEX diffractometer | Bruker SMART APEX diffractometer | Bruker SMART APEX diffractometer |
Absorption correction | Empirical (using intensity measurements) [multiscan absorption correction using APEX2 (Bruker, 2007)] | Empirical (using intensity measurements) [multiscan absorption correction using APEX2 (Bruker, 2007)] | Empirical (using intensity measurements) [multiscan absorption correction using APEX2 (Bruker, 2007)] |
Tmin, Tmax | 0.441, 0.748 | 0.441, 0.748 | 0.298, 0.746 |
No. of measured, independent and observed [I > 2σ(I)] reflections | 12328, 924, 853 | 12842, 1026, 984 | 13739, 1091, 1027 |
Rint | 0.041 | 0.026 | 0.048 |
(sin θ/λ)max (Å−1) | 0.677 | 0.680 | 0.681 |
Refinement | |||
R[F2 > 2σ(F2)], wR(F2), S | 0.028, 0.055, 1.21 | 0.014, 0.035, 1.14 | 0.017, 0.042, 1.20 |
No. of reflections | 924 | 1026 | 1091 |
No. of parameters | 71 | 71 | 71 |
Δρmax, Δρmin (e Å−3) | 0.78, −0.95 | 0.39, −0.58 | 0.62, −0.89 |
Computer programs: APEX2 (Bruker, 2007), SHELXS97 (Sheldrick, 2008), SHELXL97 (Sheldrick, 2008), DIAMOND (Brandenburg, 1999), WinGX (Farrugia, 2012).
Na2Ge4O9 | Na2Ge4O9 | K2Ge4O9 | K2Ge4O9 | Rb2Ge4O9 | Rb2Ge4O9 | |
This study | a | This study | b | This study | c | |
a (Å) | 11.3216 (10) | 11.3234 (12) | 11.8461 (4) | 11.84(?) | 12.1008 (6) | 12.08 (7) |
c (Å) | 9.6946 (18) | 9.6817 (9) | 9.8009 (4) | 9.80(?) | 9.8722 (4) | 9.86 (5) |
A-site | ||||||
A—O1 (Å) | 2.443 (4) | 2.446 (2) | 2.740 (1) | 2.753 (15) | 2.888 (1) | 2.88 (5) |
A—O2 (Å) | 2.442 (4) | 2.442 (2) | 2.738 (1) | 2.749 (15) | 2.877 (1) | 2.85 (4) |
A—O2 (Å) | 2.722 (4) | 2.733 (2) | 2.755 (1) | 2.734 (15) | 2.823 (1) | 2.81 (3) |
A—O3 (Å) | 2.633 (4) | 2.615 (3) | 2.847 (1) | 2.839 (16) | 2.967 (1) | 2.93 (3) |
A—O4 (Å) | 2.860 (4) | 2.847 (3) | 2.959 (1) | 2.960 (14) | 2.953 (1) | 2.98 (3) |
A—O4 (Å) | 2.589 (4) | 2.595 (3) | 2.813 (1) | 2.781 (14) | 2.953 (1) | 2.90 (4) |
A—O5 (Å) | 2.589 (4) | 2.584 (3) | 2.831 (1) | 2.826 (17) | 3.013 (1) | 2.87 (4) |
A—O> (Å) | 2.611 (4) | 2.609 (3) | 2.812 (1) | 2.806 (16) | 2.925 (1) | 2.89 (4) |
S (v.u.)d | 0,84 | 0.84 | 1.14 | 1.15 | 1.18 | 1.30 |
Ge1 octahedron | ||||||
<Ge1—O> (Å) | 1.920 (3) | 1.918 (2) | 1,908 (1) | 1.894 (15) | 1.907 (1) | 1.94 (4) |
<O···O> (Å) | 2.714 (3) | 2.711 (2) | 2.698 (1) | 2.678 (16) | 2.697 (1) | 2.74 (4) |
Volume (Å3) | 9.40 | 9.37 | 9,25 | 9.05 | 9.24 | 9.65 |
OAVe (°2) | 8.20 | 9.09 | 2.19 | 1.47 | 0.61 | 5.93 |
OQEf | 1.0022 | 1.0025 | 1.0006 | 1.0004 | 1.0002 | 1.0016 |
S (v.u.)d | 3.77 | 3.79 | 3.89 | 4.05 | 3.91 | 3.60 |
Ge2 tetrahedron | ||||||
Ge2—O1 | 1.757 (3) | 1.754 (2) | 1.7428 (1) | 1.741 (15) | 1.7346 (1) | 1.69 (3) |
Ge2—O1 | 1.795 (3) | 1.792 (2) | 1.789 (1) | 1.796 (14) | 1.7878 (1) | 1.81 (5) |
<Ge2—O> (Å) | 1.776 (3) | 1.773 (2) | 1.766 (1) | 1.768 (14) | 1.761 (1) | 1.75 (5) |
<O···O> (Å) | 2.883 (3) | 2.878 (2) | 2.871 (1) | 2.874 (15) | 2.864 (1) | 2.84 (5) |
Volume (Å3) | 2.77 | 2.75 | 2.76 | 2.76 | 2.74 | 2.69 |
TAVg (°2) | 111.10 | 110.5 | 74.78 | 79.09 | 66.41 | 67.25 |
TQEh | 1.0260 | 1.0217 | 1.0169 | 1.0179 | 1.0149 | 1.0148 |
S (v.u.)d | 3.71 | 3.78 | 3.82 | 3.80 | 3.87 | 3.87 |
Ge3 tetrahedron | ||||||
Ge3—O2 | 1.780 (3) | 1.781 (2) | 1.782 (1) | 1.786 (14) | 1.784 (1) | 1.78 (5) |
Ge3—O3 | 1.732 (3) | 1.734 (1) | 1.723 (1) | 1.734 (15) | 1.720 (1) | 1.72 (3) |
Ge3—O4 | 1.727 (3) | 1.732 (2) | 1.730 (1) | 1.737 (13) | 1.723 (1) | 1.74 (4) |
Ge3—O5 | 1.758 (3) | 1.760 (2) | 1.767 (1) | 1.766 (17) | 1.768 (1) | 1.81 (3) |
<Ge3—O> | 1.749 (3) | 1.752 (2) | 1.750 (1) | 1.756 (16) | 1.749 (1) | 1.76 (4) |
<O···O> | 2.839 (3) | 2.843 (2) | 2.843 (1) | 2.851 (16) | 2.842 (1) | 2.86 (4) |
Volume (Å3) | 2.61 | 2.62 | 2.64 | 2.65 | 2.64 | 2.69 |
TAVg (°2) | 153.16 | 152.95 | 123.14 | 134.61 | 109.78 | 121.69 |
TQEh | 1.0360 | 1.0360 | 1.0287 | 1.0317 | 1.0256 | 1.0292 |
S (v.u.)d | 3.99 | 3.97 | 3.98 | 3.92 | 4.00 | 4.03 |
Ge4 octahedron | ||||||
Ge4—O3 | 1.883 (3) | 1.879 (2) | 1.886 (1) | 1.878 (15) | 1.891 (1) | 1.90 (3) |
Ge4—O4 | 1.862 (3) | 1.862 (1) | 1.865 (1) | 1.882 (13) | 1.870 (1) | 1.90 (4) |
<Ge4—O> | 1.872 (3) | 1.872 (2) | 1.876 (1) | 1.881 (15) | 1.880 (1) | 1.90 (4) |
<O···O> | 2.648 (3) | 2.644 (2) | 2.653 (1) | 2.659 (15) | 2.659 (1) | 2.68 (4) |
Volume (Å3) | 8.75 | 8.72 | 8.79 | 8.86 | 8.86 | 9.09 |
OAVe (°2) | 1.54 | 1.28 | 1.50 | 2.17 | 1.87 | 1.94 |
OQEf | 1.0005 | 1.0004 | 1.0004 | 1.0006 | 1.0006 | 1.0005 |
S (v.u.)d | 4.29 | 4.31 | 4.25 | 4.19 | 4.19 | 4.00 |
< Ge, Gei (°) | 31.40 (6) | 31.47 (3) | 22.16 (4) | 21.8 (6) | 17.29 (4) | 18.4 (2) |
< Ge, (001)j (°) | 18.28 (5) | 18.2 (3) | 12.82 (4) | 12.6 (6) | 10.00 (4) | 10.6 (2) |
Ge4—O4—Ge3 (°) | 125.40 (8) | 125.3 (1) | 131.21 (4) | 129.7 (7) | 134.75 (4) | 131 (2) |
Ge4—O3—Ge3 (°) | 119.48 (8) | 119.4 (1) | 127.99 (4) | 127.6 (9) | 132.64 (4) | 133 (2) |
Ge1—O1—Ge2 (°) | 117.20 (8) | 117.81 (9) | 123.62 (4) | 124.4 (8) | 127.80 (4) | 127 (2) |
Ge2—O2—Ge3 (°) | 123.80 (8) | 122.9 (1) | 127.25 (4) | 126.5 (8) | 129.08 (4) | 127 (2) |
Ge3—O5—Ge3 (°) | 122.50 (8) | 122.45 (6) | 127.88 (4) | 127.6 (8) | 130.59 (4) | 124 (2) |
References: (a) Fleet & Muthupari (1998); (b) Völlenkle & Wittmann (1971); (c) Goreaud & Raveau (1976). Notes: (d) S = bond-valence sum (Brese & O'Keeffe, 1991) in valence units (v.u.); (e) octahedral angle variance OAV = Σi=1n(Θi - 90)2/11 (Robinson et al., 1971); (f) octahedral quadratic elongation OQE = Σi=16(li/lo)2/6, where lo is the centre-to-vertex distance for a regular octahedron whose volume is equal to that of the undistorted octahedron with bond length li (Robinson et al., 1971); (g) tetrahedral angle variance TAV = Σi=1n(Θi - 109.47)2/5 (Robinson et al., 1971); (h) tetrahedral quadratic elongation TQE = Σi=14(li/lt)2/4, where lt is the centre-to-vertex distance for a regular tetrahedron whose volume is equal to that of the undistorted tetrahedron with bond length li (Robinson et al., 1971); (i) the angle between the planes through atoms Ge2–Ge2–Ge3 of two neighbouring [Ge3O9] rings; (j) the angle between a plane through atoms Ge2–Ge2–Ge3 of a [Ge3O9] ring and the (001) plane. |