research papers
Mg(H_{2}O)_{2}[TeO_{2}(OH)_{4}]: a polytypic structure with a twomode disordered stacking arrangement
^{a}XRay Center, TU Wien, Getreidemarkt 9, 1060 Vienna, Austria, ^{b}Institute of Mineralogy and Petrography, University of Innsbruck, Innrain 52, 1060 Innsbruck, Austria, and ^{c}Institute of Chemical Technologies and Analytics, TU Wien, Getreidemarkt 9/164SC, 1060 Vienna, Austria
^{*}Correspondence email: bstoeger@mail.tuwien.ac.at
Crystals of the hydrous magnesium orthotellurate(VI) Mg(H_{2}O)_{2}[TeO_{2}(OH)_{4}] were grown by slow diffusion of an aqueous MgCl_{2} solution into a KOH/Te(OH)_{6} solution immobilized in gelatin. The is built of sheets of nearly regular cornersharing [MgO_{6}] and [TeO_{6}] octahedra. Half of the bridging O atoms are connected to disordered H atoms, which are located in rhomboidal voids (long and short diameters of ∼5.0 and ∼2.5 Å, respectively) of these layers. Moreover, the Te^{VI} atom connects to two OH^{−} ions and the Mg^{II} atom to two H_{2}O molecules. The OH^{−} ions and H_{2}O molecules connect adjacent layers forming a disordered hydrogenbonding network. In a given layer, an adjacent layer may be positioned in four ways, which can be characterized by one of two origin shifts and one of two orientations with respect to [100]. The crystals feature a disordered stacking arrangement, leading to rods of diffuse scattering in the diffraction pattern. The is explained by application of the order–disorder (OD) theory. Different models are compared and the diffuse scattering is evaluated with calculations. The of subsequent origin shifts is ∼ −0.33, whereas the orientation of the layers is essentially random. Determining the latter is particularly difficult owing to a small contribution to the diffraction pattern and virtually indistinguishable diffraction patterns for pairs of correlations with the same absolute value. On longer standing in a glass vial, an ordered polytype forms.
Keywords: polytypism; order–disorder (OD) theory; disorder; diffuse scattering; homometry.
1. Introduction
Orthotellurates(VI) of alkaline earth metals with general formula M_{2}M′[Te^{VI}O_{6}] bear interesting crystalchemical and physicochemical aspects, and a number of these phases and their solid solutions are structurally well characterized (Prior et al., 2005; Fu et al., 2008). The structures of nearly all alkaline earth metal tellurates (except Be) with a single M^{II} cation and the general formula M^{II}_{3}[Te^{VI}O_{6}] have been elucidated [M = Mg: Schulz & Bayer (1971); M = Ca: Hottentot & Loopstra (1981); M = Sr, Ba: Stöger et al. (2010)]. The structures of these tellurates are characterized by rigid, practically regular, octahedral [TeO_{6}]^{6−} units. Ca_{3}[TeO_{6}] (P2_{1}/n, Z = 2), Sr_{3}[TeO_{6}] (, Z = 32) and Ba_{3}[TeO_{6}] (I4_{1}/a, Z = 80) are hettotypes of the double perovskite structure type, where the M^{II} atom occupies two positions with distinctly different coordination spheres. The ionic radius of Mg^{II}, on the other hand, is incompatible with the large voids required by the double perovskite and therefore Mg_{3}[TeO_{6}] (, Z = 2) crystallizes in a different structure type, isotypic with Mn_{3}[TeO_{6}] (Weil, 2006).
During our ongoing studies of hydrous derivatives of M_{3}[TeO_{6}] phases with M = Mg, Ca, Sr, Ba we obtained single crystals of the title compound, Mg(H_{2}O)_{2}[TeO_{2}(OH)_{4}], with a unique So far, hydrous alkaline earth tellurates have only been described for Ba (Weil et al., 2016). We report here on the and description of the polytypic structure as well as on thermal behavior of Mg(H_{2}O)_{2}[TeO_{2}(OH)_{4}].
2. Experimental
2.1. Synthesis and crystal growth
Crystals of Mg(H_{2}O)_{2}[TeO_{2}(OH)_{4}] were grown in gelatin using a gel diffusion technique (Heinisch, 1996). Three gelatin sheets (∼4.5 g) were dissolved in a solution of KOH (4.34 g, 85%wt) and Te(OH)_{6} (7.89 g) in water (300 ml). From this mixture, 25 ml of the solution were introduced into a large test tube. After solidification, the gel was covered with 10 ml of a neutral gelatin solution, prepared by dissolving one gelatin sheet (∼1.5 g) in water (100 ml). After solidification of the second gelatin layer, it was covered with Mg^{II} solution (10 ml, 0.5%wt) which was obtained by dissolving MgCl_{2}·6H_{2}O (4.16 g) in water (100 ml). The test tube was sealed with wrapping film and kept at 295 K for one month. Squarebipyramidal crystals of Mg(H_{2}O)_{2}[TeO_{2}(OH)_{4}] had formed at the interface of both gelatin layers. The gel was cut with a scalpel and crystals with an adequate size for singlecrystal diffraction were isolated under a polarizing microscope.
2.2. Data collection
Diffraction intensities for structure refinements were collected at room temperature using finesliced ω and φscans on a Bruker KAPPA APEX II diffractometer equipped with a CCD camera (Mo radiation, graphitemonochromated). Bragg intensities were reduced using the SAINTPlus software (Bruker, 2017). An absorption correction was applied using a multiscan approach with SADABS (Bruker, 2017) using the 4/mmm Laue group.
Inspection of the diffraction pattern (reconstructed ωscan using 0.2° rotation and 2 min exposure times, resulting in 900 measured frames. Reference frames were collected every 2 h using 1 min exposures over a 10° ω rotation. Evaluation of the 29 reference frames did not show any significant change of the intensities.
layers) revealed lines with pronounced diffuse scattering. For the quantitative analysis of the diffuse scattering, a second crystal was measured with special attention paid to minimization of artifacts on a Stoe IPDSII imageplate diffractometer using graphitemonochromated Mo radiation produced by a conventional sealed Xray tube operated at 50 kV and 40 mA. A 0.5 mm fiber optic collimator and beam stop were positioned in such a way that the free beam path in air was 30 mm long, with the crystal in the center. Compared to the default setup, this arrangement has a significant shorter air beam path and the background caused by airscattering is reduced. The sampletodetector distance was set to 100 mm. For further background correction, 46 frames were collected under the same conditions without the sample. These frames were averaged and used as background in further processing. The data collection was run as a 180°The experimentally determined background was subtracted from all measured raw data frames. Furthermore, a masking procedure was applied to flag overexposed spots. XDS (Kabsch, 2010) was used to determine the orientation matrix for further processing with a modified version of Xcavate (Estermann & Steurer, 1998; Estermann, 2001). Intensity scaling of the original 32bit images was obtained with Xcavate, and shading of nonmeasured areas and extraction of line profiles were performed with ImageJ (Abràmoff et al., 2004). Onedimensional streak profiles were extracted by manually determining the lateral center of the streaks and summing over 20 pixels segments perpendicular to the streaks.
Further diffraction experiments on a crystal (70 µm × 70 µm × 80 µm) kept for six years in gelatin at room conditions have been performed at the X06DA beamline of the Swiss Light Source (Paul Scherrer Institute, Villigen, Switzerland). Monochromated radiation of 0.7085 Å was utilized to collect 1800 data frames during a 180° rotation of the crystal (0.3 seconds per frame) using a Pilatus 2MF detector. Data collection was controlled by DA+ (Wojdyla et al., 2018), evaluation of the orientation matrix and reconstruction of the layers were performed using XDS and Xcavate.
Details of the data collections are summarized in Table 1.

2.3. Refinement
The _{2}O)_{2}[TeO_{2}(OH)_{4}] was solved using the method implemented in SUPERFLIP (Palatinus & Chapuis, 2007) and refined against F^{2} in Jana2006 (Petříček et al., 2014). Owing to disorder, the H atoms could not be located reliably and thus were not considered in the refinements. All atoms were refined using anisotropic atomic displacement parameters (ADPs). More details on different modeling and attempts are given below (§3.8).
of Mg(H2.4. Calculation of diffuse scattering
Experimental peak broadening of the onedimensional intensity profiles was estimated by fitting Gaussian distributions to sharp reflections using the least squares (LS) solver Ceres (Agarwal et al., 2020) refining the origin, reciprocal basis vector length, variance σ (all in pixels) and the individual intensities (in arbitrary units). The overall peak shape of the sharp reflections was well described by a Gaussian, only the base was better described by a Lorentz (Cauchy) distribution. Onedimensional diffuse scattering was calculated using the analytical expressions derived below. Atomic coordinates and ADPs of single layers were taken from the singlecrystal refinements. The atomic form factors were calculated using polynomial approximations tabulated in International Tables for Crystallography (Brown et al., 2006). Calculations were performed on a onedimensional grid with four times the resolution of experimental data and later downsampled to the experimental grid.
Correlation parameters were estimated using a simple coordinatedescent algorithm optimizing in turn the origin (in pixels), the length of the reciprocal basis vector (in pixels) and the correlation parameter (unitless). Each variable was determined using a goldensection search. When multiple rods were refined concurrently, a hierarchical coordinatedescent was performed. In an outer loop, the correlation parameter was refined, in an inner loop the origin and basis vector length of each rod.
The validity of such a trivial search was confirmed by noting that the loss function possesses a single local minimum in each coordinate. The process was stopped when the change in all variables fell below a threshhold of 0.001 in the respective unit. The scale factor was determined after each cycle using a simple linear leastsquares regression with unit weight, which also provided the loss function . Refinements using the weighting functions w = 1/(I_{obs})^{e} (e = 1,2), which are used in powder diffraction (Toraya, 1998), led to unreasonable peak shapes owing to an exaggerated emphasis on the intensities of `valleys' (local minima between peaks).
2.5. Thermal analysis
Simultaneous _{2}), 12 (C), 14 (N), 15 (CH_{3}), 16 (CH_{4}, O), 17 (OH), 18 (H_{2}O), 28 (N_{2}, CO), 32 (O_{2}) and 44 (CO_{2}). All measurements were performed under a flowing argon atmosphere (20 ml min^{−1}) and heating rates of 10 K min^{−1}. Base line corrections of the TG curves were carried out by measuring the empty alumina crucible prior to each measurement. Temperaturedependent powder Xray diffraction measurements (PXRD) were performed on a PANalytical X'Pert PRO diffractometer using a HTK1200 AntonPaar hightemperature oven chamber mounted on the diffractometer. Prior to the measurement, the sample was finely ground and placed on a glass ceramic (Marcor) sample holder (depth 0.5 mm). The was calibrated with a LaB_{6} standard and automatically adjusted during the measurements with a PCcontrollable alignment stage. The samples were heated under atmospheric conditions at 10 K min^{−1} to the respective measurement temperature and kept for 5 min before measurement of each step to ensure temperature stability.
(STA) measurements in the temperature range 30–900°C were performed with a ∼50 mg sample in a corundum crucible on a NETZSCH STA 449 C Jupiter system coupled with a Aeolos quadrupole mass analyzer. The quartz capillary was kept at 250°C. The measured mass signals were 2 (H3. Results and discussion
3.1. Crystal chemistry
Mg(H_{2}O)_{2}[TeO_{2}(OH)_{4}] crystallizes as composed of distinct crystallochemical layers, designated as L_{n}, where n is a sequential number (Fig. 1). The L_{n} layers possess (idealized) p4/m symmetry (Kopsky & Litvin, 2006) with a square lattice spanned by (a, b). c_{0} is the vector perpendicular to the layer planes with the length of one layer width. Henceforth, all directions and will be given with respect to the basis (a, b, c_{0}). The L_{n} layers are composed of close to regular [MO_{6}] (M = Mg, Te) octahedra, which are connected by corners forming sheets (Fig. 2).
Both octahedra are located on sites with symmetry 4/m and are tilted by ∼26° in opposite directions about [001], thus leaving rhomboidal voids with a long and a short diameter of ≃5.0 and ≃2.5 Å, respectively. The M positions are alternately occupied with Te and Mg atoms in a checkerboard pattern. The O atoms connected only to Te and Mg are labeled O1 and O2, respectively. The shared O atom is O3 (Fig. 2). The [MgO_{6}] octahedron is slightly larger than the [TeO_{6}] octahedron with an average Mg—O distance of 2.056 Å compared to the average Te—O distance of 1.929 Å. Selected distances and angles are compiled in Table 2.

The rigid conformation of the octahedral [TeO_{6}]^{6−} anion and the Te—O distances are characteristic for oxotellurates(VI). Reviews on the crystal chemistry of these compounds were given by Kratotochvíl & Jenšovský (1986), Loub (1993), Levason (1997) and Christy et al. (2016). An octahedral coordination is the most common coordination for Mg^{II} cations, and the average Mg—O distance of 2.057 Å compares well to the maximum of the distribution of Mg—O distances of 2.1 Å given in a survey on Mg—O coordination polyhedra (Blatov et al., 1999; Gagné & Hawthorne, 2016).
Bond valence sums (BVSs) are a useful tool to assign H atoms, in particular for those cases where H atoms cannot be located, e.g. in the presence of heavy atoms, from Xray diffraction data (Donnay & Allmann, 1970). Neglecting the contributions of H atoms, in the ideal case, the O atoms of H_{2}O molecules, OH^{−} ions and O^{2−} ions have total BVS of 0, 1 and 2 valence units (v.u.), respectively. Bond valence calculations based on the model of §3.8 with (Brown, 2002) using the parameters R_{o} = 1.693 Å, b = 0.37 for Mg—O and R_{o} = 1.917 Å, b = 0.37 for Te—O (Brese & O'Keeffe, 1991) result in BVSs of 0.84 v.u. (O1), 0.36 v.u. (O2) and 1.34 v.u. (O3). It has to be noted that these BVS calculations are slightly skewed by substitutional disorder of the Te and Mg atoms as well as positional disorder of the O atoms, showed by enlarged ADPs.
According to these BVSs, the Te atoms are bonded to two OH^{−} anions (O1) and the Mg atoms to two H_{2}O molecules (O2). The remaining two H atoms per formula unit are connected to two out of four bridging O3 atoms, amounting to one per rhomboidal void. Thus, the structural arrangement of the compound can be expressed with the connectivity formula . This is in agreement with crystallochemical considerations and corresponds to an electronically neutral structure. Moreover, the Te—O bond lengths distribution in the [TeO_{2}(OH)_{4}]^{2−} octahedron is in good agreement with those of other structures comprising this type of anion (Weil, 2004, 2007; Weil et al., 2017).
The larger and smaller than ideal BVSs of the H_{2}O molecules and OH^{−} anions (0.36 and 0.83 versus 0 and 1 v.u.) can be explained by the H atoms being involved in hydrogenbonding. Indeed, the distances between close O atoms [O1⋯O1 2.9316 (14) Å; O1⋯O2 2.9024 (14) Å; O2⋯O2 2.8746 (14) Å; O3⋯O3 2.525 (3) Å] strongly suggest formation of intra and interlayer O—H⋯O hydrogen bonds.
3.2. Polytypism
The origin of the L_{n+1} layer is related to the origin of the adjacent L_{n} layer by a translation of or , as indicated in Fig. 2. In these two different stacking possibilities the locations of the Te and Mg atoms are exchanged. An alternation of the two will henceforth be called Te/Mg exchange. Moreover, every L_{n} layer can appear in two orientations related by m_{〈100〉} operations. A change in orientation will be called orientation inversion. The four resulting stacking possibilities are shown in Fig. 3.
3.3. Order–disorder description
The order–disorder (OD) theory (DornbergerSchiff & GrellNiemann, 1961) has been devised to explain the common occurrence of in all classes of compounds. It is based on layers, which do not necessarily correspond to layers in the crystallochemical sense (Grell, 1984). The crucial point in an OD description is that pairs of adjacent layers are equivalent, which corresponds to the vicinity condition (VC). However, pairs of adjacent layers without [Figs. 3(a) and 3(c)] and with orientation inversion [Figs. 3(b) and 3(d)] are not equivalent and therefore violate the VC. The particular layer choice as described here is therefore not of the OD type.
An OD description can nevertheless be achieved by `slicing' the structure into two kinds of layers, designated as A^{1} and A^{2} (Fig. 1, left). The structure then belongs to a tetragonal category IV OD family built of two kinds of nonpolar (with respect to the stacking direction) layers.
family symbol reads asaccording to the notation of Grell & DornbergerSchiff (1982).
The first line of the symbol gives the name of the layers, the second their symmetry and the third one possible arrangement of adjacent layers. Note that in OD theory, layer group symbols with five directions are sometimes necessary to describe tetragonal OD
families. Here, because it is not necessary to distinguish between the [100] and [010] directions, as well as the [110] and directions, the usual symbols can be used.The A^{1} layers possess p4/m symmetry. They are built of the [MgO_{6}] and [TeO_{6}] octahedra [Fig. 4(a)] and the disordered hydrogen atom belonging to O^{2−}/OH^{−} in the rhombohedral void. The A^{2} layers are built of the OH^{−} anions (O1) and H_{2}O molecules (O2) that connect the L_{n} layers [Fig. 5(a)]. Thus, the O1 and O2 atoms are located at the layer interfaces and belong to both OD layers. The layer symmetries were deduced under the assumption of a disordered hydrogenbonding network.
The third line of the symbol indicates that, in one possible arrangement, the origins of the A^{1}_{n} and A^{2}_{n+1} layers are spaced by ra + sb + c_{0}/2. According to the stacking rules described above, (r, s) adopt the values or equivalently , i.e. the 4_{[001]} and 2_{[001]} axes of the A^{1} and A^{2} layers coincide. Note that in contrast to many other OD families, here the parameters adopt a precise value because adjacent layers share common atoms (O1 and O2) located on special positions.
The NFZ relationship (Ďurovič, 1997) is a formalism to determine the alternative stacking possibilities in a family of OD structures. It is based on the groups of those operations of the A_{n} layers that do not reverse the orientation with respect to the stacking direction (λτPOs according to the OD terminology). For the A^{1} and A^{2} layers, is p4 and pmm2, respectively. Because the adjacent layers are not equivalent, the NFZ relationship reads as , where designates the index of the of . For any pair of adjacent layers, .
For an contact, Z = N/F = [p4 : p112] = 2. Thus, given an A_{n}^{1} layer, the adjacent A_{n+1}^{2} layer can appear in two orientations (with pmma and pmmb symmetry), which are related by the fourfold rotation of the A_{n}^{1} layer. For an contact, Z = N/F = [pmm2 : p112] = 2. Given an A_{n}^{2} layer, the adjacent A_{n+1}^{1} layer can likewise appear in two positions, which in this case are related by the m_{〈100〉} reflections of the A_{n}^{2} layer.
By following these stacking rules, an infinity of _{6}] and [MgO_{6}] octahedra may be the same, or different. On the other hand (), the interlayer hydrogen bonding independent of the orientation of the octahedral sheets.
can be constructed, which are equivalent to the nonOD described in the previous section. The usefulness of the OD description does not only lie in the concise symmetry classification. It also sheds light on the crystallochemical reasons of the by splitting them into two distinct contributions. On the one hand (), the orientations of the hydrogenbonding network to both sides of the [TeO3.4. Maximum degree of order polytypes
; DornbergerSchiff & Grell, 1982). MDO cannot be decomposed into simpler i.e. into composed only of a selection of pairs, triples or any ntuples of adjacent layers. Experience shows that the majority of macroscopically ordered are of the MDO type.
of a maximum degree of order (MDO) are a central concept of OD theory (DornbergerSchiff, 1982There are two kinds of triples, namely with and without orientation inversion. Moreover, there are two kinds of triples, namely with and without Te/Mg exchange.
The combination of these triples results in four MDO polytypes:
MDO_{1}: never orientation inversion, never Te/Mg exchange, B112/m, c = 2c_{0};
MDO_{2}: always orientation inversion, never Te/Mg exchange, Pcnm, c = 2c_{0};
MDO_{3}: never orientation inversion, always Te/Mg exchange, I4_{1}/a, c = 4c_{0};
MDO_{4}: always orientation inversion, always Te/Mg exchange, , c = 4c_{0}.
All other stacking arrangements can be divided into fragments of MDO
which therefore represent the `alphabet' of an OD family.Atomic coordinates for all four MDO .
are listed in Table 3

3.5. Family structure
The _{2}O)_{2}[TeO_{2}(OH)_{4}] has F4/mmm symmetry (nonstandard setting of I4/mmm) with c = 2c_{0} (coordinates in Table 3).
of an OD family is a fictitious structure in which all stacking possibilities are realized to the same degree. It plays an important role in the elucidation of OD structures. The of Mg(HFor a fixed A^{1}_{n} layer, the adjacent A^{2}_{n+1} layer can appear in two orientations related by the 4_{[001]} operation. Each of these two orientations gives rise to two orientations of the A^{1}_{n+2} layer, which are related by the m_{〈100〉} operations of the A^{2}_{n+1} layer. Thus, in the the A^{1} layers are an equal superposition of four positions (Te/Mg disorder and orientation disorder) with c4/mmm (nonstandard setting of p4/mmm) symmetry [Fig. 4(b)].
According to analogous reasoning in the A^{2} layers of the the OH^{−} anions and H_{2}O molecules are disordered in a 1:1 ratio [Fig. 5(b)]. These disordered layers possess c4/emm (nonstandard setting of p4/nmm) symmetry.
3.6. Diffraction pattern
The diffraction pattern of Mg(H_{2}O)_{2}[TeO_{2}(OH)_{4}] features rods with sharp reflections and rods with broader reflections on top of prominent onedimensional diffuse scattering (Fig. 6). Such diffraction patterns are characteristic for with translationally equivalent layers (Jeffery, 1953; Ferraris et al., 2008), and were the inspiration for the name `OD' (Bragg reflections: order; streaks: disorder).
In Mg(H_{2}O)_{2}[TeO_{2}(OH)_{4}], the L_{n} layers are not translationally equivalent, since they can appear in two orientations. As will be shown below, in this case the reason of the rods lacking diffuse scattering lies in the particular makeup of the L_{n} layers, namely the similar size of the [MgO_{6}] and the [TeO_{6}] octahedra.
In the reciprocal basis , the F(hkν) of a polytype can be calculated as the sum of the structure factors F_{n}(hkν) of the individual L_{n} layers:
Since the translation lattices of all layers are spanned by (a, b), F_{n}(hkν) is only nonzero for . The F_{n}(hkν) can be decomposed into the contributions of the O3 atom and of the remaining atoms (Te, Mg, O1, O2):
The origin of the L_{n} layer can be written as
with and α_{0} = β_{0} = 0. Since the origin shift from L_{n} to L_{n+1} is either a/2 + c_{0} or b/2 + c_{0}, α_{n} + β_{n} is even, if and only if, n is even, which can be expressed by
with .
The Mg, Te, O1 and O2 atoms are not affected by orientation inversion, since the p4/mmm, which contains the reflection relating both orientations. These parts of the layers are therefore obtained from the L_{0} layer by translation along o_{n}. According to equations (3) and (4), F^{M}_{n} can therefore be written in terms of F^{M}_{0} as
of their (layer group) orbits isNote that since Mg, Te, O1 and O2 are located on fourfold rotation axes, their displacements are isotropic in the (001) plane and, therefore, disregarding
the reflection at [100] has no influence on their (harmonic) ADPs.The orientation of the L_{n} layer will be described by ω_{n} = 0, 1, . If O3 is located on the line, which is perfectly realized if d(Te—O3) = d(Mg—O3), ω_{n} = 1 corresponds to an additional translation of (a + b)/2 with respect to o_{n}. If the displacement of the O3 atom is likewise isotropic in the (001) plane, F^{O}_{n} can be written in terms of F^{O}_{0} as
If h + k is even, then (h + k)β_{n} and (h + k)(β_{n} + ω_{n}) are likewise even and equations (5) and (7) simplify to
and
and therefore
and
where δ is the Dirac delta distribution. Note that the last equals sign represents an abuse of notation as the given function series does not converge at any point [technically, the series converges in the distributional sense (Bricogne, 2010)].
In summary, on rods h + k even only sharp reflections are observed at ν = l/2, , where h, k and l are all even or all odd. This corresponds to the diffraction pattern of a crystal with a tetragonal Fcentered (tF) lattice with the centered reciprocal basis . These reflections correspond to the diffraction pattern of the (§3.5) and are called the family reflections. All stacking arrangements, ordered or disordered, contribute equally (proportional to their volume fraction) to these reflections, since neither α_{n}, β_{n} nor ω_{n} contribute to equation (12).
On rods h + k odd the simplifications above do not apply and diffraction intensities can appear at arbitrary positions. Bragg reflections on these rods are called characteristic reflections, because they are generated only by certain The characteristic reflections of MDO_{1/2} are located at ν = l/2, , those of MDO_{3/4} at , .
The calculations above were derived under the assumption that d(Te—O3) = d(Mg—O3), whereas the actual structure deviates from this assumption [1.906 (3) versus 2.064 (2) Å]. Moreover, ordered typically feature (Ďurovič, 1979). For example, in the MDO_{3/4} the O3 atom is located slightly off the z = 0 reflection plane of the p4/m layer symmetry (Table 3). Distinctly enlarged ADPs of the O1 and O2 atoms show that they are located on the fourfold axis (which is a twofold axis in the actual MDO_{3/4} polytypes) only on average (Fig. 7). These deviations may lead to violations of the systematic noncrystallographic absences, namely faint streaking and very weak characteristic reflections on h + k even rods.
3.7. Rods with insignificant contribution of O3
To differentiate between the effects of orientation inversion and Te/Mg exchange on the diffraction pattern, it is useful to note that the fractional coordinates x and y of the O3 atom, which is essentially the only atom affected by orientation inversion, are close to and , respectively. If the O3 atom is idealized as being located on such a position, the L_{0} layer contains up to translation two O3 atoms at and the two atoms obtained by inversion at the origin. If, moreover, the displacements of the O3 atom are considered as being isotropic in the (001) plane [], then the F^{O}_{0} is
where f^{O}(hkν) is the atomic form factor of O. Note that the is real and contains only cos terms owing to the inversion at the origin. If h is divisible by three, i.e. h = 3h′, , this expression simplifies for rods h + k odd to
(h′ is even, if and only if, h is even and therefore k − h′ is odd). The same argument can be applied to rods with k divisible by three. Thus for h or k divisible by three, equation (2) simplifies to
and equation (1) can ultimately be written as
The variable ω_{n}, which describes the orientation of the octahedra, does not affect rods h + k odd with either h or k divisible by three, and any significant diffuse scattering or Bragg reflections on these rods are due to the arrangement of the Te and Mg atoms.
To illustrate the effect in absolute terms, plots of and against ν are given in Fig. 8 for the 10ν and 12ν rods. Indeed, for 10ν the contribution of F^{O}_{0} is negligible when compared to F^{M}_{0}. On the other hand, 12ν features significant contribution at low scattering angles. In particular, this rod (including symmetry equivalents) has the highest relative contribution of F^{O}_{0}.
3.8. Classical refinements
To determine atomic coordinates and ADPs, classical independent atom model refinements were performed. In a first ). In the Te/Mg exchange and orientation inversion are realized in 50% of the layers. In principle, the O3 atom is disordered over four positions. Nevertheless, only two positions could be resolved, because the Te—O3 and Mg—O3 distances are nearly equal. Likewise, the O1 and O2 atoms could not be separated without introduction of distance restraints and therefore were refined as a single O1/O2 position.
only the family reflections were considered. Excellent reliability factors are thus obtained (Table 4On top of the lines of diffuse scattering at h + k odd are located elongated peaks where the characteristic reflections of MDO_{3/4} are expected (Fig. 6). No peaks corresponding to MDO_{1/2} were observed (though see §3.13). One has to realize that treating these peaks as Bragg reflections in classical refinements will inevitably introduce systematic errors.
Owing to the systematic nonspace group absences (§3.6), cannot differentiate between MDO_{3} (I4_{1}/a) and MDO_{4} (). Moreover, owing to diffuse scattering on rods h + k odd, intensities in violation of the Icentring are observed. Thus, classical determination is unreliable and the diffractometer software strongly suggests the I4_{1}/amd (c = 4c_{0}), which is the of a 1:1 superposition of MDO_{3} and MDO_{4}.
The first model was generated and refined using this I4_{1}/amd symmetry, where the O3 atom is disordered with a 1:1 occupation ratio about the m_{[100]} reflection plane. To achieve satisfying residuals, occupational disorder of the Te and Mg atoms (with Mg′ and Te′) had to be introduced, corresponding to a contribution of MDO_{1/2} fragments.
Based on the refined model in I4_{1}/amd, the symmetry was reduced by an index of 2 to I4_{1}/a (MDO_{3}) and (MDO_{4}), respectively. The linear parts of the lost operations were retained as the and the twin volume ratio was refined. The disordered O3 position was split in both cases into two distinct positions (O3 and O3′). The coordinates and ADPs of O3 and O3′ were constrained to be equal with respect to the m_{[100]} operation and the occupancies were refined and constrained to a sum of 1. A comparison of the refinements is given in Table 4. The of MDO_{3/4} (as opposed to MDO_{1/2}) was derived from the occupancy of Te as 2occ(Te) − 1. Likewise, the of the major domain of the MDO_{3}/MDO_{4} pair was derived as 2occ(O3) − 1 (see Appendix B).
According to these refinements, there was ∼10–15% of MDO_{1/2} present in the crystal under investigation. Estimating the MDO_{3}:MDO_{4} ratio is more difficult. According to the with the best reliability factors (), there are approximately equal amounts of MDO_{3} and MDO_{4}, which would correspond to a 50% chance of orientation inversion. The I4_{1}/a on the other hand suggests an MDO_{3}:MDO_{4} ratio of ∼2:1, which shows the difficulty of deriving these values from routine refinements. The fundamental problem is that a disordered stacking is in general not equivalent to a superposition of MDO polytypes.
Allotwinning, i.e. the association of macroscopic domains of distinct (Nespolo et al., 1999), was ruled out owing to diffuse scattering. Indeed, such models did not lead to improved reliability factors. Likewise, placing the characteristic and family reflections on different scales to avoid the Ďurovič effect (Nespolo & Ferraris, 2001) led to unreliable refinements because the ratio of and the ratio of the scales correlate (Hans et al., 2015). As will be shown below (§3.10) Te/Mg exchange does occur and thus the singlescale refinements are preferred, even though the quantification of Te/Mg exchange is inaccurate.
In summary, neither the amount of MDO_{1/2} nor the MDO_{3}:MDO_{4} ratio can be quantified reliably with routine refinements, demonstrating the inherent difficulties of structurally characterizing such compounds. Nevertheless, these refinements are crucial to determine Mg—O and Te—O distances.
3.9. Disorder model
To quantify the diffuse scattering, a simple growth model was derived from the OD interpretation given in §3.3. The crystal is described as an alternating succession of A^{1} and A^{2} OD layers. According to the OD description, pairs of adjacent OD layers are geometrically equivalent, but triples may differ. Therefore, in the simplest growth model the A_{n} layer depends on the A_{n−1} and A_{n−2} layers. Since there are two kinds of triples and two kinds of triples (§3.4), this model is fully determined by two parameters. P_{MgTe} describes the probability of without Mg/Teinversion and P_{orient} the probability of triples without orientation inversion. In some cases, it will be more convenient to express these probabilities in terms of the correlation coefficients
This twoparameter model is sufficient to describe all four MDO .
and also of equal overlays of MDO as listed in Table 5

In this trivial model, each layer triple is considered independent of the previous triple. In more refined models, the orientation–inversion probability could depend on the occurrence of Mg/Te inversion and vice versa. Additional parameters would then be required.
Growth models are conveniently expressed as Markov chains (Welberry, 2010). The above model corresponds to the fourstate Markov chain
where each step corresponds to a new triple of OD layers, which has two OD layers in common with the previous triple. This Markov chain has a period of two since an A^{1} layer is only added after every second step (and likewise for A^{2}). Such chains are developed into two (or more for higher periods) independent chains, here from the nth triple to the n+2nd triple. These two Markov chains are most conveniently expressed in terms of the crystalchemical L_{n} layers:
The first Markov chain describes the relation of the origin of the L_{n} and L_{n+1} layers and the second chain the orientation of the L_{n} layer. The chains are independent, because one OD layer triple does not depend on the previous triple. Each chain can be considered as an independent nearestneighbor model, since the o_{n} depends only on o_{n−1} and ω_{n} on ω_{n–1}.
For P_{MgTe}, P_{orient} ≠ 0,1, the Markov chains converge to the equilibrium states P(Δo_{n} = a/2) = P(Δo_{n} = b/2) = P(ω_{n} = 0) = P(ω_{n} = 1) = ½, i.e. after an infinity of layers, both origin shifts and both layer orientations are equally likely. In the following only this general case will be considered.
3.10. Diffuse scattering
To calculate the diffraction pattern of disordered structures, it is advantageous to directly calculate the intensity I(hkν) = F(hkν)^{2} in terms of pair correlations between layers (Welberry, 2010):
where an overline designates the complex conjugate. The orientation of the L_{n} layer is flipped with respect to the L_{0} layer if ω_{n} = 1. To express the of such a layer, it will be related to the of the mirrored L_{0} layer:
The hkν argument of F will henceforth be omitted for brevity.
The diffraction intensity can then be expressed in terms of probabilities:
where expresses the probability that the origins of the L_{n} and L_{n+Δn} layers are separated by Δα_{n}a/2 + Δβ_{n}b/2 + Δnc (up to a full layer lattice translation). and are the probabilities that the L_{n} and L_{n+Δn} layers possess the same, respectively opposite, orientation. , the sum over Δα and Δβ, is the paircorrelation function of layers spaced by . It should be stressed that equation (29) is only valid for an equal probability of both layer orientations (P_{orient} ≠ 0, 1, large domain size) and independence of both Markov chains. On the flip side, it is valid for more complex growth models with interactions over more than one layer width.
From the stacking rules it follows that = = 0 for Δn even and = = 0 for Δn odd. Since, as has been shown above, significant diffuse scattering is only observed on rods h + k odd, let us concentrate on these. Then, the exponential factor of the Δα = Δβ = 1 terms in equation (29) is = . Factoring out the probabilities, for Δn even we thus obtain
In analogy, for Δn odd and h + k odd, = (if k is odd h is even and viceversa) and therefore
Let us now derive the `pair distribution' probabilities. Obviously, the starting state of the growth model is P^{0,0}_{0} = 1 and P^{1,1}_{0} = 0. As has been noted above, in nondegenerate cases (P_{MgTe} ≠ 0,1), the equation (23) converges to an equilibrium state where the origin shifts Δo_{n} = a/2+ c_{0} and Δo_{n} = b/2 + c_{0} are equally likely and therefore P^{1,0}_{1} = P^{0,1}_{1} = ½. Repeated application of equation (23) to theses initial states gives the general case (see Appendix C):
Note that for negative Δn, the same reasoning applies and therefore the absolute value of Δn is used in the exponents. In analogy, according to the Markov chain equation (24) the probabilities describing the orientations are
By substituting equation (35) into equation (32) it follows that s_{Δn} = 0 for odd Δn. Note that this is only valid for the simple nearestneighbor model of equation (23). In more general growth models, these terms adopt nonzero values.
For even Δn, from equations (33) and (34) it follows that − = (c_{MgTe})^{Δn/2} and equation (31) becomes
Ultimately, the intensity on rods h + k odd therefore is [see equation (30)]
where m = Δn/2 and
By identifying two geometric series (see Appendix D), an analytical expression of I(hkν) can be given as
To avoid the unwieldy expressions in the parentheses, we will introduce the function family
which describes the shape of onedimensional diffuse scattering produced by a structure with a simple nearestneighbor correlation of −1 < c < 1 (Welberry, 2010). d_{c}(x) is generally (except for c = 0) a function with periodicity 1, featuring peaks which are sharper for increasing c. For c approaching 1, d_{c}(x) converges to a Dirac comb with sharp reflections for integer x (c → 1) or halfinteger x (c → −1). For c = 0, d_{c} is the constant function d_{0}(x) = 1.
Using d_{c}, equation (42) simplifies to
which shows that the diffuse scattering is the sum of two independent shapefunctions of the nearestneighbor correlation c_{MgTe} and c′. The factor corresponds to the (hypothetical) intensity of an superposition of both orientations of the L_{0} layer and to the intensity of the difference of the electron density of these two orientations.
Since c′ depends on the square of c_{orient} [equation (41)], c_{MgTe} and c′ are of the same sign [sgn(c_{MgTe}) = sgn(c′)]. Thus, the location of the peaks depends only on c_{MgTe}, but not on c_{orient}. For c_{MgTe} > 0, I(hkν) has peaks at ν = l/2, and for c_{MgTe} < 0 at ν = l/2 + , as is expected for MDO_{1/2}like and MDO_{3/4}like stacking arrangements, respectively.
Moreover, note that from c_{orient} < 1 follows that c′ < c_{MgTe} and therefore ordering of the orientation inversion can never lead to sharper peaks for a given c_{MgTe}, whereas its disorder can lead to more diffuse peaks.
But most remarkably, under the given assumptions (nearestneighbor model, negligible c_{MgTe}, c_{orient} ≠ ± 1) the diffuse scattering is identical for pairs of c_{orient} with the same absolute value.
3.11. Estimation of the correlation coefficients
Assuming the idealization d(Te—O3) = d(Mg—O3), orientation inversion corresponds to a translation of O3 by (a + b)/2 (see § 3.6). Using the decomposition F_{0} = F_{0}^{M}+F_{0}^{O}, then is
which for h + k odd becomes and consequently
Conveniently, as has been shown above, is negligible for rods h + k odd with h or k divisible by three (Fig. 8). Thus, these rods can be used to estimate c_{MgTe} with only a negligible contribution of c_{orient}.
Fig. 9(a) gives I(10ν) plots for different values of c_{MgTe} calculated using only the term of equation (48). To estimate c_{MgTe}, a simultaneous LS optimization was performed on the rods listed in Table 6. Fig. 9(b) shows the result of the LS optimization without convolution of the experimental peak shape for the 01ν rod. As expected, an additional convolution with the experimental peak shape results in a slightly more negative correlation c_{MgTe} (−0.353 versus −0.338). Since the refinements without convolution result generally in better fits (Table 6), we will henceforth assume the latter value.

The agreement of the experimental and calculated curves is reasonable, though not perfect as the experimental peaks are somewhat narrower. Even though a stronger negative correlation c_{MgTe} leads to narrower peaks, it is in disagreement with the strong diffuse scattering between the peaks. We suppose that the sample is composed of domains with different c_{MgTe} values, some with stronger and some with weaker correlations. A model taking into account interactions over more than the nearestneighbor can be ruled out, since such models produce valleys of different shapes (Welberry, 2010).
Given c_{MgTe}, c_{orient} can be determined from the h + k odd rods with neither h nor k divisible by three. Fig. 10(a) gives I(12ν) plots with c_{MgTe} = −0.338 derived from the 10ν rod and c_{orient} = 0, 0.9, where the contribution of the term is shown separately. The 12ν rod features the highest relative contribution of the O3 atom to the scattering intensity (§3.6). Even on this rod and with the extreme values of c_{orient}, the effect on the peak shape is rather subtle.
Independent LS optimization with fixed c_{MgTe} yielded a zero correlation of c_{orient} for all rods listed in Table 7. The of the 12ν rod is displayed in Fig. 10(b). Again, without convolution of the experimental peak broadening slightly improved residuals are obtained. However, in both cases a zero c_{orient} is derived. We conclude that the orientation of the [MgO_{6}] and [TeO_{6}] octahedra is mostly random, which means that the problem of identical diffraction for pairs of structures becomes a moot point, since the sign of c_{orient} ≈ 0 is irrelevant.

3.12. Diffuse scattering on h + k even rods
As has been argued above, pairs of structures with c_{orient} of the same absolute value produce the same diffraction intensity on h + k odd rods. Moreover, under the idealization of d(Te—O3) = d(Mg—O3) the h + k even rods are identical for all stacking arrangements. Thus, such pairs of idealized structures can be considered as homometric.
Since these assumptions are not perfectly realized, very weak diffuse scattering is likewise observed on rods h + k even (Fig. 11). In principle, this could be used to determine the sign of c_{orient}.
The diffuse scattering on these rods can be derived in analogy to §3.10. However, for h + k even the equalities exp[2πi(h/2+k/2+Δnν)] = exp[2πiΔnν] and exp[2πi(h/2+Δnν)] = exp[2πi(k/2+Δnν)] hold, since h and k are either both even or both odd. Conveniently, these terms can be generalized to exp[2πi(Δnh/2+Δnν)] for even and odd Δn (see §3.6). Factoring out the probabilities of equation (29), the leading factors in the equations analogous to equations (31) and (32) are = = 1. Ultimately, the general expression for s_{Δn} on rods h + k even is
which also holds for more general growth models. Substituting the probabilities of equations (36) and (37), the intensity I(hkν) is (setting m = Δn)
The first term corresponds to the Bragg peaks of the h + k even, equation (45) becomes and therefore
again committing the abuse of notation. ForThus, as shown in §3.6, under the assumption d(Te—O3) = d(Mg—O3), only family reflections are observed on rods h + k even. Nonequal Te—O and Mg—O distances lead to a nonvanishing and thus diffuse scattering as described in the second term of equation (51). For distinctly positive c_{orient} one would expect additional peaks on top of the family reflections and valleys between the family reflections. For negative c_{orient}, additional peaks would be observed between the family reflections. This is consistent with the MDO_{1} and MDO_{2} the lattice of the former (c_{orient} = 1) does not allow for reflections between family reflections owing to the Bcentering, whereas the latter (c_{orient} = −1) has a primitive and features systematic nonspace group absences in the idealized case, which should be observable for noticeable deviations therefrom.
In the actual diffraction pattern (Fig. 11), the minute streaks are basically structureless, confirming the low correlation c_{orient} ≈ 0. Tiny sharp spots are observed, which can however be explained by λ/2 radiation. To prove this assignment, a Mg(H_{2}O)_{2}[TeO_{2}(OH)_{4}] crystal was quickly scanned using synchrotron radiation, which confirmed the structureless diffuse scattering on rods h + k even (Fig. 12).
It has to be noted that equation (51) does not allow for a simple quantitative estimation of the diffuse scattering in cases where c_{orient} ≠ 0, because the F_{0} − factor does not only represent the deviation of the equidistance of O3 from Mg and Te, but also generally the deviation from the idealized p4/m symmetry, which certainly exists as shown by enlarged ADPs. Moreover, the origin difference between adjacent layers might deviate slightly from the ideal Δo = a/2 + c_{0}/2 or Δo = b/2 + c_{0}/2, which would likewise invalidate the reasoning in §3.5 and lead to faint diffuse scattering.
All these deviations from the idealized model can not be simply derived from singlecrystal experiments, since they will differ depending on the adjacent layers. Owing to missing structural data of all MDO
these would have to be derived by relaxation, for example with DFT methods.In any case, this is of no concern here, since there appears to be no significant c_{orient}.
In summary, we propose a model with a negative correlation of the Mg/Te stacking c_{MgTe} ≈ −0.34 and a c_{orient} ≈ 0 correlation of the orientation, with the caveat that the peak shape is not described perfectly, as the crystals might be composed of domains with varying c_{MgTe}.
3.13. Rearrangement of the over time
The synchrotron measurement described in the previous section was performed on a newly isolated crystal seasoned for six years at 15–35°C in a closed glass vial containing residual gel from the synthesis. Much to our surprise, in this experiment additional sharp characteristic reflections were observed on rods h + k odd at integer and halfinteger νvalues [Fig. 13(a)], as would be expected for MDO_{1/2} To confirm the appearance of ordered domains, a different crystal was measured inhouse and likewise featured sharp reflections with h + k odd [Fig. 13(b)], though only half as many.
For ordered MDO_{1/2} (c_{MgTe} = 1), the simplifications of §3.10 do not apply and therefore macroscopic MDO_{1} and MDO_{2} produce distinctly different intensities on rods h + k odd. The sharp reflections of the second crystal can be indexed with the Bcentered cell of the MDO_{1} polytype [h + 2ν even, Fig. 14(a)]. The location of the sharp reflections of the first crystals could in principle be explained by the MDO_{2} polytype. According to calculations, for MDO_{2} one would expect alternately strong and weak characteristic reflections at opposite positions on 12ν and rods. In the actual crystal though, the strong and weak characteristic reflections appear at the same νvalues [Fig. 14(b)], which means that the characteristic reflections are probably due to two MDO_{1} orientation states, related by a fourfold rotation.
The shape of the diffuse scattering essentially stays the same during seasoning of the crystals (Fig. 15). We conclude that the disordered domains slowly convert to ordered MDO_{1} polytypes.
We recently measured three newly isolated crystals of the same synthesis batch after an additional one year timeperiod and all of them clearly contained ordered MDO_{1} fragments. One of the crystals was twinned by fourfold rotation. The amount of diffuse scattering did not decrease significantly compared to the previous year. Thus, the kinetics and preconditions of the transition are not yet understood.
3.14. Thermal behavior
The thermal decomposition of Mg(H_{2}O)_{2}[TeO_{2}(OH)_{4}] is connected with a multistep mechanism between 30–900°C. The hydrous phase is stable up to a temperature of ∼160°C in the oven chamber (Fig. 16; PXRD), followed by an amorphization. According to the TG/DTA curves (Fig. 17; STA), the onsets of the associated mass loss and the endothermic effect due to dehydration are at ∼195°C. We ascribe the different temperatures of the Mg(H_{2}O)_{2}[TeO_{2}(OH)_{4}] stability field to the different timescales of the two measurement techniques. Whereas the temperaturedependent PXRD measurement is slow due to stepwise heating rates and long measurement times, the STA measurement is much faster with continuous heating rates and much shorter measurement times. The mass loss of ∼25% up to a temperature of 500°C is due to release of water and oxygen according to the mass spectra. The amorphous phase remains up to 510°C where first reflections appear, indicating a crystallization of new phases as shown by a twostep exothermal effect at ∼550°C, also associated with a release of small amounts of water (onset DTA first step 545°C, second step 575°C). The diffuse nature of the reflections in the stability field between ∼510°C and 570°C makes a clear assignment difficult. Besides weak reflections that could be unambiguously assigned to the formation of Mg_{3}[TeO_{6}], a relationship with trirutiletype Co[Sb_{2}O_{6}] (Reimers et al., 1989) could be derived from the strong reflections. Given the very similar ionic radii for Co^{2+}/Mg^{2+} and Sb^{5+}/Te^{6+}, respectively, this could point to possible existence of a mixedvalent Te^{4+}/Te^{6+} compound with composition Mg[Te_{2}O_{6}]. The assumption of the existence of such a mixedvalent phase is supported by the detection of oxygen in the mass analyzer during the preceding decomposition step. The assumed mixedvalent phase transforms above 570°C into a phase for which the diffraction pattern could be related to Mg[TeO_{4}] (Sleight et al., 1972), which is stable until ∼660°C. Above this temperature another phase is formed for which a relation to a known phase could not be made. Above ∼710°C only the reflections of Mg_{3}[TeO_{6}] are visible, associated with another small mass loss of ∼3% in the TG curve under further release of oxygen. Above this temperature no further mass loss is observed until 900°C. We currently cannot interpret the significant endothermal effect in the DTA curve in this temperature interval (onset 824°C). Since no further mass loss is observed here, this effect could be related either to a structural of (parts of) the remaining material or to a melting of an amorphous content thereof. Both effects cannot be related with the temperaturedependent diffraction pattern, e.g. by splitting or vanishing of reflections or a significantly broader background. It should be noted that the same material heated up to 1000°C in another experiment similar to the STA study resulted in the complete formation of a glass.
In summary, the decomposition mechanism of Mg(H_{2}O)_{2}[TeO_{2}(OH)_{4}] can be formulated as:
I. Mg(H_{2}O)_{2}[TeO_{2}(OH)_{4}]_{(s)} → amorphous material_{(s)} + H_{2}O_{(g)}, O_{(g)}
II. Amorphous material_{(s)} → `Mg[Te_{2}O_{6}]'_{(s)}+ Mg_{3}[TeO_{6}]_{(s)} + H_{2}O_{(g)}
IIIa. `Mg[Te_{2}O_{6}]'_{(s)} + Mg_{3}[TeO_{6}]_{(s)} → Mg[TeO_{4}]_{(s)} + Mg_{3}[TeO_{6}]_{(s)}
IIIb. Mg[TeO_{4}]_{(s)} + Mg_{3}[TeO_{6}]_{(s)} → unknown phase_{(s)} + Mg_{3}[TeO_{6}]_{(s)}
IV. Unknown phase_{(s)} + Mg_{3}[TeO_{6}]_{(s)} → Mg_{3}[TeO_{6}]_{(s)} + O_{2(g)}
4. Conclusion and outlook
The correlated disorder of Mg(H_{2}O)_{2}[TeO_{2}(OH)_{4}] is notable because it can be decomposed into two modes, which can be treated separately. It demonstrates the difficulties of a quantitative inherent to data sets with a significant diffuse scattering. In such cases, refinements against Bragg reflections are not sufficient and information on correlated disorder has to be derived from diffuse scattering. But even such descriptions can be ambiguous. Here, pairs of entirely different structures with opposite sign of c_{orient} produce virtually indistinguishable diffraction patterns. Using a small degree of idealization, simple analytical expressions describing the diffuse scattering can be derived, which allow for extremely fast calculations and a more thorough insight on the observed diffraction phenomena.
A crucial feature in the _{2}O)_{2}[TeO_{2}(OH)_{4}] is the hydrogenbonding network that connects adjacent layers. Its role has been ignored in this Xray study owing to disorder and the weak scattering power of the hydrogen atoms. A study using neutron diffraction might reveal a very different picture, possibly even necessitating the introduction of a third correlation parameter and different ODlayer symmetries.
of Mg(HAPPENDIX A
Symbols used
∝ : Proportional to.
≈ : Almost equal to.
σ : Variance of Gaussian distribution.
n, m, h, k, l : Integers.
x : Real.
w, e : Weighting function and exponent in weighting function.
L_{n} : Crystalchemical layer with sequential index n.
A^{1}_{n}, A^{2}_{n+1} : OD layers ( A^{1}_{n}: octahedra, A^{2}_{n+1}: hydrogenbonding network).
: Group of operations of the OD layer A_{n} not inverting the layer orientation.
: Index of the
of .a, b : Layer lattice basis vectors.
c_{0} : Vector perpendicular to layer plane with the length of one L_{n} layer width.
c : Basis vector of a specific polytype.
a, b, c_{0} : Lengths of the vectors a, b, c_{0}.
r, s : Metric parameters describing the relative positions of adjacent OD layers.
a*, b*, : Basis vectors of the to a, b, c_{0}.
h, k, ν : Reciprocal coordinates with respect to (a*, b*, ) (h, k: integers; ν: real).
F, I : Structure factor and intensity (I = F^{2}) of a polytype or disordered stacking arrangement.
s_{Δn} : Sum term in the calculation of I.
F_{n} : Structure factor of the L_{n} layer.
F^{M}_{n}, F^{O}_{n} : Contributions of the nonO3 atoms and the O3 atom to F_{n} = F^{M}+F^{O}_{n}, respectively.
: Structure factor of the L_{0} layer reflected at (100).
T_{O3} : Displacement parameter of the O3 atom.
f^{O} : Atomic form factor of O.
d(O1—O2) : Distance between atoms O1 and O2.
o_{n} : Origin of the L_{n} layer.
Δo_{n} : Origin shift from the L_{n} to the L_{n+1} layer.
α_{n}, β_{n} : Origin of the L_{n} layer in coordinates: Δo_{n} = α_{n}a/2 + β_{n}b/2.
ω_{n} : Orientation of the L_{n} layer (ω = 0,1).
Δα, Δβ, Δn : Relative origin shift between two layers Δαa + Δβb + Δ nc_{0} up to layer lattice translation (Δα, Δβ = 0, 1).
P(…) : Probability that the expression … holds.
P_{MgTe} : Probability that the origin shifts o_{n} and o_{n+1} are equal.
P_{orient} : Probability that the orientation of two adjacent layers is the same (ω_{n} equals ω_{n+1}).
c_{MgTe}, c_{orient} : Nearestneighbor correlations c_{MgTe} = 2P_{MgTe} − 1 and c_{orient} = 2P_{orient} −1.
d_{c}(x) : Shape of the diffuse scattering with nearestneighbor correlation c.
: Probability that the origin shift of the (L_{n}, L_{n+Δn}) layer pair is Δαa + Δβb + Δnc_{0}.
, : Probabilities that the L_{n} and L_{n+Δn} possess the (ω_{n} = ω_{n+Δn}) or opposite (ω_{n} ≠ ω_{n+Δn}) orientation, respectively.
R_{p} : Residuals for fitting onedimensionally diffuse scattering: .
APPENDIX B
Derivation of volume fractions from occupancies
The
of were derived from refinements using expressions of the type 2occ − 1, where occ is the occupancy of an atom. This may seem surprising and is due to the symmetry used in the refinements.Consider a superposition of MDO_{3} and MDO_{4}, which conveniently possess the same unitcell parameters. Both differ in the position of the O3 atoms, which will be considered up to fourfold rotation and translation. Representative O3 atoms of MDO_{3} and MDO_{4} are listed in the first and second row of Table 8, respectively. A 1 − x : x MDO_{3/4} superposition then possesses the O3 occupancies shown in the third row. However, when refined using the MDO_{4} symmetry (), the O3 positions equivalent in MDO_{4} are averaged, leading to the occupancies listed in the last row.

Thus, such a x)/2 which leads to x = 2occ(O3) − 1. Note that an occupancy of occ(O3) = 0 likewise corresponds to MDO_{4}, but with a different origin. An analoguous argument can be made for the Mg/Te sites.
features two O3 positions with the occupancies occ(O3) = (1±APPENDIX C
General pair distribution probabilities
For Δn even and = = 0. Double application of equation (23) then gives
Repeated substitution of equation (57) into itself leads to
Substituting the initial term P_{0}^{00} = 1:
which can be expressed in terms of Δn = 2m as
An analoguous reasoning applies to Δn odd, though with the initial terms P_{1}^{10} = P_{1}^{01} = ½.
APPENDIX D
Derivation of the peak shape induced by nearestneighbor growth models
The shape of the diffuse scattering due to nearestneighbor correlated stacking arrangements is long known [see Welberry (2010), and references therein]. It will be briefly derived here for a general nearestneighbor correlation c using geometric series.
designates the real part.
Supporting information
https://doi.org/10.1107/S2052520621006223/ra5097sup1.cif
contains datablocks family, mdo34, mdo3, mdo4. DOI:Structure factors: contains datablock family. DOI: https://doi.org/10.1107/S2052520621006223/ra5097familysup2.hkl
Structure factors: contains datablock mdo3. DOI: https://doi.org/10.1107/S2052520621006223/ra5097mdo3sup3.hkl
Structure factors: contains datablock mdo34. DOI: https://doi.org/10.1107/S2052520621006223/ra5097mdo34sup4.hkl
Structure factors: contains datablock mdo4. DOI: https://doi.org/10.1107/S2052520621006223/ra5097mdo4sup5.hkl
For all structures, data collection: Apex 3 (Bruker, 2016); cell
SAINTPlus (Bruker, 2016); data reduction: SAINTPlus (Bruker, 2016); program(s) used to solve structure: SHELXT (Sheldrick, 2015); program(s) used to refine structure: Jana 2006 (Petříček et al., 2014); molecular graphics: Mercury (Macrae et al., 2008).H_{8}MgO_{8}Te  D_{x} = 3.259 Mg m^{−}^{3} 
M_{r} = 288  Mo Kα radiation, λ = 0.71073 Å 
Tetragonal, F4/mmm  Cell parameters from 5981 reflections 
Hall symbol: f_4_2  θ = 4.0–39.1° 
a = 5.3282 (1) Å  µ = 5.17 mm^{−}^{1} 
c = 10.3363 (2) Å  T = 293 K 
V = 293.45 (1) Å^{3}  Block, colorless 
Z = 2  0.22 × 0.15 × 0.15 mm 
F(000) = 256 
Bruker KAPPA APEX II CCD diffractometer  140 reflections with I > 3σ(I) 
Radiation source: Xray tube  R_{int} = 0.029 
ω– and φ–scans  θ_{max} = 38.8°, θ_{min} = 5.8° 
Absorption correction: multiscan SADABS  h = −8→8 
T_{min} = 0.32, T_{max} = 0.46  k = −5→8 
1970 measured reflections  l = −17→16 
140 independent reflections 
Refinement on F^{2}  Hatom parameters constrained 
R[F^{2} > 2σ(F^{2})] = 0.012  Weighting scheme based on measured s.u.'s w = 1/(σ^{2}(I) + 0.0004I^{2}) 
wR(F^{2}) = 0.031  (Δ/σ)_{max} = 0.008 
S = 1.33  Δρ_{max} = 0.27 e Å^{−}^{3} 
140 reflections  Δρ_{min} = −0.45 e Å^{−}^{3} 
11 parameters  Extinction correction: BC type 1 Gaussian isotropic (Becker & Coppens, 1974) 
0 restraints  Extinction coefficient: 490 (110) 
2 constraints 
x  y  z  U_{iso}*/U_{eq}  Occ. (<1)  
Te  0  0  0  0.00996 (8)  0.5 
Mg  0  0  0  0.00996 (8)  0.5 
O1  0  0  0.19444 (18)  0.0237 (3)  
O3  −0.3318 (3)  −0.1682 (3)  0  0.0199 (5)  0.5 
U^{11}  U^{22}  U^{33}  U^{12}  U^{13}  U^{23}  
Te  0.00842 (14)  0.00842 (14)  0.01304 (14)  0  0  0 
Mg  0.00842 (14)  0.00842 (14)  0.01304 (14)  0  0  0 
O1  0.0269 (6)  0.0269 (6)  0.0174 (6)  0  0  0 
O3  0.0118 (7)  0.0118 (7)  0.0362 (11)  0.0034 (8)  0  0 
Te—Mg  0  Mg—O1^{viii}  2.0098 (18) 
Te—O3  1.9820 (17)  Mg—O3  1.9820 (17) 
Te—O3^{i}  1.9820 (17)  Mg—O3^{i}  1.9820 (17) 
Te—O3^{ii}  1.9820 (17)  Mg—O3^{ii}  1.9820 (17) 
Te—O3^{iii}  1.9820 (17)  Mg—O3^{iii}  1.9820 (17) 
Te—O3^{iv}  1.9820 (17)  Mg—O3^{iv}  1.9820 (17) 
Te—O3^{v}  1.9820 (17)  Mg—O3^{v}  1.9820 (17) 
Te—O3^{vi}  1.9820 (17)  Mg—O3^{vi}  1.9820 (17) 
Te—O3^{vii}  1.9820 (17)  Mg—O3^{vii}  1.9820 (17) 
Mg—O1  2.0098 (18)  O3—O3^{iii}  1.232 (2) 
Mg—Te—O3  0  O1—Mg—O3^{iv}  90 
Mg—Te—O3^{i}  0  O1—Mg—O3^{v}  90 
Mg—Te—O3^{ii}  0  O1—Mg—O3^{vi}  90 
Mg—Te—O3^{iii}  0  O1—Mg—O3^{vii}  90 
Mg—Te—O3^{iv}  0  O1^{viii}—Mg—O3  90 
Mg—Te—O3^{v}  0  O1^{viii}—Mg—O3^{i}  90 
Mg—Te—O3^{vi}  0  O1^{viii}—Mg—O3^{ii}  90 
Mg—Te—O3^{vii}  0  O1^{viii}—Mg—O3^{iii}  90 
O3—Te—O3^{i}  143.78 (7)  O1^{viii}—Mg—O3^{iv}  90 
O3—Te—O3^{ii}  180  O1^{viii}—Mg—O3^{v}  90 
O3—Te—O3^{iii}  36.22 (7)  O1^{viii}—Mg—O3^{vi}  90 
O3—Te—O3^{iv}  90.00 (7)  O1^{viii}—Mg—O3^{vii}  90 
O3—Te—O3^{v}  53.78 (7)  O3—Mg—O3^{i}  143.78 (7) 
O3—Te—O3^{vi}  90.00 (7)  O3—Mg—O3^{ii}  180 
O3—Te—O3^{vii}  126.22 (7)  O3—Mg—O3^{iii}  36.22 (7) 
O3^{i}—Te—O3^{ii}  36.22 (7)  O3—Mg—O3^{iv}  90.00 (7) 
O3^{i}—Te—O3^{iii}  180  O3—Mg—O3^{v}  53.78 (7) 
O3^{i}—Te—O3^{iv}  126.22 (7)  O3—Mg—O3^{vi}  90.00 (7) 
O3^{i}—Te—O3^{v}  90.00 (7)  O3—Mg—O3^{vii}  126.22 (7) 
O3^{i}—Te—O3^{vi}  53.78 (7)  O3^{i}—Mg—O3^{ii}  36.22 (7) 
O3^{i}—Te—O3^{vii}  90.00 (7)  O3^{i}—Mg—O3^{iii}  180 
O3^{ii}—Te—O3^{iii}  143.78 (7)  O3^{i}—Mg—O3^{iv}  126.22 (7) 
O3^{ii}—Te—O3^{iv}  90.00 (7)  O3^{i}—Mg—O3^{v}  90.00 (7) 
O3^{ii}—Te—O3^{v}  126.22 (7)  O3^{i}—Mg—O3^{vi}  53.78 (7) 
O3^{ii}—Te—O3^{vi}  90.00 (7)  O3^{i}—Mg—O3^{vii}  90.00 (7) 
O3^{ii}—Te—O3^{vii}  53.78 (7)  O3^{ii}—Mg—O3^{iii}  143.78 (7) 
O3^{iii}—Te—O3^{iv}  53.78 (7)  O3^{ii}—Mg—O3^{iv}  90.00 (7) 
O3^{iii}—Te—O3^{v}  90.00 (7)  O3^{ii}—Mg—O3^{v}  126.22 (7) 
O3^{iii}—Te—O3^{vi}  126.22 (7)  O3^{ii}—Mg—O3^{vi}  90.00 (7) 
O3^{iii}—Te—O3^{vii}  90.00 (7)  O3^{ii}—Mg—O3^{vii}  53.78 (7) 
O3^{iv}—Te—O3^{v}  143.78 (7)  O3^{iii}—Mg—O3^{iv}  53.78 (7) 
O3^{iv}—Te—O3^{vi}  180  O3^{iii}—Mg—O3^{v}  90.00 (7) 
O3^{iv}—Te—O3^{vii}  36.22 (7)  O3^{iii}—Mg—O3^{vi}  126.22 (7) 
O3^{v}—Te—O3^{vi}  36.22 (7)  O3^{iii}—Mg—O3^{vii}  90.00 (7) 
O3^{v}—Te—O3^{vii}  180  O3^{iv}—Mg—O3^{v}  143.78 (7) 
O3^{vi}—Te—O3^{vii}  143.78 (7)  O3^{iv}—Mg—O3^{vi}  180 
Te—Mg—O1  0  O3^{iv}—Mg—O3^{vii}  36.22 (7) 
Te—Mg—O1^{viii}  0  O3^{v}—Mg—O3^{vi}  36.22 (7) 
Te—Mg—O3  0  O3^{v}—Mg—O3^{vii}  180 
Te—Mg—O3^{i}  0  O3^{vi}—Mg—O3^{vii}  143.78 (7) 
Te—Mg—O3^{ii}  0  Te—O3—Te^{ix}  143.78 (9) 
Te—Mg—O3^{iii}  0  Te—O3—Mg  0 
Te—Mg—O3^{iv}  0  Te—O3—Mg^{ix}  143.78 (9) 
Te—Mg—O3^{v}  0  Te—O3—O3^{iii}  71.89 (11) 
Te—Mg—O3^{vi}  0  Te^{ix}—O3—Mg  143.78 (9) 
Te—Mg—O3^{vii}  0  Te^{ix}—O3—Mg^{ix}  0 
O1—Mg—O1^{viii}  180  Te^{ix}—O3—O3^{iii}  71.89 (11) 
O1—Mg—O3  90  Mg—O3—Mg^{ix}  143.78 (9) 
O1—Mg—O3^{i}  90  Mg—O3—O3^{iii}  71.89 (11) 
O1—Mg—O3^{ii}  90  Mg^{ix}—O3—O3^{iii}  71.89 (11) 
O1—Mg—O3^{iii}  90 
Symmetry codes: (i) x+1/2, y+1/2, z; (ii) −x, −y, z; (iii) −x−1/2, −y−1/2, z; (iv) −y, x, z; (v) −y−1/2, x+1/2, z; (vi) y, −x, z; (vii) y+1/2, −x−1/2, z; (viii) y, x, −z; (ix) x−1/2, y−1/2, z. 
H_{8}MgO_{8}Te  D_{x} = 3.260 Mg m^{−}^{3} 
M_{r} = 288  Mo Kα radiation, λ = 0.71073 Å 
Tetragonal, I4_{1}/amd  Cell parameters from 5981 reflections 
Hall symbol: I 4bd;2  θ = 4.0–39.1° 
a = 5.3268 (1) Å  µ = 5.18 mm^{−}^{1} 
c = 20.6747 (4) Å  T = 293 K 
V = 586.64 (1) Å^{3}  Block, colorless 
Z = 4  0.22 × 0.15 × 0.15 mm 
F(000) = 512 
Bruker KAPPA APEX II CCD diffractometer  409 reflections with I > 3σ(I) 
Radiation source: Xray tube  R_{int} = 0.040 
ω– and φ–scans  θ_{max} = 39.1°, θ_{min} = 4.0° 
Absorption correction: multiscan SADABS  h = −5→8 
T_{min} = 0.32, T_{max} = 0.46  k = −8→8 
7123 measured reflections  l = −35→33 
461 independent reflections 
Refinement on F^{2}  Hatom parameters constrained 
R[F^{2} > 2σ(F^{2})] = 0.020  Weighting scheme based on measured s.u.'s w = 1/(σ^{2}(I) + 0.0016I^{2}) 
wR(F^{2}) = 0.078  (Δ/σ)_{max} = 0.040 
S = 1.61  Δρ_{max} = 1.23 e Å^{−}^{3} 
461 reflections  Δρ_{min} = −0.90 e Å^{−}^{3} 
24 parameters  Extinction correction: BC type 1 Gaussian isotropic (Becker & Coppens, 1974) 
0 restraints  Extinction coefficient: 800 (200) 
7 constraints 
x  y  z  U_{iso}*/U_{eq}  Occ. (<1)  
Te  0  0  0  0.00979 (11)  0.954 (6) 
Mg'  0  0  0  0.00979 (11)  0.046 (6) 
Mg  0.5  0.5  0  0.0096 (6)  0.954 (6) 
Te'  0.5  0.5  0  0.0096 (6)  0.046 (6) 
O1  0  0  −0.0955 (2)  0.0242 (10)  
O2  0.5  0.5  −0.0989 (2)  0.0230 (9)  
O3  −0.3215 (8)  −0.1569 (8)  −0.00124 (16)  0.0171 (9)  0.5 
U^{11}  U^{22}  U^{33}  U^{12}  U^{13}  U^{23}  
Te  0.00852 (19)  0.00852 (19)  0.0123 (2)  0  0  0 
Mg'  0.00852 (19)  0.00852 (19)  0.0123 (2)  0  0  0 
Mg  0.0077 (10)  0.0077 (10)  0.0133 (10)  0  0  0 
Te'  0.0077 (10)  0.0077 (10)  0.0133 (10)  0  0  0 
O1  0.0150 (14)  0.046 (2)  0.0114 (12)  0  0  0 
O2  0.0141 (14)  0.038 (2)  0.0170 (14)  0  0  0 
O3  0.0078 (14)  0.0089 (15)  0.034 (2)  −0.0021 (13)  −0.0018 (9)  −0.0010 (9) 
Te—Mg'  0  Mg'—O3^{iv}  1.906 (4) 
Te—O1  1.975 (4)  Mg'—O3^{v}  1.906 (4) 
Te—O1^{i}  1.975 (4)  Mg'—O3^{vi}  1.906 (4) 
Te—O3  1.906 (4)  Mg'—O3^{vii}  1.906 (4) 
Te—O3^{ii}  1.906 (4)  Mg—Te'  0 
Te—O3^{i}  1.906 (4)  Mg—O2  2.046 (4) 
Te—O3^{iii}  1.906 (4)  Mg—O2^{i}  2.046 (4) 
Te—O3^{iv}  1.906 (4)  Mg—O3^{viii}  2.060 (4) 
Te—O3^{v}  1.906 (4)  Mg—O3^{ii}  2.060 (4) 
Te—O3^{vi}  1.906 (4)  Mg—O3^{ix}  2.060 (4) 
Te—O3^{vii}  1.906 (4)  Mg—O3^{iii}  2.060 (4) 
Mg'—O1  1.975 (4)  Mg—O3^{x}  2.060 (4) 
Mg'—O1^{i}  1.975 (4)  Mg—O3^{xi}  2.060 (4) 
Mg'—O3  1.906 (4)  Mg—O3^{xii}  2.060 (4) 
Mg'—O3^{ii}  1.906 (4)  Mg—O3^{xiii}  2.060 (4) 
Mg'—O3^{i}  1.906 (4)  O3—O3^{i}  1.241 (6) 
Mg'—O3^{iii}  1.906 (4)  
Mg'—Te—O1  0  O3—Mg'—O3^{v}  90.01 (18) 
Mg'—Te—O1^{i}  0  O3—Mg'—O3^{vi}  52.02 (18) 
Mg'—Te—O3  0  O3—Mg'—O3^{vii}  127.95 (18) 
Mg'—Te—O3^{ii}  0  O3^{ii}—Mg'—O3^{i}  142.03 (18) 
Mg'—Te—O3^{i}  0  O3^{ii}—Mg'—O3^{iii}  38.00 (18) 
Mg'—Te—O3^{iii}  0  O3^{ii}—Mg'—O3^{iv}  90.01 (18) 
Mg'—Te—O3^{iv}  0  O3^{ii}—Mg'—O3^{v}  90.01 (18) 
Mg'—Te—O3^{v}  0  O3^{ii}—Mg'—O3^{vi}  127.95 (18) 
Mg'—Te—O3^{vi}  0  O3^{ii}—Mg'—O3^{vii}  52.02 (18) 
Mg'—Te—O3^{vii}  0  O3^{i}—Mg'—O3^{iii}  178.46 (14) 
O1—Te—O1^{i}  180  O3^{i}—Mg'—O3^{iv}  127.95 (18) 
O1—Te—O3  89.23 (10)  O3^{i}—Mg'—O3^{v}  52.02 (18) 
O1—Te—O3^{ii}  89.23 (10)  O3^{i}—Mg'—O3^{vi}  90.01 (18) 
O1—Te—O3^{i}  90.77 (10)  O3^{i}—Mg'—O3^{vii}  90.01 (18) 
O1—Te—O3^{iii}  90.77 (10)  O3^{iii}—Mg'—O3^{iv}  52.02 (18) 
O1—Te—O3^{iv}  90.77 (10)  O3^{iii}—Mg'—O3^{v}  127.95 (18) 
O1—Te—O3^{v}  90.77 (10)  O3^{iii}—Mg'—O3^{vi}  90.01 (18) 
O1—Te—O3^{vi}  89.23 (10)  O3^{iii}—Mg'—O3^{vii}  90.01 (18) 
O1—Te—O3^{vii}  89.23 (10)  O3^{iv}—Mg'—O3^{v}  178.46 (14) 
O1^{i}—Te—O3  90.77 (10)  O3^{iv}—Mg'—O3^{vi}  38.00 (18) 
O1^{i}—Te—O3^{ii}  90.77 (10)  O3^{iv}—Mg'—O3^{vii}  142.03 (18) 
O1^{i}—Te—O3^{i}  89.23 (10)  O3^{v}—Mg'—O3^{vi}  142.03 (18) 
O1^{i}—Te—O3^{iii}  89.23 (10)  O3^{v}—Mg'—O3^{vii}  38.00 (18) 
O1^{i}—Te—O3^{iv}  89.23 (10)  O3^{vi}—Mg'—O3^{vii}  178.46 (14) 
O1^{i}—Te—O3^{v}  89.23 (10)  Te'—Mg—O2  0 
O1^{i}—Te—O3^{vi}  90.77 (10)  Te'—Mg—O2^{i}  0 
O1^{i}—Te—O3^{vii}  90.77 (10)  Te'—Mg—O3^{viii}  0 
O3—Te—O3^{ii}  178.46 (14)  Te'—Mg—O3^{ii}  0 
O3—Te—O3^{i}  38.00 (18)  Te'—Mg—O3^{ix}  0 
O3—Te—O3^{iii}  142.03 (18)  Te'—Mg—O3^{iii}  0 
O3—Te—O3^{iv}  90.01 (18)  Te'—Mg—O3^{x}  0 
O3—Te—O3^{v}  90.01 (18)  Te'—Mg—O3^{xi}  0 
O3—Te—O3^{vi}  52.02 (18)  Te'—Mg—O3^{xii}  0 
O3—Te—O3^{vii}  127.95 (18)  Te'—Mg—O3^{xiii}  0 
O3^{ii}—Te—O3^{i}  142.03 (18)  O2—Mg—O2^{i}  180 
O3^{ii}—Te—O3^{iii}  38.00 (18)  O2—Mg—O3^{viii}  89.29 (9) 
O3^{ii}—Te—O3^{iv}  90.01 (18)  O2—Mg—O3^{ii}  89.29 (9) 
O3^{ii}—Te—O3^{v}  90.01 (18)  O2—Mg—O3^{ix}  90.71 (9) 
O3^{ii}—Te—O3^{vi}  127.95 (18)  O2—Mg—O3^{iii}  90.71 (9) 
O3^{ii}—Te—O3^{vii}  52.02 (18)  O2—Mg—O3^{x}  90.71 (9) 
O3^{i}—Te—O3^{iii}  178.46 (14)  O2—Mg—O3^{xi}  90.71 (9) 
O3^{i}—Te—O3^{iv}  127.95 (18)  O2—Mg—O3^{xii}  89.29 (9) 
O3^{i}—Te—O3^{v}  52.02 (18)  O2—Mg—O3^{xiii}  89.29 (9) 
O3^{i}—Te—O3^{vi}  90.01 (18)  O2^{i}—Mg—O3^{viii}  90.71 (9) 
O3^{i}—Te—O3^{vii}  90.01 (18)  O2^{i}—Mg—O3^{ii}  90.71 (9) 
O3^{iii}—Te—O3^{iv}  52.02 (18)  O2^{i}—Mg—O3^{ix}  89.29 (9) 
O3^{iii}—Te—O3^{v}  127.95 (18)  O2^{i}—Mg—O3^{iii}  89.29 (9) 
O3^{iii}—Te—O3^{vi}  90.01 (18)  O2^{i}—Mg—O3^{x}  89.29 (9) 
O3^{iii}—Te—O3^{vii}  90.01 (18)  O2^{i}—Mg—O3^{xi}  89.29 (9) 
O3^{iv}—Te—O3^{v}  178.46 (14)  O2^{i}—Mg—O3^{xii}  90.71 (9) 
O3^{iv}—Te—O3^{vi}  38.00 (18)  O2^{i}—Mg—O3^{xiii}  90.71 (9) 
O3^{iv}—Te—O3^{vii}  142.03 (18)  O3^{viii}—Mg—O3^{ii}  178.57 (13) 
O3^{v}—Te—O3^{vi}  142.03 (18)  O3^{viii}—Mg—O3^{ix}  35.06 (17) 
O3^{v}—Te—O3^{vii}  38.00 (18)  O3^{viii}—Mg—O3^{iii}  144.97 (17) 
O3^{vi}—Te—O3^{vii}  178.46 (14)  O3^{viii}—Mg—O3^{x}  90.01 (17) 
Te—Mg'—O1  0  O3^{viii}—Mg—O3^{xi}  90.01 (17) 
Te—Mg'—O1^{i}  0  O3^{viii}—Mg—O3^{xii}  125.01 (17) 
Te—Mg'—O3  0  O3^{viii}—Mg—O3^{xiii}  54.97 (17) 
Te—Mg'—O3^{ii}  0  O3^{ii}—Mg—O3^{ix}  144.97 (17) 
Te—Mg'—O3^{i}  0  O3^{ii}—Mg—O3^{iii}  35.06 (17) 
Te—Mg'—O3^{iii}  0  O3^{ii}—Mg—O3^{x}  90.01 (17) 
Te—Mg'—O3^{iv}  0  O3^{ii}—Mg—O3^{xi}  90.01 (17) 
Te—Mg'—O3^{v}  0  O3^{ii}—Mg—O3^{xii}  54.97 (17) 
Te—Mg'—O3^{vi}  0  O3^{ii}—Mg—O3^{xiii}  125.01 (17) 
Te—Mg'—O3^{vii}  0  O3^{ix}—Mg—O3^{iii}  178.57 (13) 
O1—Mg'—O1^{i}  180  O3^{ix}—Mg—O3^{x}  54.97 (17) 
O1—Mg'—O3  89.23 (10)  O3^{ix}—Mg—O3^{xi}  125.01 (17) 
O1—Mg'—O3^{ii}  89.23 (10)  O3^{ix}—Mg—O3^{xii}  90.01 (17) 
O1—Mg'—O3^{i}  90.77 (10)  O3^{ix}—Mg—O3^{xiii}  90.01 (17) 
O1—Mg'—O3^{iii}  90.77 (10)  O3^{iii}—Mg—O3^{x}  125.01 (17) 
O1—Mg'—O3^{iv}  90.77 (10)  O3^{iii}—Mg—O3^{xi}  54.97 (17) 
O1—Mg'—O3^{v}  90.77 (10)  O3^{iii}—Mg—O3^{xii}  90.01 (17) 
O1—Mg'—O3^{vi}  89.23 (10)  O3^{iii}—Mg—O3^{xiii}  90.01 (17) 
O1—Mg'—O3^{vii}  89.23 (10)  O3^{x}—Mg—O3^{xi}  178.57 (13) 
O1^{i}—Mg'—O3  90.77 (10)  O3^{x}—Mg—O3^{xii}  35.06 (17) 
O1^{i}—Mg'—O3^{ii}  90.77 (10)  O3^{x}—Mg—O3^{xiii}  144.97 (17) 
O1^{i}—Mg'—O3^{i}  89.23 (10)  O3^{xi}—Mg—O3^{xii}  144.97 (17) 
O1^{i}—Mg'—O3^{iii}  89.23 (10)  O3^{xi}—Mg—O3^{xiii}  35.06 (17) 
O1^{i}—Mg'—O3^{iv}  89.23 (10)  O3^{xii}—Mg—O3^{xiii}  178.57 (13) 
O1^{i}—Mg'—O3^{v}  89.23 (10)  Te—O1—Mg'  0 
O1^{i}—Mg'—O3^{vi}  90.77 (10)  Te—O3—Mg'  0 
O1^{i}—Mg'—O3^{vii}  90.77 (10)  Te—O3—Mg^{xiv}  143.5 (2) 
O3—Mg'—O3^{ii}  178.46 (14)  Te—O3—O3^{i}  71.0 (3) 
O3—Mg'—O3^{i}  38.00 (18)  Mg'—O3—Mg^{xiv}  143.5 (2) 
O3—Mg'—O3^{iii}  142.03 (18)  Mg'—O3—O3^{i}  71.0 (3) 
O3—Mg'—O3^{iv}  90.01 (18)  Mg^{xiv}—O3—O3^{i}  72.5 (3) 
Symmetry codes: (i) y, x, −z; (ii) −x, −y, z; (iii) −y, −x, −z; (iv) y, −x, −z; (v) −y, x, −z; (vi) x, −y, z; (vii) −x, y, z; (viii) x+1, y+1, z; (ix) y+1, x+1, −z; (x) y+1, −x, −z; (xi) −y, x+1, −z; (xii) x+1, −y, z; (xiii) −x, y+1, z; (xiv) x−1, y−1, z. 
MgTeO_{8}H_{8}  D_{x} = 3.259 Mg m^{−}^{3} 
M_{r} = 288  Mo Kα radiation, λ = 0.71073 Å 
Tetragonal, I4_{1}/a  Cell parameters from 5981 reflections 
Hall symbol: I 4ad  θ = 4.0–39.1° 
a = 5.3282 (1) Å  µ = 5.17 mm^{−}^{1} 
c = 20.6725 (4) Å  T = 293 K 
V = 586.89 (1) Å^{3}  Block, colorless 
Z = 4  0.22 × 0.15 × 0.15 mm 
F(000) = 512 
Bruker KAPPA APEX II CCD diffractometer  773 independent reflections 
Radiation source: Xray tube  651 reflections with I > 3σ(I) 
Graphite monochromator  R_{int} = 0.032 
ω– and φ–scans  θ_{max} = 39.1°, θ_{min} = 4.0° 
Absorption correction: multiscan SADABS  h = −5→8 
T_{min} = 0.32, T_{max} = 0.46  k = −8→8 
7853 measured reflections  l = −35→33 
Refinement on F^{2}  Hatom parameters constrained 
R[F^{2} > 2σ(F^{2})] = 0.018  Weighting scheme based on measured s.u.'s w = 1/(σ^{2}(I) + 0.0016I^{2}) 
wR(F^{2}) = 0.068  (Δ/σ)_{max} = 0.007 
S = 1.34  Δρ_{max} = 0.93 e Å^{−}^{3} 
773 reflections  Δρ_{min} = −0.71 e Å^{−}^{3} 
28 parameters  Extinction correction: BC type 1 Gaussian isotropic (Becker & Coppens, 1974) 
0 restraints  Extinction coefficient: 960 (170) 
17 constraints 
x  y  z  U_{iso}*/U_{eq}  Occ. (<1)  
Te  0  0  0  0.00998 (8)  0.923 (4) 
Mg'  0  0  0  0.00998 (8)  0.077 (4) 
Mg  0.5  0.5  0  0.0121 (4)  0.923 (4) 
Te'  0.5  0.5  0  0.0121 (4)  0.077 (4) 
O1  0  0  0.09543 (14)  0.0247 (7)  
O2  0.5  0.5  0.09889 (15)  0.0232 (6)  
O3  −0.3216 (5)  −0.1563 (5)  −0.00202 (9)  0.0183 (6)  0.822 (14) 
O3'  −0.3216 (5)  0.1563 (5)  −0.00202 (9)  0.0183 (6)  0.178 (14) 
U^{11}  U^{22}  U^{33}  U^{12}  U^{13}  U^{23}  
Te  0.00867 (14)  0.00867 (14)  0.01258 (14)  0  0  0 
Mg'  0.00867 (14)  0.00867 (14)  0.01258 (14)  0  0  0 
Mg  0.0097 (6)  0.0097 (6)  0.0169 (7)  0  0  0 
Te'  0.0097 (6)  0.0097 (6)  0.0169 (7)  0  0  0 
O1  0.0316 (14)  0.0298 (13)  0.0128 (9)  −0.011 (3)  0  0 
O2  0.0234 (11)  0.0276 (12)  0.0185 (10)  −0.003 (3)  0  0 
O3  0.0082 (9)  0.0094 (10)  0.0374 (14)  −0.0022 (9)  −0.0056 (4)  −0.0053 (5) 
O3'  0.0082 (9)  0.0094 (10)  0.0374 (14)  0.0022 (9)  −0.0056 (4)  0.0053 (5) 
Te—Mg'  0  Mg'—O3'  1.906 (3) 
Te—O1  1.973 (3)  Mg'—O3'^{ii}  1.906 (3) 
Te—O1^{i}  1.973 (3)  Mg'—O3'^{i}  1.906 (3) 
Te—O3  1.906 (3)  Mg'—O3'^{iii}  1.906 (3) 
Te—O3^{ii}  1.906 (3)  Mg—Te'  0 
Te—O3^{i}  1.906 (3)  Mg—O2  2.044 (3) 
Te—O3^{iii}  1.906 (3)  Mg—O2^{iv}  2.044 (3) 
Te—O3'  1.906 (3)  Mg—O3^{v}  2.064 (3) 
Te—O3'^{ii}  1.906 (3)  Mg—O3^{ii}  2.064 (3) 
Te—O3'^{i}  1.906 (3)  Mg—O3^{vi}  2.064 (3) 
Te—O3'^{iii}  1.906 (3)  Mg—O3^{vii}  2.064 (3) 
Mg'—O1  1.973 (3)  Mg—O3'^{viii}  2.064 (3) 
Mg'—O1^{i}  1.973 (3)  Mg—O3'^{ix}  2.064 (3) 
Mg'—O3  1.906 (3)  Mg—O3'^{i}  2.064 (3) 
Mg'—O3^{ii}  1.906 (3)  Mg—O3'^{x}  2.064 (3) 
Mg'—O3^{i}  1.906 (3)  O3—O3'^{iii}  1.248 (4) 
Mg'—O3^{iii}  1.906 (3)  
Mg'—Te—O1  0  O3^{ii}—Mg'—O3^{i}  90.03 (12) 
Mg'—Te—O1^{i}  0  O3^{ii}—Mg'—O3^{iii}  90.03 (12) 
Mg'—Te—O3  0  O3^{ii}—Mg'—O3'  128.10 (12) 
Mg'—Te—O3^{ii}  0  O3^{ii}—Mg'—O3'^{ii}  51.83 (12) 
Mg'—Te—O3^{i}  0  O3^{ii}—Mg'—O3'^{i}  38.24 (12) 
Mg'—Te—O3^{iii}  0  O3^{ii}—Mg'—O3'^{iii}  141.85 (12) 
Mg'—Te—O3'  0  O3^{i}—Mg'—O3^{iii}  177.49 (8) 
Mg'—Te—O3'^{ii}  0  O3^{i}—Mg'—O3'  38.24 (12) 
Mg'—Te—O3'^{i}  0  O3^{i}—Mg'—O3'^{ii}  141.85 (12) 
Mg'—Te—O3'^{iii}  0  O3^{i}—Mg'—O3'^{i}  51.83 (12) 
O1—Te—O1^{i}  180  O3^{i}—Mg'—O3'^{iii}  128.10 (12) 
O1—Te—O3  91.25 (5)  O3^{iii}—Mg'—O3'  141.85 (12) 
O1—Te—O3^{ii}  91.25 (5)  O3^{iii}—Mg'—O3'^{ii}  38.24 (12) 
O1—Te—O3^{i}  88.75 (5)  O3^{iii}—Mg'—O3'^{i}  128.10 (12) 
O1—Te—O3^{iii}  88.75 (5)  O3^{iii}—Mg'—O3'^{iii}  51.83 (12) 
O1—Te—O3'  91.25 (5)  O3'—Mg'—O3'^{ii}  177.49 (8) 
O1—Te—O3'^{ii}  91.25 (5)  O3'—Mg'—O3'^{i}  90.03 (12) 
O1—Te—O3'^{i}  88.75 (5)  O3'—Mg'—O3'^{iii}  90.03 (12) 
O1—Te—O3'^{iii}  88.75 (5)  O3'^{ii}—Mg'—O3'^{i}  90.03 (12) 
O1^{i}—Te—O3  88.75 (5)  O3'^{ii}—Mg'—O3'^{iii}  90.03 (12) 
O1^{i}—Te—O3^{ii}  88.75 (5)  O3'^{i}—Mg'—O3'^{iii}  177.49 (8) 
O1^{i}—Te—O3^{i}  91.25 (5)  Te'—Mg—O2  0 
O1^{i}—Te—O3^{iii}  91.25 (5)  Te'—Mg—O2^{iv}  0 
O1^{i}—Te—O3'  88.75 (5)  Te'—Mg—O3^{v}  0 
O1^{i}—Te—O3'^{ii}  88.75 (5)  Te'—Mg—O3^{ii}  0 
O1^{i}—Te—O3'^{i}  91.25 (5)  Te'—Mg—O3^{vi}  0 
O1^{i}—Te—O3'^{iii}  91.25 (5)  Te'—Mg—O3^{vii}  0 
O3—Te—O3^{ii}  177.49 (8)  Te'—Mg—O3'^{viii}  0 
O3—Te—O3^{i}  90.03 (12)  Te'—Mg—O3'^{ix}  0 
O3—Te—O3^{iii}  90.03 (12)  Te'—Mg—O3'^{i}  0 
O3—Te—O3'  51.83 (12)  Te'—Mg—O3'^{x}  0 
O3—Te—O3'^{ii}  128.10 (12)  O2—Mg—O2^{iv}  180 
O3—Te—O3'^{i}  141.85 (12)  O2—Mg—O3^{v}  91.16 (5) 
O3—Te—O3'^{iii}  38.24 (12)  O2—Mg—O3^{ii}  91.16 (5) 
O3^{ii}—Te—O3^{i}  90.03 (12)  O2—Mg—O3^{vi}  88.84 (5) 
O3^{ii}—Te—O3^{iii}  90.03 (12)  O2—Mg—O3^{vii}  88.84 (5) 
O3^{ii}—Te—O3'  128.10 (12)  O2—Mg—O3'^{viii}  91.16 (5) 
O3^{ii}—Te—O3'^{ii}  51.83 (12)  O2—Mg—O3'^{ix}  91.16 (5) 
O3^{ii}—Te—O3'^{i}  38.24 (12)  O2—Mg—O3'^{i}  88.84 (5) 
O3^{ii}—Te—O3'^{iii}  141.85 (12)  O2—Mg—O3'^{x}  88.84 (5) 
O3^{i}—Te—O3^{iii}  177.49 (8)  O2^{iv}—Mg—O3^{v}  88.84 (5) 
O3^{i}—Te—O3'  38.24 (12)  O2^{iv}—Mg—O3^{ii}  88.84 (5) 
O3^{i}—Te—O3'^{ii}  141.85 (12)  O2^{iv}—Mg—O3^{vi}  91.16 (5) 
O3^{i}—Te—O3'^{i}  51.83 (12)  O2^{iv}—Mg—O3^{vii}  91.16 (5) 
O3^{i}—Te—O3'^{iii}  128.10 (12)  O2^{iv}—Mg—O3'^{viii}  88.84 (5) 
O3^{iii}—Te—O3'  141.85 (12)  O2^{iv}—Mg—O3'^{ix}  88.84 (5) 
O3^{iii}—Te—O3'^{ii}  38.24 (12)  O2^{iv}—Mg—O3'^{i}  91.16 (5) 
O3^{iii}—Te—O3'^{i}  128.10 (12)  O2^{iv}—Mg—O3'^{x}  91.16 (5) 
O3^{iii}—Te—O3'^{iii}  51.83 (12)  O3^{v}—Mg—O3^{ii}  177.68 (7) 
O3'—Te—O3'^{ii}  177.49 (8)  O3^{v}—Mg—O3^{vi}  90.02 (11) 
O3'—Te—O3'^{i}  90.03 (12)  O3^{v}—Mg—O3^{vii}  90.02 (11) 
O3'—Te—O3'^{iii}  90.03 (12)  O3^{v}—Mg—O3'^{viii}  125.09 (11) 
O3'^{ii}—Te—O3'^{i}  90.03 (12)  O3^{v}—Mg—O3'^{ix}  54.85 (11) 
O3'^{ii}—Te—O3'^{iii}  90.03 (12)  O3^{v}—Mg—O3'^{i}  144.87 (11) 
O3'^{i}—Te—O3'^{iii}  177.49 (8)  O3^{v}—Mg—O3'^{x}  35.21 (11) 
Te—Mg'—O1  0  O3^{ii}—Mg—O3^{vi}  90.02 (11) 
Te—Mg'—O1^{i}  0  O3^{ii}—Mg—O3^{vii}  90.02 (11) 
Te—Mg'—O3  0  O3^{ii}—Mg—O3'^{viii}  54.85 (11) 
Te—Mg'—O3^{ii}  0  O3^{ii}—Mg—O3'^{ix}  125.09 (11) 
Te—Mg'—O3^{i}  0  O3^{ii}—Mg—O3'^{i}  35.21 (11) 
Te—Mg'—O3^{iii}  0  O3^{ii}—Mg—O3'^{x}  144.87 (11) 
Te—Mg'—O3'  0  O3^{vi}—Mg—O3^{vii}  177.68 (7) 
Te—Mg'—O3'^{ii}  0  O3^{vi}—Mg—O3'^{viii}  35.21 (11) 
Te—Mg'—O3'^{i}  0  O3^{vi}—Mg—O3'^{ix}  144.87 (11) 
Te—Mg'—O3'^{iii}  0  O3^{vi}—Mg—O3'^{i}  125.09 (11) 
O1—Mg'—O1^{i}  180  O3^{vi}—Mg—O3'^{x}  54.85 (11) 
O1—Mg'—O3  91.25 (5)  O3^{vii}—Mg—O3'^{viii}  144.87 (11) 
O1—Mg'—O3^{ii}  91.25 (5)  O3^{vii}—Mg—O3'^{ix}  35.21 (11) 
O1—Mg'—O3^{i}  88.75 (5)  O3^{vii}—Mg—O3'^{i}  54.85 (11) 
O1—Mg'—O3^{iii}  88.75 (5)  O3^{vii}—Mg—O3'^{x}  125.09 (11) 
O1—Mg'—O3'  91.25 (5)  O3'^{viii}—Mg—O3'^{ix}  177.68 (7) 
O1—Mg'—O3'^{ii}  91.25 (5)  O3'^{viii}—Mg—O3'^{i}  90.02 (11) 
O1—Mg'—O3'^{i}  88.75 (5)  O3'^{viii}—Mg—O3'^{x}  90.02 (11) 
O1—Mg'—O3'^{iii}  88.75 (5)  O3'^{ix}—Mg—O3'^{i}  90.02 (11) 
O1^{i}—Mg'—O3  88.75 (5)  O3'^{ix}—Mg—O3'^{x}  90.02 (11) 
O1^{i}—Mg'—O3^{ii}  88.75 (5)  O3'^{i}—Mg—O3'^{x}  177.68 (7) 
O1^{i}—Mg'—O3^{i}  91.25 (5)  Te—O1—Mg'  0 
O1^{i}—Mg'—O3^{iii}  91.25 (5)  Te—O3—Mg'  0 
O1^{i}—Mg'—O3'  88.75 (5)  Te—O3—Mg^{xi}  143.28 (16) 
O1^{i}—Mg'—O3'^{ii}  88.75 (5)  Te—O3—O3'^{iii}  70.88 (18) 
O1^{i}—Mg'—O3'^{i}  91.25 (5)  Mg'—O3—Mg^{xi}  143.28 (16) 
O1^{i}—Mg'—O3'^{iii}  91.25 (5)  Mg'—O3—O3'^{iii}  70.88 (18) 
O3—Mg'—O3^{ii}  177.49 (8)  Mg^{xi}—O3—O3'^{iii}  72.39 (18) 
O3—Mg'—O3^{i}  90.03 (12)  Te—O3'—Mg'  0 
O3—Mg'—O3^{iii}  90.03 (12)  Te—O3'—Mg^{xii}  143.28 (16) 
O3—Mg'—O3'  51.83 (12)  Te—O3'—O3^{i}  70.88 (18) 
O3—Mg'—O3'^{ii}  128.10 (12)  Mg'—O3'—Mg^{xii}  143.28 (16) 
O3—Mg'—O3'^{i}  141.85 (12)  Mg'—O3'—O3^{i}  70.88 (18) 
O3—Mg'—O3'^{iii}  38.24 (12)  Mg^{xii}—O3'—O3^{i}  72.39 (18) 
Symmetry codes: (i) y, −x, −z; (ii) −x, −y, z; (iii) −y, x, −z; (iv) y, −x+1, −z; (v) x+1, y+1, z; (vi) y+1, −x, −z; (vii) −y, x+1, −z; (viii) x+1, y, z; (ix) −x, −y+1, z; (x) −y+1, x+1, −z; (xi) x−1, y−1, z; (xii) x−1, y, z. 
MgTeO_{8}H_{8}  D_{x} = 3.259 Mg m^{−}^{3} 
M_{r} = 288  Mo Kα radiation, λ = 0.71073 Å 
Tetragonal, I42d  Cell parameters from 5981 reflections 
Hall symbol: I 4;2bw  θ = 4.0–39.1° 
a = 5.3282 (1) Å  µ = 5.17 mm^{−}^{1} 
c = 20.6725 (4) Å  T = 293 K 
V = 586.89 (1) Å^{3}  Block, colorless 
Z = 4  0.22 × 0.15 × 0.15 mm 
F(000) = 512 
Bruker KAPPA APEX II CCD diffractometer  786 independent reflections 
Radiation source: Xray tube  676 reflections with I > 3σ(I) 
Graphite monochromator  R_{int} = 0.032 
ω– and φ–scans  θ_{max} = 39.1°, θ_{min} = 4.0° 
Absorption correction: multiscan SADABS  h = −5→8 
T_{min} = 0.32, T_{max} = 0.46  k = −8→8 
7371 measured reflections  l = −35→33 
Refinement on F^{2}  Weighting scheme based on measured s.u.'s w = 1/(σ^{2}(I) + 0.0016I^{2}) 
R[F^{2} > 2σ(F^{2})] = 0.017  (Δ/σ)_{max} = 0.034 
wR(F^{2}) = 0.058  Δρ_{max} = 0.80 e Å^{−}^{3} 
S = 1.17  Δρ_{min} = −0.70 e Å^{−}^{3} 
786 reflections  Extinction correction: BC type 1 Gaussian isotropic (Becker & Coppens, 1974) 
28 parameters  Extinction coefficient: 750 (140) 
0 restraints  Absolute structure: 314 of Friedel pairs used in the refinement 
17 constraints  Absolute structure parameter: 0.51 (9) 
Hatom parameters constrained 
x  y  z  U_{iso}*/U_{eq}  Occ. (<1)  
Te  0  0  0  0.00987 (6)  0.932 (3) 
Mg'  0  0  0  0.00987 (6)  0.068 (3) 
Mg  0.5  0.5  0  0.0111 (3)  0.932 (3) 
Te'  0.5  0.5  0  0.0111 (3)  0.068 (3) 
O1  0  0  0.09541 (12)  0.0246 (7)  
O2  0.5  0.5  0.09888 (13)  0.0231 (6)  
O3  −0.3217 (4)  −0.1561 (4)  −0.00156 (9)  0.0173 (5)  0.736 (6) 
O3'  −0.3217 (4)  0.1561 (4)  −0.00156 (9)  0.0173 (5)  0.264 (6) 
U^{11}  U^{22}  U^{33}  U^{12}  U^{13}  U^{23}  
Te  0.00857 (11)  0.00857 (11)  0.01248 (12)  0  0  0 
Mg'  0.00857 (11)  0.00857 (11)  0.01248 (12)  0  0  0 
Mg  0.0087 (5)  0.0087 (5)  0.0158 (6)  0  0  0 
Te'  0.0087 (5)  0.0087 (5)  0.0158 (6)  0  0  0 
O1  0.0316 (14)  0.0298 (14)  0.0125 (7)  0.010 (6)  0  0 
O2  0.0236 (10)  0.0275 (11)  0.0182 (8)  0.001 (11)  0  0 
O3  0.0092 (8)  0.0103 (8)  0.0325 (10)  −0.0016 (7)  −0.0055 (4)  −0.0056 (4) 
O3'  0.0092 (8)  0.0103 (8)  0.0325 (10)  0.0016 (7)  −0.0055 (4)  0.0056 (4) 
Te—Mg'  0  Mg'—O3'  1.906 (2) 
Te—O1  1.972 (2)  Mg'—O3'^{ii}  1.906 (2) 
Te—O1^{i}  1.972 (2)  Mg'—O3'^{i}  1.906 (2) 
Te—O3  1.906 (2)  Mg'—O3'^{iii}  1.906 (2) 
Te—O3^{ii}  1.906 (2)  Mg—Te'  0 
Te—O3^{i}  1.906 (2)  Mg—O2  2.044 (3) 
Te—O3^{iii}  1.906 (2)  Mg—O2^{iv}  2.044 (3) 
Te—O3'  1.906 (2)  Mg—O3^{v}  2.064 (2) 
Te—O3'^{ii}  1.906 (2)  Mg—O3^{ii}  2.064 (2) 
Te—O3'^{i}  1.906 (2)  Mg—O3^{vi}  2.064 (2) 
Te—O3'^{iii}  1.906 (2)  Mg—O3^{vii}  2.064 (2) 
Mg'—O1  1.972 (2)  Mg—O3'^{viii}  2.064 (2) 
Mg'—O1^{i}  1.972 (2)  Mg—O3'^{ix}  2.064 (2) 
Mg'—O3  1.906 (2)  Mg—O3'^{i}  2.064 (2) 
Mg'—O3^{ii}  1.906 (2)  Mg—O3'^{x}  2.064 (2) 
Mg'—O3^{i}  1.906 (2)  O3—O3'^{iii}  1.250 (3) 
Mg'—O3^{iii}  1.906 (2)  
Mg'—Te—O1  0  O3^{ii}—Mg'—O3^{i}  90.02 (10) 
Mg'—Te—O1^{i}  0  O3^{ii}—Mg'—O3^{iii}  90.02 (10) 
Mg'—Te—O3  0  O3^{ii}—Mg'—O3'  128.20 (10) 
Mg'—Te—O3^{ii}  0  O3^{ii}—Mg'—O3'^{ii}  51.75 (10) 
Mg'—Te—O3^{i}  0  O3^{ii}—Mg'—O3'^{i}  38.28 (10) 
Mg'—Te—O3^{iii}  0  O3^{ii}—Mg'—O3'^{iii}  141.77 (10) 
Mg'—Te—O3'  0  O3^{i}—Mg'—O3^{iii}  178.07 (8) 
Mg'—Te—O3'^{ii}  0  O3^{i}—Mg'—O3'  38.28 (10) 
Mg'—Te—O3'^{i}  0  O3^{i}—Mg'—O3'^{ii}  141.77 (10) 
Mg'—Te—O3'^{iii}  0  O3^{i}—Mg'—O3'^{i}  51.75 (10) 
O1—Te—O1^{i}  180  O3^{i}—Mg'—O3'^{iii}  128.20 (10) 
O1—Te—O3  90.97 (6)  O3^{iii}—Mg'—O3'  141.77 (10) 
O1—Te—O3^{ii}  90.97 (6)  O3^{iii}—Mg'—O3'^{ii}  38.28 (10) 
O1—Te—O3^{i}  89.03 (6)  O3^{iii}—Mg'—O3'^{i}  128.20 (10) 
O1—Te—O3^{iii}  89.03 (6)  O3^{iii}—Mg'—O3'^{iii}  51.75 (10) 
O1—Te—O3'  90.97 (6)  O3'—Mg'—O3'^{ii}  178.07 (8) 
O1—Te—O3'^{ii}  90.97 (6)  O3'—Mg'—O3'^{i}  90.02 (10) 
O1—Te—O3'^{i}  89.03 (6)  O3'—Mg'—O3'^{iii}  90.02 (10) 
O1—Te—O3'^{iii}  89.03 (6)  O3'^{ii}—Mg'—O3'^{i}  90.02 (10) 
O1^{i}—Te—O3  89.03 (6)  O3'^{ii}—Mg'—O3'^{iii}  90.02 (10) 
O1^{i}—Te—O3^{ii}  89.03 (6)  O3'^{i}—Mg'—O3'^{iii}  178.07 (8) 
O1^{i}—Te—O3^{i}  90.97 (6)  Te'—Mg—O2  0 
O1^{i}—Te—O3^{iii}  90.97 (6)  Te'—Mg—O2^{iv}  0 
O1^{i}—Te—O3'  89.03 (6)  Te'—Mg—O3^{v}  0 
O1^{i}—Te—O3'^{ii}  89.03 (6)  Te'—Mg—O3^{ii}  0 
O1^{i}—Te—O3'^{i}  90.97 (6)  Te'—Mg—O3^{vi}  0 
O1^{i}—Te—O3'^{iii}  90.97 (6)  Te'—Mg—O3^{vii}  0 
O3—Te—O3^{ii}  178.07 (8)  Te'—Mg—O3'^{viii}  0 
O3—Te—O3^{i}  90.02 (10)  Te'—Mg—O3'^{ix}  0 
O3—Te—O3^{iii}  90.02 (10)  Te'—Mg—O3'^{i}  0 
O3—Te—O3'  51.75 (10)  Te'—Mg—O3'^{x}  0 
O3—Te—O3'^{ii}  128.20 (10)  O2—Mg—O2^{iv}  180 
O3—Te—O3'^{i}  141.77 (10)  O2—Mg—O3^{v}  90.89 (5) 
O3—Te—O3'^{iii}  38.28 (10)  O2—Mg—O3^{ii}  90.89 (5) 
O3^{ii}—Te—O3^{i}  90.02 (10)  O2—Mg—O3^{vi}  89.11 (5) 
O3^{ii}—Te—O3^{iii}  90.02 (10)  O2—Mg—O3^{vii}  89.11 (5) 
O3^{ii}—Te—O3'  128.20 (10)  O2—Mg—O3'^{viii}  90.89 (5) 
O3^{ii}—Te—O3'^{ii}  51.75 (10)  O2—Mg—O3'^{ix}  90.89 (5) 
O3^{ii}—Te—O3'^{i}  38.28 (10)  O2—Mg—O3'^{i}  89.11 (5) 
O3^{ii}—Te—O3'^{iii}  141.77 (10)  O2—Mg—O3'^{x}  89.11 (5) 
O3^{i}—Te—O3^{iii}  178.07 (8)  O2^{iv}—Mg—O3^{v}  89.11 (5) 
O3^{i}—Te—O3'  38.28 (10)  O2^{iv}—Mg—O3^{ii}  89.11 (5) 
O3^{i}—Te—O3'^{ii}  141.77 (10)  O2^{iv}—Mg—O3^{vi}  90.89 (5) 
O3^{i}—Te—O3'^{i}  51.75 (10)  O2^{iv}—Mg—O3^{vii}  90.89 (5) 
O3^{i}—Te—O3'^{iii}  128.20 (10)  O2^{iv}—Mg—O3'^{viii}  89.11 (5) 
O3^{iii}—Te—O3'  141.77 (10)  O2^{iv}—Mg—O3'^{ix}  89.11 (5) 
O3^{iii}—Te—O3'^{ii}  38.28 (10)  O2^{iv}—Mg—O3'^{i}  90.89 (5) 
O3^{iii}—Te—O3'^{i}  128.20 (10)  O2^{iv}—Mg—O3'^{x}  90.89 (5) 
O3^{iii}—Te—O3'^{iii}  51.75 (10)  O3^{v}—Mg—O3^{ii}  178.21 (7) 
O3'—Te—O3'^{ii}  178.07 (8)  O3^{v}—Mg—O3^{vi}  90.01 (9) 
O3'—Te—O3'^{i}  90.02 (10)  O3^{v}—Mg—O3^{vii}  90.01 (9) 
O3'—Te—O3'^{iii}  90.02 (10)  O3^{v}—Mg—O3'^{viii}  125.17 (9) 
O3'^{ii}—Te—O3'^{i}  90.02 (10)  O3^{v}—Mg—O3'^{ix}  54.79 (9) 
O3'^{ii}—Te—O3'^{iii}  90.02 (10)  O3^{v}—Mg—O3'^{i}  144.80 (9) 
O3'^{i}—Te—O3'^{iii}  178.07 (8)  O3^{v}—Mg—O3'^{x}  35.24 (9) 
Te—Mg'—O1  0  O3^{ii}—Mg—O3^{vi}  90.01 (9) 
Te—Mg'—O1^{i}  0  O3^{ii}—Mg—O3^{vii}  90.01 (9) 
Te—Mg'—O3  0  O3^{ii}—Mg—O3'^{viii}  54.79 (9) 
Te—Mg'—O3^{ii}  0  O3^{ii}—Mg—O3'^{ix}  125.17 (9) 
Te—Mg'—O3^{i}  0  O3^{ii}—Mg—O3'^{i}  35.24 (9) 
Te—Mg'—O3^{iii}  0  O3^{ii}—Mg—O3'^{x}  144.80 (9) 
Te—Mg'—O3'  0  O3^{vi}—Mg—O3^{vii}  178.21 (7) 
Te—Mg'—O3'^{ii}  0  O3^{vi}—Mg—O3'^{viii}  35.24 (9) 
Te—Mg'—O3'^{i}  0  O3^{vi}—Mg—O3'^{ix}  144.80 (9) 
Te—Mg'—O3'^{iii}  0  O3^{vi}—Mg—O3'^{i}  125.17 (9) 
O1—Mg'—O1^{i}  180  O3^{vi}—Mg—O3'^{x}  54.79 (9) 
O1—Mg'—O3  90.97 (6)  O3^{vii}—Mg—O3'^{viii}  144.80 (9) 
O1—Mg'—O3^{ii}  90.97 (6)  O3^{vii}—Mg—O3'^{ix}  35.24 (9) 
O1—Mg'—O3^{i}  89.03 (6)  O3^{vii}—Mg—O3'^{i}  54.79 (9) 
O1—Mg'—O3^{iii}  89.03 (6)  O3^{vii}—Mg—O3'^{x}  125.17 (9) 
O1—Mg'—O3'  90.97 (6)  O3'^{viii}—Mg—O3'^{ix}  178.21 (7) 
O1—Mg'—O3'^{ii}  90.97 (6)  O3'^{viii}—Mg—O3'^{i}  90.01 (9) 
O1—Mg'—O3'^{i}  89.03 (6)  O3'^{viii}—Mg—O3'^{x}  90.01 (9) 
O1—Mg'—O3'^{iii}  89.03 (6)  O3'^{ix}—Mg—O3'^{i}  90.01 (9) 
O1^{i}—Mg'—O3  89.03 (6)  O3'^{ix}—Mg—O3'^{x}  90.01 (9) 
O1^{i}—Mg'—O3^{ii}  89.03 (6)  O3'^{i}—Mg—O3'^{x}  178.21 (7) 
O1^{i}—Mg'—O3^{i}  90.97 (6)  Te—O1—Mg'  0 
O1^{i}—Mg'—O3^{iii}  90.97 (6)  Te—O3—Mg'  0 
O1^{i}—Mg'—O3'  89.03 (6)  Te—O3—Mg^{xi}  143.24 (13) 
O1^{i}—Mg'—O3'^{ii}  89.03 (6)  Te—O3—O3'^{iii}  70.86 (15) 
O1^{i}—Mg'—O3'^{i}  90.97 (6)  Mg'—O3—Mg^{xi}  143.24 (13) 
O1^{i}—Mg'—O3'^{iii}  90.97 (6)  Mg'—O3—O3'^{iii}  70.86 (15) 
O3—Mg'—O3^{ii}  178.07 (8)  Mg^{xi}—O3—O3'^{iii}  72.38 (15) 
O3—Mg'—O3^{i}  90.02 (10)  Te—O3'—Mg'  0 
O3—Mg'—O3^{iii}  90.02 (10)  Te—O3'—Mg^{xii}  143.24 (13) 
O3—Mg'—O3'  51.75 (10)  Te—O3'—O3^{i}  70.86 (15) 
O3—Mg'—O3'^{ii}  128.20 (10)  Mg'—O3'—Mg^{xii}  143.24 (13) 
O3—Mg'—O3'^{i}  141.77 (10)  Mg'—O3'—O3^{i}  70.86 (15) 
O3—Mg'—O3'^{iii}  38.28 (10)  Mg^{xii}—O3'—O3^{i}  72.38 (15) 
Symmetry codes: (i) y, −x, −z; (ii) −x, −y, z; (iii) −y, x, −z; (iv) y, −x+1, −z; (v) x+1, y+1, z; (vi) y+1, −x, −z; (vii) −y, x+1, −z; (viii) x+1, y, z; (ix) −x, −y+1, z; (x) −y+1, x+1, −z; (xi) x−1, y−1, z; (xii) x−1, y, z. 
Acknowledgements
The authors thank Christine Artner for performing the crystal growth experiments which led to the title crystals. We thank Christian GierlMayer for the STA measurement and Anuschka Pauluhn for technical help during the synchrotron experiments. The thorough review of an anonymous referee helped to distinctly improve the quality of the manuscript. The authors acknowledge TU Wien Bibliothek for financial support through its Open Access Funding Programme.
Funding information
Funding for this research was provided by: Horizon 2020 Framework Programme [grant No. 730872 (project CALIPSOplus) to Hannes Krüger].
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