A Review of the Structures of Vaterite: The Impossible, the Possible

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A Review of the Structures of Vaterite: The Impossible, the Possible, and the Likely Andrew G. Christy*,# Ocean and Climate Geoscience, Research School of Earth Sciences, The Australian National University, Canberra, ACT 2601, Australia ABSTRACT: The importance of the metastable vaterite form of CaCO3 is increasingly appreciated as an intermediate between amorphous calcium carbonate and more stable crystalline polymorphs in biogenic calcification pathways. Thus, there is a need to be able to identify nanocrystals of vaterite unambiguously, despite uncertainties about its exact crystal structure. The structures that have been proposed for vaterite are all based upon a common average structure with orientationally disordered carbonate groups. However, there is much ambiguity in the literature about the actual superstructure, and hence diffraction behavior of vaterite, some misinterpretation of data, and some claims that cannot be correct. This critical review of the literature aims to resolve the issue. The orientationally disordered (2H) structure is unlikely to occur, and the most plausible structures all share a √3 × √3 superstructure in the xy plane. Most experimental evidence is consistent with occurrence of 2M and 6H polytypes, but ab initio modeling suggests that these do not occur in their highest-symmetry forms, but have space-group symmetries C121 and P3221. Stacking faults in the xy plane are very common, and frequently result in packets of 4- or 6-layer periodicity, which can be regarded as nanoscale intergrowths of 4M and 6A1 polytypes.

1. INTRODUCTION

2. THE NATURE OF THE VATERITE CRYSTAL STRUCTURE The basic nature of the vaterite crystal structure has been known for nearly a century, although there remains controversy about the details. The mineral has a hexagonal substructure with a ≈ 4.1 Å and c ≈ 8.5 Å,16,17 and is optically positive with principal refractive indices ω = 1.55 and ε = 1.65.18 This optical character implies that the strongly bound carbonate groups are oriented parallel to the z axis rather than perpendicular to it,19 as is the case for optically negative calcite and aragonite. The density of vaterite is about 2.65 g/cm3, appreciably less than those of its polymorphs calcite (2.71 g/cm3) and aragonite (2.94 g/cm3). Thus, there are no pressure and temperature conditions at which it is thermodynamically stable relative to those polymorphs. The energetics of vaterite of varying degrees of order has been calculated relative to amorphous calcium carbonate, aragonite, and calcite.20 However, vaterite grows readily in the presence of organic molecules.21 The relative instability of the vaterite polymorph of CaCO3 contrasts with that of the rare-earth borates RBO3, where a phase isotypic with vaterite22 is stable between aragonite and calcite structures for rare-earth cations of intermediate size, with the stability range widening at T > 1200 °C.23 The structure is closely related to that of bastnäsite-(Ce), CeCO3F. In fact, regular interlayering of (CeF)- and (Ca□)bearing layers (where “□” indicates a vacant anion site) along the (pseudo)hexagonal z axis produces a suite of minerals such as synchysite-(Ce), CaCe(CO3)2F, and synthetic analogues.24

1

Vaterite (μ-CaCO3) was named by Meigen (1911) after Heinrich Vater, Professor of mineralogy at Tharandt, Sachsen, Germany, who first synthesized the phase in 1894 and suggested that it was distinct from calcite and aragonite.2 It was first discovered in nature as a biomineral in gastropods3 and subsequently as an abiotic secondary mineral in altered skarns by McConnell (1960).4 The history of its characterization as a mineral is more extensively reviewed in McConnell (1960) and Lippmann (1973).5 Vaterite as a biomineral has been identified from mollusc pearls,6−8 fish otoliths,9,10 ascidians,11,12 and even human heart valves.13 As a metastable phase, it is likely to occur as a transient intermediate between an amorphous calcium carbonate (ACC) precursor phase and more stable calcite or aragonite.14 Local structure as determined from pair distribution functions and spectroscopic data for various types of ACC have been discussed in terms of priming for crystallization as vaterite rather than calcite;15 however, such an interpretation relies critically on knowledge of the local environments that are present in the vaterite structure. It is thus important to be able to identify and localize biogenic vaterite for full understanding of calcification mechanisms, and for its likely influence on incorporation of trace elements. However, detection of nanoor microcrystalline vaterite requires spatially resolved, structuresensitive techniques such as selected-area electron diffraction and associated high-resolution TEM imaging, which needs the diffraction behavior and crystal structure of vaterite to be well understood. © 2017 American Chemical Society

Received: April 4, 2017 Published: May 8, 2017 3567

DOI: 10.1021/acs.cgd.7b00481 Cryst. Growth Des. 2017, 17, 3567−3578

Crystal Growth & Design

Review

prismatic interstices which lie in layers ∥ (002). The CO3 groups occupy half of the Ca6 interstices in one layer and the other half in the next layer, so as to surround each Ca2+ ion in an approximate octahedron, while the arrangement of carbon groups is approximately hexagonal close-packed. The pseudocell of Kamhi (1963)17 is hexagonal, space group P63/mmc, with a0 = 4.13 Å, c0 = 8.49 Å, and Z = 2[CaCO3], and has CO3 groups split between three 1/3-occupied orientations around threefold axes as shown in Figure 1. The O atoms at the same z

These minerals are thus classic examples of modular materials,25 in this case forming a “polysomatic series” since the bulk composition varies depending on the relative proportions of different types of layers with distinct compositions. Vaterite is the (Ce, F)-free end-member of the polysomatic series.4,26 These minerals exhibit several different types of structural variation,27 including (i) formation of ordered macroscopic and nanophases with distinctive periodic stackings of Ca- and REE-bearing layers (polysomes); (ii) occupational and orientational order producing superstructures in the xy plane; (iii) disorder of stacking, and also reordering of stacking to produce polytypes of different symmetries but similar stoichiometry.28−31 There is scope for such structure variation even while maintaining the fixed composition CaCO3. Thus, it is not surprising that additional Bragg peaks and diffuse features in diffraction patterns of vaterite were reported by early workers.4,17 These indicate that both short- and long-range ordering occur, with that the latter produces a supercell. However, experimental studies and ab initio modeling from then up to the present have not yielded a consistent picture of the superstructure(s) formed by vaterite. In fact, 16 distinct ordered superstructures have been proposed in studies to date.4,5,26,32−34 All the structures reported from experimental refinements of diffraction data and from ab initio models share a common substructure. However, there is disagreement about the exact way in which the following structure-building mechanisms give rise to an ordered superstructure: (i) the gross location of the carbonate group within trigonal prisms defined by the Ca atoms (see below); (ii) the ordering pattern of carbonate groups within an xy layer, as indicated by orientations of the most nearly horizontal C−O vectors, and the resulting superstructure formed in the xy plane; (iii) the stacking pattern of layers along the z direction, and overall periodicity of the three-dimensional structure; (iv) slight departures from maximum possible symmetry, due primarily to small rotations of the carbonate groups. It is important to note that since vaterite does not form macroscopic crystals, the true structure can be inferred only from transmission electron microscopic imaging, ab initio modeling in order to generate and refine model structures, refinement of model structures against electron diffraction of nanocrystals and X-ray diffraction data from powders, and spectroscopic techniques (NMR and Raman, in particular). Enough data now seems to be available from a wide enough range of such sources to be able to narrow down the possibilities, and to resolve the confusion in the literature resulting both from the wide range of structures discussed for which there is now more evidence against than in favor and from spurious claims due to the misinterpretation of data. The main aim of the present study is to distinguish those structures proposed in the literature for vaterite which are likely to occur, from those which are unlikely for various reasons or are even theoretically impossible. First, we review the various types of structural variation which differentiate the different model structures that have been reported to date.

Figure 1. (a) Basic pseudocell of vaterite after Kamhi (1963),17 viewed down the z direction. (b) Perspective view approximately down [110]. Each carbonate group shows threefold orientational disorder. Large orange spheres = Ca, small black spheres = C, red spheres = O.

coordinate as C atoms are labeled O1, while the other O atoms are O2. Note that it is not possible sterically for carbonate groups to occupy trigonal prisms that are vertically above one another, since that would result in unfeasibly short distances of 2.04−2.12 Å between O2 oxygen atoms of neighboring carbonate groups, rather than the already short O2···O2 distances of ≥2.66 Å when carbonate groups are located, as per the Kamhi model (Figure 1b). Hence, a periodic structure must always have an even number of carbonate layers, although the c/2 substructure is very marked in diffraction: McConnell (1960) noted only one reflection with l = odd out of 13 indexed electron diffraction maxima.4 Bunn (1945)19 noted that the “vertical” orientation of carbonate groups with the CO3 triangle plane ∥ z is not compatible with both full long-range order and hexagonal symmetry. Therefore, the unit cell of the substructure necessarily has orientational disorder of carbonate groups. However, two horizontal C−O vectors cannot both point toward a shared edge of their trigonal prisms, since this would result in an unfeasibly short O···O distance of 1.49 Å (Figure 2a). An alternative disposition of carbonate groups that avoids this problem was proposed by Lippmann (1973), who calculated a model structure for vaterite based on the structure of high-temperature YbBO3.5,22 In this model, the carbonate groups are rotated 180° about [001] relative to those of Kamhi (Figure 2b). However, all recent ab initio models and powder refinements show the carbonate positions to be closer to those of Kamhi than those of Lippmann. In fact, even the YbBO3 structure used as an analogue of vaterite by Bradley et al. (1966)22 may be erroneous: refinements before and after that one give either the P63/mmc disordered Kamhi substructure35 or a slightly distorted P63/m variant of this.36 Because the O1

3. TYPES OF STRUCTURAL VARIATION IN VATERITE 3.1. The Disordered Substructure of Vaterite. The Ca2+ and CO32− ions of vaterite are arranged similarly to the Ni and As atoms of nickeline, NiAs, and hence inversely relative to the Ca and CO3 arrays of aragonite.26 Thus, the Ca2+ ions form a primitive-hexagonal array, and sit at the corners of trigonal 3568



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3.2. Orientational Ordering of Carbonate Groups within a Layer. Various ordering patterns have been proposed for the carbonate groups within a layer of the vaterite structure (Figure 3). In all cases, the resulting symmetry of a layer is

Figure 3. Diagrammatic representations of highest-symmetry carbonate ordering patterns in a single layer of vaterite structure. Columns ∥ z of Ca atoms at z = 0 and 1/2 are shown by orange circles, edge-on CO3 groups at z = 1/4 by narrow black wedges, with the acute angle indicating the direction of the C−O1 vector. Unit meshes are indicated by thick lines; outlines of Ca6 triangular prisms are shown with thin lines. (a) Kamhi structure of Figure 1, with disordered carbonates and layer symmetry −6m2. (b) 1 × √3 superstructure with all CO3 equivalent, Cm2m layer symmetry. (c) 1 × √3 superstructure with Pb21m layer symmetry. (d) √3 × √3 superstructure with P6̅2m layer symmetry.

lowered from the P6̅m2 symmetry of the disordered layer in Figure 3a. Ordering models have been proposed in which all carbonates in a given layer share the same orientation (Figure 3b), occur in two out of the three possible orientations (Figure 3c), or in all three (Figure 3d). Each case has a different layer symmetry. Furthermore, the possible stacking patterns for a pair of adjacent layers have different symmetries again, and the difference between single-layer and layer-pair symmetries determines a range of possible stacking operations between layers that is different for each ordering pattern, in accord with order−disorder (OD) theory.25,39−41 OD structures in which the component layers have the same composition are the special case known as polytypes.25,42 Formal OD theoretical analysis has been applied to vaterite recently, but only for the layer type of Figure 3d;38 the other superstructures would require separate treatment because of the symmetry differences. Since the energies are usually very similar for different stacking patterns in a polytypic family, the stacking sequences are readily controlled by minor or trace impurities, small environmental fluctuations, and kinetic factors. Most such families exhibit several different ordered polytypes with stacking sequences of various periodicities, and stacking disorder is observed frequently. The ordered layers of Figure 3b−d give rise to distinct series of possible polytypes, and examples of all of these have been proposed as model structures for vaterite. A polar orthorhombic 2-layer structure with layers of the type shown in Figure 3b, stacking to give an overall space group C2cm (usually cited in the setting Ama2) has been modeled in several studies.21,33,34 Layers of the type shown in Figure 3c stack to give the 2-layer orthorhombic Pbnm structure of Meyer

Figure 2. (a) Ordered √3 × √3 superstructure of the Kamhi disordered structure of Figure 1, viewed down z. Note the impossibly short distance between two O1 atoms if neighboring CO3 groups point toward the edge shared by their Ca6 prisms. (b) Alternative “Lipmann” orientation of carbonate groups.

atom is far (2.87 Å) from any Ca atoms, it would be very underbonded unless the CO3 group were strongly distorted so as to give very short C−O1 and much longer C−O2 distances; there is no spectroscopic evidence for the corresponding degree of distortion, which also detracts from the plausibility of the Lippmann model, so it will not be further considered in any detail. Avoidance of short O1···O1 distances means that strong orientational ordering of CO3 groups within a layer is to be expected. Wang et al. (2014)21 discussed the energetic unfavorability of having two neighboring C−O1 vectors both pointing toward the shared edge of their Ca6 prisms in terms of electrostatics, but the distance is so very short that Born repulsion is also important. Interatomic repulsions make it very unlikely that vaterite ever actually occurs with orientational disorder on the unit-cell scale, although the Kamhi structure was regarded as the “favored model” in recent papers.11,12,37 Makovicky (2016)38 noted that the steric problems would make twin (or other domain) boundaries within a single layer unfavorable, so that long-range order within a layer is to be expected, while the possibility of disorder remains along the layer stacking direction. 3569



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symmetry of the first layer (Figure 4b−d). In general, any oddnumbered layer may be in position A, B, or C, and any evennumbered layer in position A′, B′, or C′. Such nomenclature has been used to describe vaterite stacking sequences by Demichelis et al. (2013).43 Note that it is incorrect to refer to A−C as “orientations” (cf., De La Pierre et al.44), since each layer contains carbonate groups in all three orientations. The ABC notation locates individual layers relative to a fixed origin, and can distinguish different twin orientations of a given polytype. However, it obscures the fact that there is no special relationship between positions A and A′: the A → A′, A → B′, and A → C′ vectors all have equal status. An alternative notation that captures the overall symmetry better considers the relative azimuthal orientations in the xy plane of the successive stacking vectors. The full set of six vectors, three mapping odd-numbered layers onto the following evennumbered layer and the other three for mapping even layers onto odd, is shown in Figure 5.

(1959)26 and subsequently modeled,21,33,34 as well as the slightly distorted P212121 variant of Demichelis et al. (2012).34 The Meyer structure is the earliest model proposed for ordering of the carbonate groups in vaterite and is still frequently referred to in the literature. However, it will be seen below that the layer types of Figure 3b,c probably do not occur in vaterite, and so the Pbnm Meyer structure is unlikely to exist. The majority of recent experimental evidence supports superstructures with the ordering pattern of Figure 3d, which has a √3 × √3 unit mesh like that of Figure 2a, but the carbonate groups are ordered differently so as to avoid short O···O distances. This breaks the threefold rotation axes that were running along columns of Ca atoms ∥ z in Figure 2a. The layer of Figure 3d instead has local threefold axes that run through the empty Ca6 trigonal prisms. However, it will be seen below that such layers cannot be stacked without breaking this symmetry. 3.3. Layer Pairs in Vaterite. It will be seen that stacking two adjacent layers of the type seen in Figure 3d necessarily breaks the local hexagonal symmetry. Therefore, for most structures generated from such layers, it is more conventional to use not the hexagonal mesh with a = b = √3a0 = 7.15 Å, γ = 120° shown in that figure, but a centered rectangular unit mesh with a = 3a0 = 12.39 Å and b = √3a0. Note that a0 is the a parameter of the Kamhi subcell shown in Figure 1. Figure 4a

Figure 5. Full set of possible stacking vectors for the layer type of Figure 3d, and how they inter-relate the ABC positions of successive layers. Figure 4. (a) Centered-rectangular unit mesh for the layer of Figure 3d, showing three nonequivalent trigonal prism positions A, B, and C. Edge-on CO3 groups shown as black wedges, with the acute angle indicating the direction of the C−O1 vector. (b−d) Three locally equivalent ways to position a second layer above that of (a), with CO3 groups shown as gray wedges.

After each vector such as A → A′ in Figure 5, the next vector may be rotated 60° clockwise about the z axis (A′ → B), 60° anticlockwise about z (A′ → C), or 180° (A′ → A). These three alternatives may be symbolized as +, −, and 0, respectively. Note that a sequence of such symbols remains invariant under many choices of origin and starting orientation, although reflection in a vertical mirror plane will interconvert + and − operations. The set of six possible stacking vectors is similar to those of the polytypes of mica and similar phyllosilicates,45−47 although in those minerals, all six vectors are possible between layers, and rotations of 0° or ±120° between successive vectors are favored, while angles of 180° or ±60° are unknown or rare. In vaterite, 0° or ±120° relative rotations between successive vectors are impossible. The ABC notation superficially resembles that used in discussing close packings of spheres, but differs in that only two different stacking vectors are possible between adjacent layers of such spheres. 3.4. Maximum-Degree-of-Order Polytypes. A fundamental concept in OD theory is the “maximum-degree-of-

shows the √3 × √3 superstructure again with such a centered mesh, and the three symmetrically distinct types of empty Ca6 trigonal prism identified by the letters “A”, “B”, and “C”. In the figure, the unique empty prisms which have three C−O1 vectors all pointing directly away from them are located at the A positions; this is sufficient information to orient all the carbonate groups in the layer. In principle, it would be equally possible to have the unique prism at the B or C positions. The next layer of the structure must have its empty Ca6 prisms lying above those that are occupied by carbonate groups in Figure 4a. There are similarly three choices of origin for that layer, which analogously can be called A′, B′, and C′. The first layer of the structure may be followed by a second layer in any of these three positions, which are equivalent under the local threefold 3570



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order” (MDO) polytype. Out of the potentially infinite set of different repetitive stacking sequences that can be envisaged, a small number will be particularly regular and symmetrical, in that (i) all layer pairs are related by symmetry and (ii) all layer triplets are also related by symmetry. These are the MDO structures, and any less regular polytype can be described as an interlayering of packets of these “end member” structures. For layers of the type seen in Figure 3d and Figure 4, there are three MDO structures, corresponding to the stacking sequences 000000, +-+-+-, and ++++++. The corresponding patterns of stacking vectors are shown in Figure 6, along with identifying

Figure 6. Diagrammatic representation of stacking vector sequences for the three MDO polytypes of vaterite, with Ramsdell symbols and also stacking sequences in +/−/0 notation. Numbers indicate positions of successive layers.

Figure 7. Carbonate orientations for MDO polytypes of vaterite. Layer plane is the xy plane in all cases. Edge-on carbonate groups represented by narrow wedges, while orange circles are columns of Ca atoms normal to page. (a) 2O structure (space group Cmcm) viewed down z. Carbonate groups at z = 1/4 in black and z = 3/4 in gray. (b) 2M structure (space group C12/c1). Although every second layer is equivalent, the inclined z direction means that six layers are required before vertical superposition above the first layer is attained. Thus, this is a six-layer cuboidal cell with z height of carbonate groups as follows: red = 1/12, orange = 1/4, yellow = 5/12, green = 7/12, blue = 3 /4, and purple = 11/12. (c) Side-on view of the orthogonal cell for 2M polytype (dashed), showing the 2-layer cell with oblique z axis (solid line). (d) 6H structure (space group P6122), in which carbonates of successive layers progressively rotate around a screw hexad axis.

symbols for the resulting structures in the notation of Ramsdell (1947),48 as widely used for mica polytypes. The Ramsdell symbol consists of the number of layers in the repeat unit followed by a capital-letter abbreviation for the crystal system. For instance, 2O indicates an orthorhombic structure with a 2layer periodicity. Later extensions may distinguish multiple structures with the same crystal system and periodicity by subscripts, as used for micas45 and in the more complex case of the hydrotalcites,49 or indicate multiple periodicities in different directions or indicators of additional superstructures superimposed upon the stacking.50−53 The actual orientations of carbonate groups in these structures are shown schematically in Figure 7. The 2O structure has space group Cmcm; for the 2M structure, the normally redundant “1” elements are included in the symbol C12/c1 to render unambiguous the orientation of the diad axis and glide plane (contrast the case of 4M, below). The 6H structure has helical symmetry and can occur in two enantiomorphic mirror-image forms with space group symbols P6122 and P6522, respectively. Note that the helical stacking sequence of the 6H polytype restores hexagonal symmetry, so the hexagonal setting is used for x and y axes in Figure 7d. 3.5. Near-MDO Polytypes. Any stacking sequence of vaterite layers that is not one of those in Figures 6 and 7 will have more than one type of symmetrically distinct layer pairs, layer triplets, or both. However, there are three possible structures that are uniquely “semiregular” in that they have only one kind of layer pair or one kind of triplet. These are both of the possible 4-layer structures (4O, space group Cc2m, stacking sequence +0−0, and 4M, space group C2/c11, sequence ++ - -) and an additional 6-layer structure (6T, space group P3221, sequence +0+0+0). The stacking vector sequences for these are shown in Figure 8 and the corresponding orientation patterns of carbonate groups in Figure 9. Note that although the symmetry of the 4M structure is only monoclinic, it has a vertical z axis (β = 90°).

Figure 8. Stacking sequences for the three near-MDO polytypes.

Superstructure diffraction spots corresponding to four-layer structures have been reported several times.17,26,54 However, all 3571



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introduction of regular stacking faults (Figure 10). The MDO and non-MDO polytypes discussed above are summarized in Table 1.

Figure 9. Schematic orientation patterns of CO3 groups viewed down z for (a) 4O structure and (b) 4M structure, for which color scheme is z = 1/8; yellow, z = 3/8; green, z = 5/8; and blue, z = 7/8. (c) 6T structure. Color scheme is the same as in Figure 7c. Note that the threefold screw axes run through the red-yellow-blue carbonate columns, not through the Ca atoms.

these authors claimed it had hexagonal symmetry, which is not physically possible for a four-layer structure. The highest symmetry for a fully ordered 4-layer polytype is orthorhombic, although apparent hexagonal symmetry could be produced by superposition of multiple twin orientations of the 4O or 4M structures. Later studies11,12,33 can also be interpreted in terms of the presence of domains of 4-layer periodicity in vaterite with stacking disorder (see below). Makovicky (2016)38 described the 6T structure, and appeared to identify it with a structure of the same P3121 (or P3221 in its mirror-image twin form) space group symmetry that had been modeled previously.21,34 However, the latter structure is quite different, being a slight distortion of the 6H polytype away from its ideal P6122/P6522 symmetry (see below). The richness of possible structural behaviors in vaterite is such that there are frequently several distinct structures with similar unit cell dimensions and space group symmetries that can be conflated in this way. 3.6. Other Non-MDO Polytypes. As the number of layers in the repeating sequence increases, the number of crystallographically distinct polytypes with that periodicity rises rapidly. Apart from the 6H and 6T structures already mentioned, there are an additional 18 distinct polytypes with 6-layer periodicity, which have at least two distinct types of both layer pairs and layer triplets, and at most orthorhombic symmetry overall, although most are triclinic. Six have right-prismatic cells with β = 90°, while the rest have a 1/3a offset between successive packets of six layers, analogous to the situation shown for the 2M polytype in Figure 7c, requiring a smallest unit cell with c cos β = −a/3 and hence β ≈ 99°. Only one of these irregular structures is referred to in the literature. Mugnaioli et al. (2012)55 suggested that a non-MDO structure with six layers per cell would be a good approximation to the incommensurately modulated superstructure apparent in their electron diffraction patterns; they proposed the triclinic structure with the stacking sequence ++--+- (which we label 6A1, to distinguish it from the other 6A sequences). This structure can be derived either from the 2M or 4M polytypes by the

Figure 10. Stacking sequence of 6A1 polytype (left) with sequence of layer positions highlighted. (center) Derivation from packets of the 2M sequence by introduction of regular stacking faults (red dashes). (right) Analogous derivation from the 4M sequence. Gray dashed arrows indicate the stacking vectors that would have been present if the sequence continued that is on the other side of the fault.

Table 1. Possible Vaterite Polytypes Discussed in Texta polytype 2O 2M

figure

6H

7a 7b, 7c 7d

4O 4M

9a 9b

6T

9c

6A1

10

stacking vectors

unit cell parameters in units of a0, a0, c0

MDO 3 × √3 × 1, β ≈ γ = 90° 3 × √12 × cscβ, β ≈ 116°b, γ = 90° ++++++ 1 × 1 × 3, β = 90°, γ = 120° Near-MDO +0−0 3 × √3 × 2, β ≈ γ = 90° ++ - 3 × √3 × 2, β = 90°, γ ≈ 90° +0+0+0 1 × 1 × 3, β = 90°, γ = 120° Other Non-MDO ++ - - + - 3 × √12 × 3cscβ, β ≈ 99°c, γ ≈ 90° 00 +-

ideal space group symmetry Cmcm C12/c1 P6122

Cc2m C2/c11 P3121 C1̅

a

Unit cell edges are measured in terms of those of the Kamhi subcell (Fig. 1), a0 = 4.13 Å and c0 = 8.49 Å. bβ = tan−1(a0/c0). cβ = tan−1(a0/ 3c0.

3.7. Additional Rotations of CO3 Triangles. Apart from the 180° rotation about z that relates the Kamhi17 and Lippmann5 configurations of the carbonate anion discussed above, small tilts of the CO3 triangle have also been reported from vaterite structure models. The ideal version of the Kamhi carbonate triangle has the point symmetry 2mm (Figure 11a), although the polytypic stackings discussed above may force the 3572



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two more O2 at 2.90 Å (Figure 12a). The Ca−O bond-valence parameters of Brese and O’Keeffe56 give bond valences of,

Figure 11. Small rotations (ca. 10°) of the CO3 triangle away from the ideal Kamhi orientation. (a) The original orientation, where the local point symmetry of the CO3 triangle is 2mm. (b) Rotation about z preserves only the horizontal mirror plane. (c) Tilt about horizontal axis that preserves twofold rotation but no mirror planes. (d) Tilt about horizontal axis that preserves vertical mirror plane only.

loss of some of the symmetry elements. However, relaxation of ab initio models suggests that some superstructures are not energetically stable relative to versions in which small atomic displacements reduce the symmetry below that of the most symmetrical form possible.34 The desymmetrization is largely due to small tilts of the carbonate groups, and implies that the high-symmetry vaterite structures are rich in low-frequency lattice vibrational modes that are prone to softening. Tilting the carbonate triangle about principal axes reduces its point symmetry as follows: (i) tilt of ≪180° about [001]: maximum symmetry reduced to 11m (horizontal mirror plane), shown in Figure 11b; (ii) tilt about the C−O1 vector (e.g., [11̅0] in Figure 1): maximum symmetry is 211 (Figure 11c); (iii) tilt about an axis perpendicular to the plane of the CO3 group (e.g., [100] in the axial setting of Figure 1): maximum symmetry is 1m1 (Figure 11d); (iv) combinations of the above: no rotational or mirror symmetry. Many of the published vaterite structures show such small tilts of the carbonate triangles away from their highestsymmetry orientations. The structure models of Meyer26,32 all feature rotations of about 10° of type (i), as shown in Figure 11b, while the model of Lipmann5 has a lower-than-maximum symmetry implying a tilt of the type in Figure 11c. As noted above, none of those structures can now be regarded as likely to occur. However, the lowest-energy configurations calculated by Demichelis et al. (2012)34 are all slight displacive distortions of the 2M and 6H polytypes discussed above, in which the symmetry is reduced by small tilts of the carbonate groups. In the latter case, three desymmetrized variants of the polytype can be distinguished, which are here labeled 6H-P65, 6H-P3221, and 6H-P1121. In structures with carbonates in the Kamhi configuration and maximum possible symmetry, each Ca atom is surrounded by eight oxygen atoms: 2 × O1 at 2.26 Å, 4 × O2 at 2.44 Å, and

Figure 12. Interatomic distances for local configurations in vaterite, if carbonate groups are in maximum-symmetry orientations. (a) A portion of two carbonate layers and surrounding Ca atoms, viewed approximately down [110̅ ], showing the irregular eightfold coordination of Ca. (b) The same structure viewed down z, showing nonbonded O···O distances. Ca1 atoms are shown as darker orange spheres; Ca2 in paler color.

respectively, 0.45, 0.28, and 0.08 valence units (v.u.) for these three distances, and a reasonable bond-valence sum for Ca of 2.18 v.u. (Figure 12a). If carbonate triangles are ordered as assumed in the present study, so as to produce a √3 × √3 superstructure of the Kamhi subcell, then there are (at least) two symmetrically different types of Ca atom: the two short Ca−O1 vectors are directly opposite each other (trans) for Ca1, but are on the same side of the central atom (cis) for Ca2 (Figure 12b). Long-period or low-symmetry polytypes may have various arrangements of these trans and cis types of Ca coordination polyhedron; small rotations of CO3 groups have a strong effect on Ca−O distances, and can reduce the coordination number of adjacent Ca2+ ions from 8 to 7 or 6.34 Note that there are only six distinct CO3 groups around each Ca atom, because the Ca−O1 bonds and long Ca−O2 bonds occur in pairs, and form asymmetrical bidentate couplings between Ca and CO3. Outside an individual CO3 triangle, the only nonbonded O···O distances less than 3 Å for 3573



Review

O1 are 4 × O2 of neighboring carbonate groups at 2.97 Å, and for O2, 2 × O1 at 2.97 Å and 2 × O2 at 2.98 Å (Figure 12b).

4.2. Possible but Unlikely Structures. Model structures have been proposed in which only one or two carbonate orientations are present per layer, as in Figure 3b and c above. Such layers can give rise to whole families of polytypes that are not covered in OD treatment of Makovicky.38 However, these structures seem unlikely to occur, since they fit diffraction and spectroscopic data poorly, as shown below. A 2-layer orthorhombic Pbnm structure based on the layer type of Figure 3c was proposed by Meyer,26 but not supported by any experimental data. Attempts to fit experimental powder XRD data to this structure have yielded poor fits.21,33 The relaxed ab initio model of Medeiros et al.60 has a somewhat low density (2.606 g/cm3), while Demichelis et al.34 found this structure to spontaneously relax to an enantiomeric (P212121) distorted form with an unacceptably high density (2.744 g/ cm3). The sensitivity of unit cell parameters and density to the choice of modeling method was discussed by Wang and Becker.59 A poor weighted residuum Rwp of 22.3% was obtained in trying to refine XRD data against this model,21 with many superstructure peaks not fitted. The structure was not considered in the calculational Raman study of De La Pierre et al.,44 since all carbonate groups in it are symmetrically equivalent, inconsistent with spectroscopic data. The ab initio study of Wang and Becker59 calculated this structure to be less stable than the disordered Kamhi structure. A polar orthorhombic C2cm (≡ Ama2) 2-layer structure based upon the layer of Figure 3b was originally proposed as a subcell by Le Bail et al. (2011) as the best fit to their powder XRD data, albeit with a residuum Rwp = 13.4%, slightly worse than they obtained for refinement of the Kamhi subcell, and with an unacceptably small volume.33 Wang et al. (2014) refined this model against their synchrotron XRD data to Rwp = 17.9% with an acceptable density (2.656 g/cm3), but noted that it failed to fit many superstructure peaks.21 Furthermore, this structure was calculated by Demichelis et al. to be 15−18 kJ mol−1 less favorable than any of the other seven configurations that they modeled.34 In any case, the layer structure may not actually be correct for the sample of Le Bail et al.;33 they observed five weak superstructure reflections which they did not index and which, like those observed by Kamhi,17 were consistent with a 4-layer C supercell with tripled a and doubled c repeats, and thus of dimensions 3 × √3 × 2 relative to the Kamhi subcell; all the additional reflections can be indexed on such a cell. The cell tripling in the xy plane requires a more complex ordering pattern of carbonate groups within a layer than that of Figure 3b, but is consistent with that of Figure 3c. Thus, the layer structure proposed by Le Bail et al.33 is not actually consistent with their superstructure reflections, and it seems reasonable to regard their diffraction pattern as coming from layers with the √3 × √3 mesh of Figure 3d, in one or both of the 4M or 4O near-MDO polytypes of Figure 8, which are discussed further below. All the vaterite structures for which there is reasonable experimental support are then based on stackings of the same type of layer, with the √3 × √3 mesh. These are discussed below. 4.3. Likely Structures. The structures for which there is most supporting evidence all have layers with the threefold ordering pattern of Figure 3d. The 2M polytype of Figure 6 and Figure 7b,c above was first proposed on the basis of 3D diffraction data reconstructed from automated diffraction tomography.55 It also arose in molecular dynamics simulations for particular boundary conditions,20 and

4. IMPOSSIBLE, UNLIKELY, AND LIKELY STRUCTURES 4.1. Impossible Structures. The most extreme cases in the literature of vaterite structures that are not possible even in theory are the 4-layer polytypes of purported hexagonal symmetry.17,32,54 It is simply not possible to order the orientations of the carbonate groups in the pattern of Figure 3d while preserving hexagonal symmetry, and any apparent diffraction evidence for hexagonal symmetry would presumably be a consequence of nanoscale twinning of a monoclinic or orthorhombic structure such as those of Figure 8 above. It is possible to produce 4- as well as 2-layer polytypes with hexagonal symmetry if the carbonates are oriented as in the model of Lippmann (1973),5 but as noted above, this gives a very long Ca···O2 distance. The coordinates of Lippmann result in a Ca−O2 bond valence of only 0.19 v.u. and extreme underbonding of Ca, with a bond-valence sum of only 1.11 v.u, rendering the idealized Lippmann model implausible without considerable modification. The need for O2 to form a double bond to C would also result in extreme distortion of the carbonate group. In the absence of strong evidence for it, structures based on the Lippmann configuration can be discounted. For the steric reasons mentioned earlier, long-range ordering of carbonate groups in the xy plane is virtually inevitable, so the disordered 2H Kamhi structure of Figure 1 must also be regarded as “impossible” in practice, although again, fine-scale twinning of a 2O or 2M structure (Figure 6) may be misinterpreted to imply the existence of disordered vaterite. Molecular dynamics simulations show that the disordered structure is energetically unfavorable with respect to ordered superstructures: Wang and Becker (2012)20 calculated the disordered structure to be 11 kJ/mol more endothermic than the 6H polytype (below), and carbonate orientational ordering was a cooperative process with an activation enthalpy of about 94 kJ/mol. Sato and Matsuda (1969)57 favored the disordered structure on the basis of infrared data, but their argument in favor of 2mm symmetry for the carbonate group over lower point symmetries is not actually conclusive. Raman spectra indicate that vaterite contains multiple distinct carbonate sites of lower point symmetry;10,44 13C NMR data also shows more than one distinct C environment,58 which again is inconsistent with the disordered structure. A rather different type of “impossible” model is that in which the unit cell parameters result in a density which is unreasonably far from that expected for vaterite. This can occur for unit cell parameters that have been refined against powder diffraction data if the indexing scheme for the peaks is incorrect. It may also occur for ab initio structure models that have relaxed into an energy minimum far away from the most stable one. The cell parameters of the Kamhi subcell17 imply a density for vaterite of 2.651 g/cm3. While most of the unit cells for various structure models reported in the literature have calculated densities close to this value, the full range is 2.56− 2.78 g/cm3, and it is reasonable to assume that densities that are outside the 2.60−2.70 g/cm3 window correspond to incorrect structures. The excluded unit cell data include those for the P6522 (6H) structure of Wang and Becker,59 the C12/c1 (2M) and C1̅ (6A1) structures of Mugnaioli et al.,55 and the C2cm and P212121 (relaxed from Pbnm) structures of Demichelis et al.34 3574



Review

was found to give an adequate if not the best fit to XRD data by Wang et al.21 Mugnaioli et al.55 noted the strong distinction in intensity between strong substructure (h = 3n) and weak superstructure reflections, and that the structure has two crystallographically different types of CO3 group with different symmetries, which would give rise to the three distinct Raman stretching bands which have been noted for vaterite samples of a wide variety of origins.10 The 2M structure is isostructural with high-temperature (Y0.92Er0.08)BO3.61 It is interesting to note that at low temperatures, the boron atom in that compound links to a forth oxygen, so that B3O9 triangles are formed and the structure becomes that of pseudowollastonite2M, Ca3[Si3O9].62 The electron diffraction patterns of Mugnaioli et al.55 show superstructure reflections lying on continuous diffuse streaks ∥ c* that indicate stacking disorder, mirror symmetry about c* that indicates twinning on (001), and also diffuse maxima along the streaks. They noted that the main superstructure reflections such as 400 were apparently displaced along the c* direction by about 0.17 c* due to overlap with strong modulation peaks, which they modeled by introducing stacking faults every sixth layer to produce a commensurate approximant superstructure (the 6A1 polytype). Strong modulation spots occur near l = odd positions due to violation of the c glide symmetry, and examination of their intensity profile along the line 40l shows weak maxima spaced at 1/4 c0* and 1/3 c0* intervals, suggesting that packets of local 4-, 6-, and 8-layer periodicity occurred. Additional closely spaced peaks suggest longer periodicities of 60−200 Å, but these may represent beat frequencies between shorter periods or shape transform fringes, since they are commensurate with the crystal size. However, strong preferences for particular distances between defects, mediated by elastic strain, have been observed in other systems.63 Rietveld refinements of synchrotron powder XRD data converged for both 2M and 6A1 structure models, but with lower residua in the latter case. The 6A1 structure is discussed further, below. Although the 2M unit cell of Mugnaioli et al.55 has an implausibly high density (2.777 g/cm3), this is not the case for the cell of Wang et al.21 or for the two slightly distorted variants obtained by Demichelis et al.,34,43 after minimizing energy of their model structure. The distortions can be interpreted as due to the collapse of a low-frequency phonon mode, causing small static rotations of the carbonate groups, adjustments to Ca−O distances, and loss of the center of inversion symmetry. The C121 and C1c1 structures of Demichelis et al. were more stable than the C2/c transition state by 1.3 and 0.5 kJ/mol, respectively, the C121 structure being the second-most stable structure that they found for vaterite, after the P3221 variant of the 6H polytype.34,43 The C121 structure was also one of the two whose calculated Raman spectra fitted the experimental data well in the study of De La Pierre et al.,44 along with their P3221 structure (below). Conversely, their C1c1 variant of the 2M structure gave a very different line profile. The 6H polytype of Figure 6 and Figure 7d is enantiomorphic, with a maximum space group symmetry of P6122 or its mirror image P6522. This structure was first modeled in the molecular dynamics study of Wang and Becker,59 who found it to be the most stable ordered superstructure, although a P3221 distortion of it was extremely similar in minimum energy. Again, the modeling of Demichelis et al.34,43 implied that the 6H structure was most stable in distorted variants of lower symmetry, due to collapse of

different soft phonon modes. Their three energy-minimized structures had space groups P65, P3221, and P1121, which correspond, respectively, to softening of vibrational modes with A2, B1, or E2 symmetry and suggest that the parent structure has several such low-frequency modes on the verge of collapse. The P6522 form gave the best fit to powder XRD data out of the structure models examined by Wang et al. (2014),21 fitting the superstructure peaks well and with Rwp = 16.7%, better than 17.3% for the Kamhi structure, while conversely, Le Bail et al.33 found that the 6H structure fitted their diffraction data poorly, failing to predict observed superstructure peaks while predicting many more that were not observed. Wang et al. noted that the P3221 variant of this structure also fitted their XRD data well, and was calculated to be only ∼1 kJ/mol different in energy from the P6522 form.21 A neutron diffraction study on vaterite otoliths from the Lake Sturgeon Acipenser fulvescens favored the P6522 structure over several of the other alternatives discussed here,64 but did not consider lower-symmetry variants of the 6H polytype other than P3221. The energy-minimized models of Demichelis et al. (2012)34 show that the slight distortions to P65, P3221, and P1121 symmetry have marked effects on the coordination numbers of Ca sites (6, 6 + 7, and 6 + 7 + 8, respectively) and on the distribution of Ca−O bond distances. All of these structures were calculated to be less than 1.5 kJ/ mol more stable than the fully symmetrical P6522 polytype, with the P3221 variant most stable by a very small margin. Demichelis et al. suggested that thermally induced dynamical fluctuation between the distorted structures could occur,34 which was considered feasible since the low-symmetry variants of a given polytype can interconvert via only small rotations of carbonate groups. However, spectroscopic data do not favor this possibility. De La Pierre et al. (2014)44 found that calculated Raman spectra matched experiment about equally well for the P3221 form of this structure and the C121 variant of the 2M polytype. The P65 and P1121 distortions of the 6H structure showed distinctively different profiles in their calculated Raman spectra, which did not fit the experimental data as well. Thus, the P3221 form emerges as the best supported variant of the 6H polytype. As noted above, X-ray diffraction data can be interpreted as evidence for the presence of a 4-layer polytype (Figure 8).17,26,33,54 Electron diffraction patterns can also show weak superstructure diffraction spots which, if indexed on the Kamhi subcell, have h = 3n and l = half-integer, corresponding to a 4layer structure.7,11 Calculations using POWDER CELL65 show that the 4M structure has X-ray diffraction intensities that are much stronger for l = half-integer superstructure reflections than for l = odd, while the converse is true for 4O. Combined with the fact that the 4M structure is more readily produced by faulting of 2M, it is likely that local packets of this stacking sequence are present. The 6A 1 polytype (Figure 10) was proposed as a commensurate approximation to an incommensurate modulated variant of 2M;55 it gave a better-quality Rietveld fit to synchrotron powder XRD data than did 2M. While the density for the unit cell in that study was implausibly large, this was not the case for the relaxed and distorted variants modeled by Demichelis et al.,34,43 who found the C1̅ structure to be a transition state between two slightly distorted energy minima of C1 symmetry, which differed from each other in energy by less than 0.2 kJ/mol, and were both comparable in stability to the P1121 variant of the 6H polytype (above). However, Raman spectra calculated for the unit cells of Demichelis et al.43 did 3575



Review

not give good fits to experimental data,44 suggesting that 6A1 does not occur as a macroscopic polytype of vaterite, although small domains of it may be commonly intergrown in the 2M phase. 4.4. Intergrowth of Multiple Structures. Kabalah-Amitai et al.11,12 claimed that biogenic vaterite in the spines of the ascidian Herdmania momus consists of a coherent nanoscale intergrowth of two different structures, one the disordered 2H substructure of Kamhi17 and the other unknown, but with one lattice repeat = √3a0 and topotactically related, since it is oriented at 30° to the (100) fringes of the 2H phase. However, this appears to be a misinterpretation of their TEM data, which nevertheless provides valuable insights. In particular: (i) The beam direction is incorrectly identified in Figure 2c of Kabalah-Amitai et al.11 Although [01̅1] diffraction patterns theoretically have a pseudohexagonal geometry very similar to this one, the l = odd reflections would be superstructure peaks that should be absent or systematically much weaker than l = even spots. However, the indexed spots are all of comparable intensity, suggesting that the viewing direction is actually [001]. This is confirmed by the visibility of unindexed weak, diffuse superstructure spots in the positions expected for the √3 × √3 superstructure, and by the authors’ note that the FIB section plane was oriented nearly normal to the spine length. Thus, this is a [001] diffraction pattern of an ordered superstructure, not a [011̅ ] pattern of the 2H phase, which as noted above is effectively impossible. (ii) They interpret their image Figure 4 to show discrete nanodomains of separate phases with single sets of lattice fringes at, respectively, 2.07 and 3.63 Å; a simpler interpretation is that these are domains within a single crystal viewed down [001], that are slightly tilted so as to enhance diffraction of, respectively, (100) and (110) fringe sets (using the axial setting of Kamhi17). The relative misorientation is probably due to deformation. (iii) Their Figure 3 does not represent diffraction from two superimposed phases, as claimed: it is a classic example of a pattern from a polytypic material, showing two distinct populations of diffraction spots from a single phase. The true periodicity is not visible for the rows of reflections which have h = 3n if indexed on a cell with the √3 × √3 mesh in the xy plane; these rows arise from the “superposition structure” of all possible stackings, and feature very strong substructure reflections at a spacing corresponding to the thickness of a single layer, interspersed with “non space-group absences”.25,66 The latter are not forbidden by symmetry but arise only from small deviations of individual layers away from identicality, and can be orders of magnitude less intense than substructure reflections. Conversely, the weak spots in rows with h ≠ 3n are superstructure reflections that arise from major layer displacements and show the true stacking periodicity, and lie on diffuse streaks ∥ c* due to stacking disorder. Interestingly, the spots are displaced from the positions l = 2n ± 2/3 that would be expected for the 2M polytype, to positions close to l = 2n ± 1/ 2. These spots are compatible only with very disordered stacking, but with a strong tendency to 4-layer periodicity. Along with the corresponding high-resolution TEM images, these patterns provide the best evidence to date for such a periodicity in some vaterite.17,32,33 Thus, the TEM images and diffraction data of KabalahAmitai et al.11 do not necessarily correspond to an unknown second phase intergrown with disordered 2H vaterite, but instead can be interpreted as arising from the 2M polytype with

frequent stacking faults, producing local packets of 4M structure. Note that the 4M structure can be regarded as a periodically faulted relative of 2M, analogous to the 6A1 structure proposed to explain similar diffraction behavior55 (Figure 10). The 4M polytype more closely resembles 2M in local geometry than does 4O, and hence is more likely to be produced in this way. The samples of Le Bail et al. (2011) and Mugnaioli et al. (2012) were synthetic rather than biogenic, and prepared by different methods,33,55 so local regions of 4- and 6layer periodicity due to frequent and regularly spaced stacking faults appear to be widespread in vaterite from diverse origins. 4.5. Consensus. The experimental data in the literature are consistent with orientational ordering of carbonate groups within a layer of the vaterite structure so as to produce a √3 × √3 mesh. The resulting structures then all belong to a single OD family.38 The OD nature of the vaterite structure means that energy differences between different stacking configurations are small. The occurrence of multiple polytypes on the micro- to macroscopic scale, as well as considerable stacking disorder, are both to be expected. High-resolution TEM images of vaterite do indeed show abundant stacking faults,10,11,21,55 and the experimental data support the occurrence of several polytypes, in some cases as slightly distorted variants with lower than the maximum possible symmetry. The structures that best fit most of the ab initio modeling, diffraction, and Raman data are the 2M polytype (Figure 7b,c) with symmetry reduced to C121 due to small carbonate rotations of the type shown in Figure 11c, and the 6H structure with symmetry analogously reduced to P3221. The P3221 and C121 structures are also most favored by a recent 43Ca NMR study of vaterite.68 Regularly spaced stacking faults produce local intergrowths of non-MDO longer-period structures, particularly 4M and 6A1. Energy-minimized models show that the 6A1 polytype also undergoes small rotations of carbonate groups which eliminate the center of inversion and reduce the symmetry to C1.43 The same may also be true of the 4M structure, in which case its true symmetry would be C211 rather than C2/c11. Unit cell parameters have been refined for energy-minimized ab initio models with the 2M-C121, 6H-P3221, and 6A1-C1 structures; these are summarized in Table 2. The overall periodicity and highest-possible symmetry crystal system of a vaterite polytype may be deduced from the positions of the superstructure reflections in h ≠ 3n rows of an electron diffraction pattern or the corresponding peaks in a powder XRD pattern. Different polytypes with the same size of Table 2. Unit Cell Data from Refined Models for the Most Likely Vaterite Polytypes

3576

polytype and space group

a/Å

b/Å

c/Å

α, β, γ/°

2M - C121

12.245

7.197

9.305

90, 115.16, 90 90, 90, 120 90, 90, 120 90.46, 99.78, 90.24 90.43, 99.88, 90.29

2M - P3221

7.1239

7.1239

25.3203

2M - P3221

7.1438

7.1438

25.398

6A1 - C1

12.353

7.102

25.733

2M - C1

12.358

7.106

25.741

density/ gcm−3

ref

2.687

34

2.688

34

2.665

21

2.690

43

2.687

43



Review

(16) von Olshausen, S. Z. Kristallogr. - Cryst. Mater. 1924, 61, 463− 514. (17) Kamhi, S. R. Acta Crystallogr. 1963, 16, 770−772. (18) Johnston, J.; Merwin, H. E.; Williamson, E. D. Am. J. Sci. 1916, s4-41 (41), 473−512. (19) Bunn, C. W. Chemical crystallography; Clarendon Press: Oxford, 1946. (20) Wang, J.; Becker, U. Am. Mineral. 2012, 97, 1427−1436. (21) Wang, J.; Zhang, F.; Zhang, J.; Ewing, R. C.; Becker, U.; Cai, Z. J. Cryst. Growth 2014, 407, 78−86. (22) Bradley, W. F.; Graf, D. L.; Roth, R. S. Acta Crystallogr. 1966, 20, 283−287. (23) Levin, E. M.; Roth, R. S.; Martin, J. B. Am. Mineral. 1961, 46, 1030−1055. (24) Donnay, G.; Donnay, J. D. H. Am. Mineral. 1953, 38, 932−963. (25) Ferraris, G.; Makovicky, E.; Merlino, S. Crystallography of Modular Materials; IUCr Monographs on Crystallography, Vol. 15; Oxford University Press, 2004. (26) Meyer, H. J. Versammlungsberichte. Angew. Chem. 1959, 71, 673−681. (27) van Landuyt, J.; Amelinckx, S. Am. Mineral. 1975, 60, 351−358. (28) Wu, X.; Yang, G.; Meng, D.; Pan, Z. Mineral. Mag. 1998, 62, 55−64. (29) Ni, Y.; Post, J. E.; Hughes, J. M. Am. Mineral. 2000, 85, 251− 258. (30) Meng, D. W.; Wu, X. L.; Mou, T.; Li, D. X. Mineral. Mag. 2001, 65, 797−806. (31) Meng, D.; Wu, X.; Han, Y.; Meng, X. Earth Planet. Sci. Lett. 2002, 203, 817−828. (32) Meyer, H. J. Zeitschr. Kristallogr. 1969, 128, 183−212. (33) Le Bail, A.; Ouhenia, S.; Chateigner, D. Powder Diffr. 2011, 26, 16−21. (34) Demichelis, R.; Raiteri, P.; Gale, J. D.; Dovesi, R. CrystEngComm 2012, 14, 44−47. (35) Newnham, R. E.; Redman, M. J.; Santoro, R. P. J. Am. Ceram. Soc. 1963, 46, 253−256. (36) Hosokawa, S.; Tanaka, Y.; Iwamoto, S.; Inoué, M. J. Mater. Sci. 2008, 43, 2276−2285. (37) Greer, H. F.; Liu, M.-H.; Mou, C.-Y.; Zhou, W. CrystEngComm 2016, 18, 1585−1599. (38) Makovicky, E. Am. Mineral. 2016, 101, 1636−1641. (39) Dornberger-Schiff, K. Acta Crystallogr. 1956, 9, 593−601. (40) Ď urovič, S. In Modular Aspects of Minerals; Merlino, S., Ed.; European Mineralogical Union Notes in Mineralogy; Eö tvö s University Press: Budapest, 1997; Vol. 1, Chapter 1, pp 1−28. (41) Merlino, S. In Modular Aspects of Minerals; Merlino, S., Ed.; European Mineralogical Union Notes in Mineralogy; Eö tvö s University Press: Budapest, 1997; Vol. 1, Chapter 2, pp 29−54. (42) Christy, A. G.; Zvyagin, B. B. Advanced Mineralogy Vol. 1: Composition, Structure and Properties of Mineral Matter: Concepts, Results and Problems; Marfunin, A. S., Ed; Springer-Verlag: Berlin, Heidelberg, 1994; pp 106−124. (43) Demichelis, R.; Raiteri, P.; Gale, J. D.; Dovesi, R. Cryst. Growth Des. 2013, 13, 2247−2251. (44) De La Pierre, M.; Demichelis, R.; Wehrmeister, U.; Jacob, D. E.; Raiteri, P.; Gale, J. D.; Orlando, R. J. Phys. Chem. C 2014, 118, 27493− 27501. (45) Smith, J. V.; Yoder, H. S. Mineral. Mag. 1956, 31, 209−235. (46) Ross, M.; Takeda, H.; Wones, D. R. Science 1966, 151, 191− 193. (47) Takeda, H.; Ross, M. Am. Mineral. 1995, 80, 715−724. (48) Ramsdell, L. S. Studies on silicon carbide. Am. Mineral. 1947, 32, 64−82. (49) Bookin, A. S.; Drits, V. A. Clays Clay Miner. 1993, 41, 551−557. (50) Guinier, A.; Bokij, G. B.; Boll-Dornberger, K.; Cowley, J. M.; Ď urovič, S.; Jagodzinski, H.; Krishna, P.; de Wolff, P. M.; Zvyagin, B. B.; Cox, D. E.; Goodman, P.; Hahn, Th.; Kuchitsu, K.; Abrahams, S. C. Acta Crystallogr., Sect. A: Found. Crystallogr. 1984, 40, 399−404. (51) Christy, A. G. Am. Mineral. 1988, 73, 1134−1137.

unit cell may also be distinguished by significant differences in calculated intensities for their superstructure peaks, as in the case of the 4-layer structures discussed above. This approach is well established for other polytypic systems such as micas67 and hydrotalcites.49 However, the small deviations from ideal symmetry are more challenging to characterize. In some cases, additional weak reflections may occur due to loss of glide or screw symmetry elements. For example, 6H-P3221 may show a diagnostic 003 reflection at 8.49 Å, while reflections 002 and 004 at 12.74 and 6.37 Å might be visible for 6H-P1121; none of these can be present for 6H-P6522 or 6H-P65. However, pairs such as 6H-P6522 and 6H-P65 or 2M-C12/c1 and 2M-C121 cannot be distinguished this way, and if single crystals are not available for structure determination, then peak profiles in Raman spectra appear to be a more reliable method for structure characterization at this level of detail.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Andrew G. Christy: 0000-0002-2203-1444 Present Address #

Queensland Museum, 122 Gerler Road, Hendra, Qld 4011, Australia, and School of Earth Sciences, the University of Queensland, St Lucia, Qld 4072, Australia. Notes

The author declares no competing financial interest.

■

ACKNOWLEDGMENTS I thank Prof. Stephen M. Eggins for his support of this project through his ANU Vice-Chancellor’s Funding, Dr. Stuart J. Mills for his assistance with conversion of crystallographic data file formats, and both Dr. Mills and A/Prof. Dorrit E. Jacob, as well as two anonymous reviewers for their helpful comments on earlier drafts of this manuscript.

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REFERENCES

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A Review of the Structures of Vaterite: The Impossible, the Possible

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