As predicted and more: modulated channel occupation in YZn_5+x

Like many complex intermetallic phases, the crystal structures of REZn_5+x compounds (RE = lanthanide or Group 3 element) based on the EuMg₅ type have gradually unfolded. The original reports described a complex hexagonal structure with an unusual combination of tetrahedrally close-packed regions and open spaces, as well as observations of superstructure reflections. More recently, we reinvestigated the structure of YZn₅, reclassifying it as the EuMg_5+x-type compound YZn_5+x (x ≃ 0.2), in which disordered channels run along c through the spaces formerly considered open. In addition, DFT-chemical pressure (DFT-CP) analysis of ordered models of YZn_5+x highlighted paths for communication between neighboring channels setting the stage for superstructure formation. Herein, the experimental elucidation of this effect is presented with the synthesis and structure determination of a modulated form of YZn_5+x. By slow-cooling samples of YZn_5+x from the annealing temperature, crystals were obtained that exhibit satellite reflections with the modulation wavevector q = a* + b* + 0.3041c*. Structure solution and refinement using a (3+1)D model in superspace group P31c(σ₃)00s reveals incommensurate order in the structure's channels. Here, two Zn sites associated with the channels are present, each with discontinuous atomic domains that are slanted in the x₃x₄ plane. Their slanting corresponds to adjustments along the c axis for the presence or absence of close neighbors along that axis, while the occupation patterns of neighboring channels are shifted by of the modulation period. These features follow earlier predictions from CP analysis, highlighting how this approach can be used predictively in search of new phenomena.

1. Introduction

When encountering disorder in a newly solved crystal structure, one faces several questions. Could an ordered superstructure be obtainable under different conditions or through variations in the crystallization or synthesis methods (Etzkorn et al., 2007; Lidin et al., 2009; Okamoto et al., 2017; Becker & Leineweber, 2018)? Or may a superstructure be present, but obscured by the balanced twinning of domains related by pseudosymmetry (Feuerbacher et al., 2007)? Does the randomness suggested by the crystallographic model belie local order that is just not pronounced enough to be detectable as diffuse scattering in the data available (Frey, 1995; Larsson et al., 1996)? Together, such possibilities relate to the origins of the apparent disorder in frustration or near degeneracy (Keen & Goodwin, 2015), and the potential ways in which the occupancies or positions of atoms in one region of disorder impacts the preferences of others. Over the past ten years, the DFT-chemical pressure (CP) analysis (Hilleke & Fredrickson, 2018; Lu et al., 2021) has emerged as a theoretical approach to analyze these effects, as illustrated by its ability to reveal paths of soft atomic motion underlying the channel-like structures of the Nowotny Chimney Ladders (Lu & Fredrickson, 2019) and Fe₂Al₅ (Vinokur et al., 2019), as well as the packing issues faced by the disordered polyhedra in the quasicrystal approximant CaCd₆ (Berns & Fredrickson, 2013) and Fe₁₄Pd₁₇Al₆₉ (Peterson et al., 2020). More recently, this approach has been applied in a predictive fashion, in which CP analysis on models of the disordered EuMg_5+x-type compound YZn_5+x (x ≃ 0.2) pointed to a potential superstructure (Fredrickson et al., 2022), a proposal that we herein experimentally investigate.

Several REZn₅ compounds (RE = lanthanide or Group 3 transition metal) were originally reported by Fornasini and coworkers as adopting the EuMg₅ type (Mühlpfordt, 1970), with the observation that some crystals exhibited superstructure reflections corresponding to or × 6 supercells (Bruzzone et al., 1970; Fornasini, 1971). Intrigued by the structural relationship of these phases to the family of CP-driven variations of the CaCu₅ type (Berns & Fredrickson, 2014), as well as unusual open void spaces that appear in them, we recently reinvestigated the crystal structure of YZn₅ as a representative example (Fredrickson et al., 2022). Our structure solution revealed columns of disordered Zn atoms running along the hexagonal c axis [Fig. 1(a)], placing this compound in the EuMg_5+x type (Erassme & Lueken, 1987; Mühlpfordt, 1997), a disordered variant of the EuMg₅ type adopted not only in RE–Mg and RE–Zn systems but also observed in Ni- and Al-based intermetallics (Gladyshevskii et al., 1993; Bohdana et al., 2020).

Figure 1 The disordered form of YZn5+x (x ≃ 0.2) and its predicted superstructure from CP analysis. (a) The disordered YZn5+x structure is constructed with tetrahedrally close packed (tcp) units, Y triangles, and Y–Zn channels (yellow/red) that contain disordered Zn atoms (blue). (b) Selected CP features calculated for ordered models of YZn5+x, shown for points where neighboring Zn channels are bridged by tcp units, with the in-plane Zn positions in the channel being filled (left) or vacant (right). Black lobes correspond to directions of negative CP (where contraction is preferred), while white lobes indicate positive CP (where expansion of the structure would be favorable). (c) The superstructure predicted from the CP analysis, viewed down [001] with its basic cell outlined. Three colors are used for the channels to indicate the csuper shifts for the Zn occupation patterns between neighboring columns: yellow for Δz = 0, pink for Δz = , and green for Δz = .

While the X-ray diffraction patterns of these crystals showed no indication of a superstructure, DFT-CP analysis on a series of ordered models pointed to a mechanism for the occupation pattern of one channel to influence those of its surroundings [Fig. 1(b)]: neighboring channels in the ab plane are bridged through tetrahedrally close-packed (tcp) units (blue), which are subject to different CP effects depending on whether the adjacent Zn sites in the channel are occupied or vacant. In both cases, the occupation or vacancy of the channel atom on both sides of the tcp unit leads to an accumulation of stress, which could be relieved through a staggering of the occupation pattern of neighboring columns. Based on this scheme, we proposed a superstructure pattern [Fig. 1(c)] in which Zn atoms are distributed in an ordered fashion over two basic Zn positions along each channel to give rise to a tripling of the c repeat vector, with the patterns of neighboring channels being shifted relative to each other by one subcell.

In this article, we experimentally demonstrate and elucidate the occurrence of channel ordering in YZn_5+x. Through a simple change in the temperature treatment in the synthetic procedure—allowing the reaction to slowly cool to ambient conditions from the annealing temperature (as opposed to quenching them in ice water)—crystals are obtained that exhibit both sharp diffraction peaks for the EuMg_5+x-type basic cell and additional reflections at intermediate points in reciprocal space. While the latter correspond approximately to the expected supercell, their small number and relative weakness make solution as a conventional structure challenging. As we will see below, the structure solution and refinement proceed much more smoothly using the (3+1)D superspace approach (Janner & Janssen, 1980a,b; van Smaalen, 2007), in which the additional reflections are indexed as satellites with q-vector q = σ₁a* + σ₂b* + σ₃c* = a* + b* + 0.3041c*, and the structure is modeled as periodic in (3+1)D space with the additional dimension corresponding to the phase of a modulation function. The emerging crystal structure model confirms the general ordering pattern predicted by CP analysis, highlighting the predictive potential of CP-based models of structural effects.

2. Experimental

2.1. Synthesis

In our synthetic investigation of the Y–Zn system, elemental Y (Strem Chemicals, 99.9%) and Zn (Alfa Aesar, 99.9%) were used as starting materials. The elements were weighed out in the stoichiometric molar ratio of Y:Zn = 1:5 in an Ar-filled glove box, pressed into a pellet, and then placed into a fused silica tube, which was then evacuated and sealed. The sample was heated to 1100°C for 24 h to achieve a homogeneous melt, then annealed at 500°C for 168 h to allow for crystal growth, and finally cooled slowly to ambient temperature at a rate of 10°C h⁻¹.

2.2. Single crystal X-ray diffraction analysis

Single crystal X-ray diffraction data for YZn_5+x (x = 0.217) were collected on an Oxford Diffraction Xcalibur E diffractometer using graphite monochromated Mo Kα radiation (λ = 0.71073 Å) at ambient temperature. The collection and processing of the data set were performed using the CrysAlis Pro (v. 171.42.49; Rigaku Oxford Diffraction, 2022) software. The structure was solved with the charge flipping algorithm (Oszlányi & Sütő, 2004, 2005; Palatinus, 2004) using the program SUPERFLIP (Palatinus & Chapuis, 2007) and refined on F² using the program JANA2006 (Petříček et al., 2014). Further details regarding the refinements are given in Table 1 and the supporting information. The refined atomic coordinates and atomic displacement parameters for the basic structure are given in Tables 2 and 3, respectively.

Table 1
Crystallographic data for the modulated form YZn_5+x

Chemical formula	YZn5.217
Wavelength dispersive X-ray spectroscopy composition	YZn5.26 (3)
a (Å), c (Å)	8.8811 (1), 9.2057(1)
Volume (basic cell, Å^3)	628.812 (12)
Z	6
Modulation wavevector†	q = a* + b* + 0.3041c*
Superspace group‡
Crystal dimensions (mm)	0.193 × 0.137 × 0.092
Crystal color, habit	Metallic gray, prismatic
Data collection temperature	Ambient
Radiation source, λ (Å)	Mo Kα sealed tube, 0.71073
Absorption correction	Analytical
Min, max transmission	0.024, 0.125
θmin, θmax	3.27, 27.87
Refinement method	F^2
No. of reflections	24910
No. of unique reflections [I > 3σ(I), all]	777, 1450
Rint [I > 3σ(I), all]	5.24, 5.55
No. of parameters, constraints	90, 0
Overall reflections
R[I > 3σ(I)], Rw[I > 3σ(I)]	2.72, 7.88
R(all), Rw(all)	5.60, 8.76
S[I > 3σ(I)], S(all)	1.94, 1.53
Main reflections
R[I > 3σ(I)], Rw[I > 3σ(I)]	1.76, 6.58
R(all), Rw(all)	1.90, 6.63
Satellite reflections
R[I > 3σ(I)], Rw[I > 3σ(I)]	11.12, 18.87
R(all), Rw(all)	24.85, 23.72
Δρmax, Δρmin(e Å^−3)	4.73, −4.90

†Refined q vector without symmetry constraints: q = 0.3337(6) a* + 0.3314(6) b* + 0.3041(7) c*.
‡Symmetry operations: (x₁, x₂, x₃, x₄), (−x₂, x₁ − x₂, x₃, −x₂ + x₄), (−x₁ + x₂, −x₁, x₃, −x₁ + x₄), (−x₂, −x₁, −x₃ + ½, −x₄ + ½), (−x₁ + x₂, x₂, −x₃ + ½, x₂ − x₄ + ½), (x₁, x₁ − x₂, −x₃ + ½, x₁ − x₄ + ½), (x₂, −x₁ + x₂, −x₃, x₂ − x₄), (x₁ − x₂, x₁, −x₃, x₁ − x₄), (x₂, x₁, x₃ + ½, x₄ + ½), (x₁ − x₂, − x₂, x₃ + ½, −x₂ + x₄ + ½), (−x₁, −x₁ + x₂, x₃ + ½, −x₁ + x₄ + ½), (−x₁, −x₂, −x₃, −x₄)

Table 2
Refined atomic coordinates for the basic structure of YZn_5+x

Site	Multiplicity	x	y	z	Uequiv	Occupancy
Y	6	0.61475 (10)	0.80738 (5)	¼	0.0107 (3)	1
Zn1	6	0.86508 (10)	0.43254 (5)	¼	0.0117 (3)	1
Zn2	4			0.51174 (10)	0.0122 (3)	1
Zn3	6	1	½	½	0.0123 (6)	1
Zn4	12	0.8383 (2)	0.67858 (8)	0.41030 (6)	0.0139 (5)	1
Zn5	2	1	1	0	0.0196 (12)	0.988 (4)
Zn6	2	1	1	¼	0.0140 (9)	0.664 (4)

Table 3
Refined atomic displacement parameters for the basic structure of YZn_5+x

Site	U11	U22	U33	U12	U13	U23
Y	0.0123 (4)	0.0104 (3)	0.0102 (3)	0.00617 (19)	0	0.0067 (7)
Zn1	0.0118 (4)	0.0130 (3)	0.0099 (4)	0.0059 (2)	0	0.0002 (12)
Zn2	0.0115 (3)	0.0115 (3)	0.0134 (5)	0.00577 (16)	0	0
Zn3	0.0115 (4)	0.0122 (9)	0.0108 (4)	0.0040 (9)	−0.0017 (3)	−0.0004 (11)
Zn4	0.0162 (6)	0.0147 (4)	0.0140 (3)	0.0102 (7)	0.0002 (8)	0.0006 (2)
Zn5	0.0149 (6)	0.0149 (6)	0.029 (3)	0.0074 (3)	0	0
Zn6	0.0144 (7)	0.0144 (7)	0.013 (2)	0.0072 (4)	0	0

2.3. Powder X-ray diffraction analysis

Phase analysis of the sample was performed using powder X-ray diffraction measurements. The sample was ground to a fine powder and placed on a zero-background plate. Diffraction intensities were measured on a Bruker D8 Advance Powder Diffractometer fitted with a LYNXEYE detector, using Cu Kα radiation (λ = 1.5418 Å) at ambient temperature. An exposure time of 1.0 s per 0.010° increment was used over the 2θ range 30–80°.

2.4. Wavelength-dispersive X-ray spectroscopy

To determine the elemental composition of YZn_5+x, wavelength dispersive X-ray spectroscopy (WDS) was performed on a sample whose powder X-ray diffraction pattern showed this to be the major phase. To prepare the sample for WDS measurements, a small amount of material was suspended in a conductive ep­oxy at one end of a short segment of an aluminium tube. Once the ep­oxy had hardened, the sample was ground down to produce a flat surface and then polished to reduce the number of surface scratches using a diamond lapping film (Precision Surface International Inc., 0.5 µm). Finally, the sample was carbon-coated. WDS measurements guided by back-scattered electron images were taken with a Cameca SX-Five FE-EMPA Microprobe, using Y₂O₃, elemental Zn, elemental Si, and ZnO as standards for the Y Lα, Zn Kα, Si Kα, and O Kα transitions, respectively.

3. Results and discussion

3.1. Synthesis of incommensurately modulated YZn_5+x crystals

In our earlier investigation of the disordered YZn_5+x (x ≃ 0.2) structure, we uncovered channels of disordered Zn atoms running along the c axis, making this compound a member of the EuMg_5+x type (Fredrickson et al., 2022). We also noted, however, that a mechanism exists for the communication between the occupants of neighboring channels, opening a path to a potential ordered superstructure. Based on these interactions, we proposed a model for a superstructure. In our attempt to synthetically realize this superstructure, we carried out a modified version of our synthetic procedure. As before we pressed mixtures of the elements into pellets, brought them to a melt at 1100°C for 24 h and held them at 500°C for one week for crystal growth. This time, however, rather than quenching the samples in ice water, we introduced a controlled cooling step (10°C h⁻¹) to encourage superstructure formation. The reaction resulted in relatively large, well defined crystals, particularly in comparison to the quenched products from our previous syntheses containing the disordered YZn_5+x crystals. The crystals are metallic gray in color, with smooth surfaces, and are easy to grind into powders. In addition, the crystals exhibit high stability in air with no visible decomposition.

The powder X-ray diffraction data collected for a sample from which we picked the single crystal of YZn_5+x (x = 0.217) is presented in Fig. 2. All the major peaks are indexed to the YZn_5+x structure by reference to the single crystal data described below. Some minor peaks are assigned to a secondary phase, Y₂Zn₁₇, as well as others that appear to arise from an unknown, third phase. These results are consistent with metallographic analysis of a polished sample with back-scattered electron (BSE) imaging and WDS. BSE revealed two prominent phases in the sample: one appearing lighter, the other darker. The composition for the first of these was determined to be YZn_5.26 (3) from an average over 43 points, which corresponds closely with that expected for YZn_5+x. The second phase has the composition Y₂Zn_16.81 (8) (average over 30 points); this is assigned as the compound Y₂Zn₁₇. Needle-like domains of a third, minor phase with approximate composition YZn_2.2Si_0.6 were also occasionally observed, which apparently results from a reaction with SiO₂ from the fused silica ampoule. Altogether, though, the powder X-ray diffraction pattern and WDS analysis indicates the YZn_5+x is a major phase in the reaction product.

Figure 2 Powder X-ray diffraction pattern of the modulated form of YZn5+x (radiation: Cu Kα). Top: the experimental data plotted in a 2θ versus intensity form. Bottom: filmstrip style representation of the experimental data along with the simulated patterns for YZn5+x and Y2Zn17.

3.2. Analysis of diffraction pattern and symmetry

Crystals picked from these reaction products exhibit more complex diffraction patterns [Fig. 3(a)] than those obtained from quenching the samples from the annealing temperature. The strongest reflections can be indexed to a hexagonal cell, corresponding to the EuMg_5+x-type basic structure, with dimensions a = 8.8811 (1) Å and c = 9.2057 (1) Å. Additional weaker reflections are present in the spaces between these main peaks, as is evident in the reconstruction of the (hhl) layer of reciprocal space (Fig. 3). These new reflections correspond approximately to a supercell, but the distribution of their intensities is highly organized. They trace out rectangles, with the closest reciprocal lattice points of the basic cell being the systematically absent l = odd positions (e.g. hhl = 113, 223 etc.). A careful investigation for the positions of the superstructure reflections reveals that they are all separated from an absent main reflection by one of two vectors: q = a* + b* + 0.3041c* or q′ = −a*  − b* + 0.3041c*, as shown in Figs. 3(b) and 3(c), each of which creates a pattern with trigonal symmetry. No other reflections associated with a supercell are evident, suggesting that a superspace approach that considers the additional peaks to arise from modulations will be the most efficient route to solving the structure. In particular, a (3+1)D model (Smaalen, 2007) can be created when we assign q and q′ as belonging to separate twin domains, which are related by a sixfold rotation around c*, in-line with the hexagonal symmetry of the EuMg_5+x type basic structure. While there are several multiples of 60° that could be used to generate these twin domains, the threefold symmetry of the individuals will lead to only two symmetry-distinct crystal orientations.

Figure 3 Diffraction pattern of a YZn5+x (x = 0.217) crystal. (a) The satellite reflections exhibited along with the strong main reflections in the hhl plane. (b) Enlarged area, with the indices of the main reflections. (c) Indexation of the satellite reflections with the modulation wavevector (q = a* + b* + 0.3041c*) (red) and the corresponding vector for a twin domain (orange). Blue dotted circles denote absent main reflections.

In this approach, the experimental diffraction pattern is interpreted as the projection on 3D of a (3+1)D array of reflections, with the satellites having a component along the fourth dimension. The corresponding real space construction is a periodic structure in (3+1)D from which the physical incommensurately modulated structure is obtained by taking 3D cross-sections. The direction perpendicular to the physical cross-section is given by the a_s4 axis, which represents the superspace lattice vector for the modulation function, while the a_s1, a_s2 and a_s3 axes are counterparts to the physical a, b and c axes. They are, however, tilted out of the physical space axes in accordance with how much the phase of the modulation function changes for translations along the a, b and c repeat vectors of the basic cell.

A first step in building such a model is the assignment of the structure's (3+1)D superspace group. The q-vector q = a* + b* + 0.3041c* gives rise to a trigonal arrangement of satellite reflections, which is incompatible with the hexagonal space group P6₃/mmc of the EuMg_5+x type. In terms of potential trigonal subgroups of P6₃/mmc, we observe that the systematic absence condition associated with the c glide in P6₃/mmc, hhl: l = 2n_, still applies to the main reflections (Fig. 3), suggesting that the superspace group of the modulated structure is based on the P6₃/mmc subgroup . When we consider the full set of reflections, it is notable that the satellites only appear in the layer when l is odd, pointing to the condition hhlm: l+m = 2n. In terms of the symmetry operations of the (3+1)D superspace group, this corresponds to the c glide operation in 3D space being accompanied by a shift in the phase of the modulation (x₄) by ½. Altogether, these considerations indicate that the highest possible superspace group is P1c(σ₃)00s.

3.3. Structure solution and refinement

In the structure solution process, the charge flipping algorithm using the program Superflip converged on a (3+1)D electron density corresponding to the experimental diffraction intensities. The automated symmetry search and peak analysis of this electron density map agreed with the assignment of the superspace group as P1c(σ₃)00s, and yielded the basic framework of the EuMg_5+x-type basic structure. In addition, two atomic domains associated with the disordered channels in the EuMg_5+x type were identified, which we label Zn5 and Zn6.

The electron density features for the Zn5 and Zn6 sites show strong modulations with both occupational and positional components [Figs. 4(a) and 4(b)]. For the Zn5 site, the electron density contours in the x₃x₄ section trace out nearly linear domains centered at x₄ = 0.5 that are slanted to give a negative slope, and end on discontinuities near x₄ = 0.0 and 1.0. Similar features are found for the Zn6 site, except that its domain is centered at x₄ = 0.75, and its length along x₄ is shorter, with the width Δx₄ being about . To model the discontinuous nature of these domains, we used crenel functions (Petříček et al., 2016) centered at x₄ = 0.5 and x₄ = 0.75 for Zn5 and Zn6 respectively, with the corresponding Δx₄ values initially being set to 1.0 and , where the latter is a close rational approximant to ½ + ½σ₃ (the Δx₄ width for Zn6 needed to match the ends of the Zn5 and Zn6 domains in the physical cross-sections; see the supporting information). This is convenient as σ₃ varies from crystal to crystal (in the range of 0.304 to ∼0.331). In addition, first order positional modulations (based on Legendre polynomials tailored to the Δx₄ width of the domain) were applied to allow the atomic domains to slant in the x₃x₄ plane.

Figure 4 Contour maps of the Fourier electron density for (x3, x4) cross-sections through the Zn5 and Zn6 atomic domains. Top: the contours generated from the structure solution for the (a) Zn5 and (b) Zn6 sites show positionally modulated and discontinuous atomic domains. Bottom: the corresponding electron density maps for the refined model, in which atomic modulation functions with crenel functions are used for the (c) Zn5 and (d) Zn6 sites. The model positions for Zn5 and Zn6 are shown in gray and blue, respectively.

After setting these modulation functions for the Zn5 and Zn6 sites, in addition to first order harmonic positional waves on the remaining atoms, the refinement led quickly to a drop in the R factors for the satellite reflections. However, additional corrections were needed for a successful refinement. First, a twin component was included whose reciprocal lattice is rotated around the c axis by 60°, reflecting the hexagonal symmetry of the parent structure. Curiously, though, the twin-fraction invariably refined to approximately 0.75, regardless of which individual is integrated. As the diffraction patterns for the individuals overlap only on the main reflections, this suggested that the main reflections are systematically too strong relative to the satellites. Interpreting this as arising from disordered domains in the crystal, we used separate scale factors for the main and satellite reflections. Upon doing so, the twin fractions converged to 0.50, within two times the standard uncertainty, indicating that the crystal has balanced twinning.

In the final model, all the atoms are refined anisotropically, and first order modulations were applied to the harmonic atomic displacement parameters for all atoms but Zn6, with its relatively short and discontinuous domains. In addition, the Δx₄ widths for Zn5 and Zn6 were refined with the restraint that the edges of their atomic domains are aligned for clean transitions in the physical cross-sections, as in Δx₄[Zn6] = ½ + ½σ₃ + (1 − Δx₄[Zn5]), yielding values of 0.988 (3) and 0.664 (4), respectively. The refined model positions for the Zn5 and Zn6 sites are compared to the Fourier electron density contours in Figs. 4(c) and 4(d). In both cases, the model consists of slanted lines that align closely with the density features.

3.4. Structural interpretation of the modulated model

The nature of the modulation pattern is simple to interpret in terms of the disordered channels of the EuMg_5+x-type form of YZn_5+x described previously [Fig. 1(a)] and the (3+1)D Fourier electron densities for the corresponding atomic sites in the current structure. The Y and Zn1–Zn4 sites define the host structure of the EuMg_5+x type, and their electron density features are well described by harmonic functions with relatively small amplitudes (see Fig. S2 in the supporting information). By contrast, the Zn5 and Zn6 sites corresponding to the occupants of the channels have discontinuous atomic domains, indicating much stronger modulations. The Zn5/Zn6 sites can thus be seen as the major drivers of the modulation, with the positional displacements of the remaining atoms serving as adjustments to accommodate the resulting changes in their local environments.

The structural consequences of the Zn5/Zn6 site modulations become clearer when looking at a larger cross-section for the (x₃, x₄) plane containing them (Fig. 5). Zn5 and Zn6 domains alternate as we move from left to right on the physical axis, corresponding to the centers of the flattened octahedra and tricapped trigonal prisms whose alternation builds up the channel walls. While the Zn5 and Zn6 sites are made distinct by their association with these two different polyhedra, their domains show a high degree of correlation: their electron density features are both slanted from upper left to lower right, and following one domain down this slope brings one to right to the tip of the next atomic domain, on the other side of a small gap.

Figure 5 Connection between the (3+1)D Fourier electron density and structural features in physical space. (a) Fourier electron density contours corresponding to the Zn5 and Zn6 sites in an x3x4 section. (b) The corresponding modulated channel containing Zn5 and Zn6 atoms in physical space. (c) Modulated Zn5 and Zn6 sites in physical space laid over the Fourier electron density map of the xz plane of a 3D density constructed from the (3+1)D model (t-section). Such sections can appear somewhat messy due to the density features being broader than the model's atomic domains. The x4 offset of the cross-section off of the origin in (3+1)D space has been adjusted (t = 0.16) to reveal well separated density peaks along the channel. Vacant Zn6 positions are denoted by red dotted circles.

In fact, if one were to shrink these gaps, the Zn5 and Zn6 domains would coalesce into diagonal stripes that run with a slant relative to the a_s4 axis. Such a continuous slanted domain would encode a series of equally spaced Zn atoms whose spacing in physical space is mismatched with respect to the remainder of the structure (and determined by the angle of the slant). The high degree to which this pattern approximates a series of parallel diagonally oriented lines reveals that the Zn atoms in the channel have moved towards an equal spacing that is independent of the host lattice, as in a composite structure (Yamamoto, 1993; Rohrer et al., 2000; Sun et al., 2007). Indeed, the structure can be formally expressed as a composite structure in which the Zn6/Zn5 atoms occupying the channels are described as having their own lattice vector, c_guest, which is incommensurate to that of the host structure, c_host, with c_guest = c_host/(3 + σ₃). To perform refinements within this composite formalism, one first transforms the cell to a supercell (to obtain an axial q vector) and then assigns the reciprocal basis vectors of the guest composite part (the channel occupants) as c_guest* = 3c_host* + q_host and q_guest = 2c_host*. This corresponds to the following transformation matrix connecting the hklm indices of each reflection in terms of the host and guest composite parts:

When implemented, this provides the channel occupants with their own axis in the (3+1)D model. However, two crenel functions are still needed, representing different scenarios for the tricapped trigonal prismatic and flattened octahedral holes in the channels, and the overall R factors are quite similar to those obtained above.

While a composite model would normally suggest that all points along the host cell's channels can be populated by guest atoms, the gaps between the Zn5 and Zn6 domains suggest that certain positions are not suitable. In fact, the gaps correspond approximately in physical space to the centers of flattened Zn₃Y₃ octahedra that lie on opposite sides of the Zn6 sites. An explanation for these gaps can be discerned in the somewhat steeper slope of the Zn6 atomic domains than for the Zn5 atoms. The higher verticality here corresponds to a tighter localization around the average Zn6 position, indicating a preference for the center of the tricapped trigonal prism, where the Y–Zn contacts are shorter.

The features we have discussed so far explain the component of the modulation along the c axis of the EuMg_5+x type. The full q vector, q = a* + b* + 0.3041c*, also contains components along the perpendicular directions a* and b*, indicating that the occupation patterns of neighboring channels differ from each other. In the (3+1)D model, the channels are translationally related by steps of Δx₁ = ±1 and Δx₂ = ±1. In this construction, the non-zero components of σ₁ and σ₂ are reflected in the tilting of the a_s1 and a_s2 axes off the physical a and b axes, such that points along these axes are given by (x₁, 0, 0, +σ₁x₁ + t) and (0, x₂, 0, +σ₂x₂ + t), respectively, where t is the intercept with the a_s4 axis chosen when making the 3D physical cross-section (and is arbitrary in the case of an incommensurate structure). Since σ₁ and σ₂ are both equal to [as required by the superspace group ], moving from one channel to one of its neighbors along a_basic or b_basic results in a shift of the modulation phase by . These shifts are evident in the electron density maps for the 3D cross-section [t-maps, Fig. 5(c)], where the Zn6 vacancies in neighboring channels are offset from each other by steps of of the spacing between the vacancies along the channel.

This picture can be compared to the superstructure predicted previously from tracing paths of chemical pressure between channels through the tcp units. In this model, every third Zn6 site is vacant within each channel, creating a threefold superstructure along c. The occupation patterns of neighboring channels are connected through rhombohedral centering, with the primitive cell vectors being a_prim = a_basic + c_basic, b_prim = b_basic + c_basic, and c_prim = −a_basic − b_basic + c_basic. Transforming this arrangement to the hexagonal setting gives rise to the supercell. In Fig. 6, we plot the 3D structure of the incommensurately modulated form of YZn_5+x generated from our (3+1)D model. First in Fig. 6(a), we show a view down c, color-coding the different channels according to the relative heights of the channel occupation patterns. In this case, the a and b vectors of the supercell represent true crystallographic translations, labeled a_mod and b_mod, due to the commensurability of the modulation along these directions. This corresponds to the predicted pattern in the ab plane, as can be seen by comparison with Fig. 1(b).

Figure 6 Incommensurately modulated YZn5+x (x = 0.217) structure. (a) Overview of the structure viewed down [001]. (b) The modulated Zn channels with the one-third shift pattern (yellow, pink, and green). Vacant Zn6 positions are denoted by red dotted circles.

Next, in Fig. 6(b), we examine the occupation patterns of neighboring channels in more detail. In each channel, every third Zn6 atom is missing in this portion of the structure (red dotted circles), but as these positions are not translationally equivalent points in the basic structure (the EuMg_5+x-type unit cell has two Zn6 sites per cell along c) the overall pattern repeats every six Zn6 positions. The neighboring channels are shifted out of step by one Zn6 site, with the Zn6 coordination environments in the same orientation becoming shifted by of the approximate supercell vector along c. If the structure were to be commensurate, this would correspond in all respects to the predicted model.

The close approximation of σ₃ to raises the question of whether a commensurate model may be warranted. To answer this question, we attempted refinements in which σ₃ was set to and the structure was treated as commensurate in terms of the structure factor calculations. Regardless of which t₀ value was chosen, however, the R factors for the model were unreasonably high [overall R (I > 3σ) > 10.0 versus 2.7 for the incommensurately modulated model]. The incommensurability of the phase is further supported by the observation that the σ₃ value obtained from the diffraction patterns of different crystals can vary from 0.304 to 0.331 (in fact, some slight splitting of the satellites into closely spaced pairs of σ₃ values can be detected in the lattice reconstructions). As such, the spacing of Zn atoms along the structure's channels appears not to be locked into registry with the c parameter of the host structure.

How, then, does the incommensurately modulated structure differ from the idealized threefold superstructure along c? As we mentioned above, the σ₃ value in the (3+1)D model controls the average spacing of the Zn atoms occupying the channels, c_guest, relative to the c parameter of the host sublattice's average structure, c_host, as in c_guest = c_host/(3+σ₃). For σ₃ = , this gives rise to a commensurate arrangement with three c_host = ten c_guest, i.e. ten Zn5/Zn6 atoms in a threefold supercell of the disordered EuMg_5+x-type structure, matching that of the originally predicted superstructure with composition YZn_5.225. For lower σ₃ values, the average spacing of the channel atoms increases, leading to a lower overall Zn content of , e.g. YZn_5.217 for σ₃ = 0.3041, with each atom in the channel locking into a nearby Zn5 or Zn6 atomic domain.

4. Conclusions

In our earlier analysis of EuMg_5+x-type YZn_5+x, we identified paths of chemical pressures between its disordered channels that point to a potential ordered superstructure. In this article, we have confirmed this picture experimentally, demonstrating that, through a change to the synthetic procedure, YZn_5+x crystals with satellite reflections from channel order can be obtained. In this pattern, approximately every third Zn6 position within the channel is vacant, with the remaining occupants exhibiting displacements that serve to smooth out the spacing between them. Between neighboring channels, shifts in the pattern arise to stagger the positions of the vacancies, which we had predicted would minimize the accumulation of packing tensions along paths of inter-channel contacts. Rather than the expected supercell of the EuMg_5+x type, however, the pattern is incommensurate along the c axis, as revealed by variability in the σ₃ parameter of the q vector and the satellite intensity distributions.

These results contribute to a growing theme in the structural chemistry of intermetallics: the communication of information between different regions of a crystal structure through paths of chemical pressures (CPs). For example, the remarkably large hysteresis in the cubic to tetragonal phase transition of PtGa₂ was interpreted in terms of the tight coordination of atomic motion imposed by its CP scheme (Mitchell Warden et al., 2018). Similarly, the unusual, slanted domains of the trimer–monomer patterns in the incommensurately modulated IrSi structure (Mitchell Warden & Fredrickson, 2019), as well as the form of the low-temperature superstructure and high-temperature incommensurability in PdBi (Folkers et al., 2020), were traced to the domino-like propagation of atomic displacements that create emergent, long-range patterns. In these cases, the explanations for the detailed choreographies of atomic motions benefited from knowing the structural solution beforehand. With the example of YZn_5+x, we see more clearly how this approach can be used in a predictive fashion: beginning with CP schemes for a handful of ordered models, the Zn site occupancies in the average structure, and hints from the literature about the size of the expected supercell, an ordering pattern could be proposed, which could then be successfully realized synthetically.

An open challenge highlighted by this work is the question of whether such ordering patterns will adopt commensurate or incommensurate arrangements. Could the incommensurability in YZn_5+x have been predicted ahead of time? In this case, the CP features offer clear indications of how the occupation patterns of neighboring channels are coupled, but give less guidance on the preferred sequence along the channels. Instead, our original model used experimental Zn occupancies in the average structure, and assumed the vacancies would be distributed at regular intervals. There is room within that scheme for these intervals to be mismatched from the c period of the host structure, but there was not an obvious reason for this being advantageous. One potential hint at a driving force for more Zn6 occupancy is seen in the large positive CPs taken on by Zn5 sites neighboring a Zn6 vacancy, as they move in to smooth out the Zn–Zn distances along the channel. Incommensurability would allow the structure to tune the Zn6 vacancy concentration to balance the benefits of a sparser occupation of the channels with the energetic costs to the Zn5 sites neighboring the vacancies. It will be interesting to see whether theoretical models, based on CP analysis or other tools, can be developed to detect such situations where incommensurability provides an opportunity to balance competing factors.

Supporting information

Computing details

Data collection: CrysAlis PRO, Agilent Technologies, Version 1.171.37.35 (release 13-08-2014 CrysAlis171 .NET) (compiled Aug 13 2014,18:06:01); cell refinement: CrysAlis PRO, Agilent Technologies, Version 1.171.37.35 (release 13-08-2014 CrysAlis171 .NET) (compiled Aug 13 2014,18:06:01); data reduction: CrysAlis PRO, Agilent Technologies, Version 1.171.37.35 (release 13-08-2014 CrysAlis171 .NET) (compiled Aug 13 2014,18:06:01); program(s) used to solve structure: Petricek, V., Dusek, M. & Palatinus L. (2014). Z. Kristallogr. 229 (5), 345-352.

Palatinus L. & Chapuis G. (2007). J. Appl. Cryst. 40, 786-790.; molecular graphics: Brandenburg, K. & Putz, H. (2005). DIAMOND Version 3. Crystal Impact GbR, Postfach 1251, D-53002 Bonn, Germany..

(I)

Crystal data

YZn5.217	Dx = 6.829 Mg m^−3
Mr = 431	Mo Kα radiation, λ = 0.71073 Å
Trigonal, P31c(1/31/3γ)00s†	Cell parameters from 5676 reflections
q = 0.333333a* + 0.333333b* + 0.304100c*	θ = 3.4–27.4°
a = 8.8811 (1) Å	µ = 42.90 mm^−1
c = 9.2057 (1) Å	T = 293 K
V = 628.81 (1) Å^3	Prism, metallic grey
Z = 6	0.19 × 0.14 × 0.09 mm
F(000) = 1173

† Symmetry operations: (1) x₁, x₂, x₃, x₄; (2) −x₂, x₁−x₂, x₃, −x₂+x₄; (3) −x₁+x₂, −x₁, x₃, −x₁+x₄; (4) −x₂, −x₁, −x₃+1/2, −x₄+1/2; (5) −x₁+x₂, x₂, −x₃+1/2, x₂−x₄+1/2; (6) x₁, x₁−x₂, −x₃+1/2, x₁−x₄+1/2; (7) −x₁, −x₂, −x₃, −x₄; (8) x₂, −x₁+x₂, −x₃, x₂−x₄; (9) x₁−x₂, x₁, −x₃, x₁−x₄; (10) x₂, x₁, x₃+1/2, x₄+1/2; (11) x₁−x₂, −x₂, x₃+1/2, −x₂+x₄+1/2; (12) −x₁, −x₁+x₂, x₃+1/2, −x₁+x₄+1/2.

Data collection

Oxford Diffraction CCD diffractometer	1450 independent reflections
Radiation source: X-ray tube	777 reflections with I > 3σ(I)
Graphite monochromator	Rint = 0.056
Detector resolution: 16.1500 pixels mm^-1	θmax = 27.9°, θmin = 3.3°
Oxford Diffraction CCD scans	h = −11→11
Absorption correction: analytical CrysAlisPro, Agilent Technologies, Version 1.171.37.35 (release 13-08-2014 CrysAlis171 .NET) (compiled Aug 13 2014,18:06:01) Analytical numeric absorption correction using a multifaceted crystal model based on expressions derived by R.C. Clark & J.S. Reid. (Clark, R. C. & Reid, J. S. (1995). Acta Cryst. A51, 887-897) Empirical absorption correction using spherical harmonics, implemented in SCALE3 ABSPACK scaling algorithm.	k = −11→11
Tmin = 0.024, Tmax = 0.125	l = −12→12
24896 measured reflections

Refinement

Refinement on F^2	1 constraint
R[F^2 > 2σ(F^2)] = 0.027	Weighting scheme based on measured s.u.'s w = 1/(σ^2(I) + 0.0004I^2)
wR(F^2) = 0.088	(Δ/σ)max = 0.014
S = 1.53	Δρmax = 4.73 e Å^−3
1450 reflections	Δρmin = −4.90 e Å^−3
91 parameters	Extinction correction: B-C type 1 Gaussian isotropic (Becker & Coppens, 1974)
0 restraints	Extinction coefficient: 330 (30)

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq	Occ. (<1)
Y1	0.61475 (10)	0.80738 (5)	0.25	0.0107 (3)
Zn1	0.86508 (10)	0.43254 (5)	0.25	0.0117 (3)
Zn2	0.666667	0.333333	0.51174 (10)	0.0122 (3)
Zn3	1	0.5	0.5	0.0123 (6)
Zn4	0.8383 (2)	0.67858 (8)	0.41030 (6)	0.0139 (5)
Zn5	1	1	0	0.0196 (12)	0.988 (4)
Zn6	1	1	0.25	0.0140 (9)	0.664 (4)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Y1	0.0123 (4)	0.0104 (3)	0.0102 (3)	0.00617 (19)	0	0.0067 (7)
Zn1	0.0118 (4)	0.0130 (3)	0.0099 (4)	0.0059 (2)	0	0.0002 (12)
Zn2	0.0115 (3)	0.0115 (3)	0.0134 (5)	0.00577 (16)	0	0
Zn3	0.0115 (4)	0.0122 (9)	0.0108 (4)	0.0040 (9)	−0.0017 (3)	0.0004 (11)
Zn4	0.0162 (8)	0.0147 (4)	0.0140 (3)	0.0102 (7)	0.0002 (8)	0.0006 (2)
Zn5	0.0149 (6)	0.0149 (6)	0.029 (3)	0.0074 (3)	0	0
Zn6	0.0144 (7)	0.0144 (7)	0.013 (2)	0.0072 (4)	0	0

Geometric parameters (Å, º)

	Average	Minimum	Maximum
Y1—Y1^i	3.749 (3)	3.697 (3)	3.801 (3)
Y1—Y1^ii	3.749 (2)	3.697 (2)	3.801 (2)
Y1—Zn1^iii	3.141 (4)	3.104 (5)	3.179 (5)
Y1—Zn1^iv	3.141 (4)	3.104 (4)	3.179 (4)
Y1—Zn2^v	3.081 (2)	3.045 (2)	3.119 (2)
Y1—Zn2^vi	3.081 (2)	3.045 (2)	3.119 (2)
Y1—Zn3^iii	3.318 (2)	3.294 (3)	3.342 (3)
Y1—Zn3^iv	3.317 (3)	3.276 (4)	3.360 (4)
Y1—Zn3^vii	3.318 (2)	3.294 (3)	3.342 (3)
Y1—Zn3^viii	3.317 (2)	3.276 (3)	3.360 (3)
Y1—Zn4	3.116 (3)	3.080 (4)	3.153 (4)
Y1—Zn4^iv	3.131 (2)	3.107 (3)	3.158 (3)
Y1—Zn4^viii	3.131 (4)	3.107 (4)	3.158 (4)
Y1—Zn4^ix	3.116 (3)	3.080 (3)	3.153 (3)
Y1—Zn4^x	3.166 (2)	3.129 (3)	3.202 (3)
Y1—Zn4^vi	3.166 (2)	3.129 (3)	3.202 (3)
Y1—Zn5	3.772 (3)	3.239 (6)	4.354 (6)
Y1—Zn5^xi	3.777 (3)	3.241 (6)	4.359 (6)
Y1—Zn6	3.0096 (15)	2.954 (2)	3.047 (2)
Zn1—Zn1^iii	2.643 (4)	2.634 (6)	2.653 (6)
Zn1—Zn1^xii	2.643 (3)	2.634 (3)	2.653 (3)
Zn1—Zn2	2.852 (2)	2.831 (2)	2.874 (2)
Zn1—Zn2^xiii	2.852 (2)	2.831 (2)	2.874 (2)
Zn1—Zn3	2.525 (2)	2.515 (3)	2.534 (3)
Zn1—Zn3^xiv	2.525 (2)	2.515 (2)	2.534 (2)
Zn1—Zn4	2.745 (4)	2.701 (5)	2.788 (5)
Zn1—Zn4^xii	2.735 (4)	2.673 (5)	2.797 (5)
Zn1—Zn4^viii	2.736 (4)	2.673 (5)	2.797 (5)
Zn1—Zn4^xiv	2.744 (4)	2.701 (5)	2.788 (5)
Zn2—Zn3	2.566 (3)	2.554 (3)	2.578 (3)
Zn2—Zn3^iii	2.566 (4)	2.554 (5)	2.578 (5)
Zn2—Zn3^xii	2.566 (3)	2.554 (3)	2.578 (3)
Zn2—Zn4	2.815 (3)	2.776 (3)	2.854 (3)
Zn2—Zn4^iii	2.815 (3)	2.776 (3)	2.854 (3)
Zn2—Zn4^xii	2.815 (5)	2.776 (5)	2.854 (5)
Zn3—Zn4	2.747 (3)	2.707 (4)	2.786 (4)
Zn3—Zn4^xii	2.731 (3)	2.693 (3)	2.770 (3)
Zn3—Zn4^xv	2.746 (3)	2.707 (4)	2.786 (4)
Zn3—Zn4^xvi	2.731 (3)	2.693 (3)	2.770 (3)
Zn4—Zn5^xi	2.656 (2)	2.483 (4)	3.003 (4)
Zn4—Zn6	2.857 (3)	2.709 (4)	3.131 (4)
Zn5—Zn6	2.620 (5)	2.422 (10)	2.818 (10)
Zn5—Zn6^xvii	2.621 (5)	2.420 (10)	2.822 (10)
Y1^i—Y1—Y1^ii	60.01 (5)	58.63 (5)	61.39 (6)
Y1^i—Y1—Zn1^iii	113.38 (6)	110.88 (6)	115.85 (6)
Y1^i—Y1—Zn1^iv	53.37 (6)	52.26 (6)	54.47 (7)
Y1^i—Y1—Zn2^v	52.55 (6)	51.45 (6)	53.64 (6)
Y1^i—Y1—Zn2^vi	52.54 (6)	51.97 (6)	53.09 (6)
Y1^i—Y1—Zn3^iii	96.01 (7)	94.05 (7)	97.94 (7)
Y1^i—Y1—Zn3^iv	55.61 (7)	54.73 (7)	56.48 (7)
Y1^i—Y1—Zn3^vii	55.60 (6)	54.85 (6)	56.35 (6)
Y1^i—Y1—Zn3^viii	96.00 (8)	94.58 (9)	97.40 (9)
Y1^i—Y1—Zn4	145.17 (6)	142.80 (6)	147.48 (6)
Y1^i—Y1—Zn4^iv	97.90 (9)	95.96 (9)	99.79 (9)
Y1^i—Y1—Zn4^viii	145.34 (6)	143.41 (6)	147.26 (6)
Y1^i—Y1—Zn4^ix	97.76 (9)	95.94 (9)	99.52 (9)
Y1^i—Y1—Zn4^x	97.64 (7)	96.66 (7)	98.61 (7)
Y1^i—Y1—Zn4^vi	97.79 (7)	96.72 (6)	98.87 (6)
Y1^i—Y1—Zn5	133.47 (9)	125.50 (7)	141.41 (10)
Y1^i—Y1—Zn5^xi	133.41 (9)	125.34 (8)	141.27 (10)
Y1^i—Y1—Zn6	149.32 (7)	149.21 (8)	149.42 (7)
Y1^ii—Y1—Zn1^iii	53.37 (5)	52.26 (5)	54.47 (5)
Y1^ii—Y1—Zn1^iv	113.38 (8)	110.88 (9)	115.86 (9)
Y1^ii—Y1—Zn2^v	52.54 (4)	51.97 (4)	53.09 (4)
Y1^ii—Y1—Zn2^vi	52.55 (4)	51.45 (4)	53.64 (5)
Y1^ii—Y1—Zn3^iii	55.60 (5)	54.85 (5)	56.35 (5)
Y1^ii—Y1—Zn3^iv	96.00 (7)	94.58 (7)	97.40 (7)
Y1^ii—Y1—Zn3^vii	96.01 (6)	94.05 (6)	97.94 (7)
Y1^ii—Y1—Zn3^viii	55.61 (6)	54.73 (6)	56.48 (6)
Y1^ii—Y1—Zn4	97.76 (6)	95.94 (6)	99.52 (6)
Y1^ii—Y1—Zn4^iv	145.34 (8)	143.41 (8)	147.26 (7)
Y1^ii—Y1—Zn4^viii	97.90 (7)	95.96 (6)	99.79 (7)
Y1^ii—Y1—Zn4^ix	145.17 (7)	142.80 (8)	147.48 (7)
Y1^ii—Y1—Zn4^x	97.79 (5)	96.72 (5)	98.87 (5)
Y1^ii—Y1—Zn4^vi	97.64 (5)	96.66 (5)	98.61 (5)
Y1^ii—Y1—Zn5	133.49 (9)	125.41 (7)	141.35 (10)
Y1^ii—Y1—Zn5^xi	133.39 (9)	125.44 (8)	141.34 (10)
Y1^ii—Y1—Zn6	149.32 (8)	149.20 (9)	149.42 (8)
Zn1^iii—Y1—Zn1^iv	166.74 (9)	163.15 (9)	170.32 (10)
Zn1^iii—Y1—Zn2^v	85.35 (7)	84.07 (7)	86.60 (7)
Zn1^iii—Y1—Zn2^vi	85.35 (7)	83.94 (6)	86.73 (6)
Zn1^iii—Y1—Zn3^iii	45.93 (7)	45.34 (7)	46.50 (7)
Zn1^iii—Y1—Zn3^iv	129.33 (7)	127.91 (7)	130.74 (8)
Zn1^iii—Y1—Zn3^vii	129.32 (7)	128.04 (6)	130.53 (6)
Zn1^iii—Y1—Zn3^viii	45.92 (6)	45.52 (6)	46.32 (7)
Zn1^iii—Y1—Zn4	51.85 (6)	50.81 (5)	52.87 (5)
Zn1^iii—Y1—Zn4^iv	138.11 (10)	134.79 (10)	141.48 (9)
Zn1^iii—Y1—Zn4^viii	51.90 (6)	50.87 (8)	52.89 (8)
Zn1^iii—Y1—Zn4^ix	138.12 (10)	134.68 (10)	141.58 (10)
Zn1^iii—Y1—Zn4^x	91.19 (8)	90.52 (8)	91.88 (8)
Zn1^iii—Y1—Zn4^vi	90.86 (7)	90.51 (8)	91.23 (8)
Zn1^iii—Y1—Zn5	95.27 (7)	93.53 (7)	97.34 (7)
Zn1^iii—Y1—Zn5^xi	95.29 (7)	93.65 (6)	97.19 (7)
Zn1^iii—Y1—Zn6	95.82 (8)	94.84 (8)	97.45 (7)
Zn1^iv—Y1—Zn2^v	85.35 (8)	83.94 (9)	86.73 (9)
Zn1^iv—Y1—Zn2^vi	85.35 (8)	84.07 (9)	86.60 (9)
Zn1^iv—Y1—Zn3^iii	129.32 (9)	128.04 (9)	130.53 (9)
Zn1^iv—Y1—Zn3^iv	45.92 (6)	45.52 (7)	46.32 (7)
Zn1^iv—Y1—Zn3^vii	45.93 (6)	45.34 (6)	46.50 (6)
Zn1^iv—Y1—Zn3^viii	129.33 (9)	127.91 (9)	130.74 (9)
Zn1^iv—Y1—Zn4	138.12 (7)	134.68 (8)	141.58 (7)
Zn1^iv—Y1—Zn4^iv	51.90 (8)	50.87 (8)	52.89 (8)
Zn1^iv—Y1—Zn4^viii	138.11 (7)	134.79 (7)	141.48 (7)
Zn1^iv—Y1—Zn4^ix	51.85 (8)	50.81 (8)	52.87 (8)
Zn1^iv—Y1—Zn4^x	90.86 (8)	90.51 (8)	91.23 (8)
Zn1^iv—Y1—Zn4^vi	91.19 (7)	90.52 (7)	91.88 (7)
Zn1^iv—Y1—Zn5	95.30 (6)	93.65 (6)	97.19 (7)
Zn1^iv—Y1—Zn5^xi	95.27 (6)	93.54 (6)	97.34 (7)
Zn1^iv—Y1—Zn6	95.82 (7)	94.84 (8)	97.49 (6)
Zn2^v—Y1—Zn2^vi	90.78 (6)	89.55 (7)	92.00 (7)
Zn2^v—Y1—Zn3^iii	47.11 (9)	46.48 (9)	47.73 (9)
Zn2^v—Y1—Zn3^iv	47.11 (6)	46.39 (7)	47.83 (7)
Zn2^v—Y1—Zn3^vii	107.89 (9)	106.15 (10)	109.61 (10)
Zn2^v—Y1—Zn3^viii	107.89 (7)	106.54 (7)	109.22 (7)
Zn2^v—Y1—Zn4	92.88 (7)	91.79 (8)	93.95 (8)
Zn2^v—Y1—Zn4^iv	93.08 (7)	91.87 (7)	94.26 (7)
Zn2^v—Y1—Zn4^viii	136.46 (9)	134.45 (9)	138.42 (9)
Zn2^v—Y1—Zn4^ix	136.45 (11)	134.22 (11)	138.63 (11)
Zn2^v—Y1—Zn4^x	53.53 (5)	52.63 (5)	54.47 (5)
Zn2^v—Y1—Zn4^vi	144.31 (7)	143.17 (8)	145.42 (8)
Zn2^v—Y1—Zn5	171.51 (12)	160.61 (14)	179.92 (10)
Zn2^v—Y1—Zn5^xi	97.30 (11)	86.99 (9)	109.10 (14)
Zn2^v—Y1—Zn6	134.29 (15)	126.64 (15)	143.05 (15)
Zn2^vi—Y1—Zn3^iii	107.89 (6)	106.15 (6)	109.61 (6)
Zn2^vi—Y1—Zn3^iv	107.89 (10)	106.54 (10)	109.22 (11)
Zn2^vi—Y1—Zn3^vii	47.11 (6)	46.48 (6)	47.73 (6)
Zn2^vi—Y1—Zn3^viii	47.11 (9)	46.39 (9)	47.83 (9)
Zn2^vi—Y1—Zn4	136.45 (9)	134.22 (8)	138.63 (8)
Zn2^vi—Y1—Zn4^iv	136.46 (11)	134.45 (11)	138.42 (10)
Zn2^vi—Y1—Zn4^viii	93.08 (8)	91.87 (7)	94.26 (7)
Zn2^vi—Y1—Zn4^ix	92.88 (7)	91.79 (6)	93.95 (6)
Zn2^vi—Y1—Zn4^x	144.31 (7)	143.17 (7)	145.42 (7)
Zn2^vi—Y1—Zn4^vi	53.53 (5)	52.63 (6)	54.47 (6)
Zn2^vi—Y1—Zn5	97.40 (11)	87.07 (8)	109.23 (14)
Zn2^vi—Y1—Zn5^xi	171.59 (11)	160.78 (14)	180
Zn2^vi—Y1—Zn6	134.41 (15)	126.73 (15)	143.17 (16)
Zn3^iii—Y1—Zn3^iv	84.02 (6)	83.36 (6)	84.68 (7)
Zn3^iii—Y1—Zn3^vii	149.18 (7)	146.40 (7)	151.92 (8)
Zn3^iii—Y1—Zn3^viii	87.85 (6)	86.93 (6)	88.76 (6)
Zn3^iii—Y1—Zn4	50.13 (5)	49.34 (4)	50.86 (4)
Zn3^iii—Y1—Zn4^iv	106.17 (6)	104.55 (5)	107.81 (5)
Zn3^iii—Y1—Zn4^viii	90.88 (7)	89.53 (7)	92.23 (8)
Zn3^iii—Y1—Zn4^ix	159.08 (7)	156.23 (7)	161.91 (6)
Zn3^iii—Y1—Zn4^x	50.04 (6)	49.23 (6)	50.88 (6)
Zn3^iii—Y1—Zn4^vi	136.43 (8)	136.14 (7)	136.69 (7)
Zn3^iii—Y1—Zn5	128.97 (9)	123.36 (11)	134.23 (9)
Zn3^iii—Y1—Zn5^xi	77.97 (8)	70.86 (7)	87.10 (11)
Zn3^iii—Y1—Zn6	104.76 (12)	100.06 (12)	111.81 (12)
Zn3^iv—Y1—Zn3^vii	87.85 (8)	86.93 (8)	88.76 (8)
Zn3^iv—Y1—Zn3^viii	149.17 (9)	146.99 (10)	151.35 (10)
Zn3^iv—Y1—Zn4	106.06 (6)	104.33 (5)	107.83 (5)
Zn3^iv—Y1—Zn4^iv	50.31 (7)	49.32 (7)	51.26 (7)
Zn3^iv—Y1—Zn4^viii	158.89 (8)	156.33 (8)	161.52 (8)
Zn3^iv—Y1—Zn4^ix	90.92 (7)	89.78 (8)	92.03 (8)
Zn3^iv—Y1—Zn4^x	49.75 (7)	49.06 (6)	50.47 (6)
Zn3^iv—Y1—Zn4^vi	136.76 (9)	135.97 (8)	137.54 (8)
Zn3^iv—Y1—Zn5	128.97 (7)	122.86 (9)	133.63 (5)
Zn3^iv—Y1—Zn5^xi	77.95 (8)	70.70 (7)	87.02 (11)
Zn3^iv—Y1—Zn6	104.86 (11)	100.07 (12)	111.43 (10)
Zn3^vii—Y1—Zn3^viii	84.02 (7)	83.36 (6)	84.68 (6)
Zn3^vii—Y1—Zn4	159.08 (7)	156.23 (7)	161.91 (7)
Zn3^vii—Y1—Zn4^iv	90.88 (7)	89.53 (7)	92.23 (7)
Zn3^vii—Y1—Zn4^viii	106.17 (6)	104.55 (6)	107.81 (7)
Zn3^vii—Y1—Zn4^ix	50.13 (6)	49.34 (5)	50.86 (5)
Zn3^vii—Y1—Zn4^x	136.43 (10)	136.14 (9)	136.69 (9)
Zn3^vii—Y1—Zn4^vi	50.04 (6)	49.23 (6)	50.88 (6)
Zn3^vii—Y1—Zn5	78.05 (8)	70.89 (7)	87.17 (10)
Zn3^vii—Y1—Zn5^xi	129.03 (7)	123.48 (9)	134.29 (6)
Zn3^vii—Y1—Zn6	104.85 (11)	100.11 (11)	111.92 (11)
Zn3^viii—Y1—Zn4	90.92 (7)	89.78 (9)	92.03 (9)
Zn3^viii—Y1—Zn4^iv	158.89 (7)	156.33 (8)	161.52 (7)
Zn3^viii—Y1—Zn4^viii	50.31 (6)	49.32 (6)	51.26 (6)
Zn3^viii—Y1—Zn4^ix	106.06 (6)	104.33 (6)	107.83 (6)
Zn3^viii—Y1—Zn4^x	136.76 (9)	135.97 (9)	137.54 (9)
Zn3^viii—Y1—Zn4^vi	49.75 (5)	49.06 (5)	50.47 (5)
Zn3^viii—Y1—Zn5	78.03 (8)	70.74 (6)	87.10 (11)
Zn3^viii—Y1—Zn5^xi	129.02 (9)	122.98 (10)	133.69 (6)
Zn3^viii—Y1—Zn6	104.95 (12)	100.13 (12)	111.54 (11)
Zn4—Y1—Zn4^iv	86.57 (8)	82.11 (8)	91.16 (8)
Zn4—Y1—Zn4^viii	56.40 (7)	56.13 (7)	56.65 (7)
Zn4—Y1—Zn4^ix	112.96 (8)	108.32 (9)	117.76 (9)
Zn4—Y1—Zn4^x	56.53 (8)	55.70 (8)	57.40 (8)
Zn4—Y1—Zn4^vi	112.35 (9)	111.82 (8)	112.89 (9)
Zn4—Y1—Zn5	81.26 (9)	73.79 (10)	88.92 (8)
Zn4—Y1—Zn5^xi	43.82 (4)	41.14 (4)	46.62 (5)
Zn4—Y1—Zn6	55.48 (9)	52.68 (9)	62.42 (10)
Zn4^iv—Y1—Zn4^viii	112.61 (9)	108.16 (9)	117.23 (10)
Zn4^iv—Y1—Zn4^ix	56.40 (6)	56.13 (5)	56.65 (6)
Zn4^iv—Y1—Zn4^x	56.38 (7)	55.79 (6)	57.00 (7)
Zn4^iv—Y1—Zn4^vi	112.43 (6)	111.61 (6)	113.31 (7)
Zn4^iv—Y1—Zn5	81.07 (9)	73.41 (11)	88.56 (7)
Zn4^iv—Y1—Zn5^xi	43.76 (7)	41.14 (6)	46.48 (7)
Zn4^iv—Y1—Zn6	55.35 (10)	52.64 (10)	62.07 (11)
Zn4^viii—Y1—Zn4^ix	86.57 (8)	82.11 (8)	91.17 (8)
Zn4^viii—Y1—Zn4^x	112.43 (9)	111.61 (10)	113.31 (10)
Zn4^viii—Y1—Zn4^vi	56.38 (7)	55.79 (7)	57.00 (7)
Zn4^viii—Y1—Zn5	43.78 (4)	41.14 (4)	46.48 (5)
Zn4^viii—Y1—Zn5^xi	81.14 (9)	73.56 (10)	88.67 (7)
Zn4^viii—Y1—Zn6	55.41 (9)	52.64 (9)	62.19 (10)
Zn4^ix—Y1—Zn4^x	112.35 (7)	111.82 (7)	112.89 (7)
Zn4^ix—Y1—Zn4^vi	56.53 (6)	55.70 (6)	57.40 (6)
Zn4^ix—Y1—Zn5	43.84 (7)	41.14 (6)	46.62 (7)
Zn4^ix—Y1—Zn5^xi	81.33 (9)	73.94 (10)	89.04 (7)
Zn4^ix—Y1—Zn6	55.55 (10)	52.68 (10)	62.55 (10)
Zn4^x—Y1—Zn4^vi	162.15 (7)	161.82 (8)	162.50 (9)
Zn4^x—Y1—Zn5	118.29 (11)	106.15 (14)	128.18 (9)
Zn4^x—Y1—Zn5^xi	43.76 (11)	34.01 (8)	56.02 (14)
Zn4^x—Y1—Zn6	80.95 (15)	73.61 (15)	88.58 (15)
Zn4^vi—Y1—Zn5	43.86 (11)	34.10 (8)	56.18 (13)
Zn4^vi—Y1—Zn5^xi	118.40 (11)	106.31 (14)	128.29 (8)
Zn4^vi—Y1—Zn6	81.06 (15)	73.73 (15)	88.71 (15)
Zn5—Y1—Zn5^xi	74.58 (12)	59.90 (11)	82.39 (10)
Zn5—Y1—Zn6	40.98 (15)	39.87 (14)	42.58 (16)
Zn5^xi—Y1—Zn6	40.96 (15)	39.86 (14)	42.56 (16)
Y1^xviii—Zn1—Y1^xii	73.28 (6)	72.09 (7)	74.52 (7)
Y1^xviii—Zn1—Zn1^iii	113.38 (13)	111.49 (16)	115.24 (17)
Y1^xviii—Zn1—Zn1^xii	173.36 (15)	171.37 (18)	175.34 (18)
Y1^xviii—Zn1—Zn2	115.42 (8)	114.75 (11)	116.06 (10)
Y1^xviii—Zn1—Zn2^xiii	115.42 (8)	114.81 (9)	116.02 (9)
Y1^xviii—Zn1—Zn3	70.73 (7)	70.10 (7)	71.39 (7)
Y1^xviii—Zn1—Zn3^xiv	70.74 (8)	70.62 (8)	70.87 (8)
Y1^xviii—Zn1—Zn4	63.86 (8)	63.12 (10)	64.61 (10)
Y1^xviii—Zn1—Zn4^xii	123.81 (9)	122.32 (9)	125.32 (9)
Y1^xviii—Zn1—Zn4^viii	63.60 (8)	62.76 (10)	64.44 (10)
Y1^xviii—Zn1—Zn4^xiv	124.15 (9)	123.37 (9)	124.95 (8)
Y1^xii—Zn1—Zn1^iii	173.33 (12)	171.37 (11)	175.34 (12)
Y1^xii—Zn1—Zn1^xii	113.34 (13)	111.49 (14)	115.24 (14)
Y1^xii—Zn1—Zn2	115.41 (9)	114.81 (11)	116.02 (11)
Y1^xii—Zn1—Zn2^xiii	115.41 (10)	114.75 (9)	116.06 (10)
Y1^xii—Zn1—Zn3	70.74 (9)	70.62 (9)	70.87 (9)
Y1^xii—Zn1—Zn3^xiv	70.73 (10)	70.10 (10)	71.39 (10)
Y1^xii—Zn1—Zn4	124.17 (8)	123.37 (8)	124.95 (8)
Y1^xii—Zn1—Zn4^xii	63.59 (10)	62.76 (10)	64.44 (10)
Y1^xii—Zn1—Zn4^viii	123.83 (9)	122.32 (9)	125.32 (9)
Y1^xii—Zn1—Zn4^xiv	63.85 (9)	63.12 (9)	64.61 (9)
Zn1^iii—Zn1—Zn1^xii	60.00 (11)	59.64 (12)	60.36 (12)
Zn1^iii—Zn1—Zn2	62.41 (7)	61.58 (8)	63.22 (8)
Zn1^iii—Zn1—Zn2^xiii	62.39 (8)	61.78 (7)	63.01 (7)
Zn1^iii—Zn1—Zn3	110.86 (11)	109.82 (12)	111.90 (12)
Zn1^iii—Zn1—Zn3^xiv	110.85 (12)	110.60 (11)	111.09 (11)
Zn1^iii—Zn1—Zn4	60.99 (10)	59.95 (9)	62.01 (9)
Zn1^iii—Zn1—Zn4^xii	111.05 (11)	110.22 (11)	111.86 (11)
Zn1^iii—Zn1—Zn4^viii	61.32 (11)	60.97 (11)	61.66 (11)
Zn1^iii—Zn1—Zn4^xiv	110.76 (10)	109.67 (9)	111.86 (9)
Zn1^xii—Zn1—Zn2	62.40 (6)	61.78 (7)	63.01 (7)
Zn1^xii—Zn1—Zn2^xiii	62.40 (6)	61.58 (7)	63.22 (7)
Zn1^xii—Zn1—Zn3	110.85 (9)	110.60 (11)	111.09 (11)
Zn1^xii—Zn1—Zn3^xiv	110.85 (10)	109.82 (10)	111.90 (10)
Zn1^xii—Zn1—Zn4	110.78 (11)	109.67 (11)	111.86 (11)
Zn1^xii—Zn1—Zn4^xii	61.32 (10)	60.97 (9)	61.66 (9)
Zn1^xii—Zn1—Zn4^viii	111.05 (11)	110.22 (11)	111.86 (11)
Zn1^xii—Zn1—Zn4^xiv	60.97 (10)	59.95 (11)	62.01 (11)
Zn2—Zn1—Zn2^xiii	115.31 (8)	114.95 (8)	115.66 (8)
Zn2—Zn1—Zn3	56.62 (7)	56.49 (8)	56.74 (8)
Zn2—Zn1—Zn3^xiv	171.90 (10)	171.44 (11)	172.38 (10)
Zn2—Zn1—Zn4	60.36 (8)	59.38 (8)	61.33 (8)
Zn2—Zn1—Zn4^xii	60.45 (10)	59.91 (11)	61.00 (11)
Zn2—Zn1—Zn4^viii	114.74 (14)	113.79 (13)	115.69 (12)
Zn2—Zn1—Zn4^xiv	114.45 (11)	113.57 (10)	115.34 (10)
Zn2^xiii—Zn1—Zn3	171.91 (10)	171.44 (10)	172.38 (10)
Zn2^xiii—Zn1—Zn3^xiv	56.62 (7)	56.49 (7)	56.74 (7)
Zn2^xiii—Zn1—Zn4	114.46 (13)	113.57 (12)	115.34 (12)
Zn2^xiii—Zn1—Zn4^xii	114.74 (11)	113.79 (9)	115.69 (8)
Zn2^xiii—Zn1—Zn4^viii	60.46 (8)	59.91 (7)	61.00 (7)
Zn2^xiii—Zn1—Zn4^xiv	60.35 (9)	59.38 (9)	61.33 (9)
Zn3—Zn1—Zn3^xiv	131.45 (9)	130.88 (11)	132.04 (11)
Zn3—Zn1—Zn4	62.67 (8)	61.93 (10)	63.37 (10)
Zn3—Zn1—Zn4^xii	62.40 (7)	61.81 (9)	63.00 (9)
Zn3—Zn1—Zn4^viii	121.36 (12)	121.01 (7)	121.71 (7)
Zn3—Zn1—Zn4^xiv	121.41 (14)	120.92 (15)	121.89 (15)
Zn3^xiv—Zn1—Zn4	121.41 (13)	120.92 (11)	121.89 (11)
Zn3^xiv—Zn1—Zn4^xii	121.36 (15)	121.01 (15)	121.71 (15)
Zn3^xiv—Zn1—Zn4^viii	62.41 (9)	61.81 (9)	63.00 (9)
Zn3^xiv—Zn1—Zn4^xiv	62.66 (8)	61.93 (10)	63.38 (10)
Zn4—Zn1—Zn4^xii	114.16 (8)	113.66 (8)	114.66 (7)
Zn4—Zn1—Zn4^viii	65.18 (10)	64.63 (11)	65.75 (11)
Zn4—Zn1—Zn4^xiv	171.36 (10)	171.22 (11)	171.53 (10)
Zn4^xii—Zn1—Zn4^viii	172.00 (11)	170.85 (12)	173.13 (12)
Zn4^xii—Zn1—Zn4^xiv	65.19 (10)	64.63 (8)	65.75 (8)
Zn4^viii—Zn1—Zn4^xiv	114.16 (9)	113.66 (8)	114.66 (8)
Y1^v—Zn2—Y1^xix	74.92 (6)	74.13 (6)	75.73 (6)
Y1^v—Zn2—Y1^xvi	74.93 (6)	74.13 (5)	75.73 (5)
Y1^v—Zn2—Zn1	167.72 (9)	167.44 (8)	167.99 (9)
Y1^v—Zn2—Zn1^iii	114.42 (10)	114.14 (11)	114.71 (12)
Y1^v—Zn2—Zn1^xii	114.42 (8)	113.88 (7)	114.96 (7)
Y1^v—Zn2—Zn3	137.02 (7)	136.68 (8)	137.38 (7)
Y1^v—Zn2—Zn3^iii	71.29 (9)	71.00 (8)	71.57 (9)
Y1^v—Zn2—Zn3^xii	71.28 (9)	70.80 (9)	71.78 (9)
Y1^v—Zn2—Zn4	124.43 (12)	123.93 (13)	124.95 (13)
Y1^v—Zn2—Zn4^iii	64.77 (6)	63.28 (7)	66.25 (7)
Y1^v—Zn2—Zn4^xii	124.69 (10)	123.78 (11)	125.62 (11)
Y1^xix—Zn2—Y1^xvi	74.94 (6)	74.13 (6)	75.73 (6)
Y1^xix—Zn2—Zn1	114.43 (11)	113.88 (12)	114.96 (12)
Y1^xix—Zn2—Zn1^iii	167.72 (12)	167.44 (11)	167.99 (11)
Y1^xix—Zn2—Zn1^xii	114.42 (12)	114.14 (11)	114.71 (12)
Y1^xix—Zn2—Zn3	71.29 (9)	70.80 (9)	71.78 (9)
Y1^xix—Zn2—Zn3^iii	137.03 (8)	136.68 (8)	137.38 (8)
Y1^xix—Zn2—Zn3^xii	71.29 (10)	71.00 (10)	71.57 (10)
Y1^xix—Zn2—Zn4	124.71 (8)	123.79 (8)	125.61 (8)
Y1^xix—Zn2—Zn4^iii	124.44 (9)	123.93 (9)	124.95 (9)
Y1^xix—Zn2—Zn4^xii	64.75 (9)	63.28 (8)	66.25 (8)
Y1^xvi—Zn2—Zn1	114.42 (7)	114.14 (7)	114.71 (7)
Y1^xvi—Zn2—Zn1^iii	114.42 (11)	113.88 (11)	114.96 (11)
Y1^xvi—Zn2—Zn1^xii	167.72 (9)	167.44 (9)	167.99 (9)
Y1^xvi—Zn2—Zn3	71.29 (8)	71.00 (8)	71.57 (9)
Y1^xvi—Zn2—Zn3^iii	71.29 (10)	70.80 (10)	71.78 (10)
Y1^xvi—Zn2—Zn3^xii	137.03 (7)	136.68 (7)	137.38 (7)
Y1^xvi—Zn2—Zn4	64.75 (6)	63.28 (6)	66.25 (7)
Y1^xvi—Zn2—Zn4^iii	124.71 (13)	123.79 (13)	125.62 (13)
Y1^xvi—Zn2—Zn4^xii	124.44 (11)	123.93 (10)	124.95 (10)
Zn1—Zn2—Zn1^iii	55.21 (7)	54.94 (6)	55.48 (6)
Zn1—Zn2—Zn1^xii	55.21 (6)	54.94 (7)	55.48 (7)
Zn1—Zn2—Zn3	55.24 (7)	55.01 (7)	55.47 (7)
Zn1—Zn2—Zn3^iii	103.39 (11)	103.17 (9)	103.63 (9)
Zn1—Zn2—Zn3^xii	103.40 (11)	102.58 (11)	104.21 (10)
Zn1—Zn2—Zn4	57.92 (8)	56.91 (7)	58.93 (7)
Zn1—Zn2—Zn4^iii	102.95 (6)	101.69 (6)	104.24 (6)
Zn1—Zn2—Zn4^xii	57.72 (9)	56.34 (8)	59.09 (9)
Zn1^iii—Zn2—Zn1^xii	55.21 (9)	54.94 (9)	55.48 (9)
Zn1^iii—Zn2—Zn3	103.40 (9)	102.58 (8)	104.21 (8)
Zn1^iii—Zn2—Zn3^iii	55.24 (10)	55.01 (10)	55.47 (11)
Zn1^iii—Zn2—Zn3^xii	103.39 (10)	103.17 (9)	103.63 (9)
Zn1^iii—Zn2—Zn4	57.72 (8)	56.34 (8)	59.09 (8)
Zn1^iii—Zn2—Zn4^iii	57.91 (8)	56.91 (8)	58.93 (8)
Zn1^iii—Zn2—Zn4^xii	102.97 (10)	101.69 (12)	104.24 (11)
Zn1^xii—Zn2—Zn3	103.39 (11)	103.17 (11)	103.63 (11)
Zn1^xii—Zn2—Zn3^iii	103.39 (11)	102.58 (12)	104.21 (12)
Zn1^xii—Zn2—Zn3^xii	55.24 (7)	55.01 (7)	55.47 (7)
Zn1^xii—Zn2—Zn4	102.97 (6)	101.69 (6)	104.24 (6)
Zn1^xii—Zn2—Zn4^iii	57.70 (10)	56.34 (9)	59.09 (9)
Zn1^xii—Zn2—Zn4^xii	57.92 (10)	56.91 (10)	58.93 (10)
Zn3—Zn2—Zn3^iii	119.83 (11)	118.35 (10)	121.30 (10)
Zn3—Zn2—Zn3^xii	119.83 (16)	118.35 (15)	121.30 (15)
Zn3—Zn2—Zn4	61.17 (8)	60.51 (7)	61.83 (7)
Zn3—Zn2—Zn4^iii	158.18 (6)	157.07 (7)	159.34 (7)
Zn3—Zn2—Zn4^xii	60.80 (9)	60.40 (8)	61.20 (8)
Zn3^iii—Zn2—Zn3^xii	119.81 (13)	118.35 (13)	121.30 (14)
Zn3^iii—Zn2—Zn4	60.80 (9)	60.40 (9)	61.20 (9)
Zn3^iii—Zn2—Zn4^iii	61.15 (10)	60.51 (10)	61.83 (10)
Zn3^iii—Zn2—Zn4^xii	158.19 (7)	157.07 (7)	159.34 (7)
Zn3^xii—Zn2—Zn4	158.19 (6)	157.07 (7)	159.34 (7)
Zn3^xii—Zn2—Zn4^iii	60.79 (10)	60.40 (9)	61.20 (9)
Zn3^xii—Zn2—Zn4^xii	61.16 (9)	60.51 (7)	61.83 (7)
Zn4—Zn2—Zn4^iii	109.56 (12)	108.33 (12)	110.80 (12)
Zn4—Zn2—Zn4^xii	109.58 (9)	108.33 (10)	110.80 (10)
Zn4^iii—Zn2—Zn4^xii	109.56 (10)	108.33 (10)	110.80 (10)
Y1^xviii—Zn3—Y1^xii	68.80 (6)	68.11 (6)	69.50 (6)
Y1^xviii—Zn3—Y1^xix	178.95 (9)	178.35 (13)	180
Y1^xviii—Zn3—Y1^xvi	111.20 (7)	110.26 (8)	112.13 (9)
Y1^xviii—Zn3—Zn1	63.34 (5)	62.71 (4)	63.95 (4)
Y1^xviii—Zn3—Zn1^xv	116.67 (5)	116.14 (4)	117.18 (4)
Y1^xviii—Zn3—Zn2	118.40 (8)	117.32 (10)	119.46 (11)
Y1^xviii—Zn3—Zn2^xv	61.61 (6)	61.22 (6)	62.00 (6)
Y1^xviii—Zn3—Zn4	61.31 (6)	60.50 (7)	62.12 (7)
Y1^xviii—Zn3—Zn4^xii	117.76 (7)	116.54 (6)	118.96 (6)
Y1^xviii—Zn3—Zn4^xv	118.68 (8)	117.64 (9)	119.76 (9)
Y1^xviii—Zn3—Zn4^xvi	62.25 (6)	60.59 (6)	63.88 (6)
Y1^xii—Zn3—Y1^xix	111.19 (7)	110.26 (9)	112.13 (9)
Y1^xii—Zn3—Y1^xvi	179.75 (5)	179.59 (7)	180
Y1^xii—Zn3—Zn1	63.35 (8)	62.73 (7)	63.95 (7)
Y1^xii—Zn3—Zn1^xv	116.66 (9)	115.92 (10)	117.41 (10)
Y1^xii—Zn3—Zn2	118.39 (7)	117.90 (5)	118.90 (5)
Y1^xii—Zn3—Zn2^xv	61.61 (6)	61.05 (5)	62.15 (5)
Y1^xii—Zn3—Zn4	117.91 (7)	116.48 (6)	119.33 (5)
Y1^xii—Zn3—Zn4^xii	61.08 (7)	60.31 (7)	61.88 (7)
Y1^xii—Zn3—Zn4^xv	62.09 (7)	60.68 (6)	63.52 (6)
Y1^xii—Zn3—Zn4^xvi	118.93 (8)	117.88 (7)	119.95 (7)
Y1^xix—Zn3—Y1^xvi	68.81 (6)	68.11 (6)	69.50 (6)
Y1^xix—Zn3—Zn1	116.67 (5)	116.14 (4)	117.18 (4)
Y1^xix—Zn3—Zn1^xv	63.34 (5)	62.71 (4)	63.95 (4)
Y1^xix—Zn3—Zn2	61.61 (6)	61.22 (6)	62.00 (6)
Y1^xix—Zn3—Zn2^xv	118.38 (8)	117.32 (11)	119.47 (11)
Y1^xix—Zn3—Zn4	118.70 (8)	117.64 (9)	119.76 (9)
Y1^xix—Zn3—Zn4^xii	62.22 (6)	60.59 (6)	63.88 (6)
Y1^xix—Zn3—Zn4^xv	61.30 (6)	60.50 (7)	62.12 (7)
Y1^xix—Zn3—Zn4^xvi	117.75 (7)	116.54 (6)	118.96 (6)
Y1^xvi—Zn3—Zn1	116.65 (9)	115.92 (10)	117.41 (10)
Y1^xvi—Zn3—Zn1^xv	63.34 (8)	62.73 (7)	63.95 (7)
Y1^xvi—Zn3—Zn2	61.61 (6)	61.05 (5)	62.15 (5)
Y1^xvi—Zn3—Zn2^xv	118.39 (7)	117.90 (5)	118.90 (5)
Y1^xvi—Zn3—Zn4	62.09 (7)	60.68 (6)	63.52 (6)
Y1^xvi—Zn3—Zn4^xii	118.92 (8)	117.88 (7)	119.95 (7)
Y1^xvi—Zn3—Zn4^xv	117.91 (7)	116.48 (6)	119.33 (5)
Y1^xvi—Zn3—Zn4^xvi	61.08 (7)	60.31 (7)	61.88 (7)
Zn1—Zn3—Zn1^xv	179.32 (11)	178.93 (16)	180
Zn1—Zn3—Zn2	68.14 (6)	67.86 (4)	68.42 (4)
Zn1—Zn3—Zn2^xv	111.87 (6)	111.09 (8)	112.63 (8)
Zn1—Zn3—Zn4	62.58 (9)	61.86 (7)	63.30 (7)
Zn1—Zn3—Zn4^xii	62.58 (9)	61.25 (9)	63.90 (9)
Zn1—Zn3—Zn4^xv	117.42 (10)	116.46 (8)	118.40 (8)
Zn1—Zn3—Zn4^xvi	117.44 (10)	116.35 (14)	118.51 (14)
Zn1^xv—Zn3—Zn2	111.86 (6)	111.09 (9)	112.63 (8)
Zn1^xv—Zn3—Zn2^xv	68.14 (6)	67.86 (4)	68.42 (4)
Zn1^xv—Zn3—Zn4	117.41 (10)	116.46 (8)	118.40 (8)
Zn1^xv—Zn3—Zn4^xii	117.42 (10)	116.35 (14)	118.51 (14)
Zn1^xv—Zn3—Zn4^xv	62.59 (9)	61.86 (7)	63.30 (7)
Zn1^xv—Zn3—Zn4^xvi	62.56 (9)	61.25 (9)	63.90 (9)
Zn2—Zn3—Zn2^xv	179.52 (9)	179.25 (13)	180
Zn2—Zn3—Zn4	63.89 (8)	63.28 (10)	64.50 (10)
Zn2—Zn3—Zn4^xii	64.10 (10)	63.77 (10)	64.43 (9)
Zn2—Zn3—Zn4^xv	116.11 (9)	116.00 (8)	116.23 (7)
Zn2—Zn3—Zn4^xvi	115.89 (12)	115.55 (10)	116.24 (10)
Zn2^xv—Zn3—Zn4	116.11 (9)	116.00 (8)	116.23 (7)
Zn2^xv—Zn3—Zn4^xii	115.89 (12)	115.55 (10)	116.24 (10)
Zn2^xv—Zn3—Zn4^xv	63.88 (8)	63.28 (10)	64.50 (10)
Zn2^xv—Zn3—Zn4^xvi	64.10 (10)	63.77 (10)	64.43 (9)
Zn4—Zn3—Zn4^xii	114.23 (9)	113.09 (11)	115.38 (11)
Zn4—Zn3—Zn4^xv	179.57 (6)	179.32 (10)	180
Zn4—Zn3—Zn4^xvi	65.78 (9)	65.27 (10)	66.25 (10)
Zn4^xii—Zn3—Zn4^xv	65.77 (9)	65.27 (10)	66.25 (10)
Zn4^xii—Zn3—Zn4^xvi	178.50 (8)	177.65 (11)	180
Zn4^xv—Zn3—Zn4^xvi	114.21 (9)	113.09 (11)	115.37 (11)
Y1—Zn4—Y1^xviii	110.55 (7)	107.15 (7)	113.85 (7)
Y1—Zn4—Y1^xvi	121.30 (10)	120.58 (10)	122.05 (10)
Y1—Zn4—Zn1	111.24 (8)	109.40 (7)	113.14 (8)
Y1—Zn4—Zn1^iii	64.55 (8)	62.95 (8)	66.21 (8)
Y1—Zn4—Zn2	117.37 (9)	115.08 (8)	119.72 (9)
Y1—Zn4—Zn3	165.46 (9)	162.30 (8)	168.73 (9)
Y1—Zn4—Zn3^iii	68.78 (10)	67.31 (10)	70.29 (10)
Y1—Zn4—Zn5^xi	81.31 (12)	69.70 (11)	90.75 (11)
Y1—Zn4—Zn6	60.13 (6)	56.78 (5)	61.30 (7)
Y1^xviii—Zn4—Y1^xvi	120.81 (9)	119.90 (9)	121.75 (9)
Y1^xviii—Zn4—Zn1	64.24 (8)	62.61 (8)	65.91 (8)
Y1^xviii—Zn4—Zn1^iii	111.04 (8)	108.95 (8)	113.18 (8)
Y1^xviii—Zn4—Zn2	116.87 (12)	113.87 (12)	119.97 (12)
Y1^xviii—Zn4—Zn3	68.37 (9)	66.62 (9)	70.18 (9)
Y1^xviii—Zn4—Zn3^iii	165.40 (10)	161.86 (11)	169.11 (10)
Y1^xviii—Zn4—Zn5^xi	81.02 (11)	69.53 (11)	90.59 (9)
Y1^xviii—Zn4—Zn6	59.99 (7)	56.50 (6)	61.24 (8)
Y1^xvi—Zn4—Zn1	115.01 (9)	113.78 (9)	116.26 (9)
Y1^xvi—Zn4—Zn1^iii	115.28 (7)	114.51 (7)	116.06 (7)
Y1^xvi—Zn4—Zn2	61.70 (5)	60.40 (5)	63.02 (5)
Y1^xvi—Zn4—Zn3	67.85 (8)	66.97 (8)	68.75 (8)
Y1^xvi—Zn4—Zn3^iii	68.02 (7)	66.64 (7)	69.39 (7)
Y1^xvi—Zn4—Zn5^xi	80.80 (17)	63.59 (13)	100.7 (2)
Y1^xvi—Zn4—Zn6	130.14 (14)	122.16 (16)	135.17 (13)
Zn1—Zn4—Zn1^iii	57.69 (8)	56.34 (7)	59.08 (8)
Zn1—Zn4—Zn2	61.72 (7)	60.34 (8)	63.14 (8)
Zn1—Zn4—Zn3	54.75 (8)	53.74 (9)	55.79 (9)
Zn1—Zn4—Zn3^iii	102.02 (10)	99.75 (10)	104.38 (10)
Zn1—Zn4—Zn5^xi	144.13 (13)	131.07 (17)	150.67 (11)
Zn1—Zn4—Zn6	108.50 (13)	103.82 (12)	113.72 (14)
Zn1^iii—Zn4—Zn2	61.82 (8)	60.78 (7)	62.89 (7)
Zn1^iii—Zn4—Zn3	101.87 (12)	99.81 (10)	104.01 (11)
Zn1^iii—Zn4—Zn3^iii	55.01 (8)	53.91 (7)	56.14 (7)
Zn1^iii—Zn4—Zn5^xi	144.72 (17)	132.1 (2)	151.75 (16)
Zn1^iii—Zn4—Zn6	108.61 (14)	103.87 (12)	114.50 (16)
Zn2—Zn4—Zn3	54.94 (8)	53.87 (9)	56.05 (9)
Zn2—Zn4—Zn3^iii	55.09 (11)	54.49 (11)	55.73 (12)
Zn2—Zn4—Zn5^xi	142.49 (18)	126.51 (14)	163.6 (2)
Zn2—Zn4—Zn6	168.15 (15)	164.28 (13)	174.94 (16)
Zn3—Zn4—Zn3^iii	108.33 (11)	106.97 (11)	109.74 (11)
Zn3—Zn4—Zn5^xi	111.95 (11)	104.08 (10)	120.60 (11)
Zn3—Zn4—Zn6	125.73 (9)	125.06 (9)	126.08 (9)
Zn3^iii—Zn4—Zn5^xi	112.41 (13)	104.75 (12)	121.20 (14)
Zn3^iii—Zn4—Zn6	126.45 (12)	125.50 (13)	126.89 (11)
Zn5^xi—Zn4—Zn6	55.94 (15)	48.99 (16)	59.16 (15)
Y1—Zn5—Y1^xviii	86.92 (9)	71.31 (4)	102.71 (4)
Y1—Zn5—Y1^iv	86.92 (8)	71.31 (3)	102.71 (5)
Y1—Zn5—Y1^xvii	169.66 (11)	157.77 (3)	179.88 (3)
Y1—Zn5—Y1^xx	92.52 (4)	90.99 (4)	93.11 (4)
Y1—Zn5—Y1^xxi	92.52 (4)	91.01 (3)	93.12 (4)
Y1—Zn5—Zn4^xi	124.84 (15)	97.79 (4)	157.15 (5)
Y1—Zn5—Zn4^viii	55.30 (9)	45.94 (7)	64.66 (8)
Y1—Zn5—Zn4^ix	54.96 (7)	45.79 (5)	64.44 (6)
Y1—Zn5—Zn4^vi	55.28 (6)	45.26 (5)	60.73 (4)
Y1—Zn5—Zn4^xxii	122.46 (9)	107.69 (6)	128.56 (8)
Y1—Zn5—Zn4^xxiii	122.81 (8)	107.81 (5)	129.22 (6)
Y1—Zn5—Zn6	49.21 (5)	42.308 (19)	55.60 (8)
Y1—Zn5—Zn6^xvii	123.69 (6)	115.60 (2)	130.37 (6)
Y1^xviii—Zn5—Y1^iv	86.92 (9)	71.31 (4)	102.71 (5)
Y1^xviii—Zn5—Y1^xvii	92.52 (4)	91.01 (3)	93.12 (4)
Y1^xviii—Zn5—Y1^xx	169.66 (12)	157.77 (4)	179.87 (4)
Y1^xviii—Zn5—Y1^xxi	92.52 (5)	90.99 (4)	93.11 (5)
Y1^xviii—Zn5—Zn4^xi	55.30 (8)	45.94 (5)	64.66 (7)
Y1^xviii—Zn5—Zn4^viii	54.96 (8)	45.79 (6)	64.44 (8)
Y1^xviii—Zn5—Zn4^ix	124.84 (15)	97.79 (6)	157.15 (5)
Y1^xviii—Zn5—Zn4^vi	122.46 (8)	107.69 (6)	128.56 (6)
Y1^xviii—Zn5—Zn4^xxii	122.81 (9)	107.81 (6)	129.22 (7)
Y1^xviii—Zn5—Zn4^xxiii	55.28 (7)	45.26 (5)	60.73 (6)
Y1^xviii—Zn5—Zn6	49.21 (5)	42.31 (3)	55.60 (8)
Y1^xviii—Zn5—Zn6^xvii	123.69 (6)	115.60 (3)	130.37 (7)
Y1^iv—Zn5—Y1^xvii	92.52 (4)	90.99 (4)	93.11 (4)
Y1^iv—Zn5—Y1^xx	92.52 (5)	91.01 (5)	93.12 (5)
Y1^iv—Zn5—Y1^xxi	169.66 (11)	157.77 (3)	179.87 (3)
Y1^iv—Zn5—Zn4^xi	54.96 (9)	45.79 (6)	64.44 (8)
Y1^iv—Zn5—Zn4^viii	124.84 (15)	97.79 (4)	157.15 (5)
Y1^iv—Zn5—Zn4^ix	55.30 (8)	45.94 (5)	64.66 (7)
Y1^iv—Zn5—Zn4^vi	122.81 (9)	107.81 (6)	129.22 (8)
Y1^iv—Zn5—Zn4^xxii	55.28 (6)	45.26 (5)	60.73 (4)
Y1^iv—Zn5—Zn4^xxiii	122.46 (8)	107.69 (6)	128.56 (6)
Y1^iv—Zn5—Zn6	49.21 (5)	42.308 (19)	55.60 (8)
Y1^iv—Zn5—Zn6^xvii	123.69 (6)	115.60 (2)	130.37 (6)
Y1^xvii—Zn5—Y1^xx	86.80 (9)	71.10 (4)	102.45 (4)
Y1^xvii—Zn5—Y1^xxi	86.80 (8)	71.10 (3)	102.45 (5)
Y1^xvii—Zn5—Zn4^xi	55.24 (6)	45.08 (5)	60.73 (4)
Y1^xvii—Zn5—Zn4^viii	122.39 (9)	107.37 (6)	128.56 (8)
Y1^xvii—Zn5—Zn4^ix	122.75 (8)	107.51 (5)	129.22 (6)
Y1^xvii—Zn5—Zn4^vi	124.64 (15)	97.48 (4)	156.62 (5)
Y1^xvii—Zn5—Zn4^xxii	55.25 (9)	45.83 (7)	64.57 (8)
Y1^xvii—Zn5—Zn4^xxiii	54.90 (7)	45.67 (5)	64.34 (6)
Y1^xvii—Zn5—Zn6	123.75 (6)	115.81 (2)	130.31 (6)
Y1^xvii—Zn5—Zn6^xvii	49.16 (5)	42.170 (19)	55.66 (8)
Y1^xx—Zn5—Y1^xxi	86.80 (9)	71.10 (4)	102.45 (5)
Y1^xx—Zn5—Zn4^xi	122.39 (8)	107.37 (5)	128.56 (6)
Y1^xx—Zn5—Zn4^viii	122.75 (9)	107.51 (6)	129.22 (7)
Y1^xx—Zn5—Zn4^ix	55.24 (7)	45.08 (5)	60.73 (6)
Y1^xx—Zn5—Zn4^vi	55.25 (8)	45.83 (5)	64.57 (7)
Y1^xx—Zn5—Zn4^xxii	54.90 (8)	45.67 (6)	64.34 (8)
Y1^xx—Zn5—Zn4^xxiii	124.64 (15)	97.48 (6)	156.62 (5)
Y1^xx—Zn5—Zn6	123.75 (6)	115.81 (3)	130.31 (7)
Y1^xx—Zn5—Zn6^xvii	49.16 (5)	42.17 (3)	55.66 (8)
Y1^xxi—Zn5—Zn4^xi	122.75 (9)	107.51 (6)	129.22 (8)
Y1^xxi—Zn5—Zn4^viii	55.24 (6)	45.08 (5)	60.73 (4)
Y1^xxi—Zn5—Zn4^ix	122.39 (8)	107.37 (6)	128.56 (6)
Y1^xxi—Zn5—Zn4^vi	54.90 (9)	45.67 (6)	64.34 (8)
Y1^xxi—Zn5—Zn4^xxii	124.64 (15)	97.48 (4)	156.62 (5)
Y1^xxi—Zn5—Zn4^xxiii	55.25 (8)	45.83 (5)	64.57 (7)
Y1^xxi—Zn5—Zn6	123.75 (6)	115.81 (2)	130.31 (6)
Y1^xxi—Zn5—Zn6^xvii	49.16 (5)	42.170 (19)	55.66 (8)
Zn4^xi—Zn5—Zn4^viii	108.48 (12)	91.05 (6)	120.00 (10)
Zn4^xi—Zn5—Zn4^ix	108.48 (11)	91.05 (6)	120.00 (8)
Zn4^xi—Zn5—Zn4^vi	161.05 (18)	142.56 (5)	179.74 (6)
Zn4^xi—Zn5—Zn4^xxii	68.02 (9)	64.03 (8)	70.17 (9)
Zn4^xi—Zn5—Zn4^xxiii	68.03 (8)	64.03 (7)	70.17 (7)
Zn4^xi—Zn5—Zn6	65.35 (9)	55.48 (3)	77.18 (12)
Zn4^xi—Zn5—Zn6^xvii	101.78 (9)	87.25 (4)	114.85 (12)
Zn4^viii—Zn5—Zn4^ix	108.48 (12)	91.05 (8)	120.00 (9)
Zn4^viii—Zn5—Zn4^vi	68.03 (9)	64.03 (9)	70.17 (9)
Zn4^viii—Zn5—Zn4^xxii	161.05 (18)	142.56 (5)	179.74 (6)
Zn4^viii—Zn5—Zn4^xxiii	68.02 (9)	64.03 (8)	70.17 (9)
Zn4^viii—Zn5—Zn6	65.35 (9)	55.48 (3)	77.18 (12)
Zn4^viii—Zn5—Zn6^xvii	101.78 (9)	87.25 (4)	114.85 (12)
Zn4^ix—Zn5—Zn4^vi	68.02 (8)	64.03 (7)	70.17 (7)
Zn4^ix—Zn5—Zn4^xxii	68.03 (9)	64.03 (8)	70.17 (9)
Zn4^ix—Zn5—Zn4^xxiii	161.05 (19)	142.56 (6)	179.74 (6)
Zn4^ix—Zn5—Zn6	65.35 (9)	55.48 (5)	77.18 (12)
Zn4^ix—Zn5—Zn6^xvii	101.78 (10)	87.25 (4)	114.85 (12)
Zn4^vi—Zn5—Zn4^xxii	108.41 (12)	90.81 (6)	120.00 (10)
Zn4^vi—Zn5—Zn4^xxiii	108.41 (11)	90.81 (6)	120.00 (8)
Zn4^vi—Zn5—Zn6	101.87 (10)	87.57 (4)	114.73 (11)
Zn4^vi—Zn5—Zn6^xvii	65.32 (8)	55.31 (3)	77.32 (13)
Zn4^xxii—Zn5—Zn4^xxiii	108.41 (12)	90.81 (7)	120.00 (9)
Zn4^xxii—Zn5—Zn6	101.87 (10)	87.57 (4)	114.73 (11)
Zn4^xxii—Zn5—Zn6^xvii	65.32 (8)	55.31 (3)	77.32 (13)
Zn4^xxiii—Zn5—Zn6	101.87 (10)	87.57 (4)	114.73 (12)
Zn4^xxiii—Zn5—Zn6^xvii	65.32 (9)	55.31 (5)	77.32 (13)
Zn6—Zn5—Zn6^xvii	180	180	180
Y1—Zn6—Y1^xviii	119.33 (5)	118.01 (5)	120.00 (5)
Y1—Zn6—Y1^iv	119.33 (6)	118.01 (6)	120.00 (6)
Y1—Zn6—Zn4	64.39 (7)	60.80 (6)	66.65 (7)
Y1—Zn6—Zn4^xviii	149.04 (13)	135.97 (6)	164.23 (8)
Y1—Zn6—Zn4^iv	64.67 (6)	61.43 (5)	66.87 (6)
Y1—Zn6—Zn4^xi	148.83 (13)	135.78 (7)	163.97 (7)
Y1—Zn6—Zn4^viii	64.63 (8)	61.38 (7)	66.86 (8)
Y1—Zn6—Zn4^ix	64.35 (6)	60.75 (5)	66.65 (6)
Y1—Zn6—Zn5	89.82 (7)	81.83 (5)	97.82 (3)
Y1—Zn6—Zn5^xi	89.94 (7)	81.94 (4)	97.93 (4)
Y1^xviii—Zn6—Y1^iv	119.33 (5)	118.01 (6)	120.00 (5)
Y1^xviii—Zn6—Zn4	64.67 (7)	61.43 (6)	66.87 (7)
Y1^xviii—Zn6—Zn4^xviii	64.39 (8)	60.80 (7)	66.65 (8)
Y1^xviii—Zn6—Zn4^iv	149.04 (13)	135.97 (7)	164.23 (8)
Y1^xviii—Zn6—Zn4^xi	64.63 (7)	61.38 (6)	66.86 (7)
Y1^xviii—Zn6—Zn4^viii	64.35 (8)	60.75 (7)	66.65 (8)
Y1^xviii—Zn6—Zn4^ix	148.83 (13)	135.78 (8)	163.97 (8)
Y1^xviii—Zn6—Zn5	89.82 (7)	81.83 (5)	97.82 (3)
Y1^xviii—Zn6—Zn5^xi	89.94 (7)	81.94 (4)	97.93 (4)
Y1^iv—Zn6—Zn4	149.04 (13)	135.97 (6)	164.23 (8)
Y1^iv—Zn6—Zn4^xviii	64.67 (8)	61.43 (8)	66.87 (8)
Y1^iv—Zn6—Zn4^iv	64.39 (6)	60.80 (6)	66.65 (6)
Y1^iv—Zn6—Zn4^xi	64.35 (7)	60.75 (7)	66.65 (7)
Y1^iv—Zn6—Zn4^viii	148.83 (13)	135.78 (7)	163.97 (7)
Y1^iv—Zn6—Zn4^ix	64.63 (7)	61.38 (6)	66.86 (7)
Y1^iv—Zn6—Zn5	89.82 (7)	81.83 (5)	97.82 (3)
Y1^iv—Zn6—Zn5^xi	89.94 (7)	81.94 (4)	97.93 (4)
Zn4—Zn6—Zn4^xviii	95.71 (11)	89.00 (7)	104.65 (9)
Zn4—Zn6—Zn4^iv	95.71 (11)	89.00 (7)	104.65 (8)
Zn4—Zn6—Zn4^xi	129.21 (8)	127.89 (8)	129.83 (8)
Zn4—Zn6—Zn4^viii	62.13 (5)	59.99 (5)	63.42 (5)
Zn4—Zn6—Zn4^ix	128.65 (7)	127.04 (7)	129.47 (7)
Zn4—Zn6—Zn5	120.95 (7)	113.95 (5)	125.81 (4)
Zn4—Zn6—Zn5^xi	58.87 (7)	54.03 (4)	65.74 (4)
Zn4^xviii—Zn6—Zn4^iv	95.71 (11)	89.00 (8)	104.65 (9)
Zn4^xviii—Zn6—Zn4^xi	62.13 (5)	59.99 (5)	63.42 (5)
Zn4^xviii—Zn6—Zn4^viii	128.65 (10)	127.04 (9)	129.47 (10)
Zn4^xviii—Zn6—Zn4^ix	129.21 (8)	127.89 (8)	129.83 (8)
Zn4^xviii—Zn6—Zn5	120.95 (7)	113.95 (5)	125.81 (4)
Zn4^xviii—Zn6—Zn5^xi	58.87 (7)	54.03 (4)	65.74 (4)
Zn4^iv—Zn6—Zn4^xi	128.65 (7)	127.04 (7)	129.47 (7)
Zn4^iv—Zn6—Zn4^viii	129.21 (8)	127.89 (8)	129.83 (8)
Zn4^iv—Zn6—Zn4^ix	62.13 (7)	59.99 (7)	63.42 (7)
Zn4^iv—Zn6—Zn5	120.95 (8)	113.95 (6)	125.81 (5)
Zn4^iv—Zn6—Zn5^xi	58.87 (8)	54.03 (5)	65.74 (5)
Zn4^xi—Zn6—Zn4^viii	95.59 (11)	88.89 (7)	104.48 (9)
Zn4^xi—Zn6—Zn4^ix	95.59 (11)	88.89 (7)	104.48 (8)
Zn4^xi—Zn6—Zn5	58.78 (7)	53.96 (4)	65.60 (4)
Zn4^xi—Zn6—Zn5^xi	121.04 (7)	114.09 (5)	125.88 (4)
Zn4^viii—Zn6—Zn4^ix	95.59 (11)	88.89 (8)	104.48 (9)
Zn4^viii—Zn6—Zn5	58.78 (7)	53.96 (4)	65.60 (4)
Zn4^viii—Zn6—Zn5^xi	121.04 (7)	114.09 (5)	125.88 (4)
Zn4^ix—Zn6—Zn5	58.78 (8)	53.96 (5)	65.60 (5)
Zn4^ix—Zn6—Zn5^xi	121.04 (8)	114.09 (5)	125.88 (5)
Zn5—Zn6—Zn5^xi	180	180	180

Symmetry codes: (i) −x₂+1, x₁−x₂+1, x₃, −x₂+x₄; (ii) −x₁+x₂, −x₁+1, x₃, −x₁+x₄; (iii) −x₂+1, x₁−x₂, x₃, −x₂+x₄; (iv) −x₁+x₂+1, −x₁+2, x₃, −x₁+x₄; (v) −x₁+1, −x₂+1, −x₃+1, −x₄; (vi) x₂, x₁, x₃−1/2, x₄+1/2; (vii) −x₂+1, −x₁+2, −x₃+1/2, −x₄+1/2; (viii) −x₁+x₂+1, x₂, −x₃+1/2, x₂−x₄+1/2; (ix) x₁, x₁−x₂+1, −x₃+1/2, x₁−x₄+1/2; (x) x₂, −x₁+x₂+1, −x₃+1, x₂−x₄; (xi) −x₂+2, −x₁+2, −x₃+1/2, −x₄+1/2; (xii) −x₁+x₂+1, −x₁+1, x₃, −x₁+x₄; (xiii) −x₂+1, −x₁+1, −x₃+1/2, −x₄+1/2; (xiv) x₁, x₁−x₂, −x₃+1/2, x₁−x₄+1/2; (xv) −x₁+2, −x₂+1, −x₃+1, −x₄; (xvi) x₁−x₂+1, x₁, −x₃+1, x₁−x₄; (xvii) −x₁+2, −x₂+2, −x₃, −x₄; (xviii) −x₂+2, x₁−x₂+1, x₃, −x₂+x₄; (xix) x₂, −x₁+x₂, −x₃+1, x₂−x₄; (xx) x₂, −x₁+x₂+1, −x₃, x₂−x₄; (xxi) x₁−x₂+1, x₁, −x₃, x₁−x₄; (xxii) x₁−x₂+1, −x₂+2, x₃−1/2, −x₂+x₄+1/2; (xxiii) −x₁+2, −x₁+x₂+1, x₃−1/2, −x₁+x₄+1/2.