Order-disorder (OD) structures of Rb₂Zn(TeO₃)(CO₃)·H₂O and Na₂Zn₂Te₄O₁₁

3.1.1 OD layers

The crystal structure of Rb₂Zn(TeO₃)(CO₃)·H₂O is built up of an alternation of two kinds of crystal-chemical layers, A¹ and A², extending parallel to (100) (Figure 1). The A¹ layers ([TeO₃]²⁻, Rb1⁺, Zn²⁺) possess higher (pseudo)-symmetry than the A² (Rb2⁺, CO₃²⁻, H₂O) layers, which leads to an ambiguity in the arrangement of the layers as described below. The sequence number of the layer is indicated in the subscript: … A¹ _n A²_n+1A¹_n+2A²_n+3….

Figure 1:

The crystal structure of Rb₂Zn(TeO₃)(CO₃)·H₂O viewed down [010]. Layer names are indicated to the right, alternative stacking arrangements are omitted. Radii for all atoms are fixed at an arbitrary value.

The geometric parameters of the following description are derived from a model ignoring the minor stacking arrangement (“phantom atoms”). The Te1 atom is coordinated by three oxygen atoms with almost the same distance (1.860(2)–1.872(2) Å; Table 2). The resulting [TeO₃]²⁻ polyhedron (Figure 2) has the commonly observed trigonal-pyramidal shape. The BVS was calculated to 4.04 valence units (v.u.) and 3.94 v.u. using the revised parameters [28]. The [TeO₃]²⁻ units are isolated from each other but are connected with three [ZnO₄]⁶⁻ units by corner-sharing.

Figure 2:

Coordination polyhedra of the Te, Zn and Rb atoms in the structure of Rb₂Zn(TeO₃)(CO₃)·H₂O. Displacement ellipsoids are drawn at the 90% probability level. Symmetry codes refer to Table 2.

The Zn1 atoms are coordinated by four oxygen atoms forming a slightly distorted tetrahedron (Figure 2). The BVS of Zn1 is 2.00 v.u.. The [ZnO₄]⁶⁻ units are isolated from each other but share three of their corners with [TeO₃]²⁻ groups and the fourth corner (O4) with the carbonate group. In this way, ²_∞[ZnTeO₃] layers are formed extending parallel to (100). One side of this layer is connected to a layer of Rb1 atoms, the other side with the carbonate groups. This leads to two ²_∞[ZnTeO₃] layer pairs with a Rb1 layer situated in-between – the A¹ layer (Figure 3a).

Figure 3:

The (a) A¹ and (b) A² layers of Rb₂Zn(TeO₃)(CO₃)·H₂O projected on the layer plane (100). The (pseudo) symmetry operations are indicated using the standard graphical symbols [29]. Atomic radii are as in Figure 1.

The Rb1 position is coordinated by ten oxygen atoms with distances ranging from 2.961(2) to 3.240(4) Å. The BVS of Rb1 was calculated to 1.06 v.u.. Rb2 is surrounded by six oxygen atoms with bond lengths in the range of 2.894(4)–3.063(4) Å and three additional oxygen atoms at distances of 3.510(4)–3.596(4) Å resulting in a [6+3] coordination. The BVS amounts to 0.90 v.u. considering the six closer oxygen atoms only. This value increases to 0.99 v.u. when the three more distant oxygen contacts are also taken into account.

C1, O4, O5 and O6 form the carbonate ion (Figure 3b). O4 is part of the ²_∞[ZnTeO₃] layer and is positioned at a farther distance from C1 than O5 and O6, both of which do not have many other close bonding partners in their vicinity. The O7 position corresponds to the crystal water molecule, as the BVS of O7 amounts to only 0.19 v.u..

3.1.2 OD groupoid family

The structure belongs to a category IV OD family [22] of two kinds of non-polar layers. The OD groupoid layer symbol reads as

according to the notation of Grell and Dornberger-Schiff [30]. The first line in the symbol indicates the layer names, the second line the (idealized) symmetry of the layers. Accordingly, the A¹ layer (Figure 3a) possesses idealized A(1)2/m1 and the A² layer (Figure 3b) P(1)2₁/c1 symmetry. In the OD notation, capital Bravais symbols indicate the three-dimensionality of the layers (as opposed to the translation lattice) and parentheses give the direction missing translation.

The third line of the symbol gives one possible relative position of the two adjacent layers, viz. the origins of A¹ and A² are connected by the vector rb+a₀ (Figure 3). The variable r is used instead of a fixed value, because OD groupoid families abstract from metric parameters in analogy to space group types. In this particular OD groupoid family (monoclinic point group with monoclinic axis parallel to the layer plane) no second parameter is needed. The translational component in the [001] direction is fixed for all stacking possibilities and reflected by the inclination of the a₀ vector [31].

In Rb₂Zn(TeO₃)(CO₃)·H₂O the parameter r adopts the value 0, which is expressed by the symbol

and has consequences on the possible arrangements of the layers, as shown in the next section.

3.1.3 NFZ relationship

The NFZ relationship determines the number of stacking possibilities of a layer contact given a particular set of metric parameters. It reads as Z = N/F = [ _n : _n ∩ _n+1]. _n is the group of operations of the A _n layer that do not invert the orientation of the layer with respect to the stacking direction. For A¹ layers _n  = A(1)m1 and for A² layers _n  = P(1)c1. _n ∩ _n+1 is the subgroup of _n of those operations that also apply to the adjacent layer.

Since r = 0, the c glide reflection planes of the A¹ and A² layers overlap. Therefore, in all cases, _n ∩ _n+1 = P(1)c1. [ _n : _n ∩ _n+1] is the index of _n ∩ _n+1 in _n . Ultimately, we obtain for A¹ _n → A²_n+1 contacts Z = [A(1)m1:P(1)c1] = 4/2 = 2 and for A² _n → A¹_n+1 contacts Z = [P(1)c1:P(1)c1] = 2/2 = 1. In other words, given an A¹ layer, the adjacent A² layers can be placed in two possible ways whereas for an A² layer there is only one way of placing the A¹ layer.

The two possibilities of placing the A² layers are given by the coset decomposition of P(1)c1 in A(1)m1. In particular, they are related by the A-centering (translation along b/2+c/2) of the A¹ layers. These two possibilities are denominated as A²⁺ and A²⁻.

3.1.4 Polytypes

The crucial point of the NFZ relationship is that the pairs of generated layers are geometrically equivalent. Thus, an infinity of polytypes, with arbitrary distributions of the A²⁺ and A²⁻ positions can in principle be constructed, which are all locally equivalent. If interatomic interactions over more than one layer-width are neglected, these polytypes are also energetically equivalent.

However, experience shows that the stacking in ordered polytypes are often particularly simple. This can be explained by weak interactions over more than one layer width, which favor one of the geometrically non-equivalent triples of adjacent layers. Polytypes that cannot be decomposed into even simpler polytypes (containing only a subset of the triples, quadruples, and generally n-tuples) are said to be of a maximum degree of order (MDO) [32, 33].

In Rb₂Zn(TeO₃)(CO₃)·H₂O there are two MDO polytypes:

1. MDO₁: … A¹A²⁺A¹A²⁺ …, P2₁/c, a = 2a₀ (Figure 4a)

2. MDO₂: … A¹A²⁺A¹A²⁻ …, I2/c, a = 4a₀ (Figure 4b)

Figure 4:

Schematic representations of the local symmetry of (a) the MDO₁, (b) the MDO₂ polytypes, and (c) the family structure. Triangles are red on one and blue on the other side or grey if they are symmetric by reflection at the drawing plane. A translational component of b/2 is represented by darker colors. Symmetry elements valid for the whole structure are drawn in red. The grey backdrop indicates the unit cell.

They play a special role in OD theory, because all polytypes can be decomposed into fragments of MDO polytypes.

The family structure of an OD family is a fictitious polytype, which corresponds to an equal overlay of all stacking possibilities. The family structure in Rb₂Zn(TeO₃)(CO₃)·H₂O has A2/m symmetry with a = 2a₀ [Figure 4c]. The A(1)2/m1 symmetry of the A¹ layers is valid for the whole structure and the symmetry of the A² layers is increased to A(1)2/m1 by ‘multiplying in’ the A-centering of the A¹ layers. Such an overlay of two A² layers is called A^2±.

3.1.5 Diffraction

The diffraction pattern of Rb₂Zn(TeO₃)(CO₃)·H₂O (Figure 5) is composed of distinct rods with sharp reflections and rods with more diffuse reflections, as it is characteristic for OD structures with translationally equivalent layers [34]. The sharp reflection peaks are located on rods k+l even and correspond to the diffraction pattern of the A(1)2/m1 family structure. All polytypes, ordered or disordered, contribute equally to these family reflections. The peaks resulting from diffuse reflections on the k+l odd rods are called characteristic reflections, because they are characteristic for an individual polytype. In the crystal under investigation, the characteristic reflections correspond to the MDO₂ polytype. Thus A²⁺A¹A²⁻ MDO₂ triples are more common than A²⁺A¹A²⁺ (or the equivalent A²⁻A¹A²⁻) MDO₁ triples. However, the diffuseness of the characteristic reflections means that there is also a tangible number of the latter, i.e., a high stacking fault probability.

Figure 5:

(h2l)^∗ plane of Rb₂Zn(TeO₃)(CO₃)·H₂O. Reciprocal lattice vectors are given with respect to the smallest common sublattice.

The occupancy (occ) ratio of the two orientations of the A₂ layers refined to 86.12:13.88(17), which would correspond to 2occ−1 ≈ 72% of MDO₂ and 28% MDO₁ (note that refining a pure MDO₁ crystal in the MDO₂ setting would result in a 1:1 overlay of the A₂ layers). However, such quantitative interpretations are treacherous, since random stackings are not equivalent to an overlay of MDO polytypes. Moreover, the broad peaks lead to systematic errors in the intensity evaluation, the ‘Ďurovič effect’ [35]. A proper evaluation of the stacking probabilities would require the modelling of the diffuse scattering.

3.2.1 OD-description

Na₂Zn₂Te₄O₁₁ can be considered as a category I OD structure built of one kind of non-polar layers A _n , where n is a sequential number. The layers are built of a ²_∞[Te₄O₁₁]⁶⁻ network and adopt (idealized) orthorhombic P2₁2(2) symmetry (Figure 6a). In principle, the Na and Zn atoms form a crystal-chemically separate layer (Figure 6b). However, since they are located on (or in very close vicinity to) sites that fulfill the symmetry of both adjacent ²_∞[Te₄O₁₁]⁶⁻ layers, they can be considered from an OD point of view as being located at the interface between the OD layers.

Figure 6:

Layers parallel to (001) in the crystal structure of Na₂Zn₂Te₄O₁₁ centred at z = 0 (a) and z = ⅛ (b). Te atoms are green, O atoms red, Zn atoms blue and Na atoms yellow. Radii for all atoms are fixed at an arbitrary value of 0.2 Å.

3.2.2 Crystal chemistry

For the four unique Te sites, two types of [TeO _x ] polyhedra are present in the structure of Na₂Zn₂Te₄O₁₁. Te1 and Te3 are coordinated by four oxygen atoms (Table 3) in a bisphenoidal shape, which can be derived from a trigonal bipyramid with the 5s ² lone pair (ψ) occupying an equatorial position. Te2 and Te4 are surrounded by three oxygen atoms and form a trigonal pyramid (Figure 7). The BVS of the Te atoms are 3.95 (Te1), 4.11 (Te2), 4.06 (Te3) and 4.07 (Te4) v.u.. Using the revised parameters [28], BVS of 4.03 (Te1), 4.04 (Te2), 4.06 (Te3) and 3.99 (Te4) v.u. are obtained.

Figure 7:

[TeO₃]²⁻ and [TeO₄]⁴⁻ coordination polyhedra in the crystal structure of Na₂Zn₂Te₄O₁₁. Displacement ellipsoids are drawn at the 90% probability level. Symmetry codes refer to Table 3.

The [TeO₃]²⁻ and [TeO₄]⁴⁻ units are connected to each other by a common corner forming linear [Te₄O₁₁]⁶⁻ units. Such [Te₄O₁₁]⁶⁻ units have already been described several times [5], usually in compounds with the composition M^III₂Te₄O₁₁. Using the notation introduced by Christy et al. [5], the Te–O units are denoted as (Δ–◊–◊–Δ). Additionally, each Te atom has another oxygen contact at distances between 2.734(5) and 2.808(5) Å. Through these contacts, each [Te₄O₁₁]⁶⁻ unit is linked to two neighbouring units resulting in ¹_∞[Te₈O₂₂]¹²⁻ (double)chains oriented either parallel to [100] or [010].

Each of the four Zn atoms is surrounded by four oxygen atoms between 1.925(5) and 2.007(6) Å (Table 3) forming an isolated and distorted [ZnO₄]⁶⁻ tetrahedron (Figure 8); the O–Zn–O angles are in a range of 101.6(2)°–126.8(2)°. The BVS of the Zn atoms are 2.02 (Zn1), 1.89 (Zn2), 1.94 (Zn3) and 2.00 (Zn4) v.u.. The Na atoms are coordinated by seven oxygen atoms at distances of 2.287(6)–2.962(5) Å (Figure 9). The [NaO₇]¹³⁻ polyhedra are connected to each other by the longest contacts to form chains oriented parallel to [001]. The BVS of the Na sites are 1.07 (Na1) and 0.99 (Na2) v.u..

Figure 8:

[ZnO₄]⁶⁻ tetrahedra in the crystal structure of Na₂Zn₂Te₄O₁₁. Displacement ellipsoids are drawn at the 90% probability level. Symmetry codes refer to Table 3.

Figure 9:

[NaO₇]¹³⁻ coordination polyhedra in the crystal structure of Na₂Zn₂Te₄O₁₁. Displacement ellipsoids are drawn at the 90% probability level. Symmetry codes refer to Table 3.

3.2.3 OD groupoid family

As it is the case for Rb₂Zn(TeO₃)(CO₃)·H₂O, a crucial factor of the OD description is the presence of different translation lattices of adjacent layers. In the former, the differing lattices were due to the different Bravais classes of the layers group. In the conventional setting, the basis vectors of the layer lattices are, however, identical. In contrast, adjacent layers in Na₂Zn₂Te₄O₁₁ possess different basis vectors when expressed in the conventional setting.

The smallest common sublattice^[1] of the lattices of all A _n layers in Na₂Zn₂Te₄O₁₁ is a rectangular lattice spanned by (a, b) with a = b = |a| = |b| = 15.2949(3) Å. The vector c₀ is perpendicular to (001) and has the length of one layer width (c₀ = |c₀| = 4.6946(2) Å).

The translation lattices of the A_2n (even) layers are spanned by (a, b/2) and of the A_2n+1 (odd) layers by (a/2, b), i.e., there are additional centering vectors at (0, ½, 0) ^T and (½, 0, 0) ^T , respectively, which will be expressed by the Bravais symbols X and X′. It follows that the orientations of the even and odd A _n layers are related by a rotation of ±90° about c₀. Owing to the orthorhombic layer group, and therefore a rectangular translation lattice, this can only be realized if a = b and γ = 90°. In particular, in the conventional (here primitive) setting of the P2₁2(2) layer group, the cell parameters must fulfill a = 2b. This represents a non-crystallographic layer lattice restriction, which may appear in OD structures of layers of non-equal lattices [36].

The OD groupoid family symbols for OD structures of layers of one kind list the layer symmetry and one possible set of σ-POs relating adjacent layers [22]. This notation has not been worked out for cases such as Na₂Zn₂Te₄O₁₁, where layers possess different lattices. However, the five placed symbols for tetragonal OD groupoid families can be adapted for this case. We suggest the following symbol

The first line is usually not part of the symbol but given here for convenience. It indicates the directions of the operations in the lines below. The second line gives the operations of a single layer (λ-POs), including the Bravais centering (X stands for the (0, ½, 0) ^T centering vector, see above). In the third line, the partial operations mapping a layer on an adjacent layer (σ-POs) are listed. The point group of the OD groupoid family (the group generated by the linear parts of all POs) is 4 ‾ 2 m [31].

As for the previously discussed case of Rb₂Zn(TeO₃)(CO₃)·H₂O, the metric parameters (r, s) adopt special values. Here, r = s = 0, which is due to the Na and Zn atoms that are located at the layer interfaces. This means that the axes of the 2_[001] operations of adjacent layers have to overlap. As it can be seen in Figure 10, this is the case for half of the 2_[001] axes. Again, this can be represented by the symbol

Figure 10:

The A_2n (X) and A_2n+1 (X′) layers of Na₂Zn₂Te₄O₁₁. The (pseudo)-symmetry operations are indicated using the standard graphical symbols [29]. Additional translational symmetry is indicated by the dashed unit cells. Na and Zn atoms were omitted, as they are half-assigned to two adjacent layers.

3.2.4 NFZ relationship

The NFZ relationship again is used to derive the stacking possibilities in the case of Na₂Zn₂Te₄O₁₁. For even and odd A _n layers _n  = X11(2) and _n  = X′11(2), respectively. In both cases _n ∩ _n+1 = P11(2) because half of the 2_[001] axes of adjacent layers coincide (Figure 10). Thus, given the position of an even A _n layer, there are Z = [X11(2):P11(2)] = 2 possibilities of placing the adjacent layers. The ambiguity arises owing to the different translation lattices. The origin of the adjacent (odd) layers may be located at nc₀ or b/2+nc₀, which corresponds to the X-centering vector. Z = 2 can also be obtained from the before-mentioned fact that only half of the 2_[001] axes are shared by adjacent layers.

The same argument can be made for odd A _n layers: Z = [X′11(2):P11(2)] = 2 and the origin of the adjacent (even) layers can be located at nc₀ or a/2+nc₀. The two possible locations of the A _n layers will be designated as A _n ⁺ and A _n ⁻, respectively.

3.2.5 Polytypes

There are two kinds of triples A _n A_n+1A_n+2 of consecutive layers: with the A _n and A_n+2 possessing the same (e.g., A⁺A^?A⁺) or different origins when projected on (001) (e.g., A⁺A^?A⁻). The two MDO polytypes are each formed by only one kind of triple:

1. MDO₁: … A⁺A⁺A⁺A⁺ …, P 4 ‾ 2 1 c , c = 2c₀

2. MDO₂: … A⁺A⁺A⁻A⁻ …, I 4 ‾ 2 d , c = 4c₀

All other polytypes can be decomposed into fragments of MDO₁ and MDO₂. In the family structure, even and odd n layers are an equal superposition of two layers related by translation along a/2 and b/2, respectively. Such a layer, which can be written as A^±, has P22(2) symmetry with the lattice basis (a/2, b/2). Adjacent layers are related by 4 ‾ and the overall symmetry of the family structure is P 4 ‾ 2 c with the lattice basis (a/2, b/2, 2c₀).

3.2.6 Family structure

First measurements on smaller crystals of Na₂Zn₂Te₄O₁₁ did not immediately reveal the actual stacking sequence as only strong reflections of the family structure were observed. The asymmetric unit of the family structure ( P 4 ‾ 2 c ; a = b = 7.6625(2) Å, c = 9.4336(3) Å, V = 553.88(3) Å^{3 [2]}) contains eight sites (one Te, two Zn, one Na and four O) located on five different Wyckoff positions. Te1, O1, O2 and O4 are located on the general 8n position. Zn1 lies on Wyckoff position 2f (site symmetry 4 ‾ ..), Zn2 on 2e (site symmetry 4 ‾ ..), Na1 on 4m (site symmetry 2..) and O3 on the 4i position (site symmetry .2.). As there is only one Te site, no distinction between the tellurium atoms with CN3 and CN4 exists anymore.

The oxygen sites, which are responsible for the various layer orientations, are only half occupied in this structure model since the different arrangements are all projected into a smaller unit cell. The O4 site, which corresponds to O1 and O7 in the main structure, has a very short distance of 0.714(11) Å to its own symmetry-equivalent and can therefore only be half occupied. The same can be said for the O3 site (corresponding to the O4 and O10 sites), which has a distance of 2.822(12) Å to its own symmetry-equivalent.

3.2.7 Diffraction

As has been discussed for Rb₂Zn(TeO₃)(CO₃)·H₂O, OD structures with translationally equivalent layers have characteristic diffraction patterns, where on rows with family reflections only these reflections are observed. No streaking or characteristic reflections are expected if the layers are perfectly translationally equivalent. In the case of Na₂Zn₂Te₄O₁₁, even and odd n layers are not translationally equivalent. However, since the Fourier transform is linear, the same argument can be applied when considering the two substructures of the even- and odd A _n layers only.

When indexing with (a^∗, b^∗, c ₀ ^∗) on rods h and k even, only sharp reflections are observed at l = n/2, n ∈ ℤ, corresponding to the family structure (c = 2c₀) (Figure 11a). The characteristic reflections originate from one of the two substructures (A _n with n = even and n = odd, respectively). The even A _n layers contribute to diffraction intensities of the odd-h rods and the odd A _n layers to the odd-k rods.

Figure 11:

(a) (4kl)^∗ plane of Na₂Zn₂Te₄O₁₁, (b) (3kl)^∗ plane of Na₂Zn₂Te₄O₁₁. Reciprocal lattice vectors are given with respect to the smallest common sublattice.

As listed in Table 4, sharp diffraction spots would be expected for MDO₁ at all h and k values, with l = n/2 (n ∈ ℤ) as the only restriction. This is not the case as only the family reflections are present on the h, k = even rods with no intensity in between. In the characteristic h or k = odd planes (Figure 11b) there are also reflections at l = n/4, which are in contradiction with a basis vector 2c₀.

For the MDO₂ stacking order, reflections are expected at h+k+4l = 2n, n ∈ ℤ. All observed reflections can be explained by the unit cell of the MDO₂ polytype, making it the preferred stacking order (Figure 12).

Figure 12:

The preferred (MDO₂) stacking arrangement of Na₂Zn₂Te₄O₁₁ in a projection along [010]. Te atoms are green, O atoms red, Zn atoms blue and Na atoms yellow. Atomic radii were fixed to an arbitrary value of 0.2 Å for clarity.

However, the group of reflections at h, k = odd, l = n/2 (n ∈ ℤ) (with respect to a^∗, b^∗, c ₀ ^∗), is systematically absent. Referring to (a^∗, b^∗, c^∗) of the MDO₂ polytype (Table 4; c = 4c ₀ ) this means that in the l = even planes all characteristic reflections are absent (only the family reflections remain), while in the l = odd planes the characteristic reflections are present. Again, this can be explained by the individual presence of the two substructures of the even- and odd A _n layers. The A_2n (n ∈ ℤ) substructure only allows the k = even reflections since its b is half of the MDO₂ unit cell. The B-centring of the substructure unit cell results in the condition h+l = 2n (n ∈ ℤ). In the l = even planes, where the absences appear, this results in the condition h = 2n (n ∈ ℤ) since l is already even. Therefore, only the reflections of the family structure (h, k, l = even) are present for the A_2n sublattice in the l = 2n (n ∈ ℤ) planes. The same reasoning is valid for the A_2n+1 sublattice, for which a summary is given in Table 5.

In the actual diffraction pattern, the characteristic h or k odd reflections are very weak (Figure 11b) since the deviation from the family structure is only due to the ordering of the O1, O4, O7 and O10 atoms. Distinct maxima are located at the positions expected for MDO₂, which means that the MDO₂ triples are preferred over MDO₁ triples. As in the case of Rb₂Zn(TeO₃)(CO₃)·H₂O, the reflection peaks are very broad and located on streaks of continuous diffuse scattering.

As discussed before, a refinement with different scale factors for characteristic and family reflections in JANA2006 [21] resulted in a ratio of 2.37. If P is the probability of an MDO₂ triple appearing and Q the probability of MDO₁, then the remaining intensity ratio should be P − Q P + Q = 1 2.37 . Given the condition of P+Q = 1, a value of P = 0.711 is obtained, which is very similar to the ratio of occupancy factors of oxygen atom (O1/4/7/10) and corresponding shadow atoms from the initial refinement (0.714/0.286(4)). P can also be interpreted as the probability that two adjacent layers are connected by a 4 ‾ + operation if the layers below have already been connected by a 4 ‾ + operation. It has to be noted again that interpretation of occupancies is treacherous in the case of intensities derived from diffuse scattering.

The fact that there is a preferred stacking order can be explained by the influence of the Na atoms. The Zn atoms are located exactly or almost exactly between the layers on sites with higher symmetry and do not have any bonds towards the “critical” O1/O7 and O4/O10 sites which create the differences between the A _n ⁺ and A _n ⁻ layers. In fact, the Zn atoms form bonds to all other oxygen sites except these four.

The Na1 and Na2 positions are located farther away from the idealized high-symmetric sites. Each Na site forms bonds towards both O4 and O10 and to one of O1 and O7. The deviation from the idealized position is the largest in the z-direction. The Na-atoms located in the gap between two ¹_∞[Te₈O₂₂]¹²⁻ chains are pulled closer towards the layer while the ones directly above and below the chains are pushed away and located closer to the next layer. This is responsible for the fact that two subsequent layers A _n and A_n+1 can either be positioned straight above each other or be translated by a/2 or b/2 but not translated by a/4+b/4. Furthermore, Na2 also has a dislocation of 0.115 Å from the idealized high-symmetric position within the plane. These small dislocations of the Na sites lead to a desymmetrization of the ideal OD-symmetry [37] and therefore are the presumable cause for the existence of the preference for the MDO₂ stacking over MDO₁.