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Tilting structures in inverse perovskites, M 3Tt O (M = Ca, Sr, Ba, Eu; Tt = Si, Ge, Sn, Pb) Jurgen ¨ Nuss, Claus Muhle, ¨ Kyouhei Hayama, Vahideh Abdolazimi and Hidenori Takagi

Acta Cryst. (2015). B71, 300–312

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Acta Cryst. (2015). B71, 300–312

J¨urgen Nuss et al. · Tilting structures in inverse perovskites

research papers Tilting structures in inverse perovskites, M3TtO (M = Ca, Sr, Ba, Eu; Tt = Si, Ge, Sn, Pb) ISSN 2052-5206

a a b a Ju ¨ rgen Nuss, * Claus Mu ¨hle, Kyouhei Hayama, Vahideh Abdolazimi and a,b,c Hidenori Takagi

a

Received 19 December 2014 Accepted 26 March 2015

Edited by M. Dusek, Academy of Sciences of the Czech Republic, Czech Republic Keywords: inverse perovskites; tetrelide oxides; phase transitions; multiple twinning; reticular merohedry; pseudo-symmetry. CCDC references: 1056257; 1062668; 1062669; 1062670; 1062671; 1062672; 1062673; 1062674; 1062675; 1062676; 1062677; 1062678; 1062679; 1062680; 1062681; 1062682; 1062683 Supporting information: this article has supporting information at journals.iucr.org/b

Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany, bDepartment of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, 113-0033 Tokyo, Japan, and cInstitute for Functional Matter and Quantum Technologies, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany. *Correspondence e-mail: [email protected]

Single-crystal X-ray diffraction experiments were performed for a series of inverse perovskites, M3TtO (M = Ca, Sr, Ba, Eu; Tt = tetrel element: Si, Ge, Sn, Pb) in the temperature range 500–50 K. For Tt = Sn, Pb, they crystallize as an ‘ideal’ perovskite-type structure (Pm3 m, cP5); however, all of them show distinct anisotropies of the displacement ellipsoids of the M atoms at room temperature. This behavior vanishes on cooling for M = Ca, Sr, Eu, and the structures can be regarded as ‘ideal’ cubic perovskites at 50 K. The anisotropies of the displacement ellipsoids are much more enhanced in the case of the Ba compounds. Finally, their structures undergo a phase transition at  150 K. They change from cubic to orthorhombic (Ibmm, oI20) upon cooling, with slightly tilted OBa6 octahedra, and bonding angles O—Ba—O ’ 174 (100 K). For the larger Ba2+ cations, the structural changes are in agreement with smaller tolerance factors (t) as defined by Goldschmidt. Similar structural behavior is observed for Ca3TtO. Smaller Tt4 anions (Si, Ge) introduce reduced tolerance factors. Both compounds Ca3SiO and Ca3GeO with cubic structures at 500 K, change into orthorhombic (Ibmm) at room temperature. Whereby, Ca3SiO is the only representative within the M3TtO family where three polymorphs can be found within the temperature range 500–50 K: Pm3 m–Ibmm–Pbnm. They show tiny differences in the tilting of the OCa6 octahedra, expressed by O—Ca—O bond angles of 180 (500 K),  174 (295 K) and 170 (100 K). For larger M (Sr, Eu, Ba), together with smaller Tt (Si, Ge) atoms, pronounced tilting of the OM6 octahedra, and bonding angles of O—M—O ’ 160 (295 K) are observed. They crystallize in the anti-GdFeO3 type of structure (Pbnm, oP20), and no phase transitions occur between 500 and 50 K. The observed phase transitions are all accompanied by multiple twinning, in terms of pseudo-merohedry or reticular pseudo-merohedry.

1. Introduction

# 2015 International Union of Crystallography

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Perovskite oxides have been extensively studied; they can produce an incredibly wide variety of phases with totally different functions (capacitor, piezoelectric, insulator, metallic conductor, catalyst, superconductor, giant magneto-resistance). Perovskite is the maximum multifunctional structure, which predestines this family as a major playground for investigating structure–property relations (Bhalla et al., 2000). The inverse perovskites, crystallizing in the anti-perovskite type of structure, with the reverse occupancy of cations and anions, M3XY (M = metal; X = metalloid; Y = B, C, N, O), exhibit a growing variety of interesting physical properties and functions, such as superconductivity (He et al., 2001), a nearzero temperature coefficient of resistivity (Chi et al., 2001), negative thermal expansion (Takenaka & Takagi, 2005; Iikubo et al., 2008), giant magneto-resistance (Kamishima et al., 2000),

http://dx.doi.org/10.1107/S2052520615006150

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Acta Cryst. (2015). B71, 300–312

research papers magnetostriction (Asano et al., 2008) and piezomagnetic effects (Gomonaj & L’Vov, 1992), and magneto-caloric effects (Wang et al., 2010) due to the strong correlation between lattice, spin and charge degrees of freedom (Wang et al., 2013). These compounds have also attracted attention more recently as predicted candidates for topological insulators (Sun et al., 2010; Lee et al., 2013) and three-dimensional Dirac electron systems (Kariyado & Ogata, 2011, 2012). However, systematic studies of their detailed structures have been lacking so far. Structural investigations of the inverse perovskites containing oxygen and tetrel elements, M3TtO (M = Ca, Sr, Ba, Eu, Yb; Tt = Si, Ge, Sn, Pb), have been reported by Widera & Scha¨fer (1980, 1981; M3SnO and M3PbO),1 Ro¨hr (1995; Ca3GeO), Huang & Corbett (1998; Eu3SiO, Eu3GeO, Ca3SiO) and Kirchner et al. (2006; Eu3SnO). More systematic structural investigations of the complete series were reported by Tu¨rck (1994) and Velden & Jansen (2004), mostly based on powder diffraction studies. The series of isoelectronic metal nitrides, M3PnN (Pn = P, As, Sb, Bi), was investigated and reviewed by Niewa (2013). Another class of inverse perovskitic compounds (M3XY) contain rare-earth metals (M), together with X = Al, In, Tl and Y = B, C, N, O (Zhao et al., 1995; Kirchner et al., 2006) or transition metal elements, such as Mn3XN (X = Cu, Zn, Ga; Bertaut et al., 1968; Fruchart et al., 1971). In general, the Sn, Pb and Sb, Bi compounds of the series M32+Tt4O2– or M32+Pn3N3 form the ‘ideal’ cubic antiperovskite structure, while the Si, Ge and P, As members were reported to crystallize with orthorhombic symmetry, with antiGdFeO3-type structure. An electron count for this entire set of compounds indicates that they are all electron precise, belonging to Zintl phases, and the Zintl–Klemm concept (Zintl, 1939; Klemm, 1958; Nesper, 2014) helps understand their crystal structures. Accordingly they are supposed to exhibit insulating or semiconducting properties (Huang & Corbett, 1998; Niewa, 2013). For M3XY members, containing rare-earth or transition metals (M) together with X = Al, Ga, In, Tl, Cu, Zn and Y = B, C, N, O, the non-metallic elements Y can be viewed as interstitial atoms inserted in the octahedral hole of a parent metal framework of the Cu3Au type (Zhao et al., 1995), all of them showing metallic conductivity. Although the M3TtO (M = Ca, Sr, Ba; Tt = Sn, Pb) series of compounds are structurally Zintl phases, the completely filled p-orbitals of the Tt4– anions, and the empty d-orbitals of M2+ cations are very close in energy (Kariyado & Ogata, 2011, 2012). Indeed the valence and conductance bands almost touch or slightly overlap and the family of M3TtO representatives are therefore supposed to be narrow-gap semiconductors or semimetals (with eventually three-dimensional Dirac electrons). An almost zero or ‘negative’ band gap is known to be extremely sensitive to structural distortions. Therefore, to understand the novel electronic states in M3TtO compounds, detailed crystal structures over a wide temperature range should be determined. Here we report tempera1 These materials were first described as binary intermetallic compounds, such as ‘Ca3Pb’ (Helleis et al., 1963), because oxygen as an interstitial component was overlooked during structure determination.

Acta Cryst. (2015). B71, 300–312

ture-dependent single-crystal structure investigations on inverse perovskites, M3TtO (M = Ca, Sr, Ba, Eu; Tt = Si, Ge, Sn, Pb).

2. Experimental 2.1. Synthesis

The alkaline earth metals, and europium, which were used as starting materials, and the reaction products are sensitive to air and moisture. Thus, all operations were performed in a dried argon atmosphere (Schlenk technique or glovebox with H2O, O2 < 0.1 p.p.m.; MBraun GmbH, Mu¨nchen, Germany). In order to remove impurities (oxides and hydrides) in the educts, the alkaline earth metals and europium metal were distilled at 1090 K in a dynamic vacuum of 105 mbar. The ternary compounds M3TtO (M = Ca, Sr, Ba, Eu; Tt = Si, Ge, Sn, Pb) were synthesized in  1 g batches from stoichiometric amounts of the M metal together with SnO, PbO, Si/SiO2, Ge/GeO2 in sealed tantalum ampoules. The following temperature schedule was applied: 295 ! 1370 K (100 K h1, subsequent annealing at 1370 K for 100 h); 1370 ! 1070 K (50 K h1, subsequent annealing at 1070 K for 20 h); 1070 ! 295 K (100 K h1). It has turned out that a small excess of the M metals (3–5%) in the educts increases the crystal sizes and quality, and precludes the formation of MO oxides as impurities. The excess M metals can be removed by distillation in dynamic vacuum afterwards. 2.2. X-ray diffraction

Crystals suitable for single-crystal X-ray diffraction were selected under high viscosity oil, and mounted with some grease on a loop made of Kapton foil (Micromounts2, MiTeGen, Ithaca, NY). Diffraction data were collected at 50, 100, 150, 200, 250 and 295 K with a SMART APEXII CCD Xray diffractometer (Bruker AXS, Karlsruhe, Germany), equipped with an N-Helix low-temperature device (Oxford Cryosystems; Cakmak et al., 2009). Diffraction measurements between 100 and 500 K were also collected with a SMART APEXI CCD X-ray diffractometer (Bruker AXS, Karlsruhe, Germany), equipped with Cryostream 700 Plus cooling/ heating device (Oxford Cryosystems). Both cooling devices have temperature stability and an accuracy of 0.1 K. The reflection intensities were integrated with the SAINT subprogram in the Bruker Suite software package (Bruker AXS, 2013). Owing to the often very small size of the crystals, the twinning issue, and in order to employ a uniform way of handling the data, multi-scan absorption corrections were applied using either SADABS (Sheldrick, 2012b) or TWINABS (Sheldrick, 2012a). The structures were refined by a full-matrix least-squares fit with the SHELXL software package (Sheldrick, 2008) or with JANA2006 (Petrˇı´cˇek et al., 2006). The latter code allows data setup and refinement, even if two or more different twin operations are involved. Experimental details are given in Tables 1–4; atomic coordinates are in the supporting information. Ju¨rgen Nuss et al.

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research papers Table 1 Crystal data, data collection, and refinement details of M3SnO representatives (M = Ca, Sr, Eu, Ba). For all structures: Cubic, Pm3 m, Z = 1. Experiments were carried out at 295 K with Mo K radiation using a SMART APEX II, Bruker AXS diffractometer. Absorption was corrected for by multi-scan methods, SADABS (Sheldrick, 2012b). Refinement was on 6 parameters with 0 restraints.

Crystal data Chemical formula Mr ˚) a (A ˚ 3) V (A  (mm1) Crystal size (mm) Data collection Tmin, Tmax No. of measured, independent and observed [I > 2(I)] reflections Rint ˚ 1) (sin /)max (A Refinement R[F2 > 2(F2)], wR(F2), S No. of reflections ˚ 3) max, min (e A

Ca3SnO

Sr3SnO

Eu3SnO

Ba3SnO

Cs3SnO 254.93 4.827 (3) 112.44 (18) 8.91 0.12  0.10  0.06

Sr3SnO 397.55 5.1394 (18) 135.75 (14) 33.70 0.10  0.07  0.03

Eu3SnO 590.57 5.077 (3) 130.9 (2) 40.00 0.06  0.04  0.02

Ba3SnO 546.71 5.444 (3) 161.3 (2) 21.75 0.08  0.06  0.05

0.175, 0.275 2145, 85, 85

0.074, 0.167 2627, 99, 98

0.084, 0.166 2477, 93, 89

0.094, 0.167 3126, 114, 108

0.026 0.842

0.035 0.843

0.024 0.836

0.023 0.842

0.008, 0.022, 1.38 85 0.39, 0.38

0.010, 0.035, 1.36 99 0.50, 0.43

0.012, 0.033, 1.12 93 1.10, 0.60

0.016, 0.067, 1.44 114 1.00, 0.62

Computer programs: Bruker Suite software package (Bruker AXS, 2013), SHELXL (Sheldrick, 2008).

Table 2 Crystal data, data collection, and refinement details of M3PbO representatives (M = Ca, Sr, Eu, Ba). For all structures: Cubic, Pm3 m, Z = 1. Experiments were carried out at 295 K with Mo K radiation using a SMART APEX II, Bruker AXS diffractometer. Absorption was corrected for by multi-scan methods, SADABS (Sheldrick, 2012b). Refinement was on 6 parameters with 0 restraints.

Crystal data Chemical formula Mr ˚) a (A ˚ 3) V (A  (mm1) Crystal size (mm) Data collection Tmin, Tmax No. of measured, independent and observed [I > 2(I)] reflections Rint ˚ 1) (sin /)max (A Refinement R[F2 > 2(F2)], wR(F2), S No. of reflections ˚ 3) max, min (e A

Ca3PbO

Sr3PbO

Eu3PbO

Ba3PbO

Ca3PbO 343.43 4.8402 (7) 113.39 (5) 40.39 0.14  0.08  0.05

Sr3PbO 486.05 5.151 (3) 136.6 (2) 59.67 0.12  0.12  0.04

Eu3PbO 679.07 5.0910 (19) 131.95 (15) 66.79 0.40  0.08  0.06

Ba3PbO 635.21 5.489 (7) 165.4 (7) 42.85 0.05  0.03  0.02

0.068, 0.167 2184, 86, 86

0.025, 0.111 1691, 101, 101

0.028, 0.110 2520, 93, 93

0.134, 0.275 3016, 117, 116

0.036 0.846

0.059 0.846

0.042 0.833

0.043 0.845

0.012, 0.030, 1.21 86 1.23, 1.07

0.018, 0.037, 1.24 101 1.74, 1.72

0.011, 0.028, 1.13 93 0.95, 1.06

0.012, 0.033, 1.22 117 0.95, 0.74

Computer programs: Bruker Suite software package (Bruker AXS, 2013), SHELXL (Sheldrick, 2008).

3. Results and discussion The title compounds were synthesized by reacting the alkaline earth metals and europium metal with the tetrel oxides, SnO, PbO, Si/SiO2 and Ge/GeO2, in arc-welded tantalum ampoules. The grey metallic products are sensitive to air and moisture. 3.1. M3SnO and M3PbO (M = Ca, Sr, Eu, Ba)

According to the single-crystal X-ray diffraction analyses, all tetrelide oxides of the series M3TtO (M = Ca, Sr, Eu, Ba; Tt

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= Sn, Pb) crystallize as inverse cubic perovskites at room temperature (space group Pm3 m), and their cell parameters ˚ (see Tables 1 and 2). The are in the range a = 4.8–5.5 A analyses basically confirm the results from the literature (Widera & Scha¨fer, 1980, 1981; Tu¨rck, 1994; Velden & Jansen, 2004; Kirchner et al., 2006). Fig. 1 shows the structure with the M atoms located at the 3d site (12, 0, 0), Tt atoms at 1b (12, 12, 12), and O atoms at 1a (0, 0, 0). The O atoms are in the centres of undistorted M6 octahedra, which are condensed to a threedimensional arrangement by sharing common corners. The

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research papers Table 3 Crystal data, data collection, and refinement details of Ba3SnO, Ba3PbO, Eu3SiO and Eu3GeO. For all structures: Orthorhombic, Z = 4. Experiments were carried out at 100 K with Mo K radiation using a SMART APEX II, Bruker AXS diffractometer. Refinement was with 0 restraints.

Crystal data Chemical formula Mr Space group ˚) a, b, c (A ˚ 3) V (A  (mm1) Crystal size (mm) Data collection Absorption correction Tmin, Tmax No. of measured, independent and observed [I > 2(I)] reflections Rint ˚ 1) (sin /)max (A Refinement R[F2 > 2(F2)], wR(F2), S No. of reflections No. of parameters Twin volume fractions V1–V6 ˚ 3) max, min (e A

Ba3SnO

Ba3PbO

Eu3SiO

Eu3GeO

Ba3SnO 546.7 Ibmm 7.676 (1), 7.676 (1), 10.8560 (13) 639.65 (14) 21.95 0.08  0.06  0.05

Ba3PbO 635.2 Ibmm 7.693 (2), 7.693 (2), 10.880 (3) 643.9 (3) 44.03 0.05  0.03  0.02

Eu3SiO 499.97 Pbnm 7.0138 (7), 7.0383 (7), 9.950 (1) 491.19 (9) 37.90 0.28  0.25  0.07

Eu3GeO 544.5 Pbnm 7.0448 (4), 7.0448 (4), 9.9628 (6) 494.45 (5) 43.37 0.05  0.04  0.02

Multi-scan SADABS 0.094, 0.167 5161, 2658, 2514

Multi-scan SADABS 0.132, 0.275 6195, 3139, 3005

Multi-scan TWINABS 0.031, 0.166 13 755, 3442, 3386

Multi-scan SADABS 0.035, 0.108 10 331, 2934, 2886

0.027 0.848

0.035 0.847

0.078 0.835

0.066 0.811

0.029, 0.035, 1.19 2658 24 0.092 (5), 0.079 (2), 0.262 (2), 0.242 (2), 0.086 (2), 0.239 (2) 0.99, 1.22

0.032, 0.035, 1.17 3139 24 0.076 (7), 0.261 (3), 0.147 (2), 0.119 (3), 0.255 (3), 0.142 (3) 1.37, 2.23

0.049, 0.140, 1.13 3442 30 0.708 (3), 0.292 (3) 0, 0, 0, 0

0.049, 0.063, 1.46 2934 34 0.344 (4), 0.108 (1), 0.319 (2), 0.089 (2), 0.043 (3), 0.097 (3) 2.87, 4.17

6.54, 4.35

Computer programs: Bruker Suite software package (Bruker AXS, 2013), SHELXL (Sheldrick, 2008), JANA2006 (Petrˇı´cˇek et al., 2006), SADABS (Sheldrick, 2012b), TWINABS (Sheldrick, 2012a)

O—M—O bonding angles are fixed by the crystal symmetry to 180 , and the Tt atom is located at the centre of the unit cell, surrounded by 12 M atoms in the shape of a cuboctahedron;

these are the basic structural elements of the perovskite type of structure. Anisotropy of the atomic displacements of the metal atoms M is observed, where U22 is always larger than

Figure 2 Figure 1 Perspective view of the undistorted anti-perovskite structure of M3TtO compounds (M = Ca, Sr, Ba, Eu; Tt = Sn, Pb). Acta Cryst. (2015). B71, 300–312

Quotient of the displacement parameters of the M atoms in M3TtO compounds, U22/U11 as a function of temperature (M = Ca, Sr, Eu; Tt = Sn, Pb). Ju¨rgen Nuss et al.

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research papers Table 4 Crystal data, data collection, and refinement details of Ca3SiO and Ca3GeO. Experiments were carried out with Mo K radiation using a SMART APEX I, Bruker AXS diffractometer. Absorption was corrected for by multi-scan methods, SADABS (Sheldrick, 2012b). Refinement was with 0 restraints.

Crystal data Chemical formula Mr Crystal system, space group Temperature (K) ˚) a, b, c (A ˚ 3) V (A Z  (mm1) Crystal size (mm) Data collection Tmin, Tmax No. of measured, independent and observed [I > 2(I)] reflections Rint ˚ 1) (sin /)max (A Refinement R[F2 > 2(F2)], wR(F2), S No. of reflections No. of parameters Twin volume fractions V1–V6 ˚ 3) max, min (e A

Ca3SiO

Ca3GeO

Ca3SiO 164.33 Orthorhombic, Pbnm

Ca3GeO 208.83 Orthorhombic, Ibmm

Orthorhombic, Ibmm

Cubic, Pm3 m

Cubic, Pm3 m

100 6.660 (2), 6.646 (2), 9.411 (3) 416.5 (2) 4 4.05 0.02  0.02  0.01

295 6.6679 (16), 6.6679 (16), 9.430 (2) 419.26 (18) 4 4.02

500 4.741 (6), 4.741 (6), 4.741 (6) 106.6 (4) 1 3.95

100 6.6761 (7), 6.6761 (7), 9.4414 (5) 420.81 (7) 4 10.73 0.03  0.02  0.02

500 4.7452 (13), 4.7452 (13), 4.7452 (13) 106.85 (9) 1 10.56

0.188, 0.272 6539, 1020, 818

0.143, 0.273 2722, 1444, 991

0.171, 0.272 1699, 73, 58

0.192, 0.273 3170, 1650, 1601

0.194, 0.272 1694, 74, 73

0.073 0.811

0.041 0.814

0.045 0.810

0.020 0.814

0.030 0.809

0.071, 0.196, 1.10

0.090, 0.087, 1.40

0.020, 0.041, 1.12

0.026, 0.029, 1.05

0.016, 0.041, 1.26

1020 29 0.890 (4), 0, 0, 0.110 (4), 0, 0

1444 19 0.248 (9), 0, 0.252 (6), 0252 (6), 0.248 (6), 0

73 6 –

74 6 –

2.11, 1.46

0.97, 1.19

0.45, 0.53

1650 24 0.023 (6), 0.067 (3), 0.435 (3), 0.228 (3), 0.247 (3), 0 0.37, 0.36

0.40, 0.24

Computer programs: Bruker Suite software package (Bruker AXS, 2013), SHELXL (Sheldrick, 2008), JANA2006 (Petrˇı´cˇek et al., 2006).

U11. This is not unusual, because the thermal motion is expected to be largest perpendicular to the M—O bonding direction. A noticeable feature is that the ratio q = U22/U11 continuously increases when going from smaller to larger cations, Ca2+ (q ’ 1.5), Sr2+ (q ’ 1.8), Eu2+ (q ’ 1.9), Ba2+ (q ’ 2.2). Temperature-dependent single-crystal X-ray measurements between 50 and 295 K show a decrease of U22/ U11 to approximately q = 1 for all Ca, Sr and Eu representatives when lowering the temperature, such that the structures can be regarded as more or less isotropic at 50 K, in agreement with an ‘ideal’ perovskite structure, see Fig. 2. The temperature-dependent behaviour is different for Ba3SnO and Ba3PbO: a distinct increase of q at low temperatures is observed, and the value of U22/U11 ’ 20 for Ba3SnO at 50 K indicates a pronounced deviation from an ‘ideal’ perovskite, see Fig. 3. The O—Ba—O bonding angles are maybe frozen-in at angles different to 180 , and thus the local symmetry may differ from an ‘ideal’ perovskite structure for Ba3SnO and Ba3PbO at low temperatures. Indeed, careful examination of the diffraction data revealed the presence of diffuse scattering at room temperature. The intensity profiles sharpened upon cooling, and became much more intense below 150 K for both Ba3SnO and Ba3PbO. The diffraction patterns could be indexed and integrated on the

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Figure 3 Quotient of the displacement parameters of the Ba atoms in Ba3TtO compounds U22/U11 as a function of temperature (Tt = Sn, Pb). All data are from refinements based on the assumption of an ‘ideal’ perovskite structure. The insert shows a Ba6O octahedron in Ba3SnO, with displacement ellipsoids drawn at the 75% probability level, at 150 K.

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research papers basis of a cubic F-cell with doubled a-axis in comparison to the ˚ (Ba3SnO) and a = ‘ideal’ perovskite structure, a = 10.856 (4) A ˚ (Ba3PbO), and with quite reasonable internal R10.880 (5) A values for the cubic Laue symmetry (m3 m), Rint = 0.045 and 0.058, respectively (100 K data). No peak-splitting was observed with Mo K radiation. Fig. 4 shows the hhl layers of the reciprocal space of a Ba3SnO crystal at 295 K (right) and 50 K (left), constructed pixel by pixel from the original CCD frames using the precession module of the Bruker Suite software package (Bruker AXS, 2013). This procedure allows a detailed exploration of the reciprocal space, without restrictions to integer hkl values (Nuss et al., 2007; Nuss & Jansen, 2007). All attempts to refine the structures using a cubic space group were discarded due to internal contradictions: either the displacement parameters were much too high or split positions were needed in order to obtain reasonable results. Instead, different twinning scenarios, associated with the symmetry reduction during phase transitions were developed from analogous observations in oxide perovskites ABO3.2 In the huge family of ‘perovskites’ an enormous number of distortion variants of the ‘ideal’ perovskite structure are known (Glazer, 1972; Woodward, 1997). The cubic perovskite, e.g. SrTiO3, is the aristotype of this structure family, and the space groups of all derivatives are subgroups of Pm3 m. The group–subgroup relations can be visualized by Ba¨rnighausen trees; Fig. 5 shows such a family tree of perovskite derivatives with tilted octahedra (Ba¨rnighausen, 1980; Bock & Mu¨ller, 2002). Starting from the aristotype (Pm3 m), arrows point to the subgroups, and t and k indicate whether the transformation to the subgroup is of the type translationengleich or klassengleich, the number following indicates the index of symmetry reduction, and finally the cell transformations and origin shifts are given (Mu¨ller, 2013). During phase transitions with a symmetry reduction of the type translationengleich, twinning formation occurs and the index of symmetry reduction indicates the number of differently oriented twin domains. For example, the group–subgroup relation between Pm3 m (e.g. SrTiO3) and Pbnm (e.g. GdFeO3) includes two consecutive steps of symmetry reduction (Pm3 m ! P4/mmm ! Cmmm) of the type translationengleich (t3 and t2), which can be associated with multiple twinning in terms of ‘twins of twins’,3 resulting in a second-order twin with six (3  2) twin domains and twin multiplicity of six (Nespolo, 2004). The next two transitions (Cmmm ! Ibmm ! Pbnm) are of the type klassengleich (k2 and k2). Effectively 12 = 3  2  2 (Ibmm) and 24 = 3  2  2  2 (Pbnm) domain states occur, six of which are orientation domains (twin domains), the others are translational or antiphase domain states where the domain states differ only in location but not in orientation (Janaovec

Figure 4 Reciprocal layers hhl of Ba3SnO at 295 K (right) and 50 K (left). The directions of the reciprocal axes are with respect to the cubic cell settings.

& Prˇı´vratska´, 2003). Thus, each of the six twin domains consist of two (Ibmm) and four (Pbnm) antiphase domains, whereby only the former, not the latter, have an effect on diffraction experiments.

Figure 5 2

If no specific literature is given, the terminology used with respect to the twinning of crystals follows the notation given in the International Tables of Crystallography (Hahn & Klapper, 2003) and related literature (Nespolo, 2015). 3 A term initially used by Ba¨rnighausen for the reduction of symmetry in two consecutive steps of the type translationengleich accompanied by multiple twinning due to a phase transition (Henke, 2003). Acta Cryst. (2015). B71, 300–312

Group–subgroup relations between space groups of perovskite derivatives with tilted octahedra. The space groups marked with white boxes allow the diffraction pattern of the low-temperature modifications of Ba3Sn(Pn)O, and the one marked with grey boxes are valid for Eu3Si(Ge)O, to be determined in principle. A more detailed ‘symmetry tree’ for perovskites is given by Ba¨rnighausen (1980) and Bock & Mu¨ller (2002). Ju¨rgen Nuss et al.

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research papers The observed diffraction patterns of Ba3Sn(Pb)O can be indexed according to the Bravais lattices and space-group symmetries marked with white boxes in Fig. 5, in principle. Each represented space group has its own set of twin operations (transformation matrices), which differ for different paths of symmetry reduction. The diffraction data were processed with respect to the cell settings and space groups shown in Fig. 5, and least-squares refinements were performed for all of them. A comparison emphasized the refinement in Ibmm as the best solution with respect to final R-values, displacement parameters and appropriate twin volume fractions, for both Ba3SnO and Ba3PbO (see Table 3 and the supporting information). In summary, the diffraction data are affected by twinning, and the cubic F-cell is the twin lattice Ltw ˚ with or coincidence site lattice (atw = btw = ctw ’ 10.8 A reflection indices HKL), because of the accidental concordance atw = c = 21/2  a = 21/2  b. The real structural symmetry is orthorhombic (Ibmm) and there are six possibilities (3  2) of how to set up such an orthorhombic unit cell Li (lattice of the ith individual). In terms of the unit cells of the twin components (Li), the basis vectors of the twin lattice (Ltw) are transformed by (ai, bi, ci) = (atw, btw, ctw)  Mi, and the reflection indices by (hikili) = (HKL)  Mi, using the following transformation matrices, see equation (1).

1 1 1  1 0 2 2 2 0 2 1 0 0 1 M 1 ¼ 12 M ¼ 0 2 2 1 1 0 0 0 1 2 2 1 1 0 0 1 0 2 2 M 3 ¼ 12  12 0 M 4 ¼  12 12 0 1 1 0 0 1 0 2 2 1 1  0 0 1 2 2 0 1 0 M 5 ¼ 0 0 1 M 6 ¼ 12 1 21 1 1 2 2 0 0 2 2 ð1Þ The twin operations (N), in the setting of the twin lattice (Ltw), can be derived by decomposition of the point-group symmetry of the twin (m3 m) into the symmetry of the individual with point-group mmm. Equation (2) shows one set of possible twin operations with respective twin elements, which are used for further transformations. 1 0 0 0 0 1 ^ ^ N 1 ¼ 0 1 0 ¼ 1 N 2 ¼ 1 0 0 ¼ 3þ ½1 1 1 0 0 1 0 1 0 0 1 0 0 1 0 ^ ^ N 3 ¼ 0 0 1 ¼ 3 N 4 ¼ 1 0 0 ¼ 4þ ½1 1 1 ½0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 ^ ^ N 5 ¼ 0 0 1 ¼ 2½0 1 1 N 6 ¼ 0 1 0 ¼ 4 ½0 1 0 0 1 0 1 0 0 ð2Þ The twin operations (N) in the setting of the twin lattice (Ltw) correspond to the twin operations (T) in the setting of the individual (Li), they are connected by Tij = Mj  Ni (j = 1, 2 or . . . 6). Equation (3) shows the twin operations (Ti) with respective twin elements for j = 1 (Ti = M1  Ni). 1 1 1 0 0  2 2 1 ^ ^ 1 1 T 1 ¼ 0 1 0 ¼ 1 T 2 ¼  2 2 1 ¼ 3þ ½0 2 1 0 0 1 1 1 0 2 2 1 1 0 1 0 1 2 2 ^ ^ T 3 ¼  12  12 1 ¼ 3 T 4 ¼ 1 0 0 ¼ 4þ ½2 0 1 ½0 0 1 0 0 1 1 1 0 2 2 1 1 1 1  2 2 1 2 2 1 ^ ^ 1 1 1 1 1 ¼ 4 T 5 ¼ 2  2 1 ¼ 2½1 1 1 T 6 ¼ 2 2 ½1 1 0 1 1 1 1 0 0 2 2 2 2 ð3Þ

Figure 6 Group–subgroup relations between ideal perovskites Ba3SnO and Eu3SiO at 100 K, including the atomic coordinates and their evolution starting from those of the aristotype. Boxes contain the Wykoff symbol, element and atomic coordinates x; y; z.

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Note that Ti are symmetry operations of the ‘pseudo’tetragonal I-lattice (c = 21/2  a = 21/2  b) and they lead to integer indices hikili, exclusively. The special relations among the cell parameters lead to crystals twinned by pseudomerohedry, composed of up to six domains, meaning all HKL reflections are superpositions of reflections hikili (i = 1–6), and

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research papers the eigensymmetry of the twin lattice is cubic (m3 m) with twin index [j] = 1, which is the volume ratio of the primitive unit cells of the twin lattice and of the ‘untwinned’ lattice of the individual (Hahn & Klapper, 2003; Klapper & Hahn, 2010). For data processing, matrix M1 (1) was used to transform the reflection indices HKL (F-type) into h1k1l1 (I-type). In a second step the twinning matrices Ti (3) were used to transform the orthorhombic unit cells (Li) into each other, inclusive of corresponding reflection indices by (ai, bi, ci) = (a1, b1, c1)  Ti, and (hikili) = (h1k1l1)  Ti, respectively. Atomic coordinates for the starting model were calculated by following the path of symmetry reduction, see Fig. 6 (Wondraschek & Mu¨ller, 2008; Mu¨ller, 2013), and twin volume fractions of 1/6 for all six domains were used as starting parameters for the refinement with JANA2006 (Petrˇı´cˇek et al., 2006). The final crystal structure was obtained by refining all six twin volume fractions, atomic coordinates and anisotropic displacement parameters against all intensity data. The twin volume fractions differ significantly from the ideal value of 1/6 (Table 3). This results in distinct improved R values, wRall = 0.035 against 0.066 for Ba3SnO and wRall = 0.035 against 0.053 for Ba3PbO, for the refinements without and with fixed twin volume fractions. These discrepancies should not be incurred

if the origin of twinning is a phase transition (transformation twin). We conclude that the sensitivity of the compounds (impurities) and the method of performing the measurements (non-uniform stress) lead to a perturbation during nucleation, and the crystals under investigation can be regarded as transformation twins superimposed by growth twins. Details of the refinement of Ba3SnO and Ba3PbO, at 100 K, are summarized in Table 3. Fig. 6 shows the atomic coordinates of Ba3SnO and how they arise from those of the aristotype; Fig. 7 (top) shows the projection of the crystal structure. A comparison between the atomic coordinates of Ba3Sn(Pb)O and the idealized values indicates that the structural changes are very small, in comparison to the ‘ideal’ cubic perovskite structure, which are caused by tiny shifts of the Ba atoms (Figs. 6 and 7). As a consequence, the OBa6 octahedra are tilted, and the O—Ba—O angles show distinct deviations from 180 , with the average angles at 50 K being 173.5 and 174.1 for Ba3SnO and Ba3PbO, respectively. The three-dimensional arrangement of corner-shared OBa6 octahedra can be classified as a two-tilt system (Glazer, 1975). The coherent rotation of the octahedra during the phase transition takes place along two out of the three octahedral axes, namely the two within the ab-plane (Fig. 7, top). The space group Ibmm does not allow tilting around the c-axis. Note that there is still a temperature-dependent behaviour, especially for Ba3PbO where the O—Ba—O bonding angles change by  2.5 between 50 and 150 K (see Table 5). 3.2. Eu3SiO and Eu3GeO

Figure 7 Projection of the crystal structures of Ba3SnO at 50 K (top), and Eu3SiO at 295 K (bottom) along the [110] direction (right), and the [001] direction (left). Acta Cryst. (2015). B71, 300–312

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Eu3SiO and Eu3GeO are reported as being isotypic (Tu¨rck, 1994), similar to Ba3SiO and Ba3GeO (Huang & Corbett, 1998), all with the anti-GdFeO3 type of structure (Pbnm, oP20). While the crystal structure of Eu3SiO was determined from a twinned crystal (Tu¨rck, 1994), no structural data are available for Eu3GeO. Diffraction data from crystals suitable for single-crystal X-ray diffraction were collected at 295 and 100 K. They could be indexed and integrated on the basis of a cubic primitive unit cell (cP), a = ˚ (Eu3SiO), and a = 9.936 (1) A ˚ with 9.963 (1) A (Eu3GeO) internal R-values for cubic Laue symmetry (m3 m), Rint = 0.090 and 0.094, respectively (100 K data). No peak-splitting is observed with Mo K radiation. Fig. 8 shows the hk0 layers of the reciprocal space,

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research papers Table 5

Selected bonding angles ( ) at different temperatures T (K). Angle

Ba3SnO

Ba3PbO

Eu3SiO

Eu3GeO

Ca3SiO

Ca3GeO

T

O—M1—O

– 180 173.82 (6) 172.77 (2) 172.01 (2) – 180 175.41 (5) 174.81 (2) 174.22 (2)

– 180 175.34 (4) 173.96 (4) 172.82 (3) – 180 177.22 (4) 175.67 (3) 174.84 (2)

– 160.06 (2) – 159.43 (3) – – 159.84 (2) – 158.95 (2) –

– 160.82 (6) – 159.96 (5) – – 160.81 (4) – 159.89 (3) –

180 174.1 (3) – 169.56 (7) – 180 173.44 (8) – 170.89 (5) –

180 176.7 (3) – 170.93 (4) – 180 176.08 (5) – 173.69 (3) –

500 295 150 100 50 500 295 150 100 50

O—M2—O

at 295 K, of a Eu3GeO and a Eu3PbO crystal, in comparison. Again, following the path of symmetry reduction, similar to the low-temperature structure of Ba3Sn(Pb)O (see the Ba¨rnighausen tree shown in Fig. 5), the space group P42/nmc with respective cell setting is the only one that allows the diffraction data of Eu3Si(Ge)O to be reproduced if a crystal ‘untwinned’ or twinned by pseudo-merohedry is assumed. P42/mcm and Pbnm are also valid, if twinning by reticular pseudo-merohedry is applicable.4 All three space groups are marked with grey boxes in Fig. 5. Each of the lattice symmetries has its own twinning scenario, with different twin operations, which has to be handled properly in order to correct the intensities which may or may not be affected by the twinning. The diffraction data were processed with respect to the three possible lattice symmetries; the refinements confirm previous results (Tu¨rck, 1994), see Table 3 and the supporting information: Both compounds Eu3SiO and Eu3GeO crystallize in the anti-GdFeO3 type of structure with space group Pbnm; the relation of the cell parameters is c = 21/2  a = 21/2  b, accidentally. Space-group symmetry Ibmm is formed as an intermediate step in the symmetry reduction path from the ‘ideal’ perovskite (Pm3 m) to the gadolinium ferrate structure (Pbnm). Furthermore, because Pbnm is a direct subgroup of Ibmm of type klassengleich, with the same cell dimensions (Figs. 5 and 6), the transformation matrices Mi (1), Ni (2), and Ti (3) are valid. Note that when twinning by reticular merohedry occurs the twin operation T is not a symmetry operation of the lattice symmetry, and thus leads to fractional as well as to integer indices. Three integer indices represent coincident (overlapping) reflections, whereas the occurrence of at least one fractional index indicates that this reflection is non-overlapping. As a consequence, it is advisable to use the coordinate system of the twin lattice as a reference together with the transformation matrices Mi, respectively, Ni. This has the practical advantage that all reflections, overlapping and nonoverlapping ones, of all domains Li (i = 1–6) appear with integer indices hikili, and no intensity data are lost during multiple transformations. 4 Friedel (1926) has introduced the term ‘twinning by reticular merohedry’ (macles par me´rie´drie re´ticulaire) and also ‘reticular pseudo-merohedry’ (pseudo-me´rie´drie re´ticulaire), which require partial coincidence (exact) or pseudo-coincidence (accidental) of two or more superimposed lattices (Hahn & Klapper, 2003; Klapper & Hahn, 2012).

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Note that M4 = M1  T4, and therefore L1 and L4 are related by twinning with pseudo-merohedry; the same is valid for L2,6 and L3,5. There are three domains twinned by merohedry which are again twinned by reticular merohedry. The twin lattice index [j] = 2 indicates twinning by reticular merohedry (Friedel, 1926; Hahn & Klapper, 2003; Klapper & Hahn, 2012). Four different types of HKL reflections, based on the twin lattice, can be distinguished. If all reflections are considered (Iall), then 14 belongs to all six twin domains: these are the indices with the reflection conditions H + K, H + L and K + L = 2n, which are the F-indices of the twin lattice or Iindices of the individual lattice (h + k + l = 2n). One quarter of the reflections belong to the domains L1,4, one to L2,6 and another to L3,5 exclusively, without coincidence with the respective other domains. The practical implementation of these considerations was taken into account by using the matrices Ni to set up the six orientation domains on the basis of the twin lattice by trans-

Figure 8 Reciprocal layers hk0 of Eu3PbO crystal (right), and Eu3GeO crystal (left), at 295 K. The directions of the reciprocal axes are with respect to the cubic cell settings.

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research papers forming the reflection indices HKL (P-type) into Hi0 Ki0 Li0. In a second step the matrix M1 was used to transform the reflection indices of each of the six twin components in the setting of the orthorhombic unit cells (Li) by (hikili) = (Hi0 Ki0 Li0 )  M1. This approach makes sure that no data are lost during data processing. For Eu3GeO, the atomic coordinates of Ba3GeO (Huang & Corbett, 1998) were used as a starting model for the refinement with JANA2006. The final crystal structure was obtained by refining all six twin volume fractions, atomic coordinates and anisotropic displacement parameters against all intensity data. Furthermore, the cell parameters were constrained by c = 21/2  a = 21/2  b. For Eu3SiO, it turned out that only L1 and L2 have meaningful twin volume fractions, and no twinning by merohedry (only twinning by reticular merohedry) was observed for the crystal under investigation. Therefore, T1 and T2 were used exclusively for the final data integration and a multi-component HKLF5 file was processed with TWINABS, usable for structure refinement with SHELXL, where the lattice parameters a, b, c were refined independently. Details of the refinement of Eu3SiO and Eu3GeO, at 100 K, are summarized in Table 3. Fig. 6 shows the atomic coordinates of Eu3SiO and how they arise from those of the aristotype; Fig. 7 shows the projection of the crystal structure. Both Eu3SiO and Eu3GeO show a large deviation from the ‘ideal’ perovskite structure, as there are distinct shifts of the Eu atoms. This is especially true for the Eu2 position which significantly differs from the special position (14, 14, 0), see Fig. 6. The average O—Eu—O bonding angles, which are a measure of the tilting of the OEu6 octahedra, are 159.1 and 159.8 for the Si and Ge representatives, respectively. The octahedra are twisted around all three octahedral axes, and the two structures can be described as three-tilt systems (Glazer, 1975).

and thus the data could be indexed and integrated on the basis of an orthorhombic unit cell, and the space group Pbnm could easily be determined from the reflection conditions. The structure was refined with SHELXL using the atomic coordinates of Eu3GeO as starting parameters; introducing twinning by pseudo-merohedry slightly improves the final R-values (see Table 4). Finally, both Ca3SiO and Ca3GeO undergo a phase transition upon heating, at  500 and  350 K, respectively. Their structures change into cubic primitive ones ˚ (Ca3SiO), and a = 4.745 (1) A ˚ (Ca3GeO), with a = 4.741 (6) A at 500 K. The high-temperature modifications exhibit the ‘ideal’ cubic anti-perovskite-type structure (Pm3 m), and final refinements also show evidence for anisotropies of the calcium atomic displacements (U22/U11 ’ 3 at 500 K), similar to the other cubic representatives of the Ba3Sn(Pb)O series. Fig. 9 shows the reciprocal layers h2l (systematic absences do not affect the second layer) of a Ca3SiO crystal at 100, 295 and 500 K. The atomic coordinates of all three polymorphs and how their symmetries are connected (Pm3 m, 500 K – Ibmm, 295 K – Pbnm, 100 K) are given in Fig. 10 in terms of a Ba¨rnighausen tree. The O—Ca—O bonding angles in calcium silicide oxide and germanide oxide strongly depend on the temperature (see Table 5). These angles change from 180 at high temperatures to  173.7 (Ca3SiO) and 176.3 (Ca3GeO) at room temperature, and finally to  170.4 and  172.8 at 100 K, respectively. Ca3SiO changes its space-group symmetry from Ibmm to Pbnm when lowering the temperature, while Ca3GeO stays in Ibmm down to 100 K.

3.3. Ca3SiO and Ca3GeO

For the crystal structures of both Ca3SiO and Ca3GeO contradicting reports are found in the literature. Their roomtemperature structures are described as either ‘ideal’ cubic inverse perovskites, Pm3 m (Ro¨hr, 1995; Huang & Corbett, 1998) or as orthorhombic with an anti-GdFeO3-type structure, Pbnm (Tu¨rck, 1994; Velden & Jansen, 2004). In a first step, single-crystal X-ray diffraction experiments were performed at room temperature. In contrast to previous reports, the intensity data appear similar to the lowtemperature pattern of e.g. Ba3SnO, and could be indexed and integrated in an appropriate way on the basis of a cubic F-cell ˚ (Ca3SiO), and a = 9.4414 (5) A ˚ (cF), a = 9.430 (2) A (Ca3GeO). The data were processed and refined as described for Ba3SnO at low temperatures, assuming a crystal twinned by pseudo-merohedry, orthorhombic symmetry and the space group Ibmm. Details of the structure refinements are listed in Table 4. Since no additional superstructure reflections were observed upon cooling, the 100 K data are listed for Ca3GeO. In the case of Ca3SiO, additional reflections, observed at 200 K and below, indicate a phase transition. Surprisingly, the crystal under investigation was not twinned by reticular merohedry Acta Cryst. (2015). B71, 300–312

Figure 9 Reciprocal layers h2l of the Ca3SiO crystal at different temperatures. The directions of the reciprocal axes are with respect to the orthorhombic cell settings. Ju¨rgen Nuss et al.

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research papers Table 6 Tolerance factor (t) of the inverse perovskites M3TtO (M = Ca, Sr, Ba, Eu; Tt = Si, Ge, Sn, Pb), calculated from ionic radii r, given in parentheses ˚ ), rO2 = 1.26 A ˚. (A M3TtO

Si (2.04)

Ge (2.08)

Sn (2.23)

Pb (2.25)

Ca (1.14) Sr (1.32) Eu (1.31) Ba (1.49)

0.937 0.921 0.922 0.908

0.949 0.932 0.933 0.918

0.993 0.973 0.974 0.957

0.999 0.978 0.979 0.962

3.4. Goldschmidt tolerance factor for inverse perovskites

Depending on M and Tt, different distortion variants with different crystal symmetries are observed for the inverse perovskites of the series M3TtO (M = Ca, Sr, Ba, Eu; Tt = Si, Ge, Sn, Pb). An indicator for the stability and distortion of perovskites (ABO3) was introduced by Goldschmidt (Goldschmidt, 1926; Goldschmidt et al., 1926). The tolerance factor t simply takes into account that the three different types of ions have contact with each other, and therefore the condition rA + rO = 21/2(rB + rO) yield the ideal cubic perovskite when using the ionic radii, r. This t-value can also be adapted to the inverse perovskites by replacing the respective radii, see equation (4). t¼

21=2

r A þ rO rTt þ rM ^
¼
1=2  r B þ rO 2  r O þ rM

ð4Þ

Since the ionic radii for O2 and M2+ are generally known (Shannon, 1976; Shannon & Prewitt, 1969), the radii for Tt4 have to be calculated from the binary phases M2Tt (M = Ca, Sr, Ba; Tt = Si, Ge, Sn, Pb) for coordination number nine (Eisenmann et al., 1972; Eckerlin et al., 1961; Bruzzone & Franceschi, 1978; Turban & Scha¨fer, 1973), as a rough estimation. The tolerance factors t, together with the used ionic radii, are listed in Table 6 for the whole series M3TtO (M = Ca, Sr, Ba, Eu; Tt = Si, Ge, Sn, Pb).

The size effect of the two ions M2+ and Tt4 influences the deviation from cubic symmetry in opposite directions. Going from the smaller Ca2+ ions to the larger Ba2+, the tolerance factor t slightly decreases and the cubic structure becomes destabilized. A consequence of the M3Sn(Pb)O representatives is that the displacement becomes more anisotropic for larger M2+ cations, and finally the structure changes from cubic (Pm3 m) to orthorhombic (Ibmm) for Ba3SnO and Ba3PbO at low temperatures. The other way is when going from larger Pb4 to smaller Si4, the tolerance factor clearly decreases and nearly all Si and Ge representatives crystallize in the antiGdFeO3-type structure (Pbnm) at room temperature. Owing to the small Ca2+ cation the behaviour of Ca3SiO and Ca3GeO lies somewhere between. While Ca3GeO changes from a cubic (Pm3 m) to an orthorhombic (Ibmm) structure upon cooling ( 350 K), Ca3SiO is the only representative within the M3TtO family where all three polymorphs can be found within the temperature range 500 to 50 K: Pm3 m – Ibmm – Pbnm. Fig. 11 shows a qualitative structure field map (temperature versus tolerance factor t), indicating the stability regions of the three different distortion variants. All the results can be summarized systematically as a function of tolerance factor t. Cubic symmetry is observed for  0.965 < t < 1, for smaller values the structure changes into orthorhombic, first Ibmm ( 0.935 < t <  0.965) and finally Pbnm ( 0.9 < t <  0.935). Near the threshold values the developed structures strongly depend on the temperature, as was shown for Ca3Si(Ge)O or Ba3Sn(Pb)O. From the roughly estimated data points (black dots, Fig. 11), Ca3GeO might show an additional phase transition (Ibmm ! Pbnm) on cooling at around 50 K, on one hand. On the other hand, the germanides of strontium and europium are expected to transform into higher symmetric structures (Pbnm ! Ibmm) on heating to  500 K. It would

Figure 11

Figure 10 Group–subgroup relations between Ca3SiO polymorphs at different temperatures, see Fig. 6 for comparison.

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Qualitative structure field map (temperature versus tolerance factor t) showing the stability regions of different distortion variants of the inverse perovskites M3TtO (M = Ca, Sr, Ba, Eu; Tt = Si, Ge, Sn, Pb). Black dots represent roughly estimated temperatures of phase transitions, derived from temperature-dependent single-crystal X-ray diffraction experiments.

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research papers be beneficial to investigate the respective compounds in the these temperature regions in more detail. The structural phase diagram as a function of temperature in inverse perovskites is different from that of oxide perovskites (ABO3). In the latter ones, the cubic structure is preferred for 0.89 < t < 1, and distorted (orthorhombic) variants such as GdFeO3 are observed for 0.8 < t < 0.89 (Mu¨ller, 2006). The contrast may occur naturally, because the basic assumption for Goldschmidt’s tolerance factor (packing ions with an ideal sphere shape) is not fully applicable for both perovskites and inverse perovskites. The inverse perovskites are Zintl phases, with all contributing ions having closed-shell configurations, and the model of ideal spheres should work well to encircle the structural stability regions. Nevertheless, they are in no way purely ionic compounds and, due to the large polarizabilty of the highly charged Tt4 anions, covalent bonding interactions have to be taken into account, which lead to an aspherical orbital shape. In the oxide perovskites (ABO3) the involved elements are ionic. However, partially filled d-orbitals of the transition metals on the octahedraly coordinated B-site give rise to anisotropic orbital shape. Despite those constraints, as a guideline, Goldschmidt’s tolerance factor is quite useful to understand the structural chemistry of those compounds.

4. Conclusion The series of M3TtO compounds (M = Ca, Sr, Ba, Eu; Tt = Si, Ge, Sn, Pb) that crystallize in anti-perovskite-related structures have an inverted cation and anion distribution in comparison to the perovskites ABO3 (inverse-perovskites M32+Tt4O2 versus perovskites O32A2+B4+). Despite the contrasting chemical environments, they show the same structural diversities with respect to the tilting of the octahedral arrangements, and similar to the perovskites the deviation from the cubic symmetry strongly depends on the radii (r) of the ions involved. The M3TtO compounds with Tt = Sn and Pb crystallize as ‘ideal’ inverse perovskites at room temperature. However, all of them show distinct anisotropies of the displacement ellipsoids of the M atoms, which increase with the size of the cations and are therefore largest for the barium representatives. While this anisotropic behaviour vanishes upon cooling for M = Ca, Sr, Eu, the Ba compounds undergo a phase transition at  150 K from cubic to orthorhombic (Ibmm). The O—Ba—O bonding angles, which are a direct measure of the tilting of the OBa6 octahedra, change from 180 to  174 . For the representatives containing the lighter tetrel elements silicon and germanium, two cases can be distinguished. In combination with the larger cations Sr2+, Eu2+ and Ba2+, orthorhombic structures with anti-GdFeO3-type structures (Pbnm) have been developed. Herein, pronounced tilting of the OM6 octahedra, with O—M—O angles of  160 , is observed. For the smaller calcium, an orthorhombic symmetry (Ibmm) is found at room temperature. While Ca3GeO shows a phase transition at 350 K in the cubic perovskite-type structure, all three polymorphs are observed for Ca3SiO in the Acta Cryst. (2015). B71, 300–312

temperature range 50–500 K (Pbnm, Ibmm and Pm3 m), each one with different octahedral tilting. The O—Ca—O bonding angles are  170 (100 K),  174 (295 K) and 180 (500 K). There seems to be a limit to the distortion possible within the space-group symmetry Ibmm, with an average O—M—O angle of  172 . For larger distortions, the system has to go from the two-tilt system (Ibmm) to a three-tilt system (Glazer, 1975), which can only be realised by lowering the crystal symmetry: Ibmm ! Pbnm. The undistorted ‘ideal’ anti-perovskite structure, realised in e.g. Sr3SnO and Sr3PbO, is the aristotype and the space groups of the distortion variants such as Eu3GeO (Pbnm) and Ca3GeO (Ibmm) are subgroups of Pm3 m. The group– subgroup relations of the translationengleich type lead to multiple twinning, and the crystals are either twinned by pseudo-merohedry or reticular pseudo-merohedry with up to six twin domains. Therefore, handling of the diffraction data with respect to twinning is the key requirement for structure refinement and to obtaining meaningful crystallographic data. In spite of only tiny deviations from the aristotype, the structural details reported in this work are expected to be of great importance for a number of physical properties since, in particular, the electronic and vibrational behaviours are dependent on the detailed structures of the compounds. The physical properties of the reported compounds, and how they depend on the constituent elements together with the resulting structural deviations, will be reported elsewhere in due course.

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electronic reprint

Acta Cryst. (2015). B71, 300–312