The incommensurate composite YxOs4B4 (x = 1.161)

Stöger, B.; Sologub, O.; Salamakha, L.

doi:10.1107/S205252062400982X

research papers

STRUCTURAL SCIENCE
CRYSTAL ENGINEERING
MATERIALS

ISSN: 2052-5206

Volume 80| Part 6| December 2024|

https://doi.org/10.1107/S205252062400982X

Open

access

The incommensurate composite Y_xOs₄B₄ (x = 1.161)

Berthold Stöger,^a ^* Oksana Sologub ^b and Leonid Salamakha ^b

^aX-Ray Centre, TU Wien, Getreidemarkt 9, 1060 Vienna, Austria, and ^bInstitute of Solid State Physics, TU Wien, Wiedner Hauptstraße 8–10, 1040 Vienna, Austria
^*Correspondence e-mail: bstoeger@mail.tuwien.ac.at

Edited by M. de Boissieu, SIMaP, France (Received 29 January 2024; accepted 8 October 2024; online 31 October 2024)

Y_xOs₄B₄ (x = 1.161) crystallizes as a tetragonal incommensurate composite of columns of Y atoms extending along [001] in an Os₄B₄ framework. The structure was refined using the superspace approach. The basic structure of the Y subsystem can be idealized as having I4/mmm symmetry, with a crystallographically unique Y atom located on the 4/mmm position. The actual superspace symmetry is P4₂/nmc(00σ₃)s0s0. The Y atoms feature only subtle positional modulation in the [001] direction. The Os₄B₄ subsystem [P4₂/ncm(00σ₃)00ss superspace symmetry] is built of columns of edge-sharing Os₄ tetrahedra extending along [001] and B₂ dumbbells. The Os₄ tetrahedra feature pronounced positional modulation with a distinct variation of the Os—Os bond lengths. Modulation of the B₂ dumbbells is best described as a rotation about the [001] axis.

Keywords: borides; superconducting alloys; incommensurate composites; superspace approach.

B-IncStrDB reference: wgToiDUSNfn

CCDC reference: 2389397

1. Introduction

The borides RM₄B₄ (R = rare earth metal, M = transition metal) constitute a family of compounds exhibiting a rich variety of interesting physical properties (superconductivity, ferromagnetism, re-entrant superconductivity) (Johnston & Braun, 1982 ; Maple et al., 1982 ). Depending on the R and M constituents, these compounds crystallize in four structure types, CeCo₄B₄ (space group P4₂/nmc) (Ku'zma & Bilonizhko, 1972 ), LuRu₄B₄ (space group I4₁/acd) (Johnston, 1977 ), LuRh₄B₄ (space group Ccca) (Yvon & Johnston, 1982 ) and NdCo₄B₄ (space group P4₂/n) (Ku'zma & Bilonizhko, 1978 ), which can be considered as a family of polytypes (Yvon & Johnston, 1982). The common structural units of all these structures are B—B dumbbells, M₄ tetrahedra and columns of R atoms.

Despite the spectacular properties of reported RM₄B₄ compounds, the borides with M = Os have not been studied sufficiently. The NdCo₄B₄ structure was proved for ROs₄B₄ (R = La–Nd, Sm); a different but unknown structural variant has been suggested for compounds of 1–4–4 stoichiometry occurring in the R–Os–B system when R is a heavy rare earth element or yttrium (Rogl, 1979 ; Ku, 1980 ).

In continuation of our studies of the Y–Os–B system (Sologub et al., 2007 ), we explored the B-rich corner and obtained phases with composition Y_xOs₄B₄ with x close to one. Careful examination of single-crystal X-ray diffraction data revealed the structural complexities of this compound, reminiscent of those found in related boride systems with other transition metals (Zavalij et al., 1994 ; Bezinge et al., 1985 ).

Structurally, Y_xOs₄B₄ belongs to a family of compounds with a structure derived from NdCo₄B₄. These R_xM₄B₄ compounds crystallize in tetragonal commensurate or incommensurate composite structures (van Smaalen, 2007 ) of R columns contained in an M₄B₄ framework. The incommensurability of Y_xOs₄B₄ has been observed by Zavalij et al. (1994) but no structure refinement has been published up to now. Other known members of the family are various iron borides R_x(Fe₄B₄) (Bezinge et al., 1985) and the manganese borides Pr_x(Mn₄B₄) and Pr_x(Re₄B₄) (Zavalij et al., 1994).

To the best of our knowledge, refinements of these structures have always been restricted to either individual subsystems or to commensurate approximants. Here, we report on the synthesis of Y_xOs₄B₄ and the refinement of its incommensurate structure using the superspace approach (Wolff et al., 1981 ).

2. Experimental

2.1. Synthesis

A sample with a total weight of 0.5 g was synthesized by arc melting appropriate amounts of the constituent elements in stoichiometry ∼1:4:4 under a Ti-gettered high-purity argon atmosphere on a water-cooled copper hearth. Pieces of yttrium (ChemPur, Germany, 99.9 mass%), crystalline boron (ChemPur, Germany, 99.8 mass%) and re-melted pellets of compacted osmium powder (Sigma–Aldrich, USA, 99.9 mass%) were used as starting materials. The arc-melted button was cut into pieces, from which a larger portion of alloy was wrapped in tantalum foil and vacuum-sealed in a quartz tube for annealing at 800°C for 240 h. Crystals were isolated via mechanical fragmentation of the annealed alloy. Single-crystal X-ray intensity data were collected for the specimen of best quality, which was verified via preliminary inspection on a four-circle Bruker APEX II diffractometer (CCD detector, κ geometry, Mo Kα radiation).

2.2. Single-crystal diffraction

Intensity data from a tiny crystal of the title compound were collected at 300 K in a dry stream of nitrogen on a Stoe STADIVARI diffractometer system equipped with a Mo Kα micro-source and a DECTRIS Eiger CdTe hybrid photon-counting (HPC) detector. Data were processed using X-AREA (Stoe & Cie GmbH, 2021 ). Data reduction was performed as a 3+1-dimensional modulated structure with satellites up to the second order. Corrections for absorption effects were applied using the multi-scan approach followed by a spherical absorption correction implemented in LANA (Stoe & Cie GmbH, 2021). An initial model was derived from the published data of related boride phases. The structure was refined with JANA2006 (Petříček et al., 2014 ). Data collection and refinement details are compiled in Table 1 and an overview of the cell parameters and symmetries of the two subsystems is given in Table 2.

Table 1
Data collection and refinement details for Y_1.161Os₄B₄

Crystal data
Chemical formula	B₄Os₄Y_1.161
M_r	907.3
Crystal system	Tetragonal
Temperature (K)	300
Radiation type	Mo Kα
ρ_calc (g cm⁻³)	13.2532
μ (mm⁻¹)	128.424
Crystal shape, color	Fragment, black
Crystal size (mm)	0.08 × 0.05 × 0.02

Data collection
Diffractometer	Stoe STADIVARI
Absorption correction	Multi-scan
T_min, T_max	0.035, 0.187
No. of measured, independent and observed [I > 3σ(I)] reflections	16701, 1363, 1144
R_int	0.0720
(sinθ/λ)_max (Å⁻¹)	0.84

Refinement
R_obs, wR(F²)_obs, R_all, wR(F²)_all
All	0.0321, 0.1108, 0.0390, 0.1182
Main	0.0295, 0.1119, 0.0323, 0.1161
First order	0.0316, 0.1062, 0.0373, 0.1112
Second order	0.0455, 0.1173, 0.0722, 0.1329
No. of parameters	48
Δρ_max, Δρ_min (e Å⁻³)	−3.73, 3.94
Extinction (Gaussian)	97 (17)

Table 2
Cell parameters and symmetry

	Subsystem 1 (Os₄B₄)	Subsystem 2 (Y)
Composition	B₄Os₄	Y
Superspace group	P4₂/ncm(00σ₃)00ss	P4₂/nmc(00σ₃)s0s0
Origin	$[\overline{4}]$	$[\overline{4} m2]$ , shifted by $[{{1} \over {4}}]$ in the x₄ direction
W	I	$[\left ( \matrix { 1 & 0 & 0 & 0 \cr 0 & 1 & 0 & 0 \cr 0 & 0 & 1 & 1 \cr 0 & 0 & 1 & 0 } \right )]$
a, c (Å)	7.4495 (3), 4.0967 (2)	7.4495 (3), 3.5276 (2)
q	0.161335 (10)c*	0.861078 (10)c*
V (Å³)	227.347 (17)	195.764 (17)
Z	2	2

3. Results and discussion

3.1. Indexing of the diffraction pattern

Y_xOs₄B₄ is an incommensurate composite of two tetragonal subsystems, Os₄O₄ and Y. Let $[({\bf a}^{*}, {\bf b}^{*}, {\bf c}^{*}_{\rm {OsB}})^{\rm T}]$ and $[({\bf a}^{*}, {\bf b}^{*}, {\bf c}^{*}_{\rm {Y}})^{\rm T}]$ be the reciprocal bases of these two subsystems, which share the a* and b* basis vectors. Note that, by convention, we write reciprocal bases as columns, as indicated by the superscript T, since they transform as contravariant tensors (Sands, 2002 ). In a classical treatment of an incommensurate composite, the reflections would be indexed using the four-dimensional basis $[({\bf a}^{*}, {\bf b}^{*}, {\bf c}^{*}_{\rm {OsB}}, {\bf c}^{*}_{\rm {Y}})^{\rm T}]$ . The first three basis vectors correspond to the reciprocal basis of Os₄B₄, which is used as the `reference system' because it contributes significantly more to the diffraction intensity than the Y subsystem.

The satellite order of the hklm_Y reflection (subscript Y since $[{\bf a}^{*}_{4}]$ = $[{\bf c}^{*}_{\rm {Y}}]$ ) is then given by $[\min(|l|, |m|)]$ . However, the observed diffraction intensities suggested a different integration approach. Since c_OsB/c_Y ≃ $[{{7} \over {6}}]$ , the structure can be approximated by a sixfold superstructure with respect to the Os₄B₄ subsystem, or sevenfold with respect to the Y subsystem. Fig. 1 gives the average reflection intensity by |l| when integrated in such a super-cell setting. The main reflections of the Os₄B₄ subsystem are marked in blue and those of the Y subsystem in yellow. The figure shows that |l| = 6n ± 1 reflections, which are located next to the Os₄B₄ subsystem's main reflections (|l| = 6n), are in general stronger than the Y subsystem's main reflections (|l| = 7n).

Figure 1
Average reflection intensities plotted versus |l| when integrated as a sixfold (Os₄B₄) or sevenfold (Y) superstructure. Yellow (Os₄B₄) and blue (Y) indicate main reflections of the corresponding subsystem. The |l| = 21 and |l| = 28 reflections discussed in the text are marked by arrows.

Therefore, we applied a change of reciprocal basis to $[({\bf a}^{*}, {\bf b}^{*}, {\bf c}^{*}_{\rm {OsB}}, {\bf q})^{\rm T}]$ with

$[{\bf q} = {\bf c}^{*}_{\rm {Y}} - {\bf c}^{*}_{\rm { OsB}} \eqno(1)]$

according to the following scheme.

Expressed with respect to $[{\bf c}^{*}_{\rm {OsB}}]$ , we obtain q = $[\sigma_{3} {\bf c}^{*}_{\rm {OsB}}]$ , with σ₃ = 0.16134 (8) ≃ $[{{1} \over {6}}]$ . We used this cell and modulation vector and integrated with satellites up to the second order. The residuals listed in Table 1 are given with respect to this satellite order. Reflection indices with respect to this basis will be given with a subscript q.

Note that the two indexing approaches are in fact different. For example, the l = 28 superstructure reflections are hk04_Y main reflections with respect to the $[({\bf a}^{*}, {\bf b}^{*}, {\bf c}^{*}_{\rm {OsB}}, {\bf c}^{*}_{\rm {Y}})^{\rm T}]$ basis or $[hk 5 \overline{2}_{q}]$ second-order satellites with respect to $[({\bf a}^{*}, {\bf b}^{*}, {\bf c}^{*}_{\rm {OsB}}, {\bf q})^{\rm T}]$ . These are indeed different reflections in reciprocal superspace: when transforming $[hk 5 \overline{2}_{q}]$ back, one obtains the indexes $[hk 7 \overline{2}_{\rm Y}]$ , i.e. still a second-order satellite. Since the intensity of this reflection is dominated by the second-order satellite, it is reasonable to integrate using q as the modulation wavevector.

This highlights the fundamental problem of reflection overlap: hk04_Y and $[hk 5 \overline{2}_{q}]$ will both contribute to the integrated reflection intensity. However, the JANA2006 software can detect close reflections and will treat these intensities as the sum of two reflections. From a refinement and symmetry point of view, these two settings are therefore equivalent.

Likewise, the weak reflections halfway between the Os₄B₄ main reflections (hkl3_q and $[hkl \overline{3}_{q}]$ ) could not be properly resolved into the m = 3 and m = −3 components. When including these reflections in the integration, the reliability factors of the refinements (and also of the main and low-order satellite reflections) worsened. An integration using the sixfold supercell (with respect to the Os₄Y₄ system) and transformation into the incommensurate cell likewise led to distinctly worsened reliability factors. Ultimately, we therefore only used satellites up to the second order. This means that the hk03_Y and $[hk 0 \overline{3}_{\rm Y}]$ main reflections of the Y subsystem were not included in the refinements (see Fig. 1). However, this appears acceptable, given all the other Y subsystem main reflections are included (even if indirectly).

The reflection overlap likewise affects the evaluation of the length of the q vector and consequently the evaluation of the chemical composition. Different lengths were obtained from different data reduction strategies (e.g. different maximum satellite order or independent integration of both subsystems).

The length given here was derived from the integration used for the refinements (with satellites up to the second order). It has to be stressed, however, that the actual uncertainty in the length of q is larger than that determined by the integration software. In all integration attempts, σ₃ was distinctly smaller than $[{{1} \over {6}}]$ and inspection of the images showed splitting or enlargement of satellites in the c* direction, confirming the incommensurate character of the structure.

3.2. Superspace embedding

Each subsystem of an incommensurate composite is modulated with a modulation wave corresponding to the periodicity of the other subsystem(s). Thus, it is useful to embed the structure in superspace by analogy with classical incommensurately modulated structures. The superspace then has a 3+d-dimensional (here d = 1) superspace group symmetry. However, as noted by van Smaalen (1991 ) and Zeiner & Janssen (2003 ), the peculiar situation arises that, when transforming from one subsystem to the other, non-equivalent superspace groups may be obtained. In fact, in Y_xOs₄B₄, the Os₄B₄ subsystem has P4₂/ncm(00σ₃)00ss and the Y subsystem has P4₂/nmc(00σ₃)s0s0 superspace group symmetry (note the interchanging of the direction of the n glides and the m reflections). The origin of the Y subsystem is moved by $[{{1} \over {4}}]$ in the x₄ direction with respect to the Os₄B₄ subsystem.

Fig. 2 schematizes (not to scale) an (x₃, x₄) section of superspace of such a refinement. In particular, it shows the embedding of a Y atom (gray lines) and an Os atom (blue lines), assuming the absence of positional modulation and point-shaped atoms. The real-space structure is given by a t = const section perpendicular to a₄, as indicated by a dotted line. The Os lines, since they belong to the `reference system', are parallel to a₄ and spaced by c_OsB. The Y line is inclined in such a way that the Y atoms in a t = const section are spaced by c_Y.

Figure 2
A diagram of the superspace embedding of Y_xOs₄B₄ in superspace without positional modulation. The Y (gray) and Os (blue) atoms are represented by lines projected on the (x₃, x₄) plane.

The relative slope of the Y line is −(1 + σ₃). Note that traditionally one would use the vector c_OsB − (1 + σ₃)a₄ as the third basis vector a₃, since then the Y lines extend parallel to a₃. Here we chose a₃ = c_OsB − σ₃a₄, since this is the dual basis of the reciprocal basis used for integration (Section 3.1). In a sense, this is the reduced supercell, as the angle between a₃ and a₁, a₂ is minimized, resulting in less-skewed unit cells in superspace sections.

The slope of the Y lines is automatically implemented by the JANA2006 software when relating the reciprocal basis of the Y subsystem $[({\bf a}^{*}, {\bf b}^{*}, {\bf c}^{*}_{\rm Y}, {\bf c}^{*}_{\rm OsB})^{\rm T}]$ to the refinement basis $[({\bf a}^{*}, {\bf b}^{*}, {\bf c}^{*}_{\rm OsB}, {\bf q})^{\rm T}]$ using the W matrix. From equation (1) it follows that $[{\bf c}^{*}_{\rm Y}]$ = $[{\bf c}^{*}_{\rm OsB} + {\bf q}]$ . Thus, the basis of the Y subsystem is given as

$[\eqalignno{ \left ( \matrix { {\bf a}^{*} \cr {\bf b}^{*} \cr {\bf c}^{*}_{\rm Y} \cr {\bf c}^{*}_{\rm OsB}} \right ) = & \, \left ( \matrix{ 1 & 0 & 0 & 0 \cr 0 & 1 & 0 & 0 \cr 0 & 0 & 1 & 1 \cr 0 & 0 & 1 & 0 } \right ) \left ( \matrix{ {\bf a}^{*} \cr {\bf b}^{*} \cr {\bf c}^{*}_{\rm OsB} \cr {\bf q} } \right ) &(2) \cr = & \, {\bf W} ({\bf a}^{*}, {\bf b}^{*}, {\bf c}^{*}_{\rm OsB}, {\bf q})^{\rm T}. &(3)}]$

3.3. Basic structures

The two tetragonal subsystems share the a and b lattice basis vectors [a = b = 7.4495 (3) Å]. The space-group symmetries of the basic structures are obtained by ignoring the x₄ component of the respective superspace symmetry. Accordingly, the basic structure of the Os₄B₄ subsystem [Z = 2, c_OsB = 4.09669 (15) Å] features P4₂/ncm symmetry (we use origin choice 1, the origin in $[\overline{4}]$ ). It comprises one crystallographically unique Os and one B atom, both located on the 8i Wyckoff position, i.e on the m_〈110〉 reflection planes. The Os atoms form columns of edge-sharing Os₄ tetrahedra, generated by the 4₂ screw rotations (Fig. 3). The shared edge represents the shortest Os—Os distance [2.6713 (3) Å] of the structure. Four symmetry-equivalent Os—Os contacts [2.7863 (4) Å] form the non-shared edges of the tetrahedra. The rods are connected by 2.8240 (3) Å Os—Os bonds [dotted line in Fig. 3(b)], whereby these bonds are centered on 2_〈110〉 rotation axes. The B atoms form B₂ dumbbells [B—B bond length of 1.707 (8) Å] located on ..2/m positions (Fig. 3). Note that the given geometric parameters were derived by refining against the main reflection of the Os₄B₄ subsystem, which technically results in the average structure, not the basic structure.

Figure 3
The P4₂/ncm basic structure of the Os₄B₄ subsystem viewed (a) down [001] and (b) down [100]. Os₄ units are represented by blue tetrahedra and B atoms by pink spheres of arbitrary radius. The positions of the Y atoms in the Y subsystem are indicated by gray spheres in panel (a).

The Y subsystem (Z = 2, c_Y = 3.5276 Å) is built of columns of Y atoms centered in the channels of the Os₄B₄ subsystem at x = y = 0 [Fig. 3(a)]. The basic structure has P4₂/nmc symmetry (origin choice 1 in $[\overline{4} m2]$ ). Note again that this is a different space-group type from the OsB subsystem, with the directions of the c and m reflections inverted. One crystallographically unique Y molecule is located on the $[\overline{4} m2]$ position at x = y = z = 0.

The orbit of an atom is the set of all atoms generated by application of the space group symmetry. If, assuming spherical atoms, this set features higher symmetry than the space group, the orbit is non-characteristic (Engel et al., 1984 ). The orbit of the Y atom (located on the $[\overline{4}m2]$ position of P4₂/nmc) is non-characteristic with I4/mmm symmetry. Thus, if the Y atoms are considered as point charges or centrosymmetric electron distributions, the basic structure of the Y subsystem possesses I4/mmm symmetry, as has been reported earlier (Zavalij et al., 1994). Due to the I centering, every second Y column is translated by c_Y/2 in a checkerboard pattern, as indicated by darker shading in Fig. 3(a).

However, the interaction with the Os₄B₄ subsystem must not be neglected and the actual time- and space-averaged electron density of the Y position might not be centrosymmetric. This can be expressed by anharmonic atomic displacement parameters (ADPs). For the sake of argument, we might assume that the averaged electron density is tetrahedral in shape, which is compatible with the $[\overline{4} m2]$ site symmetry. The Y rods at x = y = 0 and x = y = $[{{1} \over {2}}]$ are then inverted with respect to [001] and not related by a translation, i.e. the I centering is lost. A similar argument can be made for the positional modulation of the Y atom.

The chemical composition of Y_xOs₄B₄ is determined by the fraction of the unit-cell volumes of the basic structures of the subsystems: x = c_OsB/c_Y = 1.161 ≃ $[{{7} \over {6}}]$ .

3.4. Modulation of the Y subsystem

According to the superspace group symmetry, the Y atom may be positionally modulated along the c direction of the basic structure. No positional modulation is possible in the (a, b) plane. Since we expected at least a slight influence of the Os₄B₄ subsystem on the Y subsystem, the modulation of the x₃ coordinate was modeled with harmonics up to the fourth order. Although such high-order harmonics might seem excessive for a data set with satellites up to the second order, they correspond to merely two modulation parameters, since only odd harmonics of even order [sin(2n2πt_Y), $[n \in {\bb N}^{+}]$ ] are allowed. Note that the argument to the modulation function depends only on t_Y, because Y is located at the origin of the basic structure. The subscript indicates that t_Y is given with respect to the Y system. In the remainder of this work t will be given with respect to the Os₄B₄ system.

The residuals on the satellite reflections improved when including the fourth-order parameter, and the refined value is more than five times its standard uncertainty [x₃ = 0.0052 (5) sin(4πt_Y) + 0.0035 (6) sin(8πt_Y)]. Moreover, when including fourth-order terms, the range of Y—Y distances along the c axis (Fig. 4) match better with those of a commensurate superstructure refinement.

Figure 4
A t plot of Y—Y distances when modeling modulation of the z coordinate of Y with up to (red) second-order and (black) fourth-order harmonics. Here, in contrast to all other t plots in this work, the t coordinate is given with respect to the Y subsystem.

The absolute positional modulation is very subtle and barely visible in superspace sections (Fig. 5). Modulation of the ADPs did not improve the refinement and therefore the ADPs of Y were modeled as constant over internal space.

Figure 5
A 6 Å wide (x₃, x₄) section of superspace centered on the Y atom located at x₁ = x₂ = 0. The refined position of the Y atom is given by gray lines. Contours are drawn at the 20 e Å⁻³ level.

3.5. Modulation of the Os₄B₄ subsystem

Positional modulation of Os was modeled with harmonics up to the fourth order and that of B with harmonics up to the second order. The amplitudes of even higher harmonics refined to less than twice their standard uncertainties. As for Y, owing to symmetry, refinement of fourth-order Os harmonics corresponds to fewer parameters than one might expect. For first- and third-order harmonics, only displacement perpendicular to the m_〈110〉 reflection planes is possible, corresponding to two parameters per harmonic (one amplitude and one phase) versus six parameters for an atom on the general position (two per spatial dimension). For the second-order and fourth-order harmonics, displacement is possible parallel to the m_〈110〉 planes, corresponding to four parameters. Modulation of the Os ADPs was modeled with harmonics up to the second order. Modulation of the B ADPs did not improve the fit and was therefore omitted. An overview of the employed modulation parameters in given in Table 3.

Table 3
Employed modulation parameters

	Position		ADP
Atom	Maximum order	Number	Maximum order	Number
Os	4	12	2	12
B	2	6	–	–
Y	4	2	–	–

Displacement perpendicular to m_〈110〉 is significantly more pronounced than in the plane (Fig. 6). No signs of discontinuity of the modulation functions were observed in the superspace electron density (Fig. 7).

Figure 6
The modulated Y_xOs₄B₄ structure viewed down [001]. In the Os columns, only the shared edges between adjacent tetrahedra are shown for clarity. The m_〈110〉 planes on which the Os and B atoms are located in the basic structure are shown by the usual symbols. The gray background indicates the section shown later in Fig. 13.

Figure 7
The 2 Å wide superspace sections in the (x₁ + x₂, x₄) plane centered on (a) the Os and (b) the B atom, showing displacement away from the m_[110] reflection plane on which the Os and B atoms are located in the basic structure. Contours are drawn at levels of (a) 50 e Å⁻³ and (b) 2 e Å⁻³.

The modulation of the B₂ dumbbells is best described as a rotation about the $[{{1} \over {4}} {{1} \over {4}} z]$ axis and symmetry equivalents (Fig. 6), leading to practically constant B—B bond lengths (Fig. 8). In contrast, the Os₄ tetrahedra show significant distortions (Fig. 9). In particular, the shortest Os—Os bonds, which correspond to the shared edges between Os₄ tetrahedra, vary from 2.627 (4) to 2.740 (4) Å (red curve in Fig. 9).

Figure 8
A t plot of B—B distances of the B₂ dumbbells.

Figure 9
A t plot of Os—Os distances (red: edge-linked adjacent Os₄ tetrahedra, green: remaining tetrahedron edges, black: bond connecting tetrahedra in different columns).

The B₂ dumbbells connect adjacent Os columns. Fig. 10(a) shows two such adjacent Os columns and the connecting B₂ dumbbells. The t dependence of the corresponding bond lengths is given in Fig. 10(b). Both figures use the same color coding of the Os—B bonds. In general, each B atom is coordinated to four Os atoms. Two bonds connect Os atoms in two adjacent Os₄ tetrahedra in the same Os column (green in Fig. 10). A further bond is formed to the edge connecting these two tetrahedra. Owing to the positional modulation of the edge, the bond `switches' in internal space between the two Os atoms of the edge (represented by red and black in Fig. 10). Finally, a fourth bond connects to the second Os column (yellow in Fig. 10).

Figure 10
(a) Two adjacent Os columns connected by B₂ dumbbells. Shared edges between Os₄ tetrahedra are represented by thick lines and other Os—Os bonds by thin lines. (b) A t plot of Os—B distances. Color codes are discussed in the main text.

3.6. Interactions of the subsystems

The pronounced modulation of the Os₄B₄ subsystem is certainly due to the interaction with the Y columns. Each Os and each B atom are connected to two Y columns (gray background in Fig. 6). Fig. 11 gives a comparison of the Os—Y distances in the modulated structure and the hypothetical structure without positional modulation. In the actual structure [Fig. 11(a)], Os is in general coordinated to two Y atoms with approximately equal distances. In small parts of the internal space there are three close Y atoms. There, as expected, the Os—Y distances are slightly longer. Coordination in the hypothetical non-modulated structure [Fig. 11(b)] is chemically less reasonable, with generally one very short (<3 Å) and one or two distant Y atoms. The coordination of the B atoms is similar, with regions of the internal space where B is close to two and other regions where it is close to three Y atoms (Fig. 12). Given the minute modulation of the Y subsystem, one can conclude that the Os₄B₄ subsystem adapts to the Y subsystem but not vice versa. This explains why an integration as a modulated Os₄B₄ structure was more successful than a classical integration as an incommensurate composite structure (see Section 3.1).

Figure 11
The t plots of the Os—Y distances in (a) actual Y_xOs₄B₄ and (b) the hypothetical structure without positional modulation.

Figure 12
The t plots of the B—Y distances.

The connectivity of Os/B columns to the two adjacent Y columns (gray background in Fig. 6) is shown in Fig. 13. Note that the average positions of the Y atoms in the two Y columns are translated by c_Y/2, which corresponds to the pseudo-I centering of the basic structure of the Y subsystem.

Figure 13
A column of Os and B atoms with adjacent Y atoms (marked by the gray background in Fig. 6), viewed down $[[1\overline{1}0]]$ . Y—Os and Y—B contacts are shown up to arbitrary cut offs of 3.3 Å and 3.1 Å, respectively.

4. Conclusion and outlook

Y_xOs₄B₄ is an incommensurate composite where one subsystem (Os₄B₄) is significantly more modulated than the other (Y). Therefore, processing the raw data as if it were a regular modulated structure of the Os₄B₄ subsystem was preferred over a classical treatment as a composite system. Despite the issues of overlapping reflections, which are typical for aperiodic structures, a satisfying refinement was achieved, proving the robustness of single-crystal diffraction.

A detailed investigation of the physical properties of Y_xOs₄B₄, including density functional theory calculations, will be published in an upcoming paper.

Supporting information

B-IncStrDB reference: wgToiDUSNfn

CCDC reference: 2389397

Computing details top

(I) top

Crystal data top

B₄Os₄Y_1.161	D_x = 13.253 Mg m⁻³
M_r = 907.3	Mo Kα radiation, λ = 0.71073 Å
Tetragonal, P42/ncm(00γ)00ss†	Cell parameters from 29359 reflections
q = 0.161335c*‡	θ = 3.8–36.6°
a = 7.4495 (3) Å	µ = 128.42 mm⁻¹
c = 4.0967 (2) Å	T = 300 K
V = 227.35 (2) Å³	Fragment, black
Z = 2	0.08 × 0.05 × 0.02 × 0.05 (radius) mm
F(000) = 739

† Symmetry operations: (1) x₁, x₂, x₃, x₄; (2) −x₁, −x₂, x₃, x₄; (3) −x₂+1/2, x₁+1/2, x₃+1/2, x₄; (4) x₂+1/2, −x₁+1/2, x₃+1/2, x₄; (5) −x₁+1/2, x₂+1/2, −x₃, −x₄+1/2; (6) x₁+1/2, −x₂+1/2, −x₃, −x₄+1/2; (7) x₂, x₁, −x₃+1/2, −x₄+1/2; (8) −x₂, −x₁, −x₃+1/2, −x₄+1/2; (9) −x₁+1/2, −x₂+1/2, −x₃+1/2, −x₄; (10) x₁+1/2, x₂+1/2, −x₃+1/2, −x₄; (11) x₂, −x₁, −x₃, −x₄; (12) −x₂, x₁, −x₃, −x₄; (13) x₁, −x₂, x₃+1/2, x₄+1/2; (14) −x₁, x₂, x₃+1/2, x₄+1/2; (15) −x₂+1/2, −x₁+1/2, x₃, x₄+1/2; (16) x₂+1/2, x₁+1/2, x₃, x₄+1/2.

‡ $W^{1}=\left(\matrix{ 1 & 0 & 0 & 0 \cr 0 & 1 & 0 & 0 \cr 0 & 0 & 1 & 0 \cr 0 & 0 & 0 & 1 \cr }\right)$ ; $W^{2}=\left(\matrix{ 1 & 0 & 0 & 0 \cr 0 & 1 & 0 & 0 \cr 0 & 0 & 1 & 1 \cr 0 & 0 & 1 & 0 \cr }\right)$

Data collection top

STOE STADIVARI diffractometer	1363 independent reflections
Radiation source: Axo_Mo	1144 reflections with I > 3σ(I)
Graded multilayer mirror monochromator	R_int = 0.072
Detector resolution: 13.33 pixels mm^-1	θ_max = 36.7°, θ_min = 3.2°
Absorption correction: multi-scan STOE LANA, absorption correction by scaling of reflection intensities. J. Koziskova, F. Hahn, J. Richter, J. Kozisek, "Comparison of different absorption corrections on the model structure of tetrakis(µ₂-acetato)- diaqua-di-copper(II)", Acta Chimica Slovaca, vol. 9, no. 2, 2016, pp. 136 - 140. Afterwards a spherical absorption correction was performed within STOE LANA.	h = −12→5
T_min = 0.035, T_max = 0.187	k = −10→12
16701 measured reflections	l = −7→7

Refinement top

Refinement on F²	0 constraints
R[F² > 2σ(F²)] = 0.032	Weighting scheme based on measured s.u.'s w = 1/(σ²(I) + 0.0025000002I²)
wR(F²) = 0.118	(Δ/σ)_max = 0.030
S = 1.91	Δρ_max = 3.94 e Å⁻³
1363 reflections	Δρ_min = −3.73 e Å⁻³
48 parameters	Extinction correction: B-C type 1 Gaussian isotropic (Becker & Coppens, 1974)
0 restraints	Extinction coefficient: 97 (17)

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
Os1	0.12679 (2)	0.37321 (2)	0.61441 (6)	0.00695 (9)
B1	0.1892 (6)	0.3108 (6)	0.1114 (15)	0.0098 (11)
Y1	0	0	0	0.0129 (5)

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Os1	0.00637 (16)	0.00637 (16)	0.00811 (17)	0.00129 (8)	−0.00006 (4)	0.00006 (4)
B1	0.0103 (15)	0.0103 (15)	0.009 (2)	0.003 (2)	0.0008 (12)	−0.0008 (12)
Y1	0.0076 (3)	0.0076 (3)	0.0233 (15)	0	0	0

Geometric parameters (Å, º) top

	Average	Minimum	Maximum
Os1—Os1ⁱ	2.692 (3)	2.627 (4)	2.740 (4)
Os1—Os1ⁱⁱ	2.796 (3)	2.744 (3)	2.859 (3)
Os1—Os1ⁱⁱⁱ	2.797 (3)	2.744 (3)	2.859 (3)
Os1—Os1^iv	2.797 (3)	2.744 (3)	2.859 (3)
Os1—Os1^v	2.796 (3)	2.744 (3)	2.859 (3)
Os1—Os1^vi	2.837 (3)	2.807 (3)	2.856 (3)
Os1—B1	2.187 (10)	2.166 (10)	2.209 (10)
Os1—B1^vii	2.171 (10)	2.149 (10)	2.197 (10)
Os1—B1ⁱⁱⁱ	2.403 (13)	2.137 (14)	2.807 (14)
Os1—B1^v	2.400 (13)	2.137 (14)	2.807 (14)
Os1—B1^viii	2.157 (12)	2.093 (13)	2.221 (13)
Os1—Y1^vii	322.7 (3)	3.073 (3)	3.826 (3)
Os1—Y1^ix	414.0 (3)	3.073 (3)	5.401 (3)
Os1—Y1ⁱⁱⁱ	397.4 (3)	3.079 (3)	5.182 (3)
Os1—Y1^x	329.6 (3)	3.073 (3)	4.092 (3)
B1—B1^viii	1.758 (16)	1.750 (16)	1.769 (16)
B1—Y1	363.4 (12)	2.970 (13)	4.929 (13)
B1—Y1^vii	318.8 (12)	2.742 (13)	4.287 (13)
B1—Y1ⁱⁱⁱ	296.6 (12)	2.742 (13)	3.264 (13)
B1—Y1^x	419.3 (11)	2.780 (13)	5.453 (13)
Y1—Y1^xi	3.527 (3)	3.475 (4)	3.556 (4)
Y1—Y1^vii	3.527 (3)	3.475 (4)	3.556 (4)

Os1ⁱ—Os1—Os1ⁱⁱ	61.26 (8)	59.21 (8)	63.94 (8)
Os1ⁱ—Os1—Os1ⁱⁱⁱ	61.22 (8)	59.54 (8)	64.21 (8)
Os1ⁱ—Os1—Os1^iv	61.22 (8)	59.21 (8)	63.94 (8)
Os1ⁱ—Os1—Os1^v	61.26 (8)	59.54 (8)	64.21 (8)
Os1ⁱ—Os1—Os1^vi	154.11 (8)	150.73 (9)	158.10 (7)
Os1ⁱ—Os1—B1	106.5 (3)	104.7 (3)	108.0 (3)
Os1ⁱ—Os1—B1^vii	106.3 (3)	104.9 (3)	107.7 (3)
Os1ⁱ—Os1—B1ⁱⁱⁱ	56.3 (3)	45.3 (3)	69.2 (3)
Os1ⁱ—Os1—B1^v	56.4 (3)	45.3 (3)	69.2 (3)
Os1ⁱ—Os1—B1^viii	152.6 (3)	149.0 (3)	156.0 (3)
Os1ⁱⁱ—Os1—Os1ⁱⁱⁱ	94.31 (9)	91.64 (9)	96.40 (9)
Os1ⁱⁱ—Os1—Os1^iv	57.53 (8)	55.84 (7)	58.89 (8)
Os1ⁱⁱ—Os1—Os1^v	122.51 (11)	119.23 (11)	128.11 (11)
Os1ⁱⁱ—Os1—Os1^vi	136.51 (11)	125.81 (11)	149.04 (11)
Os1ⁱⁱ—Os1—B1	56.0 (3)	49.5 (3)	66.9 (3)
Os1ⁱⁱ—Os1—B1^vii	146.7 (3)	130.8 (3)	161.0 (3)
Os1ⁱⁱ—Os1—B1ⁱⁱⁱ	48.4 (2)	45.8 (2)	50.8 (3)
Os1ⁱⁱ—Os1—B1^v	97.4 (3)	90.0 (2)	106.8 (3)
Os1ⁱⁱ—Os1—B1^viii	96.5 (3)	89.8 (3)	103.8 (3)
Os1ⁱⁱⁱ—Os1—Os1^iv	122.44 (11)	119.23 (11)	128.11 (11)
Os1ⁱⁱⁱ—Os1—Os1^v	57.52 (8)	56.01 (8)	58.75 (8)
Os1ⁱⁱⁱ—Os1—Os1^vi	98.41 (8)	91.13 (8)	106.26 (8)
Os1ⁱⁱⁱ—Os1—B1	146.9 (3)	134.0 (3)	160.3 (3)
Os1ⁱⁱⁱ—Os1—B1^vii	56.2 (3)	48.7 (3)	67.5 (3)
Os1ⁱⁱⁱ—Os1—B1ⁱⁱⁱ	48.9 (2)	45.73 (19)	51.8 (3)
Os1ⁱⁱⁱ—Os1—B1^v	97.9 (3)	90.8 (2)	105.6 (3)
Os1ⁱⁱⁱ—Os1—B1^viii	138.4 (3)	128.0 (3)	150.7 (3)
Os1^iv—Os1—Os1^v	94.31 (9)	91.64 (9)	96.40 (9)
Os1^iv—Os1—Os1^vi	136.65 (11)	125.81 (11)	149.04 (11)
Os1^iv—Os1—B1	56.2 (3)	49.5 (3)	66.9 (3)
Os1^iv—Os1—B1^vii	146.7 (3)	130.8 (3)	161.0 (3)
Os1^iv—Os1—B1ⁱⁱⁱ	97.3 (3)	90.0 (2)	106.8 (3)
Os1^iv—Os1—B1^v	48.4 (2)	45.8 (2)	50.8 (3)
Os1^iv—Os1—B1^viii	96.5 (3)	89.8 (3)	103.8 (3)
Os1^v—Os1—Os1^vi	98.44 (8)	91.13 (8)	106.26 (8)
Os1^v—Os1—B1	147.1 (3)	134.0 (3)	160.3 (3)
Os1^v—Os1—B1^vii	56.2 (3)	48.7 (3)	67.5 (3)
Os1^v—Os1—B1ⁱⁱⁱ	97.9 (3)	90.8 (2)	105.6 (3)
Os1^v—Os1—B1^v	48.9 (2)	45.73 (19)	51.8 (3)
Os1^v—Os1—B1^viii	138.3 (3)	128.0 (3)	150.7 (3)
Os1^vi—Os1—B1	94.6 (3)	93.3 (3)	96.3 (3)
Os1^vi—Os1—B1^vii	48.9 (3)	46.8 (3)	50.7 (3)
Os1^vi—Os1—B1ⁱⁱⁱ	120.8 (3)	114.8 (3)	125.2 (3)
Os1^vi—Os1—B1^v	120.8 (3)	114.8 (3)	125.2 (3)
Os1^vi—Os1—B1^viii	49.2 (3)	48.2 (3)	50.1 (3)
B1—Os1—B1^vii	140.8 (5)	135.4 (5)	145.9 (5)
B1—Os1—B1ⁱⁱⁱ	99.5 (4)	91.7 (4)	109.7 (4)
B1—Os1—B1^v	99.6 (4)	91.7 (4)	109.7 (4)
B1—Os1—B1^viii	47.7 (4)	47.0 (4)	48.5 (4)
B1^vii—Os1—B1ⁱⁱⁱ	100.1 (4)	89.1 (4)	112.0 (4)
B1^vii—Os1—B1^v	100.2 (4)	89.1 (4)	112.0 (4)
B1^vii—Os1—B1^viii	97.1 (4)	94.4 (4)	99.5 (4)
B1ⁱⁱⁱ—Os1—B1^v	112.6 (4)	111.1 (4)	114.6 (4)
B1ⁱⁱⁱ—Os1—B1^viii	119.9 (4)	110.9 (4)	129.0 (4)
B1^v—Os1—B1^viii	119.7 (4)	110.9 (4)	129.0 (4)
Os1^xi—B1—Os1	140.7 (6)	133.8 (6)	148.4 (6)
Os1^xi—B1—Os1ⁱⁱ	75.4 (4)	66.2 (3)	81.9 (4)
Os1^xi—B1—Os1^iv	75.3 (4)	66.2 (3)	81.9 (4)
Os1^xi—B1—Os1^viii	81.9 (4)	80.2 (3)	83.5 (4)
Os1^xi—B1—B1^viii	142.1 (8)	139.0 (8)	145.6 (7)
Os1—B1—Os1ⁱⁱ	75.0 (4)	67.4 (3)	80.6 (4)
Os1—B1—Os1^iv	74.9 (4)	67.4 (3)	80.6 (4)
Os1—B1—Os1^viii	131.4 (6)	129.9 (5)	132.9 (6)
Os1—B1—B1^viii	65.3 (5)	62.4 (5)	68.1 (5)
Os1ⁱⁱ—B1—Os1^iv	67.3 (4)	65.4 (3)	68.9 (4)
Os1ⁱⁱ—B1—Os1^viii	138.8 (6)	131.3 (5)	146.1 (6)
Os1ⁱⁱ—B1—B1^viii	127.2 (7)	110.6 (7)	138.0 (6)
Os1^iv—B1—Os1^viii	138.7 (5)	131.3 (5)	146.1 (6)
Os1^iv—B1—B1^viii	127.0 (7)	110.6 (7)	138.0 (6)
Os1^viii—B1—B1^viii	67.0 (5)	64.9 (5)	69.2 (6)
Y1^xi—Y1—Y1^vii	180	180	180

Symmetry codes: (i) −x₁, −x₂+1, x₃, x₄; (ii) −x₂+1/2, x₁+1/2, x₃−1/2, x₄; (iii) −x₂+1/2, x₁+1/2, x₃+1/2, x₄; (iv) x₂−1/2, −x₁+1/2, x₃−1/2, x₄; (v) x₂−1/2, −x₁+1/2, x₃+1/2, x₄; (vi) x₂, x₁, −x₃+3/2, −x₄+1/2; (vii) x₁, x₂, x₃+1, x₄; (viii) x₂, x₁, −x₃+1/2, −x₄+1/2; (ix) x₁, x₂, x₃+2, x₄; (x) −x₂+1/2, x₁+1/2, x₃+3/2, x₄; (xi) x₁, x₂, x₃−1, x₄.

Acknowledgements

The authors acknowledge TU Wien Bibliothek for financial support through its Open Access Funding Program.

Funding information

The following funding is acknowledged: Austrian Science Fund (grant No. P31979-N36 to Oksana Sologub).

References

Bezinge, A., Braun, H. F., Muller, J. & Yvon, K. (1985). Solid State Commun. 55, 131–135. CrossRef CAS Google Scholar
Engel, P., Matsumoto, T., Steinmann, G. & Wondratschek, H. (1984). The Non-Characteristic Orbits of Space Groups. Oldenburg: De Gruyter. Google Scholar
Johnston, D. C. (1977). Solid State Commun. 24, 699–702. CrossRef CAS Google Scholar
Johnston, D. C. & Braun, H. F. (1982). Superconductivity in Ternary Compounds, Vol. II, edited by M. B. Maple & Ø. Fischer, pp. 11–55. Berlin, Heidelberg, New York: Springer. Google Scholar
Ku, H. C. (1980). PhD thesis, University of California at San Diego, California, USA. Google Scholar
Ku'zma, Y. & Bilonizhko, N. S. (1972). Sov. Phys. Crystallogr. 16, 897–898. Google Scholar
Ku'zma, Y. & Bilonizhko, N. S. (1978). Dopov. Akad. Nauk. Ukr. RSR Ser. A, 3, 276–278. Google Scholar
Maple, M. B., Hamker, H. C. & Woolf, L. D. (1982). Superconductivity in Ternary Compounds, Vol. II, edited by M. B. Maple & Ø. Fischer, pp. 99–141. Berlin, Heidelberg, New York: Springer. Google Scholar
Petříček, V., Dušek, M. & Palatinus, L. (2014). Z. Kristallogr. 229, 345–352. Google Scholar
Rogl, P. (1979). Monatsh. Chem. 110, 235–243. CrossRef CAS Google Scholar
Sands, D. E. (2002). Vectors and Tensors in Crystallography. New York: Dover Publications. Google Scholar
Smaalen, S. van (1991). Phys. Rev. B, 43, 11330–11341. CrossRef Web of Science Google Scholar
Smaalen, S. van (2007). Incommensurate Crystallography. IUCr Monographs on Crystallography, Vol. 21. Oxford University Press. Google Scholar
Sologub, O. L., Salamakha, L. P., Noël, H., Roisnel, T. & Gonçalves, A. P. (2007). J. Solid State Chem. 180, 2740–2746. CrossRef CAS Google Scholar
Stoe & Cie GmbH, (2021). X-AREA, Version 1.31.175.0, and LANA, Version 2.6.2.0. Stoe & Cie, Darmstadt, Germany. Google Scholar
Wolff, P. M. de, Janssen, T. & Janner, A. (1981). Acta Cryst. A37, 625–636. CrossRef IUCr Journals Web of Science Google Scholar
Yvon, K. & Johnston, D. C. (1982). Acta Cryst. B38, 247–250. CrossRef CAS IUCr Journals Google Scholar
Zavalij, P. Y., Mykhalenko, S. I. & Kuz'ma, Y. B. (1994). J. Alloys Compd. 203, 55–59. CrossRef CAS Google Scholar
Zeiner, P. & Janssen, T. (2003). Symmetry and Structural Properties of Condensed Matter, edited by T. Lulek, B. Lulek & A. Wal, pp. 378–392. Singapore: World Scientific Publishing. Google Scholar