research papers
Exploiting
to clarify vacancy and Al/Si ordering in mulliteaDepartamento de Física de la Materia Condensada, Facultad de Ciencia y Tecnología, Universidad del País Vasco UPV/EHU, Apartado 644, Bilbao 48080, Spain, and bDepartamento de Física Aplicada II, Facultad de Ciencia y Tecnología, Universidad del País Vasco UPV/EHU, Apartado 644, Bilbao 48080, Spain
*Correspondence e-mail: paulbenjamin.klar@ehu.eus
Synchrotron single-crystal X-ray diffraction has revealed diffuse scattering alongside sharp satellite reflections for different samples of mullite (Al4+2xSi2−2xO10−x). Structural models have been developed in (3+1)-dimensional that account for vacancy ordering and Al/Si ordering based on harmonic modulation functions. A constraint scheme is presented which explains the crystal-chemical relationships between the split sites of the average structure. The modulation amplitudes of the refinements differ significantly by a factor of ∼3, which is explained in terms of different degrees of ordering, i.e. vacancies follow the same ordering principle in all samples but to different extents. A new approach is applied for the first time to determine Al/Si ordering by combining density functional theory with the modulated volumes of the tetrahedra. The presence of Si–Si diclusters indicates that the mineral classification of mullite needs to be reviewed. A description of the of mullite must consider both the chemical composition and the degree of ordering. This is of particular importance for applications such as advanced ceramics, because the physical properties depend on the intrinsic structure of mullite.
Keywords: Al/Si ordering; vacancy ordering; ceramics; superspace; incommensurate structures.
B-IncStrDB reference: 13462ERV71a
1. Introduction
The 4+2xSi2−2xO10−x) has been a matter of investigation since the structure of the mineral sillimanite was solved (Taylor, 1928), which is chemically and structurally closely related to mullite. The main difference is the presence of oxygen vacancies with a concentration x that ranges between about 0.20 and 0.57, though it can be extended to about 0.9 depending on the synthesis conditions (Schneider et al., 2015). The composition is often expressed as the oxide ratio Al2O3:SiO2, for example 3:2 for the 3/2-mullite (x = 0.25) or 2:1 for the 2/1-mullite (x = 0.4). Fig. 1 shows a polyhedral model of mullite with the characteristic triclusters around the oxygen vacancies. In particular, the distribution of these vacancies within the is still a matter of debate. Furthermore, little is known about the distribution of Al and Si on the tetrahedral sites, though in general it is assumed that Si does not occupy tricluster tetrahedra, derived from the electrostatic bond-valence rule.
of mullite (AlIn ; Freimann & Rahman, 2001, and references therein). Maxima in the diffuse scattering visible in the hk½ section were identified as satellite reflections with a modulation wavevector q = (α0γ), which depends on the chemical composition (Cameron, 1977). The dependence of α on the vacancy concentration x is different from the dependence of γ. The latter is ½ for 3/2-, 2/1- and 5/2-mullite, but for higher x a lowering of the symmetry to monoclinic is observed as γ deviates slightly from ½. The turnover vacancy concentration is not known, but it must be close to x ≃ 0.5. α decreases approximately linearly with increasing x in the orthorhombic case and seems to be constant in the monoclinic case (Ylä-Jääski & Nissen, 1983).
apart from the main Bragg reflections, a characteristic diffuse scattering pattern can be observed, which has been explained by correlated disorder of oxygen vacancies (Welberry & Butler, 1996Agrell & Smith (1960) revealed that mullite shows either diffuse satellite reflections in hk½ sections alongside diffuse scattering, or sharp satellite reflection spots without diffuse scattering. A new terminology was suggested to distinguish between `D-mullite' and `S-mullite', which refer to samples with either diffuse or sharp satellite reflections, respectively. For more aluminous mullites, several electron diffraction studies revealed sharp satellite reflections of fifth and higher orders (Nakajima & Ribbe, 1981; Ylä-Jääski & Nissen, 1983). The observations suggested that vacancies in mullite with diffuse features are disordered, and mullites with sharp satellite reflections are ordered. A model was developed that describes mullite as a mostly disordered structure with a weak occupational modulation, implying a low degree of vacancy ordering (Birkenstock et al., 2015). In contrast, models with an ordered distribution of vacancies were suggested earlier (Saalfeld, 1979; Ylä-Jääski & Nissen, 1983; Kahn-Harari et al., 1991; Klar et al., 2017), but only the most recent of these models was refined against diffraction data, which did not show any signs of diffuse scattering but sharp first-order satellite reflections. The used different scale factors for the main and satellite reflections, assuming that the crystal consists of a component with complete vacancy ordering within a disordered polymorph, and the ordered model only accounts for the ordered component. The degree of ordering and the meaning of the model will be discussed in more detail in Sections 4.1 and 4.2.
The ; Wagner & Schönleber, 2009). A description in often marked the essential breakthrough in understanding a which could not be understood before (Elcoro et al., 2000; Izaola et al., 2007; Pinheiro & Abakumov, 2015). It is therefore surprising that it was not applied to mullite in the late 1980s or 1990s, when the tools to understand the symmetry of modulated structures were already established. In Section 4.1 it will be shown that a meaningful model can be derived from a basic analysis of the symmetry of mullite without a refinement.
approach was developed in the 1970s and established an elegant way to describe structures without lattice periodicity in a higher dimensional space, through which lattice periodicity is restored, and the observed three-dimensional is an aperiodic subspace of This approach was successfully applied to investigate and describe the modulated crystal structures of different classes of material, including minerals and pharmaceuticals (Bindi, 2008In the present study, the crystal structures of several samples are investigated using synchrotron radiation. A disordered et al. (2015) are pointed out and a new approach is used to analyse Al/Si ordering in The results allow us to understand and define the different degrees of ordering in mullite.
model is developed and explained from a crystal-chemical perspective. Similarities to and differences from the model of Birkenstock2. Experimental
2.1. Samples
Samples SA1, SA2 and SA3 originate from a batch of commercial aluminium silicate (Sigma–Aldrich), SA1 being the same crystal that was used for the work by Klar et al. (2017). From the previous study it was known that the composition of these samples corresponds to a vacancy concentration of about 0.4. In order to investigate a range of compositions, another sample labelled QG was prepared by mixing powders of γ-Al2O3 (Alfa Aesar) and amorphous SiO2 (Alfa) in a ratio of 5:2. After pressing the precursors into a pellet, the sample was kept in a flame until melting was observed and then quenched to room temperature. Crystallites that appeared single crystalline and transparent in a visible-light microscope were mounted on a polymer loop attached to a sample-holder pin.
2.2. Synchrotron single-crystal X-ray diffraction
Measurements on SA1, SA2, SA3 and QG were carried out at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France, on beamlines BM01 and ID28. The crystals were rotated about one axis (360° φ scan) and frames were recorded in 0.1° or 1.0° steps with a Dectris Pilatus 2M detector (BM01) or Pilatus 1M detector (ID28), respectively. The frames were treated with the SNBL toolbox (Dyadkin et al., 2016) and the software CrysAlisPro (Rigaku Oxford Diffraction, 2017) was used for further data reduction. Details of the measurements and data reduction are given in Table 1.
3. Results
3.1. Reciprocal-space analysis
The four measurements show the same features in q = [0.2978 (8), 0.0000 (11), 0.5000 (5)]. Besides the sharp satellite reflections, there are diffuse features in all reciprocal-space sections. However, the first-order satellite reflections are much more intense by about two orders of magnitude compared with the diffuse features and second-order satellites. The strongest diffuse features are elongated diffuse discs around strong satellite reflections. A second and a third set of satellite reflections are visible in sections perpendicular to a*, but these are not sharp and not clearly distinguishable from the diffuse scattering pattern in sections perpendicular to c*. This is shown in Fig. 2, where maxima are easily identified within the 2kl plane, but not in the plane, where streaks form an intersecting diamond grid. Close to integer values of l the grid is rather continuous, but with increasing distance from integer values of l the streak pattern dissociates into separate features that transform continuously into the diffuse discs around the first-order satellites for l = n + ½. This indicates a direct relationship between the modulation and the diffuse scattering. In sections perpendicular to b* with integer values of k, weak streaks run from intense first-order satellite reflections towards second-order satellites, i.e. they run approximately parallel to q or mx:q, but wash out before they reach the second-order satellites (Fig. 2). Interestingly, there is no section in which diffuse streaks pass through any main reflection.
namely diffuse scattering and satellite reflections. The intensity of the diffuse scattering is weakly visible as a trace for the measurements on BM01 and is clearly visible in the of SA1 measured on ID28, due to the significantly more brilliant source. Reciprocal-space sections of that measurement therefore show the highest degree of detail and are used for subsequent analysis. First- and second-order satellite reflections were observed with a corresponding modulation wavevectorPbam(a0½)0ss, as in former studies (Birkenstock et al., 2015; Klar et al., 2017); equivalent settings are discussed by Klar et al. (2017). This is used for all subsequent structure model refinements. Many main reflections show a clear splitting, suggesting that the sample is not perfectly single crystalline. In the case of SA1, the intensity ratio of the split reflections is about 4:1. However, this splitting is not observed for satellite reflections independent of the which indicates that the satellites of the two grains exhibit different intensities relative to their main reflections. The measurements on BM01 show a clear splitting of the main and satellite reflections of SA2, whereas SA3 appears to be single crystalline. The main reflections of QG show partial splitting, but the main grain is strongly dominant and no splitting of satellite reflections is detected. The existence of few reflections with no clear relationship to the of mullite indicates the presence of an impurity phase. However, second-order satellite reflections could only be detected in reciprocal-space sections of SA1.
are in agreement with the3.2. of the disordered model in (3+1)-dimensional superspace
The data set obtained on ID28 is not suitable for JANA2006 (Petříček et al., 2014). A crucial aspect of the description of mullite in is the atomic domain of O3, which is the bonding oxygen between two diclusters. As the is 2/m, the symmetry restrictions require that all sine terms of the occupational modulation function of O3 are 0, and thus as a starting model the respective cosine term is set to an arbitrary value ≠ 0. A free of the occupational modulation parameters of Al2 and Si2 was attempted, but the resulting modulation functions were not physically reasonable and different starting parameters led to different models without affecting the R factors. It is evident that the is not sensitive to possible Al/Si ordering, and therefore the parameters of Al2 and Si2 were constrained to be identical excluding the average occupancy. The ratio of the form factors of Si4+/Al3+ ranges between 1 and ∼1.16 and is ∼1.15 for sin(θmax)/λ of the measurements of this study, i.e. the reflections of the highest Bragg angles of this measurement are most sensitive to Al/Si ordering, though the contrast is still very low and apparently not sufficient to refine independent occupational modulation parameters for the respective atomic domains.
because most of the main reflections are overexposed. In the following, the of SA1 based on the measurement on BM01 is described. The was carried out with the softwareThe overall composition is forced to be stoichiometric by applying a set of constraints on the occupancies s of Al2, Si2, Al3 and O4 as a function of the parameter sO3, which is related to the vacancy concentration x = 2/3(1 − sO3). First-order harmonics for the occupational, displacive and anisotropic displacement parameter (ADP) modulation functions were refined. 40 correlations with a coefficient >0.9 between the modulation parameters of O3 and O4 and negative ADP tensors for some coordinates indicate that the model requires more constraints. By setting the U12 modulation parameters of O3 and O4 to 0, only three strong correlations are observed and all parameters converge to physically reasonable values. Although the estimated standard uncertainties of many ADP modulation parameters are rather high, the fact that the ADP modulation does not show any conspicuous behaviour is taken as an indication that the electron density is correctly modelled.
Clear relationships between the occupational modulation functions (OMFs) of Al2/Si2, Al3, O3 and O4 emerge during the shows the of the average structure and the interpretation of the split sites in terms of the presence of diclusters, triclusters and vacancies. A detailed description of the sites and their labels is given in Table 2. Split T and T* sites are modulated in an antiphase relationship to avoid face-sharing tetrahedra. As the T site consists of the atomic domains of Al2 and Si2, their OMF amplitude is half that of Al3 on the T* site. In turn, Al3 and O4 must be modulated in phase and with the same amplitude. The red, blue and yellow clusters (Fig. 3) share the same T sites, i.e. the sum of the OMFs of the oxygen split sites (O3, O4a and O4b) must be the same as the sum of the OMFs of Al2 and Si2 to avoid dangling bonds. Hence the OMF of O3 can be expressed as OMF(O3) = OMF(Al2) + OMF(Si2) − OMF(O4a) − OMF(O4b). The constraints for first-order harmonics are summarized in Table 3 and the general form for nth-order harmonics is given in the supporting information (Table S2). Even without the constraints, the fulfils the derived relationships within 1σ. The amplitude A of the OMF of Al2 is half that of Al3 [AAl2/AAl3 = 0.494 (8)]. As Al2 and Si2 are modulated identically, the overall T site has the same modulation amplitude as Al3 with a relative phase shift of ½.1 Al3 and O4 are modulated in phase with the same amplitude (AAl3/AO4 = 0.99 (8)]. From the equation for the OMF of O3, the amplitude ratio AO4/AO3 = |[1 − 2cos(απ)]−1| ≃ 5.5 is calculated, which agrees with the observed value of AO4/AO3 = 5 (3). The term 2cos(απ), with α being the first component of q, originates from the phase shift between the OMFs of O3 and its split sites O4a and O4b. Despite the large relative uncertainty in the amplitude of the OMF of O3, the is in good agreement with the expected amplitude relationships. Applying the described constraint scheme increases wR(F2) from 0.102 to 0.104, i.e. the model itself is not affected, but there is a significant improvement in the estimated standard uncertainties due to decreased correlations (Table S1). For this reason, the constrained model is preferred over free Selected atom-site parameters of the are given in Table 4 and the resulting model is described in the next section. The refinements from the other three measurements were carried out accordingly and relevant parameters are included in Table 4.
The is stable without further constraints, but the estimated standard uncertainties are greatly reduced by applying constraints on the OMFs, which are derived below. Fig. 3
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3.3. Description of the model
The refinements result in very similar x = 0.428 (4). The occupational modulation as described in the previous section is based on the requirement that the occupancy of the cations on the T site is equal to the occupancy of the central O atoms, i.e. O3, O4a and O4b. The terms 1 − OMF(T) = 1 − OMF(Tr) = 1 − OMF(O3) − OMF(O4a) − OMF(O4b) express the probability that a vacancy is present (Fig. 4). This probability is exactly the same as the occupancy of Al3, O4, Al3r and O4r, which form the tricluster environment of the vacancy. None of the OMFs reaches a value of 0 or 1, and thus diclusters, triclusters and vacancies occupy the space between octahedra with a probability that sums to 1. The probability composition for the clusters marked with black labels in Fig. 3 is included in Fig. 4. Although the occupancy merely describes a probability and the model does not require that two neighbouring sites with the same probability are simultaneously occupied, the strong correlation visible in Fig. 4 suggests that every vacancy is accompanied by two triclusters. This is also the only crystal-chemically reasonable interpretation which avoids dangling bonds and face-sharing tetrahedra. Studies of diffuse scattering confirm this tricluster environment and avoid other cluster assemblies like tetraclusters (Welberry & Butler, 1996).
models and in the following the model of SA1 is described. Important differences between the refinements are pointed out at the end of this section. The site occupancy of O3 converged to 0.357 (7), which corresponds to a vacancy concentration ofMost atoms are displaced less than 0.02 Å due to the Fig. S1). The largest displacement occurs for O1 within the ab plane by about 0.04 Å. Although Al1 and O2 are notably displaced out of the ab plane, the bond length is almost constant for any value of t (Fig. 5). The Al1—O1 distance does not vary much either and thus the volume of the octahedron changes by about ±0.001 Å3 due to modulation. The modulations of the bond lengths of T (i.e. Al2 and Si2) and Al3 are more pronounced. The Al3—O1 distance is correlated with the occupancy of Al3, i.e. O1 is closest to Al3 for t ≃ ¼, when the OMF of Al3 is at its maximum. This results in a distortion of the octahedron modulated by the presence of triclusters, diclusters and vacancies around the octahedron. The T—O3 bond length is probably related to Al/Si ordering on the T site.
(As explained in Section 3.2, the OMFs of Al2 and Si2 could not be refined directly. Here, an approach is presented to derive these modulation functions from the volume of the tetrahedron. In the following, VT,d refers to the average volume of the coordinating tetrahedron of the T site calculated from the coordinates of the average structure. As the T site is bonded to either O3, O4a or O4b, the value of VT,d is the weighted sum of three volumes VT,d,μ with a weighting factor that depends on the occupancy sμ of the respective oxygen site. VT,r refers to the volume calculated from the Al/Si ratio on that site as the sum of reference volumes VT,AlDFT and VT,SiDFT, which are derived from a density functional theory (DFT) calculation, weighted by the occupancy of the cation
The calculation of VT,r requires knowing the values of VT,AlDFT and VT,SiDFT, or alternative reference volumes. As no of mullite is available without Al/Si disordered on the T site, the reference values were taken from a DFT study of an ordered model of 2/1-mullite (Klar, Aretxabaleta et al., 2018), from which values of VT,AlDFT = 2.70 (5) Å3 and VT,SiDFT = 2.26 (4) Å3 were calculated as the averages of the volumes of ten independent AlO4 and six independent SiO4 tetrahedra, respectively.
Using equations (1) and (2) and the defined reference volumes, for SA1 the calculated volumes are VT,d = 2.5503 (19) Å3 and VT,r = 2.54 (4) Å3, i.e. the observed volumes and the volumes expected from the composition are in good agreement. The slight discrepancy can be corrected by increasing VT,AlDFT and VT,SiDFT by one-sixth of the respective This correction is used in the following steps for convenience. If the and the occupational modulation are taken into account, VT,d and VT,r are functions of t. VT,d(t) varies between 2.511 and 2.590 Å3, and VT,r(t) between 2.536 and 2.552 Å3. Both curves are plotted in Fig. 6, which shows that VT,r(t) differs substantially from VT,d(t). This is not surprising, as during the Al and Si were not distinguishable and thus the simplest model had to be used. However, new modulation functions and can be calculated from VT,d(t), so that ≃ VT,d(t)
sT(t) is the occupational modulation function of the T site, i.e. of Al2 + Si2, which is independent of Al/Si ordering. The corrected OMFs, labelled Al2# and Si2#, are shown in Fig. 7. Using these modulation functions in the results in a negligible improvement of wR(F2) by about 0.001. The significant improvement is that the occupational modulation of Al2 and Si2 is now consistent with the observed volume of the tetrahedra, as now overlaps with VT,d(t) in Fig. 6. The present approach and the resulting Al/Si ordering are discussed in Section 4.3.
The compositions of samples SA2, SA3 and QG were refined to x = 0.416 (4), 0.434 (6) and 0.426 (3), respectively. Elemental analyses of other studies of mullite report a higher uncertainty than the estimated by the fitting algorithm, and therefore the uncertainties given here might be underestimated (Birkenstock et al., 2015; Klar et al., 2017). Nevertheless, the present samples all have a comparable composition close to the vacancy concentration of 2/1-mullite, including sample QG, for which the synthesis aimed for a vacancy concentration of 0.5. Apparently the quenching conditions were not appropriate for the synthesis of 5/2-mullite and a less aluminous mullite crystallized, with a composition that is favoured according to the phase diagram (Aksay et al., 1991).
The deviations of the refined coordinates lie within the expected errors. Some ADPs, especially those of O3 and O4, differ more than other parameters, but this can be explained by the correlations between the parameters of O3 and O4 during the θmax)/λ of the measurements. The amplitude of the ADP modulation of the equivalent isotropic displacement parameter Ueq is within the 3σ range of Ueq in most cases. The similarity of the refined parameters of the basic structure confirms a consistent refinement.
and the sin(The modulation functions show the same phase relationships, which in the case of the occupational modulation are fixed by the constraints. There is one remarkable difference between the refinements, which concerns the modulation amplitudes. A comparison shows that the ratio of the amplitudes of the OMFs of the different refinements is about 3.7:2.6:1:1.5 for SA1:SA2:SA3:QG. The corresponding ratios of the significant σ). This significant difference is discussed in Section 4 in terms of different degrees of ordering.
amplitudes are 2.9 (5):1.9 (3):1:1.2 (3) (only those amplitudes were considered where at least one of the function parameters was larger than 2The et al. (2015) used a different constraint scheme, i.e. the atomic domains of Al2, Si2, Al3, O3 and O4 were set to have the same modulation amplitude. This is not reasonable, as Al2 and Si2 are on the same site, resulting in a modulation amplitude of the T site that is twice as large as the modulation amplitude of the other sites. Furthermore, O3 is modulated in phase with Al2 and Si2. This scheme suggests that a vacant O3 site is equivalent to the presence of a vacancy. This is not correct, because a vacancy requires that a total of seven sites are simultaneously vacant. The use of this constraint scheme leads to a strong ADP modulation of the O3 and O4 sites, with negative ADP tensors for some t sections and a strong of O4. The published data set of Birkenstock et al. (2015) was used for a using the constraint scheme of Table 2. The ADP modulations improved, as well as the of O4, indicating that the constraints of the present study lead to a physically correct model. Interestingly, the R factors are hardly affected. The occupational modulation amplitude of O4 and Al3 is 0.0515 (6), which is in between those of SA2 and QG.
model (SSM) published by Birkenstock4. Discussion
4.1. Comparison of ordered and disordered models
The ordered et al. (2017) was based on the sample of SA1, but the resulting models seem to be very different. This section compares the models and points out their differences and similarities, starting with an analysis of the symmetry as both were refined in the Pbam(a0½)0ss.
model presented by KlarThe vacancy distribution pattern can be derived as a function of the modulation wavevector q and the vacancy concentration x for a vacancy described by a block wavefunction centred at (0 ½ ½ 0) with 2/m. The symmetry operators of the a- and b-glide planes establish a phase shift of ½ to a neighbouring vacancy site which is therefore centred at (½ 0 ½ ½). In a similar way, a phase shift of ½ is also established between two sites that are symmetrically related by mz. As a result, two vacancies in physical space are never connected by the vectors 〈½ ½ 0〉, 〈−½ ½ 0〉 or 〈001〉. Crystal-chemical considerations and models of disorder lead to the same basic avoidance pattern (Welberry & Butler, 1996), which is intrinsic to the symmetry of Pbam(a0½)0ss. The thus reserves space for triclusters around vacancies. These phase relationships also hold for harmonic modulation functions, and then an occupancy maximum of one site implies a minimum for the symmetrically related site. Fig. 8 depicts these relationships for the in general (Fig. 8a), the ordered SSM (Fig. 8b) and the disordered SSM (Fig. 8c).
The ordered SSM naturally exhibits the simple vacancy distribution pattern, resulting in two structural block units repeating throughout the structure (Fig. 8b). In vacancy blocks, only vacancies and triclusters occur. In vacancy-free blocks, diclusters are present without any vacancies. This block pattern alternates along the a and c axes and is maintained throughout the structure.
The disordered SSM, based on first-order harmonic functions, resembles the patterns described by the ordered SSM, i.e. respective sites are likely to be vacant if there is a vacancy in the ordered SSM and a vacancy is unlikely to be present at sites where the ordered SSM prohibits a vacancy due to the above-described avoidance pattern. Thus, the vacancy distribution is approximately the superposition of the ordered SSM and an average statistical distribution of vacancies. Further aspects of the O3 domain are discussed in the supporting information in Section S4.
Due to this relationship, it is possible to use the same data sets as were used for the wR(F2) of 0.138, 0.124, 0.149 and 0.113, respectively. The ratio of the scale factor of the satellite to that of the main reflection is a measure of the degree of ordering, like the modulation amplitude discussed in the last section. Comparing these ratios relative to the ratio of SA3 gives 3.53 (4):2.40 (3):1.000 (16):1.475 (17), which is almost equivalent to the relative proportions of the corresponding modulation amplitudes of the disordered SSM in Section 3.3.
of the disordered SSM for a of the ordered SSM, although certain restrictions apply. (i) Main and satellite reflections require separate scale factors and only first-order satellite reflections may be used if only low-order satellites are present. (ii) Some parameters, especially the displacive and ADP modulation of shorter atomic domains like Al3, O3 and O4, are not refined due to instabilities or convergence to suspicious values. With these restrictions, the data sets of SA1, SA2, SA3 and QG give acceptable values forThe ordered model gives a straightforward interpretation of the origin of the modulation and of the dependence of the modulation on the composition, and a clear vacancy distribution pattern. Refinements with different scaling factors were interpreted in terms of ordered domains that are present within the mainly disordered mullite crystal (Klar et al., 2017). However, there are many indications that the observed satellite reflections in the analysed samples are not caused by the presence of ordered domains. calculations based on the DFT model used in Section 3.3 with a limit sin(θmax)/λ = 0.7 indicate that the mean of reflections of different satellite order m are 1, 0.7, 0.5, 0.2 and 0.1 for m = 1, 2, 3, 4 and 5, respectively. Hence, higher-order satellites are slightly less intense than first-order satellite reflections. In the measurement of SA1, first- and second-order satellites are present and the ratio of the mean is 1:0.01. If the sharp satellite reflections in the measurement originate from ordered domains, much stronger second-order satellites must be present. Furthermore, higher orders with m > 2 are not observed at all, even though brilliant synchrotron radiation was used. The of the disordered SSM gives very good R factors and the modulation parameters of all atomic domains are physically meaningful and realistic. In contrast, the of the ordered SSM results in worse R factors and the ADP modulation and indicate problems with the model. Therefore, the disordered SSM is a better model of the electron density in Nevertheless, both models exhibit the same underlying vacancy ordering pattern with the same distribution of maxima in the 4d electron density, due to which the ordered model can be refined with the above-mentioned restrictions. Considering all these aspects, the ordered model is essential for understanding the vacancy ordering pattern in mullite, but the disordered model is the appropriate description of the of mullite.
Precession electron diffraction tomography (PEDT) measurements on a broad range of samples with different compositions, including the commercial aluminium silicate sample, did not reveal ordered domains with high-order satellite reflections, indicating that the disordered SSM is also valid on a local scale. Details of these measurements will be published elsewhere (Klar, Palatinus & Madariaga, 2018).
4.2. Different degrees of ordering
According to the refined SSMs, the different modulation amplitudes are an intrinsic property of the samples. The modulation describes the long-range ordered deviation from the average structure with a periodicity defined by q. Correspondingly, the occupational modulation is the periodic increase and decrease of the occupancy with an amplitude that is larger in the case of SA1 and rather weak in the case of SA3. In other words, vacancies are more ordered in SA1 than in SA3. A direct comparison of the reciprocal-space sections of SA1 and SA3 supports this interpretation, because satellite reflections are clearly identified as reflections in SA1 but are weaker and more diffuse in the case of SA3 (Fig. 9). The presence of second-order satellites in the case of SA1 and their absence in all other samples also supports the result that SA1 is the most ordered sample of this study. Using the modulation amplitudes as a measure of the degree of ordering, QG is more ordered than SA3 and less ordered than SA2. The comparison of the refinements thus shows that mullite exists with different degrees of ordering.
Agrell & Smith (1960) reported on the existence of `S-mullite' with sharp satellite reflections and without diffuse scattering, in contrast with `D-mullite' with diffuse maxima as satellites and the characteristic diffuse scattering. The original samples, where S-mullite was the sample named Forster and D-mullite was sample number 58480 (Agrell & Smith, 1960; Cameron, 1977), were used in an electron diffraction study and, from their h0l sections, it seems that no second-order satellite reflections were present [cf. Figs. 3c and 3d in the work of Cameron (1977), compared with Fig. 9 of this work]. In a recent study, for which SA1 was measured with a laboratory diffractometer, diffuse scattering could not be detected (Klar et al., 2017). In contrast, Fig. 2 clearly shows sharp satellite reflections alongside diffuse scattering and weak second-order satellites. Hence, mullite with low-order satellite reflections always exhibits diffuse scattering, but its visibility in especially small samples may require an X-ray source of higher which is not commonly available in the laboratory. This also explains why Aramaki & Roy (1962) could not find `any intermediate layer lines', even `in grossly overexposed photographs' of a sample with sharp satellite reflections. The distinction into S-mullite and D-mullite should thus be avoided as they both exhibit diffuse features, and are both described by the same structural model with slightly different degrees of ordering.
All the refinements have in common that the occupancies are modulated within a small amplitude range. A physically meaningful model requires the occupancies to be modulated within a range between 0 and 1, which introduces limitations on the allowed amplitudes depending on the order of harmonics that are used to describe the modulation function. For example, the occupancy of the T site is 1 − x/2 and that of the O4 and T* sites is x/2. Therefore, the amplitude of the OMFs using only first-order harmonics is restricted to values ≤x/2. Note that all refinements discussed here are based on first-order harmonics and the amplitudes are smaller than x/2. A stronger modulation requires higher-order harmonics, which corresponds to higher-order satellite reflections. The respective constraints and hypothetical modulation functions of a more ordered mullite with third-order harmonics are presented in Section S3. In the literature, several electron diffraction studies give good examples of samples of different compositions and different orders of satellite reflections ranging from 1 to 7. The diffractograms of Cameron (1977) contain first- (Figs. 3a–3d), second- (Figs. 3e–3f) and fourth-order satellite reflections (Fig. 3g). Sayir & Farmer (1994) and Nakajima et al. (1975) reported on mullites with third-order satellite reflections. The pattern of Nakajima & Ribbe (1981) shows at least seventh-order satellite reflections for a mullite sample with monoclinic modulation wavevector q. Ylä-Jääski & Nissen (1983) reported on mullites with fifth-order satellite reflections and developed a fully ordered model, i.e. with occupancies of either 0 (unoccupied site) or 1 (occupied site), on the basis of images from high-resolution (HRTEM), achieving a good agreement between recorded and simulated images. The vacancy distribution is in full agreement with the symmetry analysis of Section 4.1 and the HRTEM model can be decomposed into vacancy blocks and vacancy-free blocks. All results of the above-mentioned studies can be explained within the picture of a range of ordering introduced in this work, i.e. higher-order satellite reflections correspond to a higher degree of vacancy ordering. On this scale, the samples of this study are only slightly ordered.
The disordered SSM exhibits a periodicity of 1b and 2c and the structure is aperiodic along a, though commensurate models of 2/1-mullite can be approximated with a periodicity of 10a. However, the presence of diffuse scattering indicates that these periodicities are violated by correlated disorder. In reciprocal-space sections perpendicular to c*, the diffuse features are rather localized close to the satellites and expand to a pattern of diamond-shaped streaks in sections close to integer values of l. The vacancy distribution pattern of the disordered SSM, especially the periodicity of 2c, is likely to be preserved to a great extent, but along a and b it apparently deviates due to a variation in composition and ordering patterns. For a highly ordered sample `the pattern of diffuse streaks […] has not been observed in the diffraction experiments' (Ylä-Jääski & Nissen, 1983), indicating that the diffuse scattering depends on the degree of ordering as well. Previous models that explained diffuse scattering did not consider different degrees of ordering (Welberry & Butler, 1996; Rahman et al., 2001). More work is needed to better understand the diffuse scattering, which is currently under investigation.
4.3. Al/Si ordering
A free T site failed, and therefore the respective modulation functions were derived from the modulation of the volume of the tetrahedron. Based on this adapted model, the Al/Si ordering can be analysed. First, the fraction of Al on the T sites is normalized with respect to the occupational modulation of the T site so that the Al fraction and the Si fraction sum up to 100%. Then, the fraction of T sites that are integrated in a tricluster is calculated. If Al–Al diclusters are avoided and Si is preferred in diclusters, the Al/Si ordering can be expressed as the occupancy of Si–Si diclusters, Al–Si diclusters and Al–Al–Al triclusters. For example, at t = 0.25, the T and Tr site are both occupied by about 66% Al and 34% Si, and about 45% of the T sites are integrated in a tricluster. With the above-described assumptions, there is a unique solution for the fraction of T–Tr pairs that form either Si–Si diclusters (13%), Al–Si diclusters (42%) or Al–Al–Al triclusters (45%). For ∼0.5 < t < ∼1, the tricluster fraction is higher than the Al fraction on one of the T sites, which requires the presence of an Si–Al–Al tricluster (Fig. 10). The presence of Si in triclusters was also identified by solid-state NMR measurements (King, 2014). The degree of ordering also affects the amplitudes of Al/Si ordering, so that for SA2, SA3 and QG the presence of Si–Al–Al triclusters is not necessary, but the general ordering pattern is the same for all samples. The phase shift of the modulation functions of subsequent diclusters in the chain of tetrahedra along the c axis is ½. Considering that vacancies are most likely to occur at t = ¼, a likely sequence is, for example, (vacancy, Si–Si, vacancy, …), (vacancy, Al–Al–Al, vacancy, …) or (Al–Si, Al–Al–Al, Al–Si, …). These results are in agreement with the DFT study of ordered mullite, which also confirms the presence of Si in triclusters (Klar, Aretxabaleta et al., 2018).
of the OMFs of Al and Si on theThe identification of Si–Si diclusters is relevant for the mineral classification of mullite, which for silicates is based on the type of network formed by SiO4 tetrahedra. In the current classification of Nickel–Strunz and Dana, the mineral mullite is classified as a nesosilicate with insular SiO4 units (Gaines et al., 1997; Strunz & Nickel, 2001). This study shows, in agreement with DFT studies (Klar, Aretxabaleta et al., 2018), that there are Si2O7 double tetrahedra groups alongside separate SiO4 tetrahedra, which makes mullite a sorosilicate with mixed SiO4 and Si2O7 groups. Therefore, we suggest that mullite be classified in Dana class 58 (instead of 52) and Nickel–Strunz class 09.BF (instead of 09.AF). Furthermore, in the classification of Dana, mullite is in the sillimanite 52.02.02a. Although there is a clear structural relationship with sillimanite, there is a similar relationship with andalusite, which in addition contains voids with a very similar geometry to the vacancy voids in mullite. Therefore, the classification in the sillimanite has an ambiguous character, which would be corrected by the suggested reclassification.
The problem of analysing Al/Si ordering on tetrahedral sites is common for many silicate families, and an equation similar to equation (3) was applied to derive the occupational modulation function from T—O bond lengths in feldspars (Angel et al., 1990; Xu et al., 2016). For both feldspar and mullite, reference values from pure SiO4 and AlO4 are not available from refinements, which is addressed by the approach of this work by deriving them from DFT models. With the of SA1, the resulting curve of VT,r(t) from equation (2) is identical to the observed VT,d(t). However, the average occupancies must be checked carefully. For example, if VT,Al and VT,Si are taken from a of sillimanite [Yang et al. (1997), VT,Al = 2.782 (3) Å3 and VT,Si = 2.192 (3) Å3], the resulting sAl2 = 0.477 (4), which deviates from the expected value of 0.5 by more than 5σ. This discrepancy is a measure of the quality of the reference volumes, and it is clear that the DFT calculation is a better source of these values. The advantage of the approach is that two different methods are combined, which must lead to a self-consistent result. The deviation from the expected occupancy is an immediate indicator to evaluate the result.
5. Summary and conclusions
Synchrotron measurements of 2/1-mullite samples were used to investigate the vacancy and Al/Si ordering of the et al., 2017). Samples SA1, SA2 and SA3 originate from the same commercial batch, and their different degrees of ordering clearly show that the samples are not homogeneous. However, the absence of higher-order satellites and the observed intensity ratio of first-order to second-order satellites in the measurements of SA1 are not in agreement with the presence of a significant number of fully ordered domains. Therefore, the new disordered model of this work is an appropriate model to describe the of mullite samples with low-order satellite reflections, and the degree of ordering is expressed by the amplitude of the occupational modulation functions. Many diffraction patterns in the literature exhibit different degrees of satellite reflections, indicating that mullite may adopt a broad range of degrees of ordering. The crystal-chemical relationships of the disordered model account for harmonic occupational modulation functions of any shape. Thus, the model of this study is the basis for describing mullite with any degree of ordering, and the stronger the ordering the more it resembles the ordered model, which represents the most ordered state.
Diffuse scattering characteristic of mullite was observed, in combination with low-order satellite reflections. A new disordered model was developed based on the of four different samples with compositions close to that of 2/1-mullite. The relationships between the occupational modulation functions of the tetrahedral cations and the O3/O4 split sites are explained by crystal-chemical considerations and symmetry. The refinements exhibit notably different amplitudes of the modulation functions concerning occupational and displacive modulations, which are discussed in the context of different degrees of ordering. analyses based on different scaling factors for main and satellite reflections allow a of the ordered model, with the interpretation that the sample is mostly disordered and few ordered domains are present in which vacancies are completely ordered (KlarDFT or et al., 2005; Chen et al., 2008; Zamani & Behdinan, 2017; Adabifiroozjaei et al., 2018), and DFT calculations use a vacancy distribution that is constrained by size considerations but not by symmetry considerations (Chen et al., 2010; Aryal et al., 2012). The different studies partly avoid Si on the T* site, but Al/Si ordering on the T site is not considered. The ordered model is a symmetry-consistent starting point for DFT calculations, which are limited to supercells with a few hundred atoms. The model is also suitable for MD simulations and disorder can easily be introduced if needed.
(MD) calculations are not possible with partially occupied atoms. Although the symmetry of mullite allows the derivation of a simple vacancy-ordering distribution, recent MD studies use a random distribution of vacancies (LacksThe results of a recent DFT study based on the ordered model were used in this study and proved to be useful to investigate the Al/Si ordering in
A relationship between the presence of Si–Si diclusters and the presence of vacancies could be established from the analysis of the modulated volumes of the tetrahedra. The approach to deriving the Al/Si ordering on tetrahedral sites in combination with DFT calculations is, in principle, applicable to any silicate system. It leads to a consistent description of Al/Si ordering, the observed tetrahedra volumes and the composition constraints. The applicability of the reference volumes derived from DFT is easily assessed by the resulting average occupancy.This work is consistent with, or explains discrepancies of, a number of former studies that applied different methods (Agrell & Smith, 1960; Cameron, 1977; Ylä-Jääski & Nissen, 1983; Kahn-Harari et al., 1991; King, 2014; Klar et al., 2017). In routine work on mullite using powder diffraction, satellite reflections are not observed and therefore not considered (Ren et al., 2018; Figueiredo et al., 2018; Lerdprom et al., 2018; Ripin et al., 2018; Sacks et al., 1991). Sayir & Farmer (1994) reported on a strong variation in the tensile strength of different samples, with second-order satellites either present or absent, indicating that different degrees of ordering may, to some extent, account for the different properties. We therefore suggest the consideration not only of the composition and microstructure of mullite samples, but also of the degree of ordering, to investigate the relationship between the intrinsic structure and physical properties. An NMR study of sintered and fused mullite shows a visible difference between the respective 29Si NMR spectra (Schmücker et al., 2005). Thus, NMR is a promising routine approach if a relationship between the degree of ordering and the NMR signal can be established. The models of this work can help to deconvolute complicated NMR signals of mullite and relate them to superspace.
Supporting information
B-IncStrDB reference: 13462ERV71a
https://doi.org/10.1107/S2052252518007467/gq5008sup1.cif
contains datablocks global, SA1, SA2, SA3, QG. DOI:Structure factors: contains datablock SA1. DOI: https://doi.org/10.1107/S2052252518007467/gq5008SA1sup2.hkl
Structure factors: contains datablock SA2. DOI: https://doi.org/10.1107/S2052252518007467/gq5008SA2sup3.hkl
Structure factors: contains datablock SA3. DOI: https://doi.org/10.1107/S2052252518007467/gq5008SA3sup4.hkl
Structure factors: contains datablock QG. DOI: https://doi.org/10.1107/S2052252518007467/gq5008QGsup5.hkl
Additional discussion, tables and figures. DOI: https://doi.org/10.1107/S2052252518007467/gq5008sup6.pdf
For all structures, data collection: CrysAlis PRO 1.171.38.46 (Rigaku OD, 2015); cell
CrysAlis PRO 1.171.38.46 (Rigaku OD, 2015); data reduction: CrysAlis PRO 1.171.38.46 (Rigaku OD, 2015); program(s) used to refine structure: Jana2006.Al4.857O9.571Si1.143 | F(000) = 156 |
Mr = 316.3 | Dx = 3.133 Mg m−3 |
Orthorhombic, Pbam(α01/2)0ss† | Synchrotron radiation, λ = 0.7231 Å |
q = 0.298800a* + 0.500000c* | Cell parameters from 2027 reflections |
a = 7.5787 (7) Å | θ = 4.1–31.9° |
b = 7.6707 (4) Å | µ = 1.08 mm−1 |
c = 2.8836 (1) Å | T = 293 K |
V = 167.64 (2) Å3 | Irregularly shaped block, colorless |
Z = 1 | 0.06 × 0.05 × 0.03 × 0.03 (radius) mm |
† Symmetry operations: (1) x1, x2, x3, x4; (2) −x1, −x2, x3, x3−x4+1/2; (3) −x1+1/2, x2+1/2, −x3, −x4+1/2; (4) x1+1/2, −x2+1/2, −x3, −x3+x4; (5) −x1, −x2, −x3, −x4; (6) x1, x2, −x3, −x3+x4+1/2; (7) x1+1/2, −x2+1/2, x3, x4+1/2; (8) −x1+1/2, x2+1/2, x3, x3−x4. |
Dectris-CrysAlisPro-abstract goniometer imported dectris images diffractometer | 678 independent reflections |
Radiation source: synchrotron | 555 reflections with I > 3σ(I) |
Synchrotron monochromator | Rint = 0.014 |
Detector resolution: 5.8 pixels mm-1 | θmax = 32.0°, θmin = 3.9° |
ω scans | h = −8→8 |
Absorption correction: multi-scan CrysAlisPro 1.171.38.46 (Rigaku Oxford Diffraction, 2015) Empirical absorption correction using spherical harmonics, implemented in SCALE3 ABSPACK scaling algorithm. | k = −11→11 |
Tmin = 0.834, Tmax = 1 | l = −4→4 |
3363 measured reflections |
Refinement on F2 | 0 restraints |
R[F2 > 2σ(F2)] = 0.037 | 33 constraints |
wR(F2) = 0.104 | Weighting scheme based on measured s.u.'s w = 1/(σ2(I) + 0.0081000002I2) |
S = 1.01 | (Δ/σ)max = 0.0003 |
678 reflections | Δρmax = 0.37 e Å−3 |
101 parameters | Δρmin = −0.33 e Å−3 |
x | y | z | Uiso*/Ueq | Occ. (<1) | |
Al1 | 0 | 0 | 0 | 0.0109 (3) | |
Al2 | 0.14896 (9) | 0.33959 (6) | 0.5 | 0.0121 (3) | 0.5 |
Si2 | 0.14896 (9) | 0.33959 (6) | 0.5 | 0.0121 (3) | 0.286 (2) |
Al3 | 0.2630 (4) | 0.2056 (3) | 0.5 | 0.0112 (7) | 0.214 (2) |
O1 | 0.35894 (15) | 0.42166 (14) | 0.5 | 0.0168 (4) | |
O2 | 0.12752 (15) | 0.21786 (19) | 0 | 0.0175 (4) | |
O3 | 0 | 0.5 | 0.5 | 0.020 (3) | 0.357 (7) |
O4 | 0.449 (2) | 0.0509 (12) | 0.5 | 0.017 (3) | 0.214 (2) |
U11 | U22 | U33 | U12 | U13 | U23 | |
Al1 | 0.0144 (7) | 0.0105 (4) | 0.0078 (4) | 0.00108 (19) | 0 | 0 |
Al2 | 0.0151 (6) | 0.0119 (4) | 0.0092 (4) | −0.00092 (17) | 0 | 0 |
Si2 | 0.0151 (6) | 0.0119 (4) | 0.0092 (4) | −0.00092 (17) | 0 | 0 |
Al3 | 0.0116 (16) | 0.0138 (10) | 0.0082 (8) | 0.0005 (9) | 0 | 0 |
O1 | 0.0215 (9) | 0.0216 (5) | 0.0073 (4) | −0.0051 (4) | 0 | 0 |
O2 | 0.0210 (8) | 0.0169 (6) | 0.0145 (4) | −0.0057 (4) | 0 | 0 |
O3 | 0.019 (7) | 0.017 (6) | 0.025 (3) | 0.000 (3) | 0 | 0 |
O4 | 0.025 (9) | 0.009 (4) | 0.016 (2) | −0.0012 (17) | 0 | 0 |
Average | Minimum | Maximum | |
Al1—O1i | 1.8931 (8) | 1.8858 (8) | 1.9007 (8) |
Al1—O1ii | 1.8931 (8) | 1.8858 (8) | 1.9007 (8) |
Al1—O1iii | 1.8931 (8) | 1.8858 (8) | 1.9007 (8) |
Al1—O1iv | 1.8931 (8) | 1.8858 (8) | 1.9007 (8) |
Al1—O2 | 1.9305 (14) | 1.9304 (14) | 1.9306 (14) |
Al1—O2v | 1.9305 (14) | 1.9304 (14) | 1.9306 (14) |
Al2—O1 | 1.7117 (14) | 1.7027 (15) | 1.7205 (15) |
Al2—O2 | 1.7256 (9) | 1.7144 (9) | 1.7367 (9) |
Al2—O2vi | 1.7256 (9) | 1.7144 (9) | 1.7367 (9) |
Al2—O3 | 1.6703 (19) | 1.635 (3) | 1.705 (3) |
Al2—O4vii | 1.781 (12) | 1.773 (13) | 1.789 (13) |
Al2—O4iv | 1.736 (16) | 1.730 (16) | 1.741 (16) |
Si2—O1 | 1.7117 (14) | 1.7027 (15) | 1.7205 (15) |
Si2—O2 | 1.7256 (9) | 1.7144 (9) | 1.7367 (9) |
Si2—O2vi | 1.7256 (9) | 1.7144 (9) | 1.7367 (9) |
Si2—O3 | 1.6703 (19) | 1.635 (3) | 1.705 (3) |
Si2—O4vii | 1.781 (12) | 1.773 (13) | 1.789 (13) |
Si2—O4iv | 1.736 (16) | 1.730 (16) | 1.741 (16) |
Al3—O1 | 1.810 (3) | 1.786 (3) | 1.834 (3) |
Al3—O2 | 1.7726 (19) | 1.761 (2) | 1.785 (2) |
Al3—O2vi | 1.7726 (19) | 1.761 (2) | 1.785 (2) |
Al3—O4 | 1.841 (15) | 1.827 (16) | 1.855 (16) |
O1i—Al1—O1ii | 99.24 (4) | 99.23 (4) | 99.26 (4) |
O1i—Al1—O1iii | 80.77 (4) | 79.99 (4) | 81.58 (4) |
O1i—Al1—O1iv | 178.83 (6) | 178.14 (5) | 180.0 (5) |
O1i—Al1—O2 | 89.55 (4) | 88.36 (4) | 90.74 (4) |
O1i—Al1—O2v | 90.45 (4) | 89.64 (4) | 91.25 (4) |
O1ii—Al1—O1iii | 178.83 (6) | 178.14 (5) | 180.0 (5) |
O1ii—Al1—O1iv | 80.77 (4) | 79.99 (4) | 81.58 (4) |
O1ii—Al1—O2 | 89.54 (4) | 88.36 (4) | 90.74 (4) |
O1ii—Al1—O2v | 90.46 (4) | 89.64 (4) | 91.25 (4) |
O1iii—Al1—O1iv | 99.24 (4) | 99.23 (4) | 99.26 (4) |
O1iii—Al1—O2 | 90.46 (4) | 89.64 (4) | 91.25 (4) |
O1iii—Al1—O2v | 89.54 (4) | 88.36 (4) | 90.74 (4) |
O1iv—Al1—O2 | 90.45 (4) | 89.64 (4) | 91.25 (4) |
O1iv—Al1—O2v | 89.55 (4) | 88.36 (4) | 90.74 (4) |
O2—Al1—O2v | 179.63 (7) | 179.401 (18) | 180.0 (5) |
O1—Al2—O2 | 106.65 (5) | 105.71 (5) | 107.62 (5) |
O1—Al2—O2vi | 106.65 (5) | 105.71 (5) | 107.62 (5) |
O1—Al2—O3 | 110.95 (12) | 109.21 (12) | 112.69 (13) |
O1—Al2—O4vii | 92.9 (5) | 91.7 (5) | 94.2 (5) |
O1—Al2—O4iv | 129.4 (4) | 127.4 (4) | 131.5 (4) |
O2—Al2—O2vi | 113.37 (6) | 111.91 (6) | 114.80 (6) |
O2—Al2—O3 | 109.57 (7) | 109.45 (7) | 109.68 (7) |
O2—Al2—O4vii | 116.95 (19) | 116.65 (18) | 117.3 (2) |
O2—Al2—O4iv | 100.3 (2) | 100.0 (2) | 100.7 (2) |
O2vi—Al2—O3 | 109.57 (7) | 109.45 (7) | 109.68 (7) |
O2vi—Al2—O4vii | 116.95 (19) | 116.65 (18) | 117.3 (2) |
O2vi—Al2—O4iv | 100.3 (2) | 100.0 (2) | 100.7 (2) |
O3—Al2—O4vii | 18.0 (5) | 17.6 (5) | 18.5 (5) |
O3—Al2—O4iv | 18.5 (4) | 18.1 (4) | 18.9 (4) |
O4vii—Al2—O4iv | 36.5 (7) | 35.7 (7) | 37.3 (7) |
O1—Si2—O2 | 106.65 (5) | 105.71 (5) | 107.62 (5) |
O1—Si2—O2vi | 106.65 (5) | 105.71 (5) | 107.62 (5) |
O1—Si2—O3 | 110.95 (12) | 109.21 (12) | 112.69 (13) |
O1—Si2—O4vii | 92.9 (5) | 91.7 (5) | 94.2 (5) |
O1—Si2—O4iv | 129.4 (4) | 127.4 (4) | 131.5 (4) |
O2—Si2—O2vi | 113.37 (6) | 111.91 (6) | 114.80 (6) |
O2—Si2—O3 | 109.57 (7) | 109.45 (7) | 109.68 (7) |
O2—Si2—O4vii | 116.95 (19) | 116.65 (18) | 117.3 (2) |
O2—Si2—O4iv | 100.3 (2) | 100.0 (2) | 100.7 (2) |
O2vi—Si2—O3 | 109.57 (7) | 109.45 (7) | 109.68 (7) |
O2vi—Si2—O4vii | 116.95 (19) | 116.65 (18) | 117.3 (2) |
O2vi—Si2—O4iv | 100.3 (2) | 100.0 (2) | 100.7 (2) |
O3—Si2—O4vii | 18.0 (5) | 17.6 (5) | 18.5 (5) |
O3—Si2—O4iv | 18.5 (4) | 18.1 (4) | 18.9 (4) |
O4vii—Si2—O4iv | 36.5 (7) | 35.7 (7) | 37.3 (7) |
O1—Al3—O2 | 100.62 (11) | 99.92 (11) | 101.35 (11) |
O1—Al3—O2vi | 100.62 (11) | 99.92 (11) | 101.35 (11) |
O1—Al3—O4 | 106.5 (5) | 105.1 (5) | 107.9 (5) |
O2—Al3—O2vi | 108.88 (17) | 107.66 (17) | 110.07 (17) |
O2—Al3—O4 | 118.47 (19) | 118.36 (18) | 118.6 (2) |
O2vi—Al3—O4 | 118.47 (19) | 118.36 (18) | 118.6 (2) |
Al1viii—O1—Al1vii | 99.21 (6) | 98.39 (6) | 100.06 (6) |
Al1viii—O1—Al2 | 129.91 (3) | 129.45 (3) | 130.38 (3) |
Al1viii—O1—Si2 | 129.91 (3) | 129.45 (3) | 130.38 (3) |
Al1viii—O1—Al3 | 121.17 (6) | 119.57 (6) | 122.76 (6) |
Al1vii—O1—Al2 | 129.91 (3) | 129.45 (3) | 130.38 (3) |
Al1vii—O1—Si2 | 129.91 (3) | 129.45 (3) | 130.38 (3) |
Al1vii—O1—Al3 | 121.17 (6) | 119.57 (6) | 122.76 (6) |
Al2—O1—Si2 | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Al2—O1—Al3 | 44.71 (10) | 44.31 (10) | 45.10 (10) |
Si2—O1—Al3 | 44.71 (10) | 44.31 (10) | 45.10 (10) |
Al1—O2—Al2ix | 121.05 (4) | 119.65 (4) | 122.44 (4) |
Al1—O2—Al2 | 121.03 (4) | 119.65 (4) | 122.44 (4) |
Al1—O2—Si2ix | 121.05 (4) | 119.65 (4) | 122.44 (4) |
Al1—O2—Si2 | 121.03 (4) | 119.65 (4) | 122.44 (4) |
Al1—O2—Al3ix | 104.14 (10) | 103.14 (10) | 105.12 (10) |
Al1—O2—Al3 | 104.12 (10) | 103.14 (10) | 105.12 (10) |
Al2ix—O2—Al2 | 113.36 (8) | 113.35 (8) | 113.36 (8) |
Al2ix—O2—Si2ix | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Al2ix—O2—Si2 | 113.36 (8) | 113.35 (8) | 113.36 (8) |
Al2ix—O2—Al3ix | 45.12 (8) | 44.78 (8) | 45.45 (9) |
Al2ix—O2—Al3 | 130.82 (11) | 130.70 (11) | 130.95 (11) |
Al2—O2—Si2ix | 113.36 (8) | 113.35 (8) | 113.36 (8) |
Al2—O2—Si2 | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Al2—O2—Al3ix | 130.82 (11) | 130.70 (11) | 130.95 (11) |
Al2—O2—Al3 | 45.11 (8) | 44.78 (8) | 45.45 (9) |
Si2ix—O2—Si2 | 113.36 (8) | 113.35 (8) | 113.36 (8) |
Si2ix—O2—Al3ix | 45.12 (8) | 44.78 (8) | 45.45 (9) |
Si2ix—O2—Al3 | 130.82 (11) | 130.70 (11) | 130.95 (11) |
Si2—O2—Al3ix | 130.82 (11) | 130.70 (11) | 130.95 (11) |
Si2—O2—Al3 | 45.11 (8) | 44.78 (8) | 45.45 (9) |
Al3ix—O2—Al3 | 108.87 (13) | 108.86 (13) | 108.87 (13) |
Al2—O3—Al2x | 179.28 (15) | 178.9 (2) | 180.0 (5) |
Al2—O3—Si2 | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Al2—O3—Si2x | 179.28 (15) | 178.9 (2) | 180.0 (5) |
Al2x—O3—Si2 | 179.28 (15) | 178.9 (2) | 180.0 (5) |
Al2x—O3—Si2x | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Si2—O3—Si2x | 179.28 (15) | 178.9 (2) | 180.0 (5) |
Al2ii—O4—Al2xi | 143.5 (9) | 142.3 (9) | 144.7 (9) |
Al2ii—O4—Si2ii | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Al2ii—O4—Si2xi | 143.5 (9) | 142.3 (9) | 144.7 (9) |
Al2ii—O4—Al3 | 105.6 (9) | 104.8 (9) | 106.5 (9) |
Al2xi—O4—Si2ii | 143.5 (9) | 142.3 (9) | 144.7 (9) |
Al2xi—O4—Si2xi | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Al2xi—O4—Al3 | 110.9 (6) | 110.2 (6) | 111.5 (6) |
Si2ii—O4—Si2xi | 143.5 (9) | 142.3 (9) | 144.7 (9) |
Si2ii—O4—Al3 | 105.6 (9) | 104.8 (9) | 106.5 (9) |
Si2xi—O4—Al3 | 110.9 (6) | 110.2 (6) | 111.5 (6) |
Symmetry codes: (i) −x1+1/2, x2−1/2, −x3, −x4+1/2; (ii) −x1+1/2, x2−1/2, −x3+1, −x4+1/2; (iii) x1−1/2, −x2+1/2, −x3, −x3+x4; (iv) x1−1/2, −x2+1/2, −x3+1, −x3+x4; (v) −x1, −x2, x3, x3−x4+1/2; (vi) x1, x2, x3+1, x4; (vii) −x1+1/2, x2+1/2, −x3+1, −x4+1/2; (viii) −x1+1/2, x2+1/2, −x3, −x4+1/2; (ix) x1, x2, x3−1, x4; (x) −x1, −x2+1, x3, x3−x4+1/2; (xi) x1+1/2, −x2+1/2, −x3+1, −x3+x4. |
Al4.832O9.584Si1.168 | F(000) = 156 |
Mr = 316.5 | Dx = 3.139 Mg m−3 |
Orthorhombic, Pbam(α01/2)0ss† | Synchrotron radiation, λ = 0.7231 Å |
q = 0.301000a* + 0.500000c* | Cell parameters from 1568 reflections |
a = 7.577 (2) Å | θ = 4.1–32.0° |
b = 7.6727 (18) Å | µ = 1.08 mm−1 |
c = 2.8804 (10) Å | T = 293 K |
V = 167.46 (8) Å3 | Irregularly shaped block |
Z = 1 |
† Symmetry operations: (1) x1, x2, x3, x4; (2) −x1, −x2, x3, x3−x4+1/2; (3) −x1+1/2, x2+1/2, −x3, −x4+1/2; (4) x1+1/2, −x2+1/2, −x3, −x3+x4; (5) −x1, −x2, −x3, −x4; (6) x1, x2, −x3, −x3+x4+1/2; (7) x1+1/2, −x2+1/2, x3, x4+1/2; (8) −x1+1/2, x2+1/2, x3, x3−x4. |
Dectris-CrysAlisPro-abstract goniometer imported dectris images diffractometer | 768 independent reflections |
Radiation source: synchrotron | 544 reflections with I > 3σ(I) |
Synchrotron monochromator | Rint = 0.020 |
ω scans | θmax = 31.9°, θmin = 4.1° |
Absorption correction: multi-scan CrysAlisPro 1.171.38.46 (Rigaku Oxford Diffraction, 2015) Empirical absorption correction using spherical harmonics, implemented in SCALE3 ABSPACK scaling algorithm. | h = −10→10 |
Tmin = 0.761, Tmax = 1 | k = −10→10 |
3079 measured reflections | l = −4→4 |
Refinement on F2 | 0 restraints |
R[F2 > 2σ(F2)] = 0.031 | 33 constraints |
wR(F2) = 0.089 | Weighting scheme based on measured s.u.'s w = 1/(σ2(I) + 0.0035999999I2) |
S = 1.08 | (Δ/σ)max = 0.0004 |
768 reflections | Δρmax = 0.35 e Å−3 |
101 parameters | Δρmin = −0.39 e Å−3 |
x | y | z | Uiso*/Ueq | Occ. (<1) | |
Al1 | 0 | 0 | 0 | 0.00863 (17) | |
Al2 | 0.14885 (5) | 0.33985 (4) | 0.5 | 0.00933 (17) | 0.5 |
Si2 | 0.14885 (5) | 0.33985 (4) | 0.5 | 0.00933 (17) | 0.2920 (19) |
Al3 | 0.2626 (2) | 0.2043 (2) | 0.5 | 0.0086 (4) | 0.2080 (19) |
O1 | 0.35912 (10) | 0.42179 (11) | 0.5 | 0.0151 (2) | |
O2 | 0.12766 (11) | 0.21705 (13) | 0 | 0.0161 (2) | |
O3 | 0 | 0.5 | 0.5 | 0.0196 (19) | 0.376 (6) |
O4 | 0.4483 (9) | 0.0466 (9) | 0.5 | 0.0114 (17) | 0.2080 (19) |
U11 | U22 | U33 | U12 | U13 | U23 | |
Al1 | 0.0085 (3) | 0.0089 (3) | 0.0085 (3) | 0.00056 (12) | 0 | 0 |
Al2 | 0.0070 (3) | 0.0107 (3) | 0.0102 (3) | −0.00055 (10) | 0 | 0 |
Si2 | 0.0070 (3) | 0.0107 (3) | 0.0102 (3) | −0.00055 (10) | 0 | 0 |
Al3 | 0.0066 (7) | 0.0109 (8) | 0.0084 (7) | 0.0005 (5) | 0 | 0 |
O1 | 0.0147 (4) | 0.0221 (4) | 0.0086 (4) | −0.0079 (3) | 0 | 0 |
O2 | 0.0159 (4) | 0.0173 (4) | 0.0150 (4) | −0.0072 (3) | 0 | 0 |
O3 | 0.008 (4) | 0.019 (4) | 0.0317 (18) | 0.0023 (18) | 0 | 0 |
O4 | 0.005 (4) | 0.011 (3) | 0.0182 (18) | 0.0004 (12) | 0 | 0 |
Average | Minimum | Maximum | |
Al1—Al1i | 2.88040 (12) | 2.8628 (2) | 2.8980 (2) |
Al1—Al1ii | 2.88040 (12) | 2.8628 (2) | 2.8980 (2) |
Al1—Al3i | 2.9137 (18) | 2.9049 (18) | 2.9224 (18) |
Al1—Al3 | 2.9135 (18) | 2.9049 (18) | 2.9224 (18) |
Al1—Al3iii | 2.9135 (18) | 2.9049 (18) | 2.9224 (18) |
Al1—Al3iv | 2.9137 (18) | 2.9049 (18) | 2.9224 (18) |
Al2—Al2i | 2.880436 (3) | 2.880405 (4) | 2.880466 (4) |
Al2—Al2ii | 2.880436 (3) | 2.880405 (4) | 2.880466 (4) |
Al2—Si2 | 0 | 0 | 0 |
Al2—Al3 | 1.3510 (18) | 1.3496 (18) | 1.3525 (18) |
Al2—Al3v | 2.8754 (17) | 2.8716 (18) | 2.8790 (18) |
Al2—Al3vi | 2.946 (3) | 2.935 (3) | 2.957 (3) |
Al2—O3 | 1.6681 (15) | 1.648 (3) | 1.687 (3) |
Si2—Al3 | 1.3510 (18) | 1.3496 (18) | 1.3525 (18) |
Si2—O3 | 1.6681 (15) | 1.648 (3) | 1.687 (3) |
Al3—Al3i | 2.880430 (11) | 2.880410 (14) | 2.880448 (14) |
Al3—Al3ii | 2.880430 (11) | 2.880410 (14) | 2.880448 (14) |
O3—O4v | 0.530 (8) | 0.521 (9) | 0.539 (9) |
O3—O4vi | 0.530 (8) | 0.521 (9) | 0.539 (9) |
O4—O4vii | 1.060 (11) | 1.051 (12) | 1.069 (12) |
Al1i—Al1—Al1ii | 180.0 (5) | 180.0 (5) | 180.0 (5) |
Al1i—Al1—Al3i | 60.38 (2) | 60.21 (2) | 60.54 (2) |
Al1i—Al1—Al3 | 119.62 (2) | 119.46 (2) | 119.79 (2) |
Al1i—Al1—Al3iii | 60.38 (2) | 60.21 (2) | 60.54 (2) |
Al1i—Al1—Al3iv | 119.62 (2) | 119.46 (2) | 119.79 (2) |
Al1ii—Al1—Al3i | 119.62 (2) | 119.46 (2) | 119.79 (2) |
Al1ii—Al1—Al3 | 60.38 (2) | 60.21 (2) | 60.54 (2) |
Al1ii—Al1—Al3iii | 119.62 (2) | 119.46 (2) | 119.79 (2) |
Al1ii—Al1—Al3iv | 60.38 (2) | 60.21 (2) | 60.54 (2) |
Al3i—Al1—Al3 | 59.25 (3) | 59.25 (3) | 59.25 (3) |
Al3i—Al1—Al3iii | 120.75 (3) | 120.47 (3) | 121.04 (3) |
Al3i—Al1—Al3iv | 179.80 (12) | 179.67 (3) | 180.0 (5) |
Al3—Al1—Al3iii | 179.80 (12) | 179.67 (3) | 180.0 (5) |
Al3—Al1—Al3iv | 120.75 (3) | 120.47 (3) | 121.04 (3) |
Al3iii—Al1—Al3iv | 59.25 (3) | 59.25 (3) | 59.25 (3) |
Al2i—Al2—Al2ii | 179.461 (19) | 179.223 (19) | 179.784 (19) |
Al2i—Al2—Si2 | 0 | 0 | 0 |
Al2i—Al2—Al3 | 90.000 (11) | 89.881 (11) | 90.119 (11) |
Al2i—Al2—Al3v | 89.996 (10) | 89.656 (10) | 90.343 (10) |
Al2i—Al2—Al3vi | 90.003 (12) | 89.698 (12) | 90.302 (12) |
Al2i—Al2—O3 | 89.999 (11) | 89.888 (11) | 90.112 (11) |
Al2ii—Al2—Si2 | 0 | 0 | 0 |
Al2ii—Al2—Al3 | 90.000 (11) | 89.881 (11) | 90.119 (11) |
Al2ii—Al2—Al3v | 89.996 (10) | 89.656 (10) | 90.343 (10) |
Al2ii—Al2—Al3vi | 90.003 (12) | 89.698 (12) | 90.302 (12) |
Al2ii—Al2—O3 | 89.999 (11) | 89.888 (11) | 90.112 (11) |
Si2—Al2—Al3 | 0 | 0 | 0 |
Si2—Al2—Al3v | 0 | 0 | 0 |
Si2—Al2—Al3vi | 0 | 0 | 0 |
Si2—Al2—O3 | 0 | 0 | 0 |
Al3—Al2—Al3v | 126.87 (9) | 126.61 (9) | 127.13 (9) |
Al3—Al2—Al3vi | 123.04 (9) | 122.90 (8) | 123.19 (8) |
Al3—Al2—O3 | 177.09 (12) | 176.71 (12) | 177.46 (12) |
Al3v—Al2—Al3vi | 110.09 (5) | 109.91 (5) | 110.26 (5) |
Al3v—Al2—O3 | 56.04 (9) | 55.89 (9) | 56.20 (9) |
Al3vi—Al2—O3 | 54.05 (9) | 53.72 (9) | 54.37 (9) |
Al2—Si2—Al3 | 0 | 0 | 0 |
Al2—Si2—O3 | 0 | 0 | 0 |
Al3—Si2—O3 | 177.09 (12) | 176.71 (12) | 177.46 (12) |
Al1—Al3—Al1ii | 59.25 (4) | 58.92 (4) | 59.58 (4) |
Al1—Al3—Al2 | 88.77 (8) | 88.67 (8) | 88.88 (8) |
Al1—Al3—Al2viii | 68.68 (4) | 68.47 (4) | 68.88 (4) |
Al1—Al3—Al2ix | 128.07 (5) | 127.98 (5) | 128.16 (5) |
Al1—Al3—Si2 | 88.77 (8) | 88.67 (8) | 88.88 (8) |
Al1—Al3—Al3i | 60.38 (4) | 60.07 (4) | 60.68 (4) |
Al1—Al3—Al3ii | 119.63 (6) | 119.20 (6) | 120.05 (6) |
Al1ii—Al3—Al2 | 88.77 (8) | 88.67 (8) | 88.88 (8) |
Al1ii—Al3—Al2viii | 68.68 (4) | 68.47 (4) | 68.88 (4) |
Al1ii—Al3—Al2ix | 128.07 (5) | 127.98 (5) | 128.16 (5) |
Al1ii—Al3—Si2 | 88.77 (8) | 88.67 (8) | 88.88 (8) |
Al1ii—Al3—Al3i | 119.63 (6) | 119.20 (6) | 120.05 (6) |
Al1ii—Al3—Al3ii | 60.38 (4) | 60.07 (4) | 60.68 (4) |
Al2—Al3—Al2viii | 153.86 (11) | 153.71 (11) | 154.00 (11) |
Al2—Al3—Al2ix | 136.23 (10) | 136.14 (10) | 136.33 (10) |
Al2—Al3—Si2 | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Al2—Al3—Al3i | 90.00 (5) | 89.85 (5) | 90.15 (5) |
Al2—Al3—Al3ii | 90.00 (5) | 89.85 (5) | 90.15 (5) |
Al2viii—Al3—Al2ix | 69.91 (4) | 69.74 (4) | 70.08 (4) |
Al2viii—Al3—Si2 | 153.86 (11) | 153.71 (11) | 154.00 (11) |
Al2viii—Al3—Al3i | 90.00 (5) | 89.79 (5) | 90.21 (5) |
Al2viii—Al3—Al3ii | 90.00 (5) | 89.79 (5) | 90.21 (5) |
Al2ix—Al3—Si2 | 136.23 (10) | 136.14 (10) | 136.33 (10) |
Al2ix—Al3—Al3i | 90.00 (5) | 89.75 (5) | 90.25 (5) |
Al2ix—Al3—Al3ii | 90.00 (5) | 89.75 (5) | 90.25 (5) |
Si2—Al3—Al3i | 90.00 (5) | 89.85 (5) | 90.15 (5) |
Si2—Al3—Al3ii | 90.00 (5) | 89.85 (5) | 90.15 (5) |
Al3i—Al3—Al3ii | 179.50 (9) | 179.34 (8) | 179.69 (9) |
Al2—O3—Al2x | 179.60 (14) | 179.37 (17) | 180.0 (5) |
Al2—O3—Si2 | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Al2—O3—Si2x | 179.60 (14) | 179.37 (17) | 180.0 (5) |
Al2—O3—O4v | 89.8 (8) | 87.1 (9) | 92.5 (8) |
Al2—O3—O4vi | 90.2 (8) | 86.9 (9) | 93.5 (9) |
Al2x—O3—Si2 | 179.60 (14) | 179.37 (17) | 180.0 (5) |
Al2x—O3—Si2x | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Al2x—O3—O4v | 90.2 (8) | 86.9 (9) | 93.5 (9) |
Al2x—O3—O4vi | 89.8 (8) | 87.1 (9) | 92.5 (8) |
Si2—O3—Si2x | 179.60 (14) | 179.37 (17) | 180.0 (5) |
Si2—O3—O4v | 89.8 (8) | 87.1 (9) | 92.5 (8) |
Si2—O3—O4vi | 90.2 (8) | 86.9 (9) | 93.5 (9) |
Si2x—O3—O4v | 90.2 (8) | 86.9 (9) | 93.5 (9) |
Si2x—O3—O4vi | 89.8 (8) | 87.1 (9) | 92.5 (8) |
O4v—O3—O4vi | 176.5 (12) | 174.6 (13) | 180.0 (5) |
O3viii—O4—O4vii | 1.7 (6) | 0.0 (5) | 2.8 (7) |
Symmetry codes: (i) x1, x2, x3−1, x4; (ii) x1, x2, x3+1, x4; (iii) −x1, −x2, x3−1, x3−x4+1/2; (iv) −x1, −x2, x3, x3−x4+1/2; (v) −x1+1/2, x2+1/2, −x3+1, −x4+1/2; (vi) x1−1/2, −x2+1/2, −x3+1, −x3+x4; (vii) −x1+1, −x2, x3, x3−x4+1/2; (viii) −x1+1/2, x2−1/2, −x3+1, −x4+1/2; (ix) x1+1/2, −x2+1/2, −x3+1, −x3+x4; (x) −x1, −x2+1, x3, x3−x4+1/2. |
Al4.868O9.566Si1.132 | F(000) = 156 |
Mr = 316.2 | Dx = 3.131 Mg m−3 |
Orthorhombic, Pbam(α01/2)0ss† | Synchrotron radiation, λ = 0.7231 Å |
q = 0.306800a* + 0.500000c* | Cell parameters from 982 reflections |
a = 7.5768 (13) Å | θ = 4.1–31.9° |
b = 7.6760 (16) Å | µ = 1.08 mm−1 |
c = 2.8833 (12) Å | T = 293 K |
V = 167.69 (8) Å3 | Irregularly shaped block |
Z = 1 |
† Symmetry operations: (1) x1, x2, x3, x4; (2) −x1, −x2, x3, x3−x4+1/2; (3) −x1+1/2, x2+1/2, −x3, −x4+1/2; (4) x1+1/2, −x2+1/2, −x3, −x3+x4; (5) −x1, −x2, −x3, −x4; (6) x1, x2, −x3, −x3+x4+1/2; (7) x1+1/2, −x2+1/2, x3, x4+1/2; (8) −x1+1/2, x2+1/2, x3, x3−x4. |
Dectris-CrysAlisPro-abstract goniometer imported dectris images diffractometer | 667 independent reflections |
Radiation source: synchrotron | 382 reflections with I > 3σ(I) |
Synchrotron monochromator | Rint = 0.030 |
ω scans | θmax = 31.6°, θmin = 3.9° |
Absorption correction: multi-scan CrysAlisPro 1.171.38.46 (Rigaku Oxford Diffraction, 2015) Empirical absorption correction using spherical harmonics, implemented in SCALE3 ABSPACK scaling algorithm. | h = −11→11 |
Tmin = 0.585, Tmax = 1 | k = −10→10 |
3299 measured reflections | l = −3→3 |
Refinement on F2 | 0 restraints |
R[F2 > 2σ(F2)] = 0.037 | 33 constraints |
wR(F2) = 0.111 | Weighting scheme based on measured s.u.'s w = 1/(σ2(I) + 0.0025000002I2) |
S = 1.44 | (Δ/σ)max = 0.0001 |
667 reflections | Δρmax = 0.43 e Å−3 |
101 parameters | Δρmin = −0.46 e Å−3 |
x | y | z | Uiso*/Ueq | Occ. (<1) | |
Al1 | 0 | 0 | 0 | 0.0089 (3) | |
Al2 | 0.14905 (5) | 0.34005 (5) | 0.5 | 0.0096 (3) | 0.5 |
Si2 | 0.14905 (5) | 0.34005 (5) | 0.5 | 0.0096 (3) | 0.283 (3) |
Al3 | 0.2624 (2) | 0.2058 (2) | 0.5 | 0.0105 (6) | 0.217 (3) |
O1 | 0.35841 (11) | 0.42165 (16) | 0.5 | 0.0166 (4) | |
O2 | 0.12754 (11) | 0.21837 (15) | 0 | 0.0173 (4) | |
O3 | 0 | 0.5 | 0.5 | 0.017 (3) | 0.349 (8) |
O4 | 0.4534 (10) | 0.0482 (9) | 0.5 | 0.0120 (19) | 0.217 (3) |
U11 | U22 | U33 | U12 | U13 | U23 | |
Al1 | 0.0128 (3) | 0.0056 (3) | 0.0082 (7) | 0.00085 (13) | 0 | 0 |
Al2 | 0.0117 (4) | 0.0073 (3) | 0.0099 (8) | −0.00061 (13) | 0 | 0 |
Si2 | 0.0117 (4) | 0.0073 (3) | 0.0099 (8) | −0.00061 (13) | 0 | 0 |
Al3 | 0.0131 (7) | 0.0114 (8) | 0.0069 (16) | 0.0008 (6) | 0 | 0 |
O1 | 0.0200 (5) | 0.0196 (5) | 0.0101 (10) | −0.0087 (3) | 0 | 0 |
O2 | 0.0211 (4) | 0.0125 (5) | 0.0183 (9) | −0.0060 (3) | 0 | 0 |
O3 | 0.019 (5) | 0.008 (4) | 0.023 (4) | −0.006 (2) | 0 | 0 |
O4 | 0.011 (4) | 0.004 (3) | 0.020 (3) | −0.0017 (17) | 0 | 0 |
Average | Minimum | Maximum | |
Al1—Al1i | 2.88330 (17) | 2.8753 (3) | 2.8913 (3) |
Al1—Al1ii | 2.88330 (17) | 2.8753 (3) | 2.8913 (3) |
Al1—Al3i | 2.9198 (19) | 2.9170 (19) | 2.9225 (19) |
Al1—Al3 | 2.9197 (19) | 2.9170 (19) | 2.9225 (19) |
Al1—Al3iii | 2.9197 (19) | 2.9170 (19) | 2.9225 (19) |
Al1—Al3iv | 2.9198 (19) | 2.9170 (19) | 2.9225 (19) |
Al2—Al2i | 2.8833098 (16) | 2.883301 (2) | 2.883318 (2) |
Al2—Al2ii | 2.8833098 (16) | 2.883301 (2) | 2.883318 (2) |
Al2—Si2 | 0 | 0 | 0 |
Al2—Al3 | 1.342 (2) | 1.341 (2) | 1.343 (2) |
Al2—Al3v | 2.8863 (18) | 2.8834 (18) | 2.8891 (18) |
Al2—Al3vi | 2.951 (3) | 2.947 (3) | 2.954 (3) |
Al2—O3 | 1.6683 (18) | 1.653 (3) | 1.683 (3) |
Si2—Al3 | 1.342 (2) | 1.341 (2) | 1.343 (2) |
Si2—O3 | 1.6683 (18) | 1.653 (3) | 1.683 (3) |
Al3—Al3i | 2.883308 (6) | 2.883302 (9) | 2.883314 (9) |
Al3—Al3ii | 2.883308 (6) | 2.883302 (9) | 2.883314 (9) |
O3—O4v | 0.512 (8) | 0.510 (8) | 0.514 (8) |
O3—O4vi | 0.512 (8) | 0.510 (8) | 0.514 (8) |
O4—O4vii | 1.024 (11) | 1.022 (12) | 1.025 (12) |
Al1i—Al1—Al1ii | 180.0 (5) | 180.0 (5) | 180.0 (5) |
Al1i—Al1—Al3i | 60.41 (2) | 60.31 (2) | 60.52 (2) |
Al1i—Al1—Al3 | 119.59 (2) | 119.48 (2) | 119.69 (2) |
Al1i—Al1—Al3iii | 60.41 (2) | 60.31 (2) | 60.52 (2) |
Al1i—Al1—Al3iv | 119.59 (2) | 119.48 (2) | 119.69 (2) |
Al1ii—Al1—Al3i | 119.59 (2) | 119.48 (2) | 119.69 (2) |
Al1ii—Al1—Al3 | 60.41 (2) | 60.31 (2) | 60.52 (2) |
Al1ii—Al1—Al3iii | 119.59 (2) | 119.48 (2) | 119.69 (2) |
Al1ii—Al1—Al3iv | 60.41 (2) | 60.31 (2) | 60.52 (2) |
Al3i—Al1—Al3 | 59.18 (3) | 59.18 (3) | 59.18 (3) |
Al3i—Al1—Al3iii | 120.82 (3) | 120.62 (4) | 121.02 (3) |
Al3i—Al1—Al3iv | 179.88 (16) | 179.79 (3) | 180.0 (5) |
Al3—Al1—Al3iii | 179.88 (16) | 179.79 (3) | 180.0 (5) |
Al3—Al1—Al3iv | 120.82 (3) | 120.62 (4) | 121.02 (3) |
Al3iii—Al1—Al3iv | 59.18 (3) | 59.18 (3) | 59.18 (3) |
Al2i—Al2—Al2ii | 179.72 (2) | 179.60 (2) | 179.90 (2) |
Al2i—Al2—Si2 | 0 | 0 | 0 |
Al2i—Al2—Al3 | 90.000 (13) | 89.947 (13) | 90.052 (13) |
Al2i—Al2—Al3v | 89.998 (12) | 89.828 (12) | 90.172 (12) |
Al2i—Al2—Al3vi | 90.002 (14) | 89.837 (13) | 90.163 (13) |
Al2i—Al2—O3 | 90.000 (13) | 89.949 (13) | 90.051 (13) |
Al2ii—Al2—Si2 | 0 | 0 | 0 |
Al2ii—Al2—Al3 | 90.000 (13) | 89.947 (13) | 90.052 (13) |
Al2ii—Al2—Al3v | 89.998 (12) | 89.828 (12) | 90.172 (12) |
Al2ii—Al2—Al3vi | 90.002 (14) | 89.837 (13) | 90.163 (13) |
Al2ii—Al2—O3 | 90.000 (13) | 89.949 (13) | 90.051 (13) |
Si2—Al2—Al3 | 0 | 0 | 0 |
Si2—Al2—Al3v | 0 | 0 | 0 |
Si2—Al2—Al3vi | 0 | 0 | 0 |
Si2—Al2—O3 | 0 | 0 | 0 |
Al3—Al2—Al3v | 126.76 (10) | 126.50 (10) | 127.03 (10) |
Al3—Al2—Al3vi | 122.95 (9) | 122.65 (9) | 123.25 (9) |
Al3—Al2—O3 | 177.19 (14) | 176.90 (14) | 177.46 (14) |
Al3v—Al2—Al3vi | 110.29 (6) | 110.26 (5) | 110.32 (5) |
Al3v—Al2—O3 | 56.05 (12) | 55.97 (12) | 56.14 (12) |
Al3vi—Al2—O3 | 54.24 (12) | 54.15 (12) | 54.32 (12) |
Al2—Si2—Al3 | 0 | 0 | 0 |
Al2—Si2—O3 | 0 | 0 | 0 |
Al3—Si2—O3 | 177.19 (14) | 176.90 (14) | 177.46 (14) |
Al1—Al3—Al1ii | 59.18 (4) | 58.97 (4) | 59.38 (4) |
Al1—Al3—Al2 | 88.85 (9) | 88.64 (9) | 89.05 (9) |
Al1—Al3—Al2viii | 68.42 (4) | 68.33 (4) | 68.51 (4) |
Al1—Al3—Al2ix | 127.70 (5) | 127.65 (5) | 127.75 (5) |
Al1—Al3—Si2 | 88.85 (9) | 88.64 (9) | 89.05 (9) |
Al1—Al3—Al3i | 60.41 (4) | 60.32 (4) | 60.51 (4) |
Al1—Al3—Al3ii | 119.59 (7) | 119.31 (7) | 119.86 (7) |
Al1ii—Al3—Al2 | 88.85 (9) | 88.64 (9) | 89.05 (9) |
Al1ii—Al3—Al2viii | 68.42 (4) | 68.33 (4) | 68.51 (4) |
Al1ii—Al3—Al2ix | 127.70 (5) | 127.65 (5) | 127.75 (5) |
Al1ii—Al3—Si2 | 88.85 (9) | 88.64 (9) | 89.05 (9) |
Al1ii—Al3—Al3i | 119.59 (7) | 119.31 (7) | 119.86 (7) |
Al1ii—Al3—Al3ii | 60.41 (4) | 60.32 (4) | 60.51 (4) |
Al2—Al3—Al2viii | 153.65 (12) | 153.45 (12) | 153.85 (12) |
Al2—Al3—Al2ix | 136.64 (11) | 136.37 (11) | 136.91 (11) |
Al2—Al3—Si2 | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Al2—Al3—Al3i | 90.00 (5) | 89.93 (5) | 90.07 (5) |
Al2—Al3—Al3ii | 90.00 (5) | 89.93 (5) | 90.07 (5) |
Al2viii—Al3—Al2ix | 69.71 (5) | 69.64 (5) | 69.78 (5) |
Al2viii—Al3—Si2 | 153.65 (12) | 153.45 (12) | 153.85 (12) |
Al2viii—Al3—Al3i | 90.00 (5) | 89.91 (5) | 90.09 (5) |
Al2viii—Al3—Al3ii | 90.00 (5) | 89.91 (5) | 90.09 (5) |
Al2ix—Al3—Si2 | 136.64 (11) | 136.37 (11) | 136.91 (11) |
Al2ix—Al3—Al3i | 90.00 (6) | 89.86 (6) | 90.14 (6) |
Al2ix—Al3—Al3ii | 90.00 (6) | 89.86 (6) | 90.14 (6) |
Si2—Al3—Al3i | 90.00 (5) | 89.93 (5) | 90.07 (5) |
Si2—Al3—Al3ii | 90.00 (5) | 89.93 (5) | 90.07 (5) |
Al3i—Al3—Al3ii | 179.74 (9) | 179.65 (10) | 179.85 (9) |
Al2—O3—Al2x | 179.9 (2) | 179.8 (2) | 180.0 (5) |
Al2—O3—Si2 | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Al2—O3—Si2x | 179.9 (2) | 179.8 (2) | 180.0 (5) |
Al2—O3—O4v | 93.7 (9) | 91.7 (9) | 95.7 (9) |
Al2—O3—O4vi | 86.3 (9) | 84.1 (9) | 88.4 (9) |
Al2x—O3—Si2 | 179.9 (2) | 179.8 (2) | 180.0 (5) |
Al2x—O3—Si2x | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Al2x—O3—O4v | 86.3 (9) | 84.1 (9) | 88.4 (9) |
Al2x—O3—O4vi | 93.7 (9) | 91.7 (9) | 95.7 (9) |
Si2—O3—Si2x | 179.9 (2) | 179.8 (2) | 180.0 (5) |
Si2—O3—O4v | 93.7 (9) | 91.7 (9) | 95.7 (9) |
Si2—O3—O4vi | 86.3 (9) | 84.1 (9) | 88.4 (9) |
Si2x—O3—O4v | 86.3 (9) | 84.1 (9) | 88.4 (9) |
Si2x—O3—O4vi | 93.7 (9) | 91.7 (9) | 95.7 (9) |
O4v—O3—O4vi | 177.6 (13) | 176.3 (14) | 180.0 (5) |
O3viii—O4—O4vii | 1.2 (7) | 0.0 (5) | 1.9 (7) |
Symmetry codes: (i) x1, x2, x3−1, x4; (ii) x1, x2, x3+1, x4; (iii) −x1, −x2, x3−1, x3−x4+1/2; (iv) −x1, −x2, x3, x3−x4+1/2; (v) −x1+1/2, x2+1/2, −x3+1, −x4+1/2; (vi) x1−1/2, −x2+1/2, −x3+1, −x3+x4; (vii) −x1+1, −x2, x3, x3−x4+1/2; (viii) −x1+1/2, x2−1/2, −x3+1, −x4+1/2; (ix) x1+1/2, −x2+1/2, −x3+1, −x3+x4; (x) −x1, −x2+1, x3, x3−x4+1/2. |
Al4.853O9.574Si1.147 | F(000) = 156 |
Mr = 316.3 | Dx = 3.134 Mg m−3 |
Orthorhombic, Pbam(α01/2)0ss† | Synchrotron radiation, λ = 0.7231 Å |
q = 0.294800a* + 0.500000c* | Cell parameters from 1039 reflections |
a = 7.577 (2) Å | θ = 4.1–31.8° |
b = 7.6738 (19) Å | µ = 1.08 mm−1 |
c = 2.8823 (10) Å | T = 293 K |
V = 167.59 (8) Å3 | Irregularly shaped block, colorless |
Z = 1 | 0.05 mm (radius) |
† Symmetry operations: (1) x1, x2, x3, x4; (2) −x1, −x2, x3, x3−x4+1/2; (3) −x1+1/2, x2+1/2, −x3, −x4+1/2; (4) x1+1/2, −x2+1/2, −x3, −x3+x4; (5) −x1, −x2, −x3, −x4; (6) x1, x2, −x3, −x3+x4+1/2; (7) x1+1/2, −x2+1/2, x3, x4+1/2; (8) −x1+1/2, x2+1/2, x3, x3−x4. |
Dectris-CrysAlisPro-abstract goniometer imported dectris images diffractometer | 797 independent reflections |
Radiation source: synchrotron | 454 reflections with I > 3σ(I) |
Synchrotron monochromator | Rint = 0.014 |
Detector resolution: 5.8 pixels mm-1 | θmax = 31.9°, θmin = 3.8° |
ω scans | h = −11→11 |
Absorption correction: multi-scan CrysAlisPro 1.171.38.46 (Rigaku Oxford Diffraction, 2015) Empirical absorption correction using spherical harmonics, implemented in SCALE3 ABSPACK scaling algorithm. | k = −10→10 |
Tmin = 0.569, Tmax = 1 | l = −4→4 |
3576 measured reflections |
Refinement on F2 | 0 restraints |
R[F2 > 2σ(F2)] = 0.025 | 33 constraints |
wR(F2) = 0.097 | Weighting scheme based on measured s.u.'s w = 1/(σ2(I) + 0.0035999999I2) |
S = 1.03 | (Δ/σ)max = 0.001 |
797 reflections | Δρmax = 0.28 e Å−3 |
101 parameters | Δρmin = −0.30 e Å−3 |
x | y | z | Uiso*/Ueq | Occ. (<1) | |
Al1 | 0 | 0 | 0 | 0.01054 (17) | |
Al2 | 0.14908 (4) | 0.33987 (4) | 0.5 | 0.01046 (17) | 0.5 |
Si2 | 0.14908 (4) | 0.33987 (4) | 0.5 | 0.01046 (17) | 0.2868 (17) |
Al3 | 0.26226 (19) | 0.2054 (2) | 0.5 | 0.0121 (4) | 0.2132 (17) |
O1 | 0.35879 (9) | 0.42178 (11) | 0.5 | 0.0165 (3) | |
O2 | 0.12793 (9) | 0.21778 (13) | 0 | 0.0173 (2) | |
O3 | 0 | 0.5 | 0.5 | 0.021 (2) | 0.360 (5) |
O4 | 0.4543 (9) | 0.0499 (8) | 0.5 | 0.0124 (16) | 0.2132 (17) |
U11 | U22 | U33 | U12 | U13 | U23 | |
Al1 | 0.0128 (3) | 0.0098 (3) | 0.0090 (3) | 0.00056 (12) | 0 | 0 |
Al2 | 0.0118 (3) | 0.0096 (3) | 0.0100 (3) | −0.00058 (10) | 0 | 0 |
Si2 | 0.0118 (3) | 0.0096 (3) | 0.0100 (3) | −0.00058 (10) | 0 | 0 |
Al3 | 0.0133 (6) | 0.0120 (7) | 0.0110 (7) | 0.0002 (5) | 0 | 0 |
O1 | 0.0193 (4) | 0.0201 (4) | 0.0099 (5) | −0.0076 (3) | 0 | 0 |
O2 | 0.0208 (3) | 0.0155 (4) | 0.0155 (4) | −0.0062 (3) | 0 | 0 |
O3 | 0.020 (4) | 0.017 (4) | 0.026 (2) | −0.0076 (18) | 0 | 0 |
O4 | 0.015 (3) | 0.006 (3) | 0.0159 (18) | −0.0020 (17) | 0 | 0 |
Average | Minimum | Maximum | |
Al1—O1i | 1.8926 (7) | 1.8890 (7) | 1.8964 (7) |
Al1—O1ii | 1.8927 (7) | 1.8890 (7) | 1.8964 (7) |
Al1—O1iii | 1.8927 (7) | 1.8890 (7) | 1.8964 (7) |
Al1—O1iv | 1.8926 (7) | 1.8890 (7) | 1.8964 (7) |
Al1—O2 | 1.9320 (10) | 1.9320 (10) | 1.9320 (10) |
Al1—O2v | 1.9320 (10) | 1.9320 (10) | 1.9320 (10) |
Al2—O1 | 1.7089 (14) | 1.7050 (14) | 1.7129 (14) |
Al2—O2 | 1.7264 (7) | 1.7217 (7) | 1.7311 (7) |
Al2—O2vi | 1.7264 (7) | 1.7217 (7) | 1.7311 (7) |
Al2—O3 | 1.669 (2) | 1.661 (4) | 1.677 (4) |
Al2—O4vii | 1.792 (8) | 1.784 (9) | 1.801 (9) |
Al2—O4iv | 1.701 (8) | 1.691 (9) | 1.711 (9) |
Si2—O1 | 1.7089 (14) | 1.7050 (14) | 1.7129 (14) |
Si2—O2 | 1.7264 (7) | 1.7217 (7) | 1.7311 (7) |
Si2—O2vi | 1.7264 (7) | 1.7217 (7) | 1.7311 (7) |
Si2—O3 | 1.669 (2) | 1.661 (4) | 1.677 (4) |
Si2—O4vii | 1.792 (8) | 1.784 (9) | 1.801 (9) |
Si2—O4iv | 1.701 (8) | 1.691 (9) | 1.711 (9) |
Al3—O1 | 1.815 (2) | 1.805 (2) | 1.825 (2) |
Al3—O2 | 1.7670 (12) | 1.7622 (12) | 1.7717 (12) |
Al3—O2vi | 1.7670 (12) | 1.7622 (12) | 1.7717 (12) |
Al3—O4 | 1.881 (8) | 1.877 (9) | 1.886 (9) |
O1i—Al1—O1ii | 99.19 (4) | 99.19 (4) | 99.19 (4) |
O1i—Al1—O1iii | 80.81 (4) | 80.49 (4) | 81.14 (4) |
O1i—Al1—O1iv | 179.52 (7) | 179.24 (4) | 180.0 (5) |
O1i—Al1—O2 | 89.47 (3) | 88.98 (3) | 89.96 (4) |
O1i—Al1—O2v | 90.53 (3) | 90.21 (3) | 90.85 (3) |
O1ii—Al1—O1iii | 179.52 (7) | 179.24 (4) | 180.0 (5) |
O1ii—Al1—O1iv | 80.81 (4) | 80.49 (4) | 81.14 (4) |
O1ii—Al1—O2 | 89.47 (3) | 88.98 (3) | 89.96 (4) |
O1ii—Al1—O2v | 90.53 (3) | 90.21 (3) | 90.85 (3) |
O1iii—Al1—O1iv | 99.19 (4) | 99.19 (4) | 99.19 (4) |
O1iii—Al1—O2 | 90.53 (3) | 90.21 (3) | 90.85 (3) |
O1iii—Al1—O2v | 89.47 (3) | 88.98 (3) | 89.96 (4) |
O1iv—Al1—O2 | 90.53 (3) | 90.21 (3) | 90.85 (3) |
O1iv—Al1—O2v | 89.47 (3) | 88.98 (3) | 89.96 (4) |
O2—Al1—O2v | 179.85 (13) | 179.74 (2) | 180.0 (5) |
O1—Al2—O2 | 106.61 (3) | 106.22 (3) | 107.01 (3) |
O1—Al2—O2vi | 106.61 (3) | 106.22 (3) | 107.01 (3) |
O1—Al2—O3 | 111.00 (14) | 110.27 (14) | 111.74 (14) |
O1—Al2—O4vii | 94.3 (2) | 93.6 (2) | 95.1 (2) |
O1—Al2—O4iv | 128.6 (2) | 127.9 (2) | 129.3 (2) |
O2—Al2—O2vi | 113.19 (5) | 112.59 (5) | 113.79 (5) |
O2—Al2—O3 | 109.68 (7) | 109.63 (7) | 109.72 (7) |
O2—Al2—O4vii | 116.59 (9) | 116.42 (9) | 116.76 (9) |
O2—Al2—O4iv | 100.91 (13) | 100.82 (13) | 100.99 (13) |
O2vi—Al2—O3 | 109.68 (7) | 109.63 (7) | 109.72 (7) |
O2vi—Al2—O4vii | 116.59 (9) | 116.42 (9) | 116.76 (9) |
O2vi—Al2—O4iv | 100.91 (13) | 100.82 (13) | 100.99 (13) |
O3—Al2—O4vii | 16.7 (3) | 16.4 (3) | 17.0 (3) |
O3—Al2—O4iv | 17.6 (3) | 17.5 (3) | 17.7 (3) |
O4vii—Al2—O4iv | 34.3 (4) | 34.1 (3) | 34.4 (3) |
O1—Si2—O2 | 106.61 (3) | 106.22 (3) | 107.01 (3) |
O1—Si2—O2vi | 106.61 (3) | 106.22 (3) | 107.01 (3) |
O1—Si2—O3 | 111.00 (14) | 110.27 (14) | 111.74 (14) |
O1—Si2—O4vii | 94.3 (2) | 93.6 (2) | 95.1 (2) |
O1—Si2—O4iv | 128.6 (2) | 127.9 (2) | 129.3 (2) |
O2—Si2—O2vi | 113.19 (5) | 112.59 (5) | 113.79 (5) |
O2—Si2—O3 | 109.68 (7) | 109.63 (7) | 109.72 (7) |
O2—Si2—O4vii | 116.59 (9) | 116.42 (9) | 116.76 (9) |
O2—Si2—O4iv | 100.91 (13) | 100.82 (13) | 100.99 (13) |
O2vi—Si2—O3 | 109.68 (7) | 109.63 (7) | 109.72 (7) |
O2vi—Si2—O4vii | 116.59 (9) | 116.42 (9) | 116.76 (9) |
O2vi—Si2—O4iv | 100.91 (13) | 100.82 (13) | 100.99 (13) |
O3—Si2—O4vii | 16.7 (3) | 16.4 (3) | 17.0 (3) |
O3—Si2—O4iv | 17.6 (3) | 17.5 (3) | 17.7 (3) |
O4vii—Si2—O4iv | 34.3 (4) | 34.1 (3) | 34.4 (3) |
O1—Al3—O2 | 100.53 (8) | 100.16 (8) | 100.90 (8) |
O1—Al3—O2vi | 100.53 (8) | 100.16 (8) | 100.90 (8) |
O1—Al3—O4 | 105.6 (2) | 105.2 (2) | 105.9 (2) |
O2—Al3—O2vi | 109.30 (10) | 108.77 (10) | 109.83 (10) |
O2—Al3—O4 | 118.67 (11) | 118.49 (11) | 118.85 (11) |
O2vi—Al3—O4 | 118.67 (11) | 118.49 (11) | 118.85 (11) |
Al1viii—O1—Al1vii | 99.18 (5) | 98.83 (5) | 99.54 (5) |
Al1viii—O1—Al2 | 129.96 (2) | 129.75 (2) | 130.17 (2) |
Al1viii—O1—Si2 | 129.96 (2) | 129.75 (2) | 130.17 (2) |
Al1viii—O1—Al3 | 121.20 (4) | 120.54 (4) | 121.86 (4) |
Al1vii—O1—Al2 | 129.96 (2) | 129.75 (2) | 130.17 (2) |
Al1vii—O1—Si2 | 129.96 (2) | 129.75 (2) | 130.17 (2) |
Al1vii—O1—Al3 | 121.20 (4) | 120.54 (4) | 121.86 (4) |
Al2—O1—Si2 | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Al2—O1—Al3 | 44.65 (6) | 44.52 (6) | 44.77 (6) |
Si2—O1—Al3 | 44.65 (6) | 44.52 (6) | 44.77 (6) |
Al1—O2—Al2ix | 121.07 (3) | 120.48 (3) | 121.64 (3) |
Al1—O2—Al2 | 121.06 (3) | 120.48 (3) | 121.64 (3) |
Al1—O2—Si2ix | 121.07 (3) | 120.48 (3) | 121.64 (3) |
Al1—O2—Si2 | 121.06 (3) | 120.48 (3) | 121.64 (3) |
Al1—O2—Al3ix | 104.03 (7) | 103.56 (7) | 104.48 (7) |
Al1—O2—Al3 | 104.02 (7) | 103.56 (7) | 104.48 (7) |
Al2ix—O2—Al2 | 113.19 (6) | 113.19 (6) | 113.19 (6) |
Al2ix—O2—Si2ix | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Al2ix—O2—Si2 | 113.19 (6) | 113.19 (6) | 113.19 (6) |
Al2ix—O2—Al3ix | 45.17 (6) | 45.05 (6) | 45.28 (6) |
Al2ix—O2—Al3 | 131.05 (7) | 131.01 (7) | 131.09 (7) |
Al2—O2—Si2ix | 113.19 (6) | 113.19 (6) | 113.19 (6) |
Al2—O2—Si2 | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Al2—O2—Al3ix | 131.05 (7) | 131.01 (7) | 131.09 (7) |
Al2—O2—Al3 | 45.17 (6) | 45.05 (6) | 45.28 (6) |
Si2ix—O2—Si2 | 113.19 (6) | 113.19 (6) | 113.19 (6) |
Si2ix—O2—Al3ix | 45.17 (6) | 45.05 (6) | 45.28 (6) |
Si2ix—O2—Al3 | 131.05 (7) | 131.01 (7) | 131.09 (7) |
Si2—O2—Al3ix | 131.05 (7) | 131.01 (7) | 131.09 (7) |
Si2—O2—Al3 | 45.17 (6) | 45.05 (6) | 45.28 (6) |
Al3ix—O2—Al3 | 109.30 (8) | 109.30 (8) | 109.30 (8) |
Al2—O3—Al2x | 179.71 (19) | 179.5 (3) | 180.0 (5) |
Al2—O3—Si2 | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Al2—O3—Si2x | 179.71 (19) | 179.5 (3) | 180.0 (5) |
Al2x—O3—Si2 | 179.71 (19) | 179.5 (3) | 180.0 (5) |
Al2x—O3—Si2x | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Si2—O3—Si2x | 179.71 (19) | 179.5 (3) | 180.0 (5) |
Al2ii—O4—Al2xi | 145.7 (5) | 145.7 (5) | 145.8 (5) |
Al2ii—O4—Si2ii | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Al2ii—O4—Si2xi | 145.7 (5) | 145.7 (5) | 145.8 (5) |
Al2ii—O4—Al3 | 103.4 (4) | 103.0 (4) | 103.8 (4) |
Al2xi—O4—Si2ii | 145.7 (5) | 145.7 (5) | 145.8 (5) |
Al2xi—O4—Si2xi | 0.0 (5) | 0.0 (5) | 0.0 (5) |
Al2xi—O4—Al3 | 110.8 (4) | 110.4 (4) | 111.3 (4) |
Si2ii—O4—Si2xi | 145.7 (5) | 145.7 (5) | 145.8 (5) |
Si2ii—O4—Al3 | 103.4 (4) | 103.0 (4) | 103.8 (4) |
Si2xi—O4—Al3 | 110.8 (4) | 110.4 (4) | 111.3 (4) |
Symmetry codes: (i) −x1+1/2, x2−1/2, −x3, −x4+1/2; (ii) −x1+1/2, x2−1/2, −x3+1, −x4+1/2; (iii) x1−1/2, −x2+1/2, −x3, −x3+x4; (iv) x1−1/2, −x2+1/2, −x3+1, −x3+x4; (v) −x1, −x2, x3, x3−x4+1/2; (vi) x1, x2, x3+1, x4; (vii) −x1+1/2, x2+1/2, −x3+1, −x4+1/2; (viii) −x1+1/2, x2+1/2, −x3, −x4+1/2; (ix) x1, x2, x3−1, x4; (x) −x1, −x2+1, x3, x3−x4+1/2; (xi) x1+1/2, −x2+1/2, −x3+1, −x3+x4. |
Footnotes
1Note that in this study the term `phase shift' always refers to the phase difference between two modulation functions in t space. The modulation functions are functions of 2πx4 with x4 = q·x + t. The physical-space section t is constant for a structure model in physical space, which represents one subspace of the model. The analysis of crystal-chemical aspects of the model requires the consideration of all the different environments of a selected site, which is an infinite number in the case of incommensurate structures. Instead of analysing the crystal chemistry of all sites in one crystal, the crystal chemistry of that site in different subspaces, i.e. different values of t, gives the same overview. Similar values of t usually describe a similar environment, but in physical space they describe two sites that are not close to each other. Therefore, t-plots give an overview of the possible crystal-chemical environments, but the analysis of a specific situation requires us to look at the same value of t.
Acknowledgements
We acknowledge the European Synchrotron Radiation Facility for provision of synchrotron radiation facilities and we would like to thank Dmitry Chernyshov and Alexei Bossak for assistance in using beamlines BM01 and ID28. We greatly appreciate experimental support by Noelia de la Pinta and Tomasz Breczewski for the synthesis of the QG sample.
Funding information
The following funding is acknowledged: Spanish Ministerio de Economia y Competitividad and FEDER funds, Secretaría de Estado de Investigación, Desarrollo e Innovación (award No. MAT2015-66441-P); Government of the Basque Country (grant No. IT-779-13). P. B. Klar thanks the Basque Government for a grant.
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