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Hydroflux syntheses and crystal structures of hydrogarnets Ba3[RE(OH)6]2 (RE = Sc, Y, Ho–Lu)

  • Ralf Albrecht , Florian Graßme , Thomas Doert and Michael Ruck EMAIL logo
Published/Copyright: October 12, 2020
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Abstract

Colorless crystals of the barium rare earth hydrogarnets Ba3[RE(OH)6]2 (RE = Sc, Y, Ho–Lu) were synthesized under hydroflux conditions with KOH at about T = 200 °C starting from the respective RE2O3 and Ba (NO3)2. Single-crystal X-ray diffraction analysis on these distorted rhombic dodecahedra revealed the cubic space group Ia3d (no. 230). The crystal structures of the hydrogarnets Ba3[RE(OH)6]2 are discussed and compared with those of other hydrogarnets. The occurrence of additional reflections, which do not fulfill the reflection conditions of Ia3d, is analyzed and described by Renninger or λ/2 effects. A correlation is established between the space group adopted by a hydrogarnet and characteristic interatomic distances. In addition, single-crystal structure data of the strontium indium hydrogarnet Sr3[In(OH)6]2 are provided.

1 Introduction

Garnets, a class of nesosilicates, are described by the general formula A32+M23+(XO4)3. In the flexible garnet structure, a variety of elements can be host on the dodecahedral A and octahedral M positions. In the majority of these structures, earth-alkaline metals and bivalent transition metals like iron(II) or manganese(II) are found on the A position [1]. On the M position, trivalent 3d transitions metals or elements of the boron group are usually situated. The tetrahedral X position is normally occupied by silicon (IV). Garnets crystallize typically in the cubic space group Ia3d with a rhombic dodecahedral shape forming a complex three-dimensional crystal structure. In hydrogarnets, like the mineral katoite, Ca3[Al(OH)6]2 = Ca3Al2(O4H4)3, the tetrahedral sites X can be partially or completely unoccupied. The charge is balanced by protonation of the four oxide ions of the silicate tetrahedron, (SiO4)4– ↔ (O4H4)4–. Hydrogarnets can easily be dehydrated and thus are useful carbon-free single-source precursors for functional oxides, e.g., with rare earth metals [2], [3], [4].

In the group of barium hydrogarnets, Ba3[Al(OH)6]2, Ba3[In(OH)6]2 and Ba3[Sc(OH)6]2 are the only known representatives [5], [6]. These compounds were synthesized by using the hydrothermal method yielding fine powders. Experiments to synthesize other barium rare earth hydrogarnets failed, so that these three compounds were assumed to be the only barium containing hydrogarnets existing [6]. A similar observation was made for strontium rare earth hydrogarnets, where a hydrothermal synthesis of hydrogarnets with rare earth metals larger than thulium was described as impossible [7].

We succeeded in synthesizing a variety of hydrogarnets using the novel hydroflux method as highly crystalline samples at comparatively low temperatures [8], [9]. Among them are strontium rare earth hydrogarnets Sr3[RE(OH)6]2 (RE = Sc, Y, Ho–Lu), of which we also obtained large and well-shaped crystals [4]. The reaction medium of the hydroflux method consists of an approximately equimolar mixture of alkali-metal hydroxide and water. The autogenous pressure is much lower than in classical hydrothermal processes, making the use of autoclaves dispensable. In addition, the hydroflux medium has a good solubility of oxides and hydroxides, which allows, among other things, the direct use of rare earth oxides instead of more expensive nitrates or sulfates. We now employed this approach to explore the apparently barren field of barium rare earth hydrogarnets. The seven hydrogarnets Ba3[RE(OH)6]2 (RE = Sc, Y, Ho–Lu) presented here considerably expand the knowledge about this compound class and provide the first single-crystal structures of barium hydroxometalates with the respective elements [10]. In addition, we provide single-crystal structure data of the strontium indium hydrogarnet Sr3[In(OH)6]2.

2 Results and discussion

2.1 Synthesis

An approximately equimolar mixture of water and diverse alkali-metal hydroxides can be utilized as a reaction medium in the hydroflux synthesis, with sodium and potassium hydroxide being the most common. In our experiments, no significant influence of the nature of these hydroxides was observed on the formation of the barium rare earth hydrogarnets. The base concentration of the hydroflux represents one of the most important reaction parameters [11]. By lowering the base concentration, the size and the quality of the crystals decreased until only a powder was obtained, which is in well accordance with the results of the hydrothermal experiments. Additionally, the product formation is notably slower when using the hydrothermal method, like in the synthesis of Ba3[In(OH)6]2 and Ba3[Sc(OH)6]2 where only small amounts of these hydrogarnets were obtained after 90 h at T = 200 °C and quantitative yields required 95 h at 400 °C [6]. In contrast, a typical hydroflux synthesis is completed after 10 h at 200 °C or even faster [12].

The syntheses of the here described barium rare earth hydrogarnets were performed in a potassium hydroxide hydroflux with a water-base molar ratio of n(H2O):n(KOH) = 1.6 in exception of Ba3[Sc(OH)6]2, where a ratio of 3 was used. Ba(NO3)2 and RE2O3 were added to the hydroflux in the stoichiometric ratio of the target compound. As both binary compounds are known to form hydroxides in alkaline aqueous solution [13], [14], the following metathesis reaction can be formulated:

3Ba(OH)2+2K3[RE(OH)6]Ba3[RE(OH)6]2+6KOH

The experiments were performed in a stainless steel autoclave with PTFE inlet considering the high corrosivity and to maintain a constant base concentration. The autoclaves were heated up to 200 °C for 10 h before slowly cooling down to room temperature. All barium hydrogarnets show signs of decomposition on their surfaces upon washing with water. Consequently, the products were washed with methanol and colorless distorted rhombic dodecahedra in single-crystal quality were obtained (Figure 1). In the case of the synthesis of Ba3[Ho(OH)6]2, large amounts of Ho(OH)3 and BaCO3 were obtained, showing parallels to the synthesis of Sr3[Ho(OH)6]2 [4]. In contrast, the remaining barium rare earth hydrogarnets were received phase pure (Figures S1 and S2, Supporting Information available online). Experiments with the remaining lanthanide oxides excluding promethium resulted in the formation of the respective hydroxides or oxide hydroxides as well as barium carbonate. In a synthesis with the same reaction parameters but starting from Sr(NO3)2 and In2O3, water-stable colorless rhombic dodecahedral crystals of Sr3[In(OH)6]2 were obtained (Figure S3, Information). Syntheses and powder X-ray analyses have already been reported for this compound [15]. Since no single-crystal diffraction data of this compound were published [10] we would like to fill also this gap.

Figure 1: Single crystals of Ba3[Lu(OH)6]2 with the typical morphology of hydrogarnets obtained from hydroflux.
Figure 1:

Single crystals of Ba3[Lu(OH)6]2 with the typical morphology of hydrogarnets obtained from hydroflux.

The thermal decomposition of Ba3[Y(OH)6]2 was investigated by thermogravimetry (TG) up to T = 1000 °C in a stream of CO2-free synthetic air (Figure S4, Supporting Information). In the thermogram, the first decomposition signal appears at about 250 °C, where three equivalents of water are quickly released. Subsequent heating to 900 °C leads in several steps to complete dehydration. The observed total mass loss of 13.9% matches the theoretical value (13.6%) for the elimination of six water molecules per formula unit. In addition, a sample of Ba3[Y(OH)6]2 was annealed at 1000 °C in air for 12 h. The powder X-ray diffraction (PXRD) pattern of the product (Figure S5, Supporting Information) revealed BaY2O4 and BaCO3. Thermal stability and decomposition products of Ba3[Y(OH)6]2 resemble those of Sr3[RE(OH)6]2 (RE = Sc, Y, Ho–Lu) [4].

2.2 Crystal structures

Typical hydrogarnets crystallize in the centrosymmetric cubic space group Ia3d (no. 230) with the structure type of katoite, Ca3[Al(OH)6]2 [16]. In contrast, the strontium rare earth hydrogarnets adopt the non-centrosymmetric space group I43d (no. 220) [4]. In the case of the hydrogarnets Ba3[RE(OH)6]2 (RE = Y, Ho–Lu, but not Sc), weak reflections that violate the reflection conditions of the space group Ia3d were found in the single-crystal X-ray diffraction patterns at 100(1) K. For example, in the 0kl plane of Ba3[Tm(OH)6]2 (Figure 2), additional reflections with odd k and l indices are present, which do not comply with the reflection condition 0kl: k, l = 2n of the space group Ia3d. The observed diffraction pattern does also not match the non-centrosymmetric cubic space group I43d of the strontium rare earth hydrogarnets. Instead, the pattern fits to I4132, which is another maximal subgroup of Ia3d in the cubic system. The group-subgroup relations [17], [18] between these three closely related structure types are displayed in a Bärnighausen tree (Figure S6, Supporting Information). However, the appearance of violating reflections is not systematic (see below) and their average I/σ(I) ratio is about one only (Figure 3 and Figure S7, Supporting Information). For comparison, in the case of the strontium rare earth hydrogarnets (I43d), the reflections that violate the reflection conditions for the a glide planes had an average I/σ(I) ratio greater than six (same diffractometer and temperature, similar crystal sizes and measuring parameters).

Figure 2: Reconstructed 0kl planes of Ba3[Sc(OH)6]2 and Ba3[Tm(OH)6]2 hydrogarnets showing different reflection densities. Note the small reflections with odd k and l indices for the thullium hydrogarnet, which do not fulfill the reflection conditions of Ia3‾$‾{3}$d.
Figure 2:

Reconstructed 0kl planes of Ba3[Sc(OH)6]2 and Ba3[Tm(OH)6]2 hydrogarnets showing different reflection densities. Note the small reflections with odd k and l indices for the thullium hydrogarnet, which do not fulfill the reflection conditions of Ia3d.

Figure 3: Ratio of the X-ray diffraction intensity and the standard deviation of selected reflection sets of Ba3[Sc(OH)6]2 and Ba3[Yb(OH)6]2 single crystals. For the ytterbium hydrogarnet, a small fraction of the reflections of a set have an unusually large I/σ ratio caused by the Renninger effect [19].
Figure 3:

Ratio of the X-ray diffraction intensity and the standard deviation of selected reflection sets of Ba3[Sc(OH)6]2 and Ba3[Yb(OH)6]2 single crystals. For the ytterbium hydrogarnet, a small fraction of the reflections of a set have an unusually large I/σ ratio caused by the Renninger effect [19].

Upon close inspection of the data we are convinced that all violating reflections can be explained either by Renninger or λ/2 effects. If several reflections meet the Laue condition simultaneously, i.e., if two or more reciprocal lattice points concurrently lie on the Ewald sphere, this can lead to multiple reflection (Umweganregung, four-beam case), which enhances the intensities of weak reflections and lowers those of strong reflections significantly [19]. Such Renninger reflections never occur outside the Bravais lattice, but can violate zonal or serial reflection conditions. Renninger reflections vanish under a slightly different measuring position, i.e., the azimuthal angle ψ, of the same reflection. The effect is typically strong in crystals of high quality (low mosaic spread).

The here investigated cubic hydrogarnets are a showcase for multiple reflection. The quality of the crystals was excellent and extinction correction was necessary. Moreover, cooling to 100 (1) K reduced the lattice vibrations. The data sets (Rσ < 1%) have a high redundancy, with one and the same reflection being measured under several ψ angles. For example, the data of Ba3[Yb(OH)6]2 include the reflection 0 1 5 and its symmetry equivalents (multiplicity 24). These reflections should be systematically absent, because of the a glide planes of Ia3d. 23 of the 142 measured data of the {0 1 5} set have Io ≥ 4 σ (Io) and thus clearly violate the reflection condition; yet the other 119 data comply with the space group (Table S1, Supporting Information). Multiple measurements of identical reflections at different measuring positions (ψ angles) yielded very different intensities (after applying all corrections). In the full data sets of the here presented barium hydrogarnets, all large deviations (FoFc)/σ(Fo) are positive and concern predominantly weak reflections. The donating reflections with decreased intensity by multiple reflection do not occur in the upper part of the (FoFc)/σ(Fo) list because their intensities and standard deviations are comparatively high in relation to the transferred intensities. Refinements in cubic subgroups of Ia3d that have no zonal reflection conditions yielded neither an improved model (better R values or smaller displacement parameters) nor removed the conspicuous positive deviations (FoFc)/σ(Fo).

Some of the reflections which do not fulfill the reflection conditions of Ia3d have Io ≥ 3σ(Io) for all measured data, e.g., {0 0 2} of Ba3[Yb(OH)6]2. Such an occurrence is very unlikely for the Renninger effect. Since these reflections are only found at small diffraction angles and their corresponding reflections 2h 2k 2l are very strong, we assume that they are caused by the λ/2 effect.

Hence, all barium rare earth hydrogarnets Ba3[RE(OH)6]2 with RE = Sc, Y, Ho–Lu crystallize with the centrosymmetric katoite structure in the space group Ia3d (Figure 4, Tables S2–S12 of the Supporting Information). This structure type was discussed in detail, e.g., for the isostructural hydrogarnets Sr3[Cr(OH)6]2 and Sr3[Rh(OH)6]2 [9]. The asymmetric unit consists of four atoms: rare earth metal (site symmetry.3; Wyckoff position 16a), barium (2.22; 24c), oxygen (1; 96h) and hydrogen (1; 96h). Within the [REO6] octahedron, the RE-O bond lengths are in average about 0.4% longer than in Sr3[RE(OH)6]2 with the exception of Ba3[Sc(OH)6]2, which has 0.1% longer bonds compared to its strontium equivalent [4]. The shape of the [BaO8] polyhedron is between a cube and a square antiprism. The Ba-O bond lengths slightly increase with the size of the rare earth metal atoms and range from 279.1(2) to 279.9(2) pm for Ba3[RE(OH)6]2 (RE = Y, Ho–Lu), while being 277.7(1) pm in the scandium compound. The [REO6] octahedra share their edges with six barium atoms, three on each side of the octahedron viewing along [111], forming a one-dimensional chain of staggered [REO6] and [Ba3O18] polyhedra parallel to each of the threefold axes (Figure S8, Supporting Information). Every [BaO8] polyhedron is part of another chain, resulting in a three-dimensional network.

Figure 4: Left: Crystal structure of Ba3[Ho(OH)6]2 projected along [111], as an example of a hydrogarnet. Ellipsoids comprise 99.99% probability densities of the atoms at 100(1) K. Hydrogen atoms are omitted for clarity. Right: [RE(OH)6]3– octahedron and (O4H4)4– tetrahedra in Ba3[Ho(OH)6]2.
Figure 4:

Left: Crystal structure of Ba3[Ho(OH)6]2 projected along [111], as an example of a hydrogarnet. Ellipsoids comprise 99.99% probability densities of the atoms at 100(1) K. Hydrogen atoms are omitted for clarity. Right: [RE(OH)6]3– octahedron and (O4H4)4– tetrahedra in Ba3[Ho(OH)6]2.

In all hydrogarnets, four adjacent hydroxide groups form an empty (O4H4)4– tetrahedron (Figure 4). The hydrogen atoms are located near faces of the polyhedron with the O-H bonds roughly bisecting the angle of the face [20]. The O (H)···O distances in the (O4H4)4– tetrahedron are long (>360 pm) and thus hydrogen bonds are supposed to be weak and not structure-directing. In the IR spectrum, the hydrogarnets Ba3[RE(OH)6]2 (RE = Sc, Y, Ho–Lu) show single infrared absorption bands in the narrow interval between 3632 and 3634 cm−1 (Figure S9, Supporting Information), which is related to the valence vibration of their hydroxide groups. Compared to the strontium rare earth hydrogarnets, the O-H valence vibrations in the barium compounds occur at higher energies, which fits well with the longer distances to the acceptor of the hydrogen bonds OD-H···OA (D: donor, A: acceptor atom of the H bridge).

2.3 Correlation between adopted space group and the ratio of interatomic distances d(M–O)/d(A–O)

Since hydrogen bonds seem to be not important for the structure, the space group of a specific hydrogarnet A3[M(OH)6]2 might depend on the size of its cations. Using the effective ionic radii tabulated by Shannon [21] did not lead to a meaningful correlation. Thus, we tested the structure immanent ratio of (average) interatomic distances d(M–O)/2d(A–O). The double weight for d(A–O) takes the occurrence of the respective interatomic distances into account (2 × [MO6] vs. 3 × [AO8] polyhedra). Table 1 lists most of the known hydrogarnets together with their space groups in the order of increasing distance ratio. Obviously, for small values of this ratio, i.e., small M3+ and large A2+ cations, the centrosymmetric space group Ia3d applies. I43d is favored at large ratios, i.e., for large M3+ and small A2+ cations. As can be expected, there is no hard threshold value. For 0.4 ≤ d(M–O)/2d(A–O) ≤ 0.415 examples for both space groups are observed.

Table 1:

Ratio of interatomic distances d(M–O)/2d(A–O) in strontium and barium hydrogarnets A3[M(OH)6]2. Unmarked compounds crystallize in the centrosymmetric katoite type (space group Ia3d), marked (underlined in gray) compounds in the non-centrosymmetric subgroup I43d.

A3[M(OH)6]2d(M–O)/2d(A–O)A3[M(OH)6]2d(M–O)/2d(A–O)
Sr3[Al(OH)6]2 [22]0.366Sr3[Sc(OH)6]2 [4]0.401
Ba3[Al(OH)6]2 [5]0.378Ba3[Er(OH)6]20.402
Sr3[Ga(OH)6]2 [23]0.381Ba3[Y(OH)6]20.403
Ba3[Sc(OH)6]20.381Ba3[Ho(OH)6]20.404
Sr3[Cr(OH)6]2 [9]0.382Sr3[In(OH)6]20.410
Ca3[Al(OH)6]2 [16]0.384Ca3[Rh(OH)6]2 [2]0.415
Sr3[Fe(OH)6]2 (own data)0.388Sr3[Lu(OH)6]2 [4]0.417
Sr3[Rh(OH)6]2 [9]0.392Sr3[Yb(OH)6]2 [4]0.419
Ba3[Lu(OH)6]20.397Sr3[Tm(OH)6]2 [4]0.420
Ba3[Yb(OH)6]20.399Sr3[Er(OH)6]2 [4]0.422
Ba3[Tm(OH)6]20.400Sr3[Ho(OH)6]2 [4]0.424
Ca3[Cr(OH)6]2 [9]0.400Sr3[Y(OH)6]2 [4]0.425

The distance ratio 0.410 of the strontium hydrogarnet Sr3[In(OH)6]2 falls into the undefined interval. Thus the structure, which had previously been described in Ia3d based on X-ray powder data [15], might be non-centrosymmetric. The single-crystal data comprised only few insignificant violations of the reflection conditions of both types of glide planes in Ia3d, which show all signs of Renninger reflections. The refinement in Ia3d converged smoothly to R1 = 0.021 and wR2 = 0.016 for all data and 19 parameters. The refinement in I43d gave R1 = 0.026 and wR2 = 0.023 for all data and 34 parameters. The positions of the heavy atoms in the non-centrosymmetric solution are both shifted by 0.4 pm or 5 standard deviations (Tables S11 and S12, Supporting Information). Their displacement parameters are identical with those of the solution in Ia3d. The oxygen atoms are shifted by 6 pm with a maximum parameter change of 13 standard deviations. Their displacement parameter Ueq is slightly reduced from 65.2(9) pm2 in Ia3d to 49(5) pm2 and 61(5) pm2 in I43d. Overall, the calculated deviations from the higher symmetry are extremely small and despite the good data quality questionable. As the description in Ia3d is excellent and there is no significant evidence for symmetry reduction, this model is favored. With this additional data point, the stability of the centrosymmetric katoite type can be confirmed up to the ratio d(M–O)/2d(A–O) = 0.415, with the single exception of Sr3[Sc(OH)6]2.

3 Conclusions

Six new barium rare earth hydrogarnets Ba3[RE(OH)6]2 (RE = Y, Ho–Lu) were synthesized by using the hydroflux method starting from the respective oxide RE2O3 and Ba(NO3)2. A KOH hydroflux was used as reaction medium resulting in large colorless single crystals. Additionally, we were able to obtain single crystals of Ba3[Sc(OH)6]2 and Sr3[In(OH)6]2, whereas syntheses in literature had yielded only microcrystalline powders. X-ray diffraction analyses on single crystals of these barium rare earth hydrogarnets revealed the space group Ia3d. Additional reflections that violate the reflection condition of this cubic space group are caused by Renninger and λ/2 effects. The distances O(H)···O are much longer than in other hydrogarnets and thus hydrogen bonding effects are negligible in Ba3[RE(OH)6]2. In analogy to other hydrogarnets, these compounds might be promising carbon-free single-source precursors for oxides, yet their thermal decomposition still has to be explored.

4 Experimental

4.1 Synthesis

The hydrogarnets were synthesized in a potassium hydroxide hydroflux. The reactions were carried out in a PTFE-lined 50 mL Berghof type DAB-2 autoclave starting from 0.3 mmol Ba(NO3)2 (VEB Laborchemie Apolda, 99%) and 0.1 mmol of RE2O3 [RE = Sc (ChemPur, 99.9%), Y (Fluka, 99.98%), Lu (Fluka, 99.99%), Yb (abcr, 99.9%), Tm (Riedel-de Haën, pure), Er (abcr, 99.9%), Ho (abcr, 99.995%)]. Water and potassium hydroxide (86%, Fisher Scientific) were added in a molar ratio of 1.6:1.0 to these compounds with the exception of Ba3[Sc(OH)6]2, where a ratio of 3 was used. The autoclaves were sealed and heated to 200 °C with 2 K min−1, held for 10 h before cooling down with a rate of 0.5 K min−1 to room temperature. The crystalline products were isolated by washing with methanol.

Sr3[In(OH)6]2 was synthesized under the same conditions as the lanthanide hydrogarnets but starting from 0.3 mmol Sr(NO3)2 and 0.1 mmol In2O3. The reaction product consists of water-stable colorless rhombic dodecahedral single crystals.

4.2 X-ray crystal structure determination

Intensity data was collected at T = 100(1) K with a four-circle diffractometer Kappa Apex2 (Bruker) equipped with a CCD detector using graphite-monochromated Mo radiation (λ = 71.073 pm). The raw data were corrected for background, Lorentz and polarization factors [24], and multi-scan absorption correction was applied [25]. The structures were solved using Shelxl [26]. Structure refinement against Fo2 with Shelxl [27] included anisotropic displacement parameters for all non-hydrogen atoms (Tables S4–S12 of the Supporting Information). The hydrogen atom position was found in the difference Fourier map, and its isotropic displacement parameter was coupled to the respective oxygen atom. Graphics of the structure were developed with the program Diamond [28].

Further details of the crystal structure determinations are available from the Fachinformationszentrum Karlsruhe, 76344 Eggenstein-Leopoldshafen, Germany, , on quoting the depository numbers listed in Table S2 (Supporting Information).

4.3 Powder X-ray diffraction

X-ray powder diffraction for phase identification and Rietveld refinement were determined at room temperature on a Stadi P diffractometer (STOE & Cie.) equipped with a curved Ge monochromator using Cu1 radiation (λ = 154.056 pm) and with a Dectris Mythen 1K detector.

4.4 Infrared spectroscopy

IR spectra were recorded using a Vertex 70 FT-IR (Bruker) spectrometer. The device was operated in the ATR mode (diamond for a measuring range of 4000–600 cm−1). The software used to evaluate the spectra was Opus 6.5 [29].

4.5 Thermal analysis

The behavior of Ba3[Y(OH)6]2 upon heating in a synthetic air flow (CO2 free) was explored between T = 25 and 1000 °C with a heating rate of 5 K min−1 using a simultaneous thermal analyzer STA 409C/CD (Netzsch). In addition, a sample of Ba3[Y(OH)6]2 was thermally decomposed at 1000 °C in a chamber furnace for 12 h.

5 Supporting information

Additional crystallographic data, powder diffraction patterns, IR spectra, group-subgroup relationships, and structure graphics are given as online supplementary material.


Dedicated to: Professor Robert Glaum on the occasion of his 60th birthday.



Corresponding author: Michael Ruck, Fakultät Chemie und Lebensmittelchemie, Technische Universität Dresden, 01062 Dresden, Germany; Max-Planck-Institut für Chemische Physik fester Stoffe, Nöthnitzer Straße 40, 01187 Dresden, Germany, E-mail:

Supporting information for this article is available online. See note at the end of the document for availability.


Award Identifier / Grant number: 501100001659

Acknowledgments

This work was financially supported by the Deutsche Forschungsgemeinschaft (project-id: 438795198).

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: This work was financially supported by the Deutsche Forschungsgemeinschaft (project-id: 438795198).

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

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Supplementary Material

The online version of this article offers supplementary material (https://doi.org/10.1515/znb-2020-0147).


Received: 2020-08-24
Accepted: 2020-09-15
Published Online: 2020-10-12
Published in Print: 2020-11-26

© 2020 Ralf Albrecht et al., published by De Gruyter, Berlin/Boston

This work is licensed under the Creative Commons Attribution 4.0 International License.

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