The structure of kaliophilite KAlSiO4, a long-lasting crystallographic problem

The elusive structure of the mineral kaliophilite has been determined by 3D electron diffraction and refined using single-crystal X-ray data. Despite its simple formula, ideally KAlSiO4, the structure of this mineral remained a mystery for over a century as a result of pseudo-symmetry and twinning, which reduce the coherent crystalline domain size to a few hundreds of nanometres.

.1. Published chemical analyses, densities and optical constants of kaliophilite (wet analyses except the last five). All values from original paper except sca88 and zam10 for which 2 secondary references (mug27 and ban31) have been used. All elements have been recalculated from oxides per 4 oxygens.   29 Si MAS-NMR spectra were obtained for various kaliophilite samples from Colle Cimino, using exceptionally long accumulation times to allow interpretation of the broad base of the peaks. The spectra of the four samples in Fig. S2.1 and Fig. S2.2 show clearly that there is a difference in the left, but not in the right part of the base. We therefore concluded that the left part is probably due to some impurity which changes from sample to sample, while the right part is inherent to kaliophilite itself.
To confirm the hypothesis, we simulated a spectrum containing six components: a central doublet due to kaliophilite (δ = -86.5 and 89.5 ppm), and four broad peaks representing an impurity at about -74 to -80 ppm, a sum of distorted kaliophilite Q 4 (4Al) environments at around -90 ppm, and two environments resulting from Si for Al substitution in kaliophilite, i.e. Q 4 (3Al) at about -96 ppm and Q 4 (0Al) at about -105 ppm. The latter two have been found earlier for Bancroft nepheline and its K-exchanged derivative (Stebbins et al., 1986;Sobrados, 1991) and are thought to be associated with alkali vacancies and twin boundaries. All peaks were found from curve fitting with some restriction for the -96 and -105 ppm positions. Fitting results are summarized in Table S2.1.
The central doublet was also observed in the 29 Si MAS-NMR spectrum of O1 and the amount of disordered Q 4 (4Al) and Q 4 (3Al) and Q 4 (0Al) environments in kaliophilite is compatible with earlier findings for nepheline and O1 (Stebbins et al., 1986;Gregorkiewitz et al., 2008). The amount of impurities, on the other hand, seems high and this is not supported by powder diffraction where crystalline impurities can be excluded at a level of about 0.4%, and amorphous impurities seem improbable. The central kaliophilite doublet (Fig. S2.3) was calculated ab initio using CASTEP (Clark et al., 2005) for the model given in Table S5.3, which contains 18 crystallographically different Si positions, space group P3. While the positions are very similar to the fitting results, intensities appear inverted with the higher peak at about -86.4 ppm. The nuclei probed in NMR are different from those in XRD, and small differences in the structures of the Colle Cimino and Pollena samples are possible.

S3. Optical microscopy and Weissenberg X-ray diffraction patterns
On the search of possibly pure single crystals, about 12 optically homogeneous microcrystals could be isolated under the petrographic microscope ( Fig. S3.1). With an approximate thickness of 40 μm, the observed interference colours indicate Δn ≃ -0.004, in agreement with literature (Table S1.1). Although clear, many crystals show lines along the needle which are probably twin planes (see also similar observations in Cellai et al., 1992), but artefacts cannot be excluded for such small dimensions. Further screening of these crystals with laboratory single crystal diffraction revealed that many of them were composed of slightly (~1º) disoriented individuals but no streaks parallel or orthogonal to c could be detected. A representative Weissenberg pattern is given in Fig. S3.2.
For one crystal, CuKα Weissenberg photographs of levels hkl: 0 ≤ l ≤ 5 showed no splitting. In order to test symmetry and twinning, this crystal was used to collect high resolution diffraction scans (Supporting Information S4).

Figure S3.2.
Level hk1 Weissenberg equi-inclination diagram obtained with Ni-filtered CuK radiation (λ = 1.5406 Å, camera R = 27.65 mm, 2º(ω)/mm translation ||c) for a kaliophilite 'single' crystal. Abscissa gives crystal rotation, ordinate gives diffraction angle. The a* repeat lies on the almost extinct line passing through the abscissa. Splitting of reflections shows that the 'crystal' is composed of essentially two individuals which are rotated by Δω < 1.4º against each other. Note that streaks are due to white radiation and not to structural features.
With the available data (about 620 reflections in 63 complete sets, about half of which had a signal/background ratio < 3), no obvious violations of Laue symmetry P6/mmm could be observed.

Crystal data
Chemical formula Al9K9Si9O36 Crystal system, space group Trigonal, P3c1

S6. Rietveld refinements using high resolution synchrotron radiation
The major steps of Rietveld refinements are resumed in Table S6.1 and can be divided in two groups: the first, from #40 to #235, starts with the raw model in space group P3c1 obtained from 3D ED and addresses various improvements until the introduction of the model in P3 obtained from single crystal refinement, and the second, from #b10 to #e20, is concerned with the correct description of peak shape (anisotropic line broadening ALB). Some representative patterns are shown at the end.
Group 1 (structure from P3c1 to P3) The starting structure, obtained from 3D ED and preliminarily refined using electron diffraction intensities, had already an ordered Al/Si distribution. Once introduced in Rietveld refinement (#40, R(F 2 ) = 0.34), it became immediately clear that there were enormous problems with peak shape, and trials to refine the structure (#58, R(F 2 ) = 0.20) showed correlations and divergence even when the tetrahedral distances (T-O and O-O) were restrained. Slight improvements could be achieved introducing shape anisotropy to model ALB and a small fraction of kalsilite as impurity to account for some extra peaks (#642, R(F 2 ) = 0.17), but results correspond to false minima recognized by lack of convergence and improbable interatomic distances.
In the meanwhile, SCXRD refinement using data from a twinned crystal allowed to obtain an improved structure in the subgroup P3. These atom parameters, introduced in Rietveld refinement without releasing them, clearly lowered the residual (#235, χ 2 = 292, R(F 2 ) = 0.131) but the inadequateness of peak shape description became now even more visible and is reflected by the high reduced χ 2 value a . The use of a microstrain model (Stephens, 1999) as an alternative to explain ALB was not successful either.
The new peak shape model was improved in two (structureless) Le Bail refinements (#c18, χ 2 = 43.1; #f21, χ 2 = 31.3), whose details will be discussed below. The overall parameters obtained in #c18 were, unchanged, fed into a final Rietveld refinement (#e20, χ 2 = 117, R(F 2 ) = 0.089) where only the scale factors of kaliophilite and a Values for χ 2 are high throughout due to the low statistical error of diffracted intensities, a fact well known for work with synchrotron radiation, but χ 2 = 292 is still 10 times higher than the minimum obtained in standard refinement (χ 2 = 30). kalsilite were allowed to refine. As expected, the global fit error χ 2 , but not the structural error R(F 2 ), dropped with respect to step #b10.
The structural residuals of the final result (#e20, R(F 2 ) = 0.089, R(F) = 0.071) compare well with their counterparts from single crystal refinement (wR2 = 0.182, R1 = 0.090), a confirmation which is independent from any bias introduced by twin laws. In addition, Rietveld refinement gave at least three new informations: (a) more representative (whole phase) and precise (synchrotron radiation) values for the unit cell parameters, (b) the detection of kalsilite as an impurity in Colle Cimino kaliophilite, and (c) a wealth of details on microstructure which is discussed below.

Interpretation of line shape and displacement (microstructure)
The description of peak shape and position was addressed, without any bias coming from crystal structure, using full pattern Le Bail refinements (#c18 and #f21 in Table S6.1) as well as single peak fittings of some selected reflections.
In experiment #c18 (χ 2 = 43.1), ALB was simulated using the hk = 3n parity rule that divides reflections into two groups with different Lorentzian contributions to the line FWHM. Inspection of the patterns (e.g. peak 311 in Fig. S6.3) shows that many broad peaks exhibit a super-Lorentzian shape which cannot be simulated in the TCH approach of FullProf. A trial to overcome this limitation was made in experiment #f21 (χ 2 = 31.3) where three groups of reflections (h -k = 3n, hk = 3n + 1 and hk = 3n -1) were defined. With 3n -1 and 3n + 1 being lattice equivalent pairs, this is a mathematical trick to create super-Lorentzians which allow an improved Le Bail fit, showing that super-Lorentzian peak shape explains most of the gap from χ 2 = 43.1 to the minimum at about χ 2 = 30, roughly expected also from NAC standard refinement.
A direct check of the parity rule hkl: hk = 3n (hkl: h -k ≠ 3n) for sharp (broad) reflections is shown in Fig.  S6.1 which compares calculated and observed total FWHMs. The observed line widths (blue dots) were obtained from well resolved reflections and clearly distribute over two branches, confirming that the parity rule holds for all reflections. The lower branch ranges from FWHM = 0.013º to 0.034º and closely follows the calculated mean (red diamonds) at the same angle, whereas the upper one (about 0.028 -0.042º) is more dispersed and values are considerably lower than their calculated counterparts, due to the super-Lorentzian form which, at constant area, has a broader base and smaller FWHM.
Outliers can in most cases be explained. As an example, in the upper branch, negative deviations from the mean are seen for various reflections with high --l indices, e.g. 213, 423, 515, 875. This indicates needle shaped domains for the 27 Å cell, elongated parallel to c in perfect agreement with HRTEM where faults parallel to c are observed b . A resulting scheme for ALB is given in the drawing at right of Fig. S6.2. Note the difference to the scheme at left which was unable to fit data (experiment #235).  Rotational ellipsoids representing anisotropic line broadening (ALB) for three different hypotheses containing a subcell defined by parity condition hk = 3n. The vector Bcosθ = FWHMcosθ is the cosθ corrected line breadth for reflections at inclination φ from axis c*. In the first case ("clay"), there are stacking faults (planes orthogonal to c*) and the two sets of reflections have the same width orthogonal to c* but different widths for φ < 90º. In the second case (corresponding to the model used in all refinements from #b10 onwards), widths are independent and different but isotropic within each set. The last case represents the most probable model for kaliophilite, where shape anisotropy (needles parallel to c) is added. Note that, here, the length of the coherent domain in subcell and full cell are likely to be similar (semiaxes c ≃ c'), in contrast to case 1 where the width of reflections perpendicular c* is common to the two sets.
The vertical shift (~0.017º) between the two branches is constant, i.e. there is a difference in the size component of line broadening while microstrain remains similar. Domain sizes were estimated using the Scherrer equation with integral breadths β obtained from both Le Bail refinement and single peak fits. For the broad reflections, after subtraction of the instrument and Lorentzian strain contributions, one obtains β = 0.050º and β = 0.045º, respectively, corresponding to apparent domain sizes of 1360 Å and 1500 Å. Such domain sizes apply essentially to hkl: hk ≠ 3n reflections with low indices (e.g. 211 and 311) and an inclination of φ ≃ 50º from c*. Remembering the approximately cylindrical shape anisotropy discussed above (Fig. S6.2), one obtains an average of 1070 Å for the thickness of the cylinders, in fair agreement with HRTEM results where the 27 Å cell is seen to extend over 200-1200 Å in ab. Fig. S6.3 shows the intensity patterns of Le Bail and single peak fits for the representative reflections 311 (hkl: hk ≠ 3n) and 410 (hkl: hk = 3n). Fig. S6.4 reports the best Rietveld fit (#e20, χ 2 = 117).
Precisions are likely to be overestimated (Herbstein, 2000;David, 2004;Tian & Billinge, 2011), but there might also be some difference in composition and microstructure. The second phenomenon seems particularly interesting in view of the evident misfit of reflection 110 (Fig. S6.5). A test was therefore designed were reflection 110 was forced to fit observation (note that this reflection has a low intensity and little weight in the global refinement, but it occurs at low angle and is well separated from anything else). A good fit of the position was obtained for a = 27.051 Å which comes nearer to the Monte Somma value.
In this refinement, peak shape parameters stood almost unchanged at their original values, but the shift parameters changed and all unit cell parameters blew up. In particular, the sine shift parameter changed from 8.6 to 56.2 mdeg, which corresponds to sample displacements of about 0.1 and 0.8 mm along the X-ray beam. The first value comes near to the result for NAC standard refinement (realized at the same date) and is within the expected limits (Gozzo et al., 2010), whereas the second value seems unreasonably high for the experimental setup at the MS-X04SA beamline.
We therefore postulate that the displacement of reflection 110 is due to details of the microstructure which are still to be resolved.