Re-refinement of sodium ammonium sulfate dihydrate at 170 K

Eckhardt, T.; Wagner, C.; Imming, P.; Seidel, R.W.

doi:10.1107/S2414314620012754

inorganic compounds

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ISSN: 2414-3146

Volume 5| Part 9| September 2020| x201275

https://doi.org/10.1107/S2414314620012754

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Re-refinement of sodium ammonium sulfate dihydrate at 170 K

Tamira Eckhardt,^a Christoph Wagner,^b Peter Imming ^a and Rüdiger W. Seidel ^a ^*

^aInstitut für Pharmazie, Wolfgang-Langenbeck-Str. 4, 06120 Halle (Saale), Germany, and ^bInstitut für Chemie, Kurt-Mothes-Str. 2, 06120 Halle (Saale), Germany
^*Correspondence e-mail: ruediger.seidel@pharmazie.uni-halle.de

Edited by M. Weil, Vienna University of Technology, Austria (Received 26 August 2020; accepted 18 September 2020; online 25 September 2020)

The title compound, sodium ammonium sulfate dihydrate (SASD), NaNH₄SO₄·2H₂O, a synthetic analogue of the mineral lecontite, is a well known ferroelectric. The crystal structure of the paraelectric phase has been re-refined at 170 K on the basis of single-crystal X-ray data, improving the previous study [Arzt & Glazer (1994 ). Acta Cryst. B50, 425–431] in terms of accuracy regarding hydrogen-atom positions and thus details of the hydrogen bonding. O—H⋯O and N—H⋯O hydrogen bonds between the principal building units [Na(OH₂)₄O₂ octahedra, SO₄ tetrahedra and ammonium cations] constitute a three-dimensional network structure.

Keywords: crystal structure; hydrogen bonding; chirality; chain structure; ferroelectric; sulfate mineral.

CCDC reference: 2032745

Structure description

The title compound, sodium ammonium sulfate dihydrate (SASD), NaNH₄SO₄·2H₂O, is the synthetic analogue of the mineral lecontite (Hawthorne et al., 2000 ), as revealed by Faust & Bloss (1963 ) through a diffractometry study of both synthetic and natural material. The crystal structure of SASD was first determined by Corazza et al. (1967 ) from equi-inclination Weissenberg photographs at room temperature. Arzt & Glazer (1994) redetermined the crystal structure at room temperature based on serial detector data. Properties of the well-known ferroelectric SASD have been widely studied (Arzt & Glazer, 1994; Fawcett et al., 1975 ; Genin & O'Reilly, 1969 ; Hilczer et al., 1991 , 1992 , 1993 ; Kanesaka & Ozaki, 1994 ; Kassem & Hedewy, 1988 ; Kloprogge et al., 2006 ; Lipinski et al., 2003 ; Lipinski & Kuriata, 2005 ; Ono et al., 1993 ; Osaka, 1978 ; Osaka & Makita, 1970 ; Ribeiro et al., 2006 ). Kloprogge et al. (2006) also reported a Rietveld refinement of the structure of SASD at room temperature, thereby confirming the results of the previous single-crystal X-ray analyses. We have now re-refined the crystal structure of the paralectric phase of SASD at 170 K on the basis of single-crystal X-ray diffraction data.

As shown in Fig. 1, the sodium cation is hexa-coordinated with a considerably distorted octahedral coordination sphere formed by four water molecules in the equatorial plane and two sulfate O atoms in the apical positions. Selected bond lengths and angles are listed in Table 1. Each of the ligands links two sodium cations in a μ-coordination mode, resulting in chains along the [100] direction with the Na cations located near to a 2₁ screw axis. Na1⋯Na1ⁱ and Na1⋯Na1ⁱⁱ are separated by 3.1317 (2) and 3.1316 (2) Å, respectively [symmetry codes: (i) x − $[{1\over 2}]$ , −y + $[{3\over 2}]$ , −z + 1; (ii) x + $[{1\over 2}]$ , −y + $[{3\over 2}]$ , −z + 1]. The chains can be described as consisting of NaO₆ octahedra sharing one face (Fig. 2) defined by atoms O1, O2 and O4. The sulfate anion exhibits the typical tetrahedral shape with an r.m.s. deviation from exact T_d symmetry of only 0.0092 Å, as calculated with MOLSYM in PLATON (Spek, 2020 ). In the chains, the SO₄ tetrahedra have one O atom in common with a pair of NaO₆ octahedra. Chain motifs are encountered in the structures of many other sulfates (Gorogotskaya & Bokii, 1973 ).

Table 1
Selected geometric parameters (Å, °)

Na1—O2ⁱ	2.3229 (15)	Na1—O4ⁱ	2.4546 (14)
Na1—O4	2.3733 (14)	S1—O5	1.4628 (14)
Na1—O2	2.4054 (16)	S1—O4	1.4721 (12)
Na1—O1	2.4087 (15)	S1—O3	1.4728 (15)
Na1—O1ⁱ	2.4389 (16)	S1—O6	1.4740 (15)

O2ⁱ—Na1—O4	111.05 (5)	O4—Na1—O4ⁱ	167.55 (5)
O2ⁱ—Na1—O2	166.61 (6)	O2—Na1—O4ⁱ	86.58 (5)
O4—Na1—O2	81.31 (5)	O1—Na1—O4ⁱ	92.00 (5)
O2ⁱ—Na1—O1	103.97 (6)	O1ⁱ—Na1—O4ⁱ	81.23 (5)
O4—Na1—O1	83.54 (5)	O5—S1—O4	109.42 (8)
O2—Na1—O1	81.97 (5)	O5—S1—O3	109.02 (11)
O2ⁱ—Na1—O1ⁱ	83.03 (5)	O4—S1—O3	110.07 (8)
O4—Na1—O1ⁱ	101.42 (5)	O5—S1—O6	109.17 (11)
O2—Na1—O1ⁱ	89.58 (5)	O4—S1—O6	109.83 (8)
O1—Na1—O1ⁱ	169.50 (3)	O3—S1—O6	109.31 (9)
O2ⁱ—Na1—O4ⁱ	81.28 (5)
Symmetry code: (i) $[x-{\script{1\over 2}}, -y+{\script{3\over 2}}, -z+1]$ .

Figure 1
Section of the crystal structure of SASD, viewed approximately along the c-axis direction towards the origin. Displacement ellipsoids are drawn at the 50% probability level. Hydrogen atoms are represented by small spheres of arbitrary radius. Thick dashed lines represent hydrogen bonds. [Symmetry codes: (i) x + $[{1\over 2}]$ , −y + $[{3\over 2}]$ , −z + 1; (ii) x − $[{1\over 2}]$ , −y + $[{3\over 2}]$ , −z + 1.]

Figure 2
The crystal structure of SASD viewed down the b-axis direction, showing the chains featuring face-sharing NaO₆ octahedra with appended SO₄ tetrahedra. Ammonium ions are omitted for clarity.

The crystal structure features hydrogen bonds of the O—H⋯O and N—H⋯O type (Table 2). The water molecules form medium–strong and nearly linear intra- and interchain hydrogen bonds to sulfate oxygen atoms. The interstices between the [Na(μ-SO₄)(μ-H₂O)₂]_n⁻ chains accommodate the ammonium cations, which form hydrogen bonds to sulfate oxygen atoms, thus establishing a three-dimensional network. The positions of the ammonium hydrogen atoms determined in the current study appear to be more accurate than those in the room-temperature structure reported by Arzt & Glazer (1994). Note that details of hydrogen bonding were not discussed in the latter report; based on the reported structure data (Arzt & Glazer, 1994), N—H distances range between 0.73 and 0.99 Å. Corazza et al. (1967) did not refine hydrogen-atom parameters in the original room-temperature structure determination but included their presumed positions in the structure-factor calculation for the final refinement of the non-hydrogen atoms. In the current study, semi-free refinement applying only similarity restraints on the 1,2-distances involving hydrogen atoms resulted in reasonable hydrogen-atom parameters and a sensible hydrogen-bonding scheme.

Table 2
Hydrogen-bond geometry (Å, °)

D—H⋯A	D—H	H⋯A	D⋯A	D—H⋯A
O1—H1A⋯O6ⁱⁱ	0.84 (2)	1.99 (2)	2.812 (2)	165 (3)
O1—H1B⋯O5ⁱⁱⁱ	0.85 (2)	1.89 (2)	2.7439 (19)	175 (3)
O2—H2A⋯O3^iv	0.81 (2)	1.98 (2)	2.781 (2)	171 (3)
O2—H2B⋯O6ⁱ	0.81 (2)	1.98 (2)	2.781 (2)	175 (3)
N1—H1C⋯O4	0.81 (2)	2.14 (2)	2.948 (2)	172 (3)
N1—H1D⋯O3^iv	0.83 (2)	2.03 (2)	2.854 (2)	172 (3)
N1—H1E⋯O3^v	0.81 (2)	2.23 (2)	2.977 (2)	152 (3)
N1—H1E⋯O5^v	0.81 (2)	2.60 (3)	3.324 (3)	149 (3)
N1—H1F⋯O5^vi	0.81 (2)	2.58 (3)	3.245 (3)	140 (3)
N1—H1F⋯O6^vi	0.81 (2)	2.13 (2)	2.897 (3)	157 (3)
Symmetry codes: (i) $[x-{\script{1\over 2}}, -y+{\script{3\over 2}}, -z+1]$ ; (ii) $[-x+1, y+{\script{1\over 2}}, -z+{\script{3\over 2}}]$ ; (iii) x, y+1, z; (iv) $[-x+{\script{1\over 2}}, -y+1, z-{\script{1\over 2}}]$ ; (v) $[x+{\script{1\over 2}}, -y+{\script{1\over 2}}, -z+1]$ ; (vi) $[x-{\script{1\over 2}}, -y+{\script{1\over 2}}, -z+1]$ .

The absolute structure of the crystal was established by anomalous-dispersion effects in the diffraction data, as indicated by a Flack x parameter close to zero with a reasonably small standard uncertainty (Table 3). The Hooft y parameter (Hooft et al., 2008 ), as calculated with PLATON, is 0.07 (2). Interestingly, the structure exhibits chirality opposite to the previously reported room temperature structures (Corazza et al., 1967; Arzt & Glazer, 1994).

Table 3
Experimental details

Crystal data
Chemical formula	NaNH₄SO₄·2H₂O
M_r	173.12
Crystal system, space group	Orthorhombic, P2₁2₁2₁
Temperature (K)	170
a, b, c (Å)	6.2001 (2), 8.1917 (3), 12.8121 (6)
V (Å³)	650.72 (4)
Z	4
Radiation type	Mo Kα
μ (mm⁻¹)	0.53
Crystal size (mm)	0.50 × 0.48 × 0.28

Data collection
Diffractometer	Stoe IPDS 2T
Absorption correction	Multi-scan [MULABS (Blessing, 1995 ) in PLATON (Spek, 2020)]
T_min, T_max	0.728, 1.117
No. of measured, independent and observed [I > 2σ(I)] reflections	14489, 1754, 1690
R_int	0.038

Refinement
R[F² > 2σ(F²)], wR(F²), S	0.023, 0.058, 1.09
No. of reflections	1754
No. of parameters	114
No. of restraints	12
H-atom treatment	All H-atom parameters refined
Δρ_max, Δρ_min (e Å⁻³)	0.20, −0.34
Absolute structure	Flack x determined using 672 quotients [(I⁺)−(I⁻)]/[(I⁺)+(I⁻)] (Parsons et al., 2013 )
Absolute structure parameter	−0.05 (2)
Computer programs: X-AREA and X-RED (Stoe & Cie, 2016 ), SHELXT (Sheldrick, 2015a ), SHELXL2018 (Sheldrick, 2015b ), DIAMOND (Brandenburg, 2018 ), enCIFer (Allen et al., 2004 ) and publCIF (Westrip, 2010 ).

Synthesis and crystallization

A crystal of the title compound suitable for single-crystal X-ray analysis was obtained unintentionally from a solution in an acetonitrile/water mixture after synthesis of an organic compound. Ammonium ions and sodium sulfate in this mixture originated from an employed reagent and the drying agent, respectively.

Refinement

Crystal data, data collection and structure refinement details are summarized in Table 3.

Structural data

CCDC reference: 2032745

Crystal structure: contains datablocks global, I. DOI: https://doi.org/10.1107/S2414314620012754/wm4138sup1.cif

Structure factors: contains datablock I. DOI: https://doi.org/10.1107/S2414314620012754/wm4138Isup2.hkl

Supporting information file. DOI: https://doi.org/10.1107/S2414314620012754/wm4138Isup3.cml

checkCIF report

Computing details top

Data collection: X-AREA (Stoe & Cie, 2016); cell refinement: X-AREA (Stoe & Cie, 2016); data reduction: X-RED (Stoe & Cie, 2016); program(s) used to solve structure: SHELXT (Sheldrick, 2015a); program(s) used to refine structure: SHELXL2018 (Sheldrick, 2015b); molecular graphics: DIAMOND (Brandenburg, 2018); software used to prepare material for publication: enCIFer (Allen et al., 2004) and publCIF (Westrip, 2010).

Ammonium sodium sulfate dihydrate top

Crystal data top

Na⁺·NH₄⁺·SO₄²⁻·2H₂O	D_x = 1.767 Mg m⁻³
M_r = 173.12	Mo Kα radiation, λ = 0.71073 Å
Orthorhombic, P2₁2₁2₁	Cell parameters from 15574 reflections
a = 6.2001 (2) Å	θ = 3.0–29.5°
b = 8.1917 (3) Å	µ = 0.53 mm⁻¹
c = 12.8121 (6) Å	T = 170 K
V = 650.72 (4) Å³	Prism, colourless
Z = 4	0.50 × 0.48 × 0.28 mm
F(000) = 360

Data collection top

STOE IPDS 2T diffractometer	1754 independent reflections
Radiation source: sealed X-ray tube, Incoatec Iµs	1690 reflections with I > 2σ(I)
Detector resolution: 6.67 pixels mm^-1	R_int = 0.038
ω scans	θ_max = 29.2°, θ_min = 3.0°
Absorption correction: multi-scan [MULABS (Blessing, 1995) in PLATON (Spek, 2020)]	h = −8→7
T_min = 0.728, T_max = 1.117	k = −11→11
14489 measured reflections	l = −17→17

Refinement top

Refinement on F²	Secondary atom site location: difference Fourier map
Least-squares matrix: full	Hydrogen site location: difference Fourier map
R[F² > 2σ(F²)] = 0.023	All H-atom parameters refined
wR(F²) = 0.058	w = 1/[σ²(F_o²) + (0.0357P)² + 0.0942P] where P = (F_o² + 2F_c²)/3
S = 1.09	(Δ/σ)_max < 0.001
1754 reflections	Δρ_max = 0.20 e Å⁻³
114 parameters	Δρ_min = −0.34 e Å⁻³
12 restraints	Absolute structure: Flack x determined using 672 quotients [(I⁺)-(I^-)]/[(I⁺)+(I^-)] (Parsons et al., 2013)
Primary atom site location: dual	Absolute structure parameter: −0.05 (2)

Special details top

Geometry. All esds (except the esd in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell esds are taken into account individually in the estimation of esds in distances, angles and torsion angles; correlations between esds in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell esds is used for estimating esds involving l.s. planes.

Refinement. Hydrogen-atom positions were located in a difference Fourier map, and their parameters were refined with standard similarity restraints on 1,2-distances for O—H and and N—H bonds (with a standard uncertainty of 0.02 Å). U_iso(H) values were refined freely.

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

	x	y	z	U_iso*/U_eq
Na1	0.08933 (10)	0.73476 (8)	0.51432 (5)	0.01968 (17)
S1	0.37520 (7)	0.41221 (5)	0.62681 (3)	0.01762 (11)
O1	0.3414 (2)	0.91744 (17)	0.59691 (11)	0.0248 (3)
H1A	0.368 (6)	0.903 (4)	0.6609 (18)	0.060 (10)*
H1B	0.351 (5)	1.020 (3)	0.587 (2)	0.050 (9)*
O2	0.3121 (2)	0.79007 (17)	0.36474 (11)	0.0228 (3)
H2A	0.303 (5)	0.721 (3)	0.320 (2)	0.040 (8)*
H2B	0.250 (5)	0.873 (3)	0.348 (2)	0.035 (7)*
O3	0.1880 (3)	0.4256 (2)	0.69708 (12)	0.0326 (3)
O4	0.3627 (2)	0.53757 (14)	0.54471 (10)	0.0212 (3)
O5	0.3756 (4)	0.25020 (16)	0.57880 (12)	0.0390 (4)
O6	0.5751 (2)	0.4335 (2)	0.68758 (12)	0.0337 (4)
N1	0.3688 (3)	0.3331 (2)	0.35538 (13)	0.0244 (3)
H1C	0.356 (5)	0.394 (3)	0.4051 (19)	0.042 (8)*
H1D	0.364 (6)	0.401 (3)	0.308 (2)	0.051 (9)*
H1E	0.483 (4)	0.284 (4)	0.354 (3)	0.056 (10)*
H1F	0.269 (4)	0.269 (3)	0.358 (3)	0.050 (9)*

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Na1	0.0170 (3)	0.0211 (3)	0.0209 (3)	0.0009 (2)	0.0010 (3)	−0.0001 (2)
S1	0.02186 (19)	0.01448 (17)	0.01652 (18)	−0.00021 (15)	0.00033 (15)	0.00224 (14)
O1	0.0334 (7)	0.0183 (6)	0.0226 (6)	−0.0006 (6)	−0.0037 (5)	−0.0011 (5)
O2	0.0253 (6)	0.0237 (6)	0.0195 (6)	0.0023 (5)	−0.0034 (5)	0.0005 (5)
O3	0.0308 (7)	0.0383 (8)	0.0288 (7)	0.0050 (6)	0.0113 (6)	0.0072 (6)
O4	0.0259 (6)	0.0170 (5)	0.0206 (5)	−0.0001 (5)	−0.0020 (5)	0.0056 (4)
O5	0.0715 (11)	0.0150 (6)	0.0304 (7)	0.0014 (8)	0.0006 (8)	−0.0004 (5)
O6	0.0311 (7)	0.0402 (8)	0.0298 (7)	−0.0088 (6)	−0.0106 (6)	0.0138 (6)
N1	0.0263 (8)	0.0266 (7)	0.0201 (7)	0.0006 (7)	0.0007 (7)	−0.0009 (6)

Geometric parameters (Å, º) top

Na1—O2ⁱ	2.3229 (15)	S1—O3	1.4728 (15)
Na1—O4	2.3733 (14)	S1—O6	1.4740 (15)
Na1—O2	2.4054 (16)	O1—H1A	0.84 (2)
Na1—O1	2.4087 (15)	O1—H1B	0.85 (2)
Na1—O1ⁱ	2.4389 (16)	O2—H2A	0.81 (2)
Na1—O4ⁱ	2.4546 (14)	O2—H2B	0.81 (2)
Na1—Na1ⁱⁱ	3.1316 (2)	N1—H1C	0.814 (19)
Na1—Na1ⁱ	3.1317 (2)	N1—H1D	0.828 (19)
Na1—H2B	2.61 (3)	N1—H1E	0.81 (2)
S1—O5	1.4628 (14)	N1—H1F	0.81 (2)
S1—O4	1.4721 (12)

O2ⁱ—Na1—O4	111.05 (5)	O1ⁱ—Na1—H2B	89.2 (6)
O2ⁱ—Na1—O2	166.61 (6)	O4ⁱ—Na1—H2B	68.8 (5)
O4—Na1—O2	81.31 (5)	Na1ⁱⁱ—Na1—H2B	59.5 (6)
O2ⁱ—Na1—O1	103.97 (6)	Na1ⁱ—Na1—H2B	104.3 (6)
O4—Na1—O1	83.54 (5)	O5—S1—O4	109.42 (8)
O2—Na1—O1	81.97 (5)	O5—S1—O3	109.02 (11)
O2ⁱ—Na1—O1ⁱ	83.03 (5)	O4—S1—O3	110.07 (8)
O4—Na1—O1ⁱ	101.42 (5)	O5—S1—O6	109.17 (11)
O2—Na1—O1ⁱ	89.58 (5)	O4—S1—O6	109.83 (8)
O1—Na1—O1ⁱ	169.50 (3)	O3—S1—O6	109.31 (9)
O2ⁱ—Na1—O4ⁱ	81.28 (5)	Na1—O1—Na1ⁱⁱ	80.48 (4)
O4—Na1—O4ⁱ	167.55 (5)	Na1—O1—H1A	118 (2)
O2—Na1—O4ⁱ	86.58 (5)	Na1ⁱⁱ—O1—H1A	112 (2)
O1—Na1—O4ⁱ	92.00 (5)	Na1—O1—H1B	127 (2)
O1ⁱ—Na1—O4ⁱ	81.23 (5)	Na1ⁱⁱ—O1—H1B	112 (2)
O2ⁱ—Na1—Na1ⁱⁱ	144.84 (5)	H1A—O1—H1B	105 (3)
O4—Na1—Na1ⁱⁱ	50.70 (4)	Na1ⁱⁱ—O2—Na1	82.93 (4)
O2—Na1—Na1ⁱⁱ	47.40 (4)	Na1ⁱⁱ—O2—H2A	118 (2)
O1—Na1—Na1ⁱⁱ	50.18 (4)	Na1—O2—H2A	113 (2)
O1ⁱ—Na1—Na1ⁱⁱ	126.60 (5)	Na1ⁱⁱ—O2—H2B	127 (2)
O4ⁱ—Na1—Na1ⁱⁱ	118.04 (5)	Na1—O2—H2B	95 (2)
O2ⁱ—Na1—Na1ⁱ	49.66 (4)	H2A—O2—H2B	111 (3)
O4—Na1—Na1ⁱ	141.29 (5)	S1—O4—Na1	129.03 (8)
O2—Na1—Na1ⁱ	117.40 (5)	S1—O4—Na1ⁱⁱ	136.05 (8)
O1—Na1—Na1ⁱ	130.11 (5)	Na1—O4—Na1ⁱⁱ	80.86 (4)
O1ⁱ—Na1—Na1ⁱ	49.34 (4)	H1C—N1—H1D	99 (3)
O4ⁱ—Na1—Na1ⁱ	48.44 (4)	H1C—N1—H1E	114 (3)
Na1ⁱⁱ—Na1—Na1ⁱ	163.71 (5)	H1D—N1—H1E	110 (3)
O2ⁱ—Na1—H2B	149.9 (5)	H1C—N1—H1F	107 (3)
O4—Na1—H2B	99.0 (5)	H1D—N1—H1F	116 (3)
O2—Na1—H2B	18.0 (5)	H1E—N1—H1F	110 (3)
O1—Na1—H2B	80.9 (6)

Symmetry codes: (i) x−1/2, −y+3/2, −z+1; (ii) x+1/2, −y+3/2, −z+1.

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O1—H1A···O6ⁱⁱⁱ	0.84 (2)	1.99 (2)	2.812 (2)	165 (3)
O1—H1B···O5^iv	0.85 (2)	1.89 (2)	2.7439 (19)	175 (3)
O2—H2A···O3^v	0.81 (2)	1.98 (2)	2.781 (2)	171 (3)
O2—H2B···O6ⁱ	0.81 (2)	1.98 (2)	2.781 (2)	175 (3)
N1—H1C···O4	0.81 (2)	2.14 (2)	2.948 (2)	172 (3)
N1—H1D···O3^v	0.83 (2)	2.03 (2)	2.854 (2)	172 (3)
N1—H1E···O3^vi	0.81 (2)	2.23 (2)	2.977 (2)	152 (3)
N1—H1E···O5^vi	0.81 (2)	2.60 (3)	3.324 (3)	149 (3)
N1—H1F···O5^vii	0.81 (2)	2.58 (3)	3.245 (3)	140 (3)
N1—H1F···O6^vii	0.81 (2)	2.13 (2)	2.897 (3)	157 (3)

Symmetry codes: (i) x−1/2, −y+3/2, −z+1; (iii) −x+1, y+1/2, −z+3/2; (iv) x, y+1, z; (v) −x+1/2, −y+1, z−1/2; (vi) x+1/2, −y+1/2, −z+1; (vii) x−1/2, −y+1/2, −z+1.

Acknowledgements

Professor Kurt Merzweiler is gratefully acknowledged for providing diffractometer time. TE would like to thank Christoph Lehmann for his support. RWS would like to thank Dr Richard Goddard and Jan Henrik Halz for helpful discussions.

Funding information

We acknowledge the financial support within the funding programme Open Access Publishing by the German Research Foundation (DFG).

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