The structure of charoite, (K,Sr,Ba,Mn)15–16(Ca,Na)32[(Si70(O,OH)180)](OH,F)4.0.nH2O, solved by conventional and automated electron diffraction

Abstract Charoite, ideally (K,Sr,Ba,Mn)15–16(Ca,Na)32[(Si70(O,OH)180)](OH,F)4.0·nH2O, a rare mineral from the Murun massif in Yakutiya, Russia, was studied using high-resolution transmission electron microscopy, selected-area electron diffraction, X-ray spectroscopy, precession electron diffraction and the newly developed technique of automated electron-diffraction tomography. The structure of charoite (a = 31.96(6) Å, b = 19.64(4) Å, c = 7.09(1) Å, β = 90.0(1)°, V = 4450(24) Å3, space group P21/m) was solved ab initio by direct methods from 2878 unique observed reflections and refined to R 1/wR 2 = 0.17/0.21. The structure can be visualized as being composed of three different dreier silicate chains: a double dreier chain, [Si6O17]10–; a tubular loop-branched dreier triple chain, [Si12O30]12–; and a tubular hybrid dreier quadruple chain, [Si17O43]18–. The silicate chains occur between ribbons of edge-sharing Ca and Na-octahedra. The chains of tetrahedra and the ribbons of octahedra extend parallel to the z axis. K+, Ba2+, Sr2+, Mn2+ and H2O molecules lie inside tubes and channels of the structure. On the basis of microprobe analyses and occupancy refinement of the cation sites, the crystal chemical formula of this charoite can be written as (Z = 1): (K13.88Sr1.0Ba0.32Mn0.36)Σ15.56(Ca25.64Na6.36)Σ32 [(Si6O11(O,OH)6)2(Si12O18(O,OH)12)2(Si17O25(O,OH)18)2](OH,F)4.0·3.18H2O.

( K 3 C a 1 0 ( C a , M 3 + ) 2 (Konev et al., 1996). The structures of all these minerals are characterized by the occurrence of various types of dreier silicate chains. Close associations, intergrowths and transformation of some minerals into others suggest their structural affinity and metasomatic origin (Rozhdestvenskaya and Nikishova, 2002). Although the structure of charoite was unknown, there was strong evidence that it belongs to the group of alkaline Ca-silicates containing tubular and/or other dreier chains (Frank-Kamenetskaya and Rozhdestvenskaya, 2004).
Despite many attempts to solve the charoite structure, or at least to construct a convincing structural model, the structure and stoichiometry have remained enigmatic, and reported data were partially inconsistent. The crystal chemical formula of charoite, inferred from X-ray powder d i f f r a c t i o n , w a s p r o p o s e d a s (K,Na) 5 (Ca,Ba,Sr) 8  (F,OH) 2 ·nH 2 O; Z = 4; monoclinic, space group Pm, P2 or P2/m with a = 19.6, b = 32.02, c =7.25 Å , b = 94.3º (Nikishova et al., 1985). Chiragov and Shirinova (2004) proposed a model of the charoite structure on the basis of a miserite-like structure in which silicate-oxygen chains are connected to double octahedral bands and form [Ca 8 (Si 12 O 30 ) 2 ] 8À structure blocks. These blocks are distributed about y = 0 and Ý and are connected to a [Si 6 O 16 ] 8À okenite band, or a [Si 6 O 15 ] 6À okenite net.
Based on the common features of alkaline Casilicate structures, X-ray powder diffraction, and on high-resolution transmission electron microscopy (HRTEM), a structural model for charoite was proposed recently by Rozhdestvenskaya et al. (2007Rozhdestvenskaya et al. ( , 2009a. According to this model, charoite is monoclinic (space group P2 1 /m) with cell parameters a = 32.296, b = 19.651, c = 7.16 Å , b = 96.3º, V = 4517 Å 3 , with a general formula K 6 À7 (Ca,Na) 18 Liebau's (1985) classification terminology for silicates, we will henceforth use the appropriate notation for such silicate radicals: [ [(Ca,Na)O 6 ] octahedra. The chains and bands extend parallel to the z axis. Potassium ions and H 2 O molecules lie at the centres of the tubular chains, and other K ions lie at the centres of eightmembered rings of SiO 4 tetrahedra.
Despite the fact that the model of (Rozhdestvenskaya et al., 2007(Rozhdestvenskaya et al., , 2009a displayed important characteristics of the charoite structure, a definitive structure determination was still missing. For such a structure determination the collection of 3D-single crystal diffraction data is essential, but the acquisition of such data is difficult owing to the very small sizes of charoite crystals. Furthermore, our recent studies have shown that the charoite specimens not only contain other silicate minerals and an amorphous phase, but also individual asbestiform microcrystals of charoite having intergrowths of different structural variants, one clearly being monoclinic, a second one being metrically pseudo-orthorhombic, but still monoclinic and a third one with a doubled a parameter (Rozhdestvenskaya et al., 2009b). Consequently, we decided that for a reliable structure investigation, data from single-phase nanocrystals were required.
Nanoscale electron diffraction provides information from single nanocrystals down to 5 nm in diameter, as the electron beam can be focused down to this size, and the strong scattering of electrons allows a reasonable signal-to-noise ratio even for nanovolumes. Conventional electron diffraction often fails to give enough reflections required for a reliable structure solution. Moreover, it is well known that dynamic effects can result in spurious intensities when they are collected from zones oriented along low-index directions. In order to increase the number of reflections collected from a nanocrystal, and at the same time reduce both the electron dose on the sample and dynamic effects, a newly developed software module called 'automated diffraction tomography' (ADT) (Kolb et al., 2007(Kolb et al., , 2008 was used. The ADT software allows a rich sampling of reciprocal space in order to obtain unambiguous cell parameters even for triclinic crystals, and gives a full 3D-visualization of the reciprocal lattice. As a sample does not need to be oriented along low index zones, dynamic effects are largely reduced. In order to reduce dynamic effects further and to improve intensity integration, ADT can be coupled with 'precession electron diffraction' (PED) (Vincent and Midgley, 1994;Own, 2005;Avilov et al., 2007) whereby almost complete quasi-kinematic intensities can be collected by electron diffraction from a nanocrystal. Mugnaioli et al. (2009) showed that by this method it is possible to solve small inorganic structures ab initio in one step, and also to detect the positions of such elements like oxygen.
In this paper we report our investigations of charoite, performed using a wide range of techniques À HRTEM, selected-area electron diffraction (SAED), microprobe analysis, energy dispersive X-ray spectroscopy (EDX), PED and ADT, which have allowed the structure of charoite to be determined for the first time.

Experimental procedures
Charoite occurs in different morphologies (Evdokimov et al., 1995;Konev et al., 1996). For our study, a charoite sample of light-violet colour was chosen in which individual asbestoslike micro-crystals are elongated along the z axis (Fig. 1).
The sample was prepared by crushing in an agate mortar, and also by focused ion beam thinning (FIB). It was studied by conventional and HRTEM and SAED using Philips EM400T, Philips Tecnai F20, Philips Tecnai F30 and JEOL 3010UHR microscopes operated at 100, 200, 300 kV and 300 kV respectively. A Tracor Voyager HP-Ge detector attached to the Tecnai F20 microscope was used for EDX analysis.
The composition of charoite was determined by microprobe analysis using a JEOL JXA8900 electron microprobe using an accelerating voltage of 15 kV and a probe current of 15 nA.
For ADT investigations, the sample was ground, suspended in ethanol and sprayed onto a carbon-coated Cu grid using the sonifier described by Mugnaioli et al. (2009). This method made it easy to disperse charoite nanocrystals so that there was no overlap at high tilt angles. The data collection was carried out using a Tecnai F30 S-TWIN transmission electron microscope equipped with a field emission gun and operating at 300 kV. A FISCHIONE tomography holder with a tilt range up to Ô70º was used.
The ADT technique allows automatic collection of reflection data sets from a nanocrystal using electron diffraction. The ADT module alternates between scanning transmission electron microscopy (STEM) imaging for crystal tracking in low-dose mode and nanoscale acquisitions (Kolb et al., 2007(Kolb et al., , 2008. The module operates in microprobe STEM mode, and includes routines which allow tracking of a crystal after each tilt step and sequential acquisition of electron diffraction patterns. A 10 mm C2 condenser aperture with a 50 nm beam on the sample was employed in order to produce quasi-parallel illumination of the sample. A mild illumination setting was used, resulting in an electron dose rate of 10À15 e Å À2 s À1 , suitable also for very beamsensitive samples. The STEM images were collected by a FISCHIONE high-angular annular dark-field detector (HAADF). The HRTEM images and electron diffraction patterns were taken with a CCD camera (14-bit GATAN 794MSC, 102461024 pixels) and acquired by Digital Micrograph (GATAN) software. The ADT tilt series were collected in 1º steps with normal parallel illumination and in PED mode (Vincent and Midgley, 1994;Own, 2005;Avilov et al., 2007). The PED was performed using the SpinningStar unit developed by NanoMEGAS Company (NanoMEGAS, 2004). The precession angle was 1º and the precession frequency was 100 Hz.
The ADAP package (Kolb et al., 2007(Kolb et al., , 2008Mugnaioli et al., 2009) written in Matlab was used for data processing, including 3D-reconstruction of diffraction space, automated cellparameter determination, reflection indexing and intensity integration. For visualization and validation of 3D-data sets, we used the newly developed ADT-3D software package written in C++ (Heil et al., 2009). The ab initio structure solution was performed by direct methods as implemented in THE STRUCTURE OF CHAROITE SIR2008, included in the package IL MILIONE (Burla et al., 2007). The least squares structure refinement was performed using the SHELXL97 (Sheldrick, 2008) and CSD (Akselrud et al., 1989) programs. Electron scattering factors for the refinement were extracted from the SIR database.

Results
Bulk sample analysis and charoite polytypism A TEM investigation of a thin foil of part of the hand specimen containing charoite, previously examined by electron microprobe, shows that it is not homogeneous: crystals of acmite, NaFeSi 2 O 6 , and K-feldspar K(Fe 0.1 Al 0.9 Si 3.0 )O 8 are also present. The charoite itself has an average chemical composition (11 analyses) of: In the charoite sample studied here, asbestiform fibres of about 200 nm diameter are almost zparallel, but the xÀy directions are differently oriented (azimuthally disoriented), and they are often separated by amorphous material (Fig. 2a,b). Analysis, by EDX using TEM, indicates that the amorphous material contains almost no K and less Ca compared with charoite ( Fig. 2c,d).
Some SAED patterns as well as some HRTEM images (and the corresponding Fourier transforms) show a doubling of the a parameter, suggesting a possible third, hitherto unknown, charoite polytype (charoite-2a, Fig. 3c; note also the weak reflections in reciprocal rows with l = 2n, corresponding to 64 Å ). Often, the three structural variants are intimately intergrown as lamellae within the single fibre, having a common axis a*. Disordered intergrowth of very fine lamellae of the polytypes shows up as diffuse h0l (l = 2n+1) rows ('charoite-d', Fig. 3d). Frequently, additional diffraction maxima with uneven distances from the main reflections could be observed in the reciprocal rows h0l with l = 2n+1. Together, with diffuse streaks along a*, they point at disorder and some kind of (100) micro-lamellae. The micro-lamellae could be caused by stacking faults with fault vector 1/2[001], but micro-twinning of charoite-96 on (100), or intergrowth of charoite-96 and (100) micro-lamellae of charoite-90, also seem to be possible. The [010]-HRTEM images show that the interfaces between lamellar intergrowths can be of various types (Fig. 4). In several [0kl] SAED patterns, except [010], additional rows of weak reflections halfway between the rows in h0l sections were observed, suggesting a doubling of the c parameter, which could be associated with ordering along the z axis of ions inside the tubular chains. From a 3D-reconstruction of reciprocal lattices by non-precessed ADT data, the two polytypes, charoite-96 and charoite-90, were easily distinguished. For both polytypes, the only systematic absence is k = 2n+1 for 0k0, arising from a 2 1 screw axis along y. The intensities extracted from the precessed ADT tilt series show unambiguously that charoite-90 is monoclinic. This observation is confirmed by single PED patterns collected along [010] and [110], which show the absence of any mirror elements orthogonal to h00 and 00l (Fig. 5).

Ab initio structure solution and ref|nement.
A structurally coherent monophase single fibre of charoite-90 was chosen for intensity collection (Fig. 6a). Two tilt series were taken from the same crystal in steps of 1º and in the ranges of 120º and 105º respectively. The tilt axes of the two tilt series were approximately orthogonal. For both series, precessed and non-precessed nanoscale electron diffraction patterns were acquired. The reconstructed 3D-reciprocal space showed a perfectly ordered structure with no disorder streaks (Fig. 6bÀd). Finally, a precessed ADT intensity dataset was obtained by merging the two tilt series. No merging factor was applied. The unit-cell parameters, data acquisition and refinement parameters are given in Table 1. The hkl reflection file is available at www.staff.unimainz.de/kolb/ADTdata/charoite90.hkl.
The structure solution was performed in space g r o u p P 2 1 / m , i n a c c o r d a n c e w i t h 4450 (24) )]/(nÀp)} 1/2 , where n is the number of reflections and p is the number of refined parameters. Rozhdestvenskaya et al. (2007Rozhdestvenskaya et al. ( , 2009a. The structure was solved ab initio by direct methods using SIR2008 (Burla et al., 2007) using a fully kinematic approach, i.e. intensities were assumed to be proportional to F hkl 2 . The best solution has R = 0.231 and an overall atom displacement parameter U overall of 0.039 Å 2 . The best solution is much better, both in terms of R value and figure of merit, than the cloud of incorrect solutions (Fig. 7). A total of 8 Ca, 20 Si and 47 O positions were correctly identified. The missing 4 Os of the framework and the channel content (3 H 2 O molecules, 7 K and 1 Sr positions) were detected in the first difference-Fourier map calculated by SHELXL97 (Sheldrick, 1997). The least squares refinements using SHELXL97 (Sheldrick, 1997) and CSD (Akselrud et al., 1989), with 2878 reflections, F hkl >4s F , converged to unweighted/weighted residuals R 1 /wR 2 = 0.173/ 0.21. The number of independent atom positions is 90, the number of free variables 707. Provisional refinement of site occupancies was also performed. The refinement was not perfect because the intensities of equivalent reflections varied too much (R eqv. = 0.176), and no geometrical restraints or constraints were applied. Nevertheless, the data were sufficiently precise to reveal the relevant topological and geometrical features of the structure, and to find sites with mixed occupancy of cations. The refined atom positions, isotropic displacement parameters and site occupancies are given in Table 2, and selected bond lengths are shown in Table 3.  (7) 0.1341 (15) (Fig. 8a). All chains run parallel the z axis. The chains are bonded by their apical O atoms to (Ca,Na)-octahedra bands of various widths, which are also parallel z (Fig. 8b). K + , Ba 2+ , Sr 2+ , Mn 2+ and H 2 O molecules are located inside the silicate chains.

Silicate chains
The dreier double chain, [Si 6 O 17 ] 10À , is of the same type as the chain found in the xonotlite structure (Fig. 9a,d). The tubular dreier triple chain, [Si 12 O 30 ] 12À (Fig. 9b,e), differs from the chains of identical stoichiometry found in the structures of frankamenite and miserite. In charoite, this tubular chain is formed by three loop-branched dreier single chains and has threefold symmetry. Such loop-branched dreier single chains are known from the structure of synthetic Li 2 Mg 2 [Si 4 O 11 ] (Czank and Bissert, 1993).
The dreier single chain is similar to that found in pectolite and the dreier double chain is of the type found in okenite. It is of interest that tetrahedron T(18) of this large tubular chain is connected to four other tetrahedra, thus its connectivity is more typical for tetrahedra in framework structures than for tetrahedra in chain silicates.
Si(20) tetrahedra join with blocks of Na(1)-Na(2) À Ca(1)-Ca(2) octahedra (Fig. 8a, 11d). Tetrahedron Si (18) is unique in that it connects to four other tetrahedra of the same chain (Fig. 8a, 9h, 11d). Horizontal [Si 2 O 7 ] 6À groups of each chain connect the separate bands of Ca-octahedra in such a way that apical O atoms of neighbouring [Si 2 O 7 ] 6À groups are joined to different apices of vertical edges of octahedra and, therefore, adjacent chains are shifted relative to each other by half a translation along the z axis (Fig. 8a).
The mean bond lengths in the tetrahedra range between 1.57 and 1.65 Å ( Table 3). The mean OÀSiÀO angles are in the range 109.1À109.4º. However, we point out that the shortest and longest bond lengths found (1.51 and 1.77 Å respectively) are probably not realistic. This large deviation from standard values is almost certainly due to the fact that the refinement is not perfect, at least when considered from the point of view of standard X-ray diffraction methods applied to standard crystals with small structures. This imperfection is also reflected in the correspondingly large e.s.d. of lattice parameters, bond lengths and angles.
Tetrahedra Si (9) exhibit free channels along the chains, which are defined by nine-and eleven-membered rings of tetrahedra, '9MR' and '11MR' respectively ( Fig. 9b,c). The double chain [Si 6 O 17 ] 10À forms a tube together with Ca(5) and Ca(6) octahedra and Si (11) and Si(12) tetrahedra (Fig. 9g). The tubes extend along the z axis. All chains also have windows of eight-membered rings '8MR' of various free diameters. The free diameter of the channels and windows has been estimated from the distance between O atoms across the ring minus 2.7 Å (assuming the ionic radius of O to be 1.35 Å ). The sizes of the free diameters are shown in Fig. 9aÀh In the '8MR' windows on the side of the basal Os of all silicate chains, the K atoms are located in the sites K(1), K(2), K(3), K(6) (c.f. Fig. 8a, 11a,b), and have nine-fold coordination. The mean bond length KÀO is 2.94 Å (Table 3). Six Os of the '8MR' within which the K atoms are located and three Os of the adjacent chain take part in the charge compensation for each of these cations, thus providing an additional link between adjacent chains. Another K atom, K(5), is located on the axis of the tubular chain [Si 12 O 30 ] 12À (Fig. 8a). It has weaker bonds (mean bond length KÀO = 3.18 Å ) in the structure, this site is not fully occupied (occ. = 0.55 K). The H 2 O(1) molecule can be located at 1/2 of the translation along the z axes from that K site in the tube cavity. This site is also not fully occupied (occ. = 0.31 O). The O of the H 2 O group is linked with K in site K(3), located in the '8MR' window; bond length K(3)ÀH 2 O(1) = 2.80 Å .
The cavity within the large tubular chain [Si 17 O 43 ] 18À is nearly empty. Only two small peaks were found in a difference-Fourier map. Taking into account the distances from these peaks to the nearest O atoms, we infer that K atoms are located in site K(7) (mean bond length K(7)ÀO = 3.08 Å ) and H 2 O molecules in site H 2 O(2) (mean bond length H 2 O(2)ÀO = 2.97 Å ) ( Table 3). The refined occupancies for K(7) are 0.12 K and 0.46 O for H 2 O(2) ( Table 2). The short distances K(7)ÀK(7) = 2.13 and H 2 O(2)ÀH 2 O(2) = 1.51 Å should be considered admissible since the sites are only partly occupied.
The interior of the tube formed by the bent ribbon [Si 6 O 17 ] 10À , the Ca(5), Ca(6) octahedra and the Si(11), Si(12) tetrahedra, has a higher occupancy than the [Si 17 O 43 ] 18À tubular chain (Fig. 8a). Three peaks were found on Fourier maps. From the distances of these peaks from nearest O atoms, we infer that Sr 2+ is the occupant (mean bond length SrÀO = 2.65 Å ). The refinement showed that the occupancy of the Sr site is 0.50 Sr, in good agreement with the microprobe analysis of charoite.  (Table 3). Finally, the remaining cations Ba 2+ , Mn 2+ and part of the K + were inferred, statistically, to occupy the K(4) site (mean bond length K(4)ÀO = 2.68 Å ) (Tables 2, 3). Thus, H 2 O molecules and the impurity cations Sr 2+ , Ba 2+ , Mn 2+ and K + , which always occur in charoite, are located inside this tube. For the partially occupied positions (especially the Sr position), we can expect ordering along the z axis, e.g. alternating 1.0 and 0.0 occupancy for Sr. Such ordering implies a doubling of the c parameter and so may be associated with the observed additional rows of weak diffraction maxima halfway between the rows h0l.
The structure of charoite is unique among alkaline Ca-silicates, in that impurity elements such as Sr, Ba and Mn are abundant; Fe and Ti can also be detected. The various colours of charoite samples depend on the varying amounts of Mn, Fe and Ti; in particular, the colour from light lilac to dark violet depends on the content of Mn 3+ cations (Evdokimov et al., 2000). In the structures of canasite and frankamenite, these impurity cations were not determined; in the structure of miserite, the rare-earth elements are located in octahedrally-coordinated sites; in the structure of Sr-agrellite, Sr 2+ substitutes for Ca 2+ at octahedrally-coordinated sites. In the structure of charoite, all impurity cations are located in the tube formed by the dreier double chain [Si 6 O 17 ] 10À , Ca(5), Ca(6) octahedra and Si(11), Si(12) tetrahedra.

Charge-balance mechanism and crystal chemical formula
The structure of charoite contains 51 O sites (excluding the H 2 O molecules). Twenty-nine O atoms make bonds with two Si atoms and their charges are completely compensated. Seven Os have a bond with one Si, and additional bonds with three octahedrally-coordinated cations. Allowing for the occupancy of the Na(1) and Na(2) octahedra by a majority of Na + (Table 2), the charges of the corresponding Os are not completely compensated. Twelve O atoms have a bond with one Si atom and additional bonds with two octahedrally-coordinated Ca 2+ ions. Therefore, we can assume some substitution of O 2À by OH À groups at these O sites. Two O atoms are bonded to one Si and form bonds with one octahedrally-coordinated Ca 2+ and with cations in the partly occupied Sr and K(4) sites. The anion in the OH forms only three bonds with cations in the Na(1) and Na(2) sites and, thus, it is likely that both OH À and F À are located in this position.
In spite of the fact that these data differ from the average result of the microprobe analysis, (K 3.28 Sr 0.26 Ba 0.08 Mn 0.09 ) S3.71 (Ca 6.94 Na 1.37 ) S8.31 [Si 17.65 O 45 ](OH,F) 1.0 , the results are well within the limits of the minimum and maximum variations in the chemical composition of the separate analyses. The calculated density based on the results of the structure refinement is 2.55 g cm À3 which is in excellent agreement with the measured density of 2.54 g cm À3 .
The fact that most of the O sites with (O,OH) are located within the (100) zigzag layers of octahedra suggests that here, only weak bonds are active. As a consequence, the silicate chains connected on opposite sides of the zigzag layers may undergo a cooperative discrete shift of one half of the z translation, which corresponds to a stacking fault with fault vector Ý[001]. This is a precondition for the occurrence of polytypism. Regular shifts on every layer in the same direction would then result in another species of charoite being monoclinic with b = 96º (charoite-96; see also Fig. 4). Shifts alternating on every second layer in the + and À directions respectively, might form the polymorph with a doubled a parameter (charoite-2a). Randomly distributed + or À shifts would result in a disordered species (charoite-d).

Summary and discussion
The structure of charoite has been solved by the new method of ADT, coupled with PED. Despite the complex structure of charoite, the large number of reflections collected by ADT (more than 2800 unique reflections) and the strong reduction of dynamic effects enabled us to solve this structure ab initio and refine it in a fully automatic way using the same software which has been used for more than forty years in X-ray crystallography. The efficiency and reproducibility of the solution path offered by ADT is a glimpse of a significant step forward in nanostructural investigations. Ongoing improvements for geometric corrections, complete 3D-reconstruction of diffraction patterns and dynamic refinement will, hopefully, lead to a fully automated structure routine for any class of nanomaterials, including organic molecules and other complex inorganic structures, such as zeolites.
The charoite structure is unique in that it consists of three different types of silicate chains; the radical [Si 17 (O,OH) 43 ] 18À builds up a new type of silicate chain À a tubular hybrid dreier quadruple chain (classification after Liebau, 1985).
We have solved the structure of polytype charoite-90 and obtained informations about other polytypes such as charoite-96 and charoite-2a. However, to understand the 'charoite story' more deeply it will be necessary to explore the polytypism in detail and solve the structure of other polytypes.
The polytypic features of charoite may be treated as in other Ca dreier chain silicate minerals (wollastonite, xonotlite, tobermorite etc.). In a more formal way, the polytypic features and one-dimensional disorder in charoite may be dealt with through the OD approach. A short account written by Prof. Stefano Merlino is given in the Appendix to this paper. case the mirror plane normal to c is lost, as well as the twofold axis || a, and valid operations for the whole structure are the l-POs m normal to b, and the two s-POs 2 1 || b and inversion centres.
It is remarkable that all the polytypes, as well as the disordered sequences, should present a common set of reflections, those with l = 2n. This set of reflections, the so-called 'family reflections', is always sharp, and in case of disordered sequences should display orthorhombic symmetry. The various polytypes give rise to distinct diffraction patterns for l = 2n+1, with continuous streaks for the disordered sequences.