Orthosilicates with glaserite-type crystal structures: Na2BaZr[SiO4]2 and Na2BaHf[SiO4]2

Yamane, H.; Funahashi, S.; Hirosaki, N.; Takeda, T.

doi:10.1107/S2056989025002956

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ISSN: 2056-9890

Volume 81| Part 5| May 2025|

https://doi.org/10.1107/S2056989025002956

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Orthosilicates with glaserite-type crystal structures: Na₂BaZr[SiO₄]₂ and Na₂BaHf[SiO₄]₂

Hisanori Yamane,^a Shiro Funahashi,^a Naoto Hirosaki ^a and Takashi Takeda ^a ^*

^aResearch Center for Electronic and Optical Materials, National Institute for Materials Science (NIMS), 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan
^*Correspondence e-mail: takeda.takashi@nims.go.jp

Edited by M. Weil, Vienna University of Technology, Austria (Received 28 February 2025; accepted 2 April 2025; online 4 April 2025)

Single crystal particles of Na₂BaZr[SiO₄]₂ [systematic name: disodium barium zirconium bis(orthosilicate)] and Na₂BaHf[SiO₄]₂ [disodium barium hafnium bis(orthosilicate)] were extracted from grain-grown polycrystals obtained by heating compacts of binary oxide mixtures at 1473 K. Single crystal X-ray diffraction analysis revealed that these are isostructural orthosilicates with a glaserite-type crystal structure, in which all sites of X, Y, M, and T in the general formula XY₂[M(TO₄)₂] are fully occupied by atoms of different elements. The crystal structures of the title compounds were refined in space group P3 under consideration of a two-component twin model. The SiO₄ tetrahedra are rotated approximately ±10.2° from the mirror plane of space group P3m around an axis parallel to [001].

Keywords: crystal structure; glaserite-type structure; orthosilicate; single-crystal X-ray diffraction; twin analysis.

CCDC references: 2440344; 2440343

1. Chemical context

Nikolova & Kostov-Kytin (2013 ) described more than 100 oxides with glaserite-type crystal structures by the general formula X_(□;1)Y_(□;2)[M(TO₄)₂] and summarized their crystal structural features: the T sites are always fully occupied by the atoms of transition metals (V, Cr, Mo, W, Re, Fe, Ru) or non-metals (Si, P, S, Se). These T atoms are fourfold coordinated by oxygen atoms to form isolated tetrahedra in the crystal structures. Silicates (T = Si), such as BaMg[SiO₄] and Ba(Ba,Sr,Ca)₂Mg[SiO₄]₂ doped with Eu²⁺, have been studied for their fluorescent properties and photochromism (Yonesaki et al., 2008 , 2011 ; Yonesaki, 2013 ; Yonezaki et al., 2018 ; Yonezaki, 2015 , 2018 , 2020 ; Yonezaki & Takei, 2016 ; Yonezaki & Yanai, 2021 ; Birkel et al., 2015 ). Recently, ferroaxial transitions of compounds with glaserite-type crystal structures were investigated, in which the space-group type changes from P $[\overline{3}]$ to P $[\overline{3}]$ m (Yamagishi et al., 2023 ). The rotation angle φ of the TO₄ tetrahedron was defined relative to the mirror plane of P $[\overline{3}]$ m, and BaCa₂Mg[SiO₄]₂ with φ = 12.5° was proposed as a potential ferroaxial transition material. For the compounds with M = Zr, Kostov-Kytin and co-workers analysed the crystal structures of Na_3–xH_1+xZr(SiO)₄·yH₂O in which water molecules are located between the ZrO₆ octahedra (Kostov-Kytin et al., 2012 , 2013 ).

In the current study, we report the synthesis and crystal structure analysis of two new orthosilicate compounds with glaserite-type crystal structure, Na₂BaZr[SiO₄]₂ and Na₂BaHf[SiO₄]₂.

2. Structural commentary

According to the classification by Nikolova & Kostov-Kytin (2013) using the general formula X_(□;1)Y_(□;2)[M(TO₄)₂] for compounds with glaserite-type crystal structures, Na₂BaZr[SiO₄]₂ and Na₂BaHf[SiO₄]₂ meet the condition X ≠ Y ≠ M ≠ T of XY₂[M(TO₄)₂] with no vacancy. The two new orthosilicates are isostructural and crystallize in the trigonal space group P $[\overline{3}]$ . The crystal structures of both silicates were refined under consideration of a two-component twin model in each case. Multiplicity, Wyckoff letter, and site symmetry are: 1, a and $[\overline{3}]$ for Ba1, 1, b and $[\overline{3}]$ for Zr1/Hf1, 2, d and 3 for Na1, Si1 and O2, and 6, g and 1 for O1. As shown in Fig. 1, slabs identified in the crystal structure are composed of M = Zr- or Hf-centred oxygen octahedra and SiO₄ tetrahedra.

Figure 1
Crystal structure of Na₂BaZr[SiO₄]₂ in a projection along [100], drawn with Zr1-centered oxygen octahedra and Si1-centered oxygen tetrahedra.

Fig. 2 shows the arrangement of oxygen atoms around each cation. Interatomic distances of Zr—O and Hf—O are 2.0667 (14) Å (×6) and 2.0600 (15) Å (×6), respectively (Tables 1 and 2), which are consistent with the sizes of Shannon's effective ionic radii [six-coordinated Zr (0.72 Å) and Hf (0.71 Å); Shannon, 1976 ]. The Si—O distances of 1.6373 (13) and 1.578 (2) Å for Na₂BaZr[SiO₄]₂ agree with those of 1.6354 (16) and 1.575 (2) Å for Na₂BaHf[SiO₄]₂. The Ba—O distances for the 12-coordinate X site are slightly longer for Na₂BaZr[SiO₄]₂ [2.8764 (15)–3.1198 (2) Å] than for Na₂BaHf[SiO₄]₂ [2.8687 (16)–3.1158 (2) Å]. The distances between O1 and Na1 at the tenfold coordination sites are slightly longer for Na₂BaM[SiO₄]₂ M = Zr [2.4461 (15)–3.1545 (19) Å] than for M = Hf [2.4423 (15)–3.145 (2) Å], while the Na1—O2 distance of 2.250 (3) Å for M = Zr is slightly shorter than that of 2.258 (3) Å for M = Hf. The bond-valence sums for Na1, Ba1, Zr1, Hf1, and Si1 calculated with the bond valence parameters provided by Gagné & Hawthorne (2015 ) are 1.00, 1.86, 4.03, and 4.11 valence units for Na₂BaM[SiO₄]₂ M = Zr, and 1.01, 1.89, 4.05, and 4.09 valence units for M = Hf. The rotation angles φ of the [SiO₄] tetrahedra are 10.18° for M = Zr and 10.15° for M = Hf (Fig. 3).

Table 1
Selected bond lengths (Å) for Na₂BaZr[SiO₄]₂

Na1—O1ⁱ	2.9791 (15)	Ba1—O2	3.1198 (2)
Na1—O1ⁱⁱ	2.4461 (15)	Zr1—O1	2.0667 (13)
Na1—O1	3.1545 (19)	Si1—O1	1.6373 (13)
Na1—O2ⁱⁱⁱ	2.250 (3)	Si1—O2	1.578 (2)
Ba1—O1	2.8764 (15)
Symmetry codes: (i) ; (ii) ; (iii) .

Table 2
Selected bond lengths (Å) for Na₂BaHf[SiO₄]₂

Na1—O1ⁱ	2.4423 (15)	Ba1—O2	3.1158 (2)
Na1—O1ⁱⁱ	2.9738 (15)	Hf1—O1	2.0600 (15)
Na1—O1	3.145 (2)	Si1—O1	1.6354 (16)
Na1—O2ⁱⁱⁱ	2.258 (3)	Si1—O2	1.575 (2)
Ba1—O1	2.8687 (16)
Symmetry codes: (i) ; (ii) ; (iii) .

Figure 2
Atomic arrangements around Na1, Ba1, Zr1, and Si1 in the crystal structure of Na₂BaZr[SiO₄]₂. Displacement ellipsoids are depicted at the 90% probability level. [Symmetry codes: (i) x, y + 1, z; (ii) −x + 1, −y + 2, −z + 1; (iii) −x + 1, −y + 1, −z + 1; (iv) −x, −y + 1, −z + 1; (v) x − y, x, −z + 1; (vi) x − y + 1, x + 1, −z + 1; (vii) y, −x + y, −z + 1; (viii) y, −x + y + 1, −z + 1; (ix) −x + y, −x + 1, z; (x) −y + 1, x − y + 1, z; (xi) x, y, z + 1; (xii) −x + y, −x, z; (xiii) −x, −y, −z; (xiv) −y, x − y, z; (xv) x − y, x, −z; (xvi) y, −x + y, −z; (xvii) −x, −y + 1, −z; (xviii) x, y − 1, z; (xix) −x + 1, −y + 1, −z; (xx) x − 1, y − 1, z; (xxi) −x, −y, −z + 1; (xxii) x + 1, y + 1, z; (xxiii) x, y, z − 1.]

Figure 3
[001] projection of the twin domains in the crystal structure of Na₂BaZr[SiO₄]₂.

The Madelung energy part of the lattice energies (MAPLE; Hoppe, 1995 ) of Na₂BaM[SiO₄]₂ calculated using VESTA (Momma & Izumi, 2011 ) are −50.170 mJ mol⁻¹ (M = Zr) and −50.260 MJ mol⁻¹ (M = Hf). These values are close to those of −49.880 MJ mol⁻¹ and −49.880 mJ mol⁻¹ with differences of 0.6% and 0.8%, respectively, as calculated from the equation BaO + MO₂ + α-Na₂Si₂O₅ = Na₂BaM[SiO₄]₂ using MAPLE values calculated from the crystal structure data for BaO (–3.510 MJ mol⁻¹; Zollweg, 1955 ), ZrO₂ (–12.740 MJ mol⁻¹; Gualtieri et al., 1996 ), HfO₂ (–12.740 MJ mol⁻¹; Pathak et al., 2020 ) and α-Na₂Si₂O₅ (–33.630 MJ mol⁻¹; Pant & Cruickshank, 1968 ).

3. Database survey

The crystal structures listed for glaserite-type silicates in the ICSD database (ICSD, 2025 ) are the high-temperature phase of Ca₂[SiO₄] (P $[\overline{3}]$ m, Z = 2, V = 194.19 Å³; Mumme et al., 1996 ), Ba₃Mg[SiO₄]₂ (P $[\overline{3}]$ m, Z = 1, V = 198.64 Å³; Iwata et al., 2009 ) and its superstructure (P $[\overline{3}]$ , Z = 3, V = 594.71 Å³; Park et al., 2009 ), BaCa₂Mg[SiO₄]₂ (P $[\overline{3}]$ , Z = 1, V = 173.31 Å³; Park et al., 2011 ), Ba_xSr_3–xMg[SiO₄]₂ (x = 0.0 – 0.5, C2, Z = 4, V = 714.9 – 723.7 Å³; x = 0.625 – 2.375, P $[\overline{3}]$ m, Z = 1, V = 181.81 – 193.46 Å³; x = 2.5 – 3.0, P $[\overline{3}]$ , Z = 3, V = 583.2 – 594.72 Å³; Yonezaki, 2015), Ba(Sr_1-xCa_x)₂Mg[SiO₄]₂ (x = 0.0 – 0.5, P $[\overline{3}]$ m, Z = 1, V = 182.76 – 178.70 Å³; x = 0.5625 – 1.0, P $[\overline{3}]$ , Z = 1, V = 177.97 – 174.00 Å³; Yonesaki et al., 2008; Yonesaki, 2013), Ba₃Mn[SiO₄] (P $[\overline{3}]$ m, Z = 1, V = 203.44 Å³; Avdeev et al., 2018 ), Na₃HZr[SiO₄]₂, P $[\overline{1}]$ , Z = 2, V = 350.12 Å³; Na₃HZr[SiO₄]₂·0.4H₂O, P $[\overline{1}]$ , Z = 2, V = 350.19 Å³; Na₃HZr[SiO₄]₂·H₂O, C2/m, Z = 4, V = 683.92 Å³; Kostov-Kytin et al., 2012).

4. Synthesis and crystallization

The starting materials were powders of Na₂O (∼80%, Sigma-Aldrich), SiO₂ (99.9% Kojundo Chemical Lab. Co., Ltd.), BaO (99.99%, Sigma-Aldrich), ZrO₂ (99%, Sigma-Aldrich), and HfO₂ (98%, Kojundo Chemical Lab. Co., Ltd.), which were weighed in a glove box in a nitrogen atmosphere, mixed in an agate mortar, and formed into disk-shaped compacts. The compacts were placed in a nickel boat and sealed in a stainless-steel container. The container was heated up to 1473 K for 3 h in a nitrogen gas flow to prevent oxidation of the stainless steel, and this temperature was maintained for 30 min. The temperature was subsequently lowered to 1073 K at a rate of −100 K h⁻¹. The power supply to the heater wire was stopped at this temperature, and the sample was allowed to cool in the furnace. The single crystal grains of Na₂BaZr[SiO₄]₂ used for single crystal X-ray diffraction data measurements were isolated from fragments of polycrystals synthesised from a starting material mixture with a metal element molar ratio of Na:Ba:Zr:Si = 2:1:1:2. Single crystal grains of Na₂BaHf[SiO₄]₂ were obtained from polycrystals prepared with a starting mixture of Na:Ba:Hf:Si = 2.2:1:1:2.2.

Semi-quantitative analysis of the Na₂BaZr[SiO₄]₂ and Na₂BaHf[SiO₄]₂ grains was performed using an energy-dispersive X-ray detector (Bruker AXS, XFlash 5010) attached to a scanning electron microscope (Hitachi High-Tech SU1510) and measurement analysis software (Bruker AXS, QUANTAX2000). The atomic ratios acquired from the analysis were Na:Ba:Hf:Si = 2.0 (3):0.6 (1):0.66 (5):2.0 (3) for Na₂BaZr[SiO₄]₂ and Na:Ba:Zr:Si = 2.3 (4):1.2 (1):1.3 (1):2.0 (1) for Na₂BaHf[SiO₄]₂, both of which correspond sufficiently with the metal element ratios of the formulae.

5. Refinement

Crystal data, data collection and structural refinement details are summarised in Table 3. In the first refinement step, the ideal model of the glaserite-type structure with space group P $[\overline{3}]$ m was adopted and the R1 values were 0.032 and 0.018 for Na₂BaM[SiO₄]₂ for M = Zr and Hf, respectively. However, the ratio of the mean-square displacements of the major and minor axes of the atomic displacement ellipsoid of O1 was 30.3 (M = Zr) and 22.4 (M = Hf). Subsequently, when the refinement was performed in P $[\overline{3}]$ removing mirror symmetry operation from P $[\overline{3}]$ m, the R1 values were 0.035 and 0.020, and the displacement ratios were still large at 27.6 and 18.2, for M = Zr and Hf, respectively. Further analysis in space group P $[\overline{3}]$ under consideration of twinning (twin matrix 010, 100, 00 $[\overline{1}]$ ) resulted in R1 values of 0.016 and 0.010 for M = Zr and Hf, respectively, and both displacement ratios were reasonable with a value of 3.2. The twin ratios of domains 1 and 2 were refined to 0.635 (4):0.365 for M = Zr and 0.623 (4):0.377 (4) for M = Hf.

Table 3
Experimental details

	Na₂BaZr[SiO₄]₂	Na₂BaHf[SiO₄]₂
Crystal data
M_r	458.72	545.99
Crystal system, space group	Trigonal, P $[\overline{3}]$	Trigonal, P $[\overline{3}]$
Temperature (K)	293	293
a, c (Å)	5.3966 (2), 7.2153 (3)	5.3889 (2), 7.1996 (2)
V (Å³)	181.98 (2)	181.07 (1)
Z	1	1
Radiation type	Mo Kα	Mo Kα
μ (mm⁻¹)	7.27	20.19
Crystal size (mm)	0.04 × 0.03 × 0.01	0.03 × 0.03 × 0.01

Data collection
Diffractometer	ROD, Synergy Custom system, HyPix-Arc 150	ROD, Synergy Custom system, HyPix-Arc 150
Absorption correction	Gaussian (CrysAlis PRO; Rigaku OD, 2023 )	Gaussian (CrysAlis PRO; Rigaku OD, 2023)
T_min, T_max	0.544, 0.817	0.641, 0.883
No. of measured, independent and observed [I > 2σ(I)] reflections	9034, 587, 543	8609, 538, 532
R_int	0.043	0.045
(sin θ/λ)_max (Å⁻¹)	0.832	0.806

Refinement
R[F² > 2σ(F²)], wR(F²), S	0.016, 0.035, 1.11	0.010, 0.024, 1.11
No. of reflections	587	538
No. of parameters	24	24
Δρ_max, Δρ_min (e Å⁻³)	0.52, −0.59	0.61, −0.60
Computer programs: CrysAlis PRO (Rigaku OD, 2023), SHELXT (Sheldrick, 2015a ), SHELXL (Sheldrick, 2015b ), VESTA (Momma & Izumi, 2011), OLEX2 (Dolomanov et al., 2009 ) and publCIF (Westrip, 2010 ).

Supporting information

CCDC references: 2440344; 2440343

Crystal structure: contains datablocks global, I, II. DOI: https://doi.org/10.1107/S2056989025002956/wm5753sup1.cif

Structure factors: contains datablock I. DOI: https://doi.org/10.1107/S2056989025002956/wm5753Isup2.hkl

Structure factors: contains datablock II. DOI: https://doi.org/10.1107/S2056989025002956/wm5753IIsup3.hkl

checkCIF report

Computing details top

Disodium barium zirconium bis(orthosilicate) (I) top

Crystal data top

Na₂BaZr[SiO₄]₂	D_x = 4.186 Mg m⁻³
M_r = 458.72	Mo Kα radiation, λ = 0.71073 Å
Trigonal, P3	Cell parameters from 5734 reflections
a = 5.3966 (2) Å	θ = 2.8–47.6°
c = 7.2153 (3) Å	µ = 7.27 mm⁻¹
V = 181.98 (2) Å³	T = 293 K
Z = 1	Plate, colourless
F(000) = 210	0.04 × 0.03 × 0.01 mm

Data collection top

ROD, Synergy Custom system, HyPix-Arc 150 diffractometer	587 independent reflections
Radiation source: Rotating-anode X-ray tube, Rigaku (Mo) X-ray Source	543 reflections with I > 2σ(I)
Mirror monochromator	R_int = 0.043
Detector resolution: 10.0000 pixels mm^-1	θ_max = 36.2°, θ_min = 2.8°
ω scans	h = −8→7
Absorption correction: gaussian (CrysAlisPro; Rigaku OD, 2023)	k = −8→8
T_min = 0.544, T_max = 0.817	l = −12→12
9034 measured reflections

Refinement top

Refinement on F²	0 restraints
Least-squares matrix: full	Primary atom site location: dual
R[F² > 2σ(F²)] = 0.016	w = 1/[σ²(F_o²) + (0.0092P)² + 0.1709P] where P = (F_o² + 2F_c²)/3
wR(F²) = 0.035	(Δ/σ)_max < 0.001
S = 1.11	Δρ_max = 0.52 e Å⁻³
587 reflections	Δρ_min = −0.59 e Å⁻³
24 parameters

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. Refined as a 2-component twin. 1. Twinned data refinement Scales: 0.635 (4) 0.365 (4)

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

	x	y	z	U_iso*/U_eq
Na1	0.333333	0.666667	0.71034 (19)	0.0167 (2)
Ba1	0.000000	0.000000	0.000000	0.01522 (7)
Zr1	0.000000	0.000000	0.500000	0.00498 (6)
Si1	0.333333	0.666667	0.24086 (9)	0.00685 (12)
O1	0.1242 (3)	0.3476 (3)	0.3269 (2)	0.0144 (3)
O2	0.333333	0.666667	0.0221 (3)	0.0222 (5)

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Na1	0.0157 (4)	0.0157 (4)	0.0188 (6)	0.00784 (18)	0.000	0.000
Ba1	0.01899 (9)	0.01899 (9)	0.00767 (9)	0.00950 (4)	0.000	0.000
Zr1	0.00436 (8)	0.00436 (8)	0.00620 (12)	0.00218 (4)	0.000	0.000
Si1	0.00661 (17)	0.00661 (17)	0.0073 (3)	0.00331 (9)	0.000	0.000
O1	0.0154 (7)	0.0082 (5)	0.0175 (6)	0.0043 (5)	0.0010 (5)	0.0046 (4)
O2	0.0287 (8)	0.0287 (8)	0.0093 (8)	0.0144 (4)	0.000	0.000

Geometric parameters (Å, º) top

Na1—Zr1ⁱ	3.4657 (6)	Ba1—O1^xvi	2.8763 (15)
Na1—Si1ⁱⁱ	3.1356 (2)	Ba1—O1	2.8763 (15)
Na1—Si1ⁱⁱⁱ	3.1356 (2)	Ba1—O2^xvii	3.1198 (2)
Na1—Si1	3.3875 (15)	Ba1—O2	3.1198 (2)
Na1—Si1^iv	3.1356 (2)	Ba1—O2^xviii	3.1198 (2)
Na1—O1^v	2.9791 (15)	Ba1—O2^xix	3.1198 (2)
Na1—O1^vi	2.4461 (15)	Ba1—O2^xiii	3.1198 (2)
Na1—O1^iv	2.4461 (14)	Ba1—O2^xx	3.1198 (2)
Na1—O1^vii	2.4461 (14)	Zr1—O1^xxi	2.0666 (13)
Na1—O1^viii	2.9791 (15)	Zr1—O1^vii	2.0666 (14)
Na1—O1ⁱⁱⁱ	2.9791 (15)	Zr1—O1	2.0667 (13)
Na1—O1	3.1545 (19)	Zr1—O1^v	2.0667 (13)
Na1—O1^ix	3.1545 (19)	Zr1—O1^xiv	2.0666 (14)
Na1—O1^x	3.1545 (19)	Zr1—O1^xii	2.0667 (13)
Na1—O2^xi	2.250 (3)	Si1—O1^x	1.6373 (13)
Ba1—O1^xii	2.8764 (15)	Si1—O1^ix	1.6373 (13)
Ba1—O1^xiii	2.8763 (15)	Si1—O1	1.6373 (13)
Ba1—O1^xiv	2.8763 (15)	Si1—O2	1.578 (2)
Ba1—O1^xv	2.8764 (15)

Si1ⁱⁱⁱ—Na1—Zr1ⁱ	160.48 (5)	Na1—Zr1—Na1ⁱⁱⁱ	77.74 (2)
Si1^iv—Na1—Zr1ⁱ	66.579 (9)	Na1—Zr1—Na1^xxi	180.0
Si1ⁱⁱ—Na1—Zr1ⁱ	66.579 (9)	Na1ⁱⁱⁱ—Zr1—Na1^xx	180.0
Si1—Na1—Zr1ⁱ	64.03 (2)	Na1^iv—Zr1—Na1ⁱⁱⁱ	102.26 (2)
Si1ⁱⁱⁱ—Na1—Si1	96.45 (3)	Na1^xviii—Zr1—Na1^iv	180.0
Si1^iv—Na1—Si1ⁱⁱⁱ	118.756 (11)	Na1^xxi—Zr1—Na1ⁱⁱⁱ	102.26 (2)
Si1^iv—Na1—Si1	96.45 (3)	Na1^xviii—Zr1—Na1^xxi	77.74 (2)
Si1ⁱⁱ—Na1—Si1^iv	118.757 (11)	Na1^iv—Zr1—Na1	77.74 (2)
Si1ⁱⁱ—Na1—Si1ⁱⁱⁱ	118.756 (11)	Na1^xviii—Zr1—Na1^xx	102.26 (2)
Si1ⁱⁱ—Na1—Si1	96.45 (3)	Na1^xviii—Zr1—Na1ⁱⁱⁱ	77.74 (2)
O1ⁱⁱⁱ—Na1—Zr1ⁱ	138.23 (4)	O1—Zr1—Na1^xviii	135.97 (4)
O1^vii—Na1—Zr1ⁱ	137.54 (5)	O1^vii—Zr1—Na1^xviii	58.86 (4)
O1^iv—Na1—Zr1ⁱ	35.96 (3)	O1^xxi—Zr1—Na1^xx	58.86 (4)
O1^viii—Na1—Zr1ⁱ	36.43 (3)	O1^vii—Zr1—Na1ⁱⁱⁱ	63.77 (5)
O1^v—Na1—Zr1ⁱ	86.58 (3)	O1^xii—Zr1—Na1^xx	135.97 (4)
O1^vi—Na1—Zr1ⁱ	85.84 (4)	O1^xxi—Zr1—Na1	116.23 (5)
O1^iv—Na1—Si1	83.70 (5)	O1^v—Zr1—Na1^xx	44.03 (4)
O1^vi—Na1—Si1	83.70 (5)	O1^v—Zr1—Na1^xviii	116.23 (5)
O1^vii—Na1—Si1ⁱⁱⁱ	31.10 (3)	O1^v—Zr1—Na1ⁱⁱⁱ	135.97 (4)
O1^vii—Na1—Si1	83.70 (5)	O1^xii—Zr1—Na1ⁱⁱⁱ	44.03 (4)
O1ⁱⁱⁱ—Na1—Si1	84.83 (4)	O1^xiv—Zr1—Na1^xviii	121.14 (4)
O1^v—Na1—Si1	84.83 (4)	O1^xii—Zr1—Na1^xviii	63.77 (5)
O1^v—Na1—Si1ⁱⁱⁱ	91.85 (3)	O1^xiv—Zr1—Na1^iv	58.86 (4)
O1ⁱⁱⁱ—Na1—Si1^iv	148.85 (3)	O1—Zr1—Na1^iv	44.03 (4)
O1^v—Na1—Si1ⁱⁱ	148.85 (3)	O1^xxi—Zr1—Na1^xviii	44.03 (4)
O1^viii—Na1—Si1	84.83 (4)	O1^xii—Zr1—Na1	121.14 (4)
O1ⁱⁱⁱ—Na1—Si1ⁱⁱ	91.85 (3)	O1^v—Zr1—Na1^iv	63.77 (5)
O1^vii—Na1—Si1ⁱⁱ	148.63 (3)	O1—Zr1—Na1ⁱⁱⁱ	58.86 (4)
O1^iv—Na1—Si1^iv	31.10 (3)	O1—Zr1—Na1^xxi	116.23 (5)
O1^viii—Na1—Si1ⁱⁱⁱ	148.85 (3)	O1^xxi—Zr1—Na1^iv	135.97 (4)
O1^vi—Na1—Si1^iv	148.63 (3)	O1^v—Zr1—Na1^xxi	121.14 (4)
O1^viii—Na1—Si1ⁱⁱ	30.93 (3)	O1^xii—Zr1—Na1^iv	116.23 (5)
O1^vii—Na1—Si1^iv	92.26 (3)	O1^xii—Zr1—Na1^xxi	58.86 (4)
O1^iv—Na1—Si1ⁱⁱ	92.26 (3)	O1^vii—Zr1—Na1^xx	116.23 (5)
O1^viii—Na1—Si1^iv	91.85 (3)	O1^vii—Zr1—Na1^iv	121.14 (4)
O1ⁱⁱⁱ—Na1—Si1ⁱⁱⁱ	30.93 (3)	O1—Zr1—Na1^xx	121.14 (4)
O1^v—Na1—Si1^iv	30.93 (3)	O1^xxi—Zr1—Na1^xxi	63.77 (5)
O1^iv—Na1—Si1ⁱⁱⁱ	148.63 (3)	O1^xiv—Zr1—Na1^xx	63.77 (5)
O1^vi—Na1—Si1ⁱⁱ	31.10 (3)	O1^vii—Zr1—Na1	44.03 (4)
O1^vi—Na1—Si1ⁱⁱⁱ	92.26 (3)	O1^xxi—Zr1—Na1ⁱⁱⁱ	121.14 (4)
O1^vii—Na1—O1ⁱⁱⁱ	56.84 (6)	O1^xiv—Zr1—Na1	135.97 (4)
O1^vi—Na1—O1^viii	56.84 (6)	O1^vii—Zr1—Na1^xxi	135.97 (4)
O1^iv—Na1—O1^vii	118.811 (19)	O1^v—Zr1—Na1	58.86 (4)
O1^iv—Na1—O1^viii	62.52 (6)	O1^xiv—Zr1—Na1ⁱⁱⁱ	116.23 (5)
O1^vi—Na1—O1ⁱⁱⁱ	62.52 (6)	O1^xiv—Zr1—Na1^xxi	44.03 (4)
O1^vi—Na1—O1^v	168.18 (9)	O1—Zr1—Na1	63.77 (5)
O1^v—Na1—O1ⁱⁱⁱ	119.197 (12)	O1^xxi—Zr1—O1^xii	92.75 (6)
O1^vii—Na1—O1^v	62.52 (6)	O1^xiv—Zr1—O1	87.25 (6)
O1^iv—Na1—O1^v	56.84 (6)	O1^xxi—Zr1—O1^xiv	92.75 (6)
O1^vii—Na1—O1^viii	168.18 (9)	O1^xxi—Zr1—O1	180.0
O1^iv—Na1—O1^vi	118.810 (18)	O1^xxi—Zr1—O1^vii	87.25 (6)
O1^viii—Na1—O1ⁱⁱⁱ	119.197 (13)	O1^xiv—Zr1—O1^v	92.75 (6)
O1^viii—Na1—O1^v	119.196 (13)	O1—Zr1—O1^xii	87.25 (6)
O1^vi—Na1—O1^vii	118.812 (19)	O1^vii—Zr1—O1^xii	92.75 (6)
O1^iv—Na1—O1ⁱⁱⁱ	168.18 (9)	O1^vii—Zr1—O1^v	87.25 (6)
O2^xi—Na1—Zr1ⁱ	115.97 (2)	O1^xxi—Zr1—O1^v	87.25 (6)
O2^xi—Na1—Si1	180.0	O1—Zr1—O1^v	92.75 (6)
O2^xi—Na1—Si1^iv	83.55 (3)	O1^xiv—Zr1—O1^xii	87.25 (6)
O2^xi—Na1—Si1ⁱⁱⁱ	83.55 (3)	O1^vii—Zr1—O1	92.75 (6)
O2^xi—Na1—Si1ⁱⁱ	83.55 (3)	O1^vii—Zr1—O1^xiv	180.0
O2^xi—Na1—O1^v	95.17 (4)	O1^v—Zr1—O1^xii	180.0
O2^xi—Na1—O1^iv	96.30 (5)	Na1ⁱⁱⁱ—Si1—Na1	83.55 (3)
O2^xi—Na1—O1^vi	96.30 (5)	Na1ⁱⁱ—Si1—Na1ⁱⁱⁱ	118.756 (11)
O2^xi—Na1—O1^vii	96.30 (5)	Na1ⁱⁱ—Si1—Na1	83.55 (3)
O2^xi—Na1—O1^viii	95.17 (4)	Na1^iv—Si1—Na1	83.55 (3)
O2^xi—Na1—O1ⁱⁱⁱ	95.17 (4)	Na1^iv—Si1—Na1ⁱⁱⁱ	118.756 (11)
O1—Ba1—O1^xiv	59.43 (4)	Na1ⁱⁱ—Si1—Na1^iv	118.757 (11)
O1^xiii—Ba1—O1^xv	59.43 (4)	Na1ⁱⁱ—Si1—Ba1ⁱ	67.716 (15)
O1—Ba1—O1^xvi	120.57 (4)	Na1^iv—Si1—Ba1	67.715 (15)
O1—Ba1—O1^xii	59.43 (4)	Na1ⁱⁱ—Si1—Ba1	157.30 (3)
O1^xiii—Ba1—O1^xiv	120.57 (4)	Na1^iv—Si1—Ba1^xxii	157.30 (3)
O1^xiv—Ba1—O1^xii	59.43 (4)	Na1^iv—Si1—Ba1ⁱ	67.716 (15)
O1^xvi—Ba1—O1^xiv	180.00 (10)	Na1—Si1—Ba1	119.151 (10)
O1^xvi—Ba1—O1^xii	120.57 (4)	Na1ⁱⁱⁱ—Si1—Ba1^xxii	67.715 (15)
O1^xiii—Ba1—O1	180.0	Na1ⁱⁱⁱ—Si1—Ba1ⁱ	157.30 (3)
O1^xiii—Ba1—O1^xii	120.57 (4)	Na1ⁱⁱⁱ—Si1—Ba1	67.715 (15)
O1^xiii—Ba1—O1^xvi	59.43 (4)	Na1—Si1—Ba1^xxii	119.151 (10)
O1—Ba1—O1^xv	120.57 (4)	Na1ⁱⁱ—Si1—Ba1^xxii	67.715 (15)
O1^xv—Ba1—O1^xii	180.00 (10)	Na1—Si1—Ba1ⁱ	119.151 (10)
O1^xiv—Ba1—O1^xv	120.57 (4)	Ba1—Si1—Ba1^xxii	98.284 (12)
O1^xvi—Ba1—O1^xv	59.43 (4)	Ba1ⁱ—Si1—Ba1	98.284 (12)
O1^xiv—Ba1—O2^xix	127.30 (5)	Ba1ⁱ—Si1—Ba1^xxii	98.284 (12)
O1^xiv—Ba1—O2^xvii	80.82 (4)	O1^x—Si1—Na1ⁱⁱⁱ	50.51 (5)
O1^xvi—Ba1—O2^xix	52.70 (5)	O1—Si1—Na1ⁱⁱⁱ	69.25 (5)
O1^xii—Ba1—O2	99.18 (4)	O1^ix—Si1—Na1ⁱⁱⁱ	149.59 (7)
O1^xv—Ba1—O2^xiii	99.18 (4)	O1^ix—Si1—Na1^iv	69.25 (5)
O1—Ba1—O2^xix	80.82 (4)	O1^ix—Si1—Na1ⁱⁱ	50.51 (5)
O1^xiii—Ba1—O2^xx	80.82 (4)	O1—Si1—Na1	67.72 (6)
O1^xii—Ba1—O2^xix	71.30 (5)	O1^x—Si1—Na1	67.72 (6)
O1^xvi—Ba1—O2^xiii	108.70 (5)	O1^x—Si1—Na1ⁱⁱ	69.25 (5)
O1^xiv—Ba1—O2^xiii	71.30 (5)	O1—Si1—Na1ⁱⁱ	149.59 (7)
O1^xiii—Ba1—O2^xix	99.18 (4)	O1—Si1—Na1^iv	50.51 (5)
O1—Ba1—O2^xx	99.18 (4)	O1^x—Si1—Na1^iv	149.59 (7)
O1^xv—Ba1—O2^xix	108.70 (5)	O1^ix—Si1—Na1	67.72 (6)
O1^xvi—Ba1—O2	71.30 (5)	O1—Si1—Ba1^xxii	134.92 (5)
O1^xvi—Ba1—O2^xx	127.30 (5)	O1—Si1—Ba1ⁱ	117.30 (5)
O1^xiv—Ba1—O2^xx	52.70 (5)	O1—Si1—Ba1	52.35 (6)
O1^xv—Ba1—O2	80.82 (4)	O1^ix—Si1—Ba1ⁱ	52.36 (6)
O1—Ba1—O2^xiii	127.30 (5)	O1^x—Si1—Ba1	117.30 (5)
O1^xiv—Ba1—O2^xviii	99.18 (4)	O1^ix—Si1—Ba1^xxii	117.30 (5)
O1^xii—Ba1—O2^xx	108.70 (5)	O1^ix—Si1—Ba1	134.92 (5)
O1^xv—Ba1—O2^xviii	127.30 (5)	O1^x—Si1—Ba1^xxii	52.35 (6)
O1^xii—Ba1—O2^xiii	80.82 (4)	O1^x—Si1—Ba1ⁱ	134.92 (5)
O1^xii—Ba1—O2^xviii	52.70 (5)	O1^ix—Si1—O1	106.53 (6)
O1^xii—Ba1—O2^xvii	127.30 (5)	O1—Si1—O1^x	106.53 (6)
O1^xiii—Ba1—O2	127.30 (5)	O1^ix—Si1—O1^x	106.53 (6)
O1—Ba1—O2	52.70 (5)	O2—Si1—Na1	180.0
O1^xiii—Ba1—O2^xvii	108.70 (5)	O2—Si1—Na1ⁱⁱ	96.45 (3)
O1^xiii—Ba1—O2^xviii	71.30 (5)	O2—Si1—Na1^iv	96.45 (3)
O1—Ba1—O2^xvii	71.30 (5)	O2—Si1—Na1ⁱⁱⁱ	96.45 (3)
O1^xiv—Ba1—O2	108.70 (5)	O2—Si1—Ba1	60.849 (10)
O1^xvi—Ba1—O2^xvii	99.18 (4)	O2—Si1—Ba1^xxii	60.849 (10)
O1—Ba1—O2^xviii	108.70 (5)	O2—Si1—Ba1ⁱ	60.849 (10)
O1^xv—Ba1—O2^xvii	52.70 (5)	O2—Si1—O1	112.28 (6)
O1^xiii—Ba1—O2^xiii	52.70 (5)	O2—Si1—O1^ix	112.28 (6)
O1^xvi—Ba1—O2^xviii	80.82 (4)	O2—Si1—O1^x	112.28 (6)
O1^xv—Ba1—O2^xx	71.30 (5)	Na1^iv—O1—Na1ⁱⁱⁱ	168.17 (9)
O2^xvii—Ba1—O2^xix	119.741 (7)	Na1^iv—O1—Ba1	89.26 (5)
O2^xvii—Ba1—O2^xx	60.259 (7)	Ba1—O1—Na1ⁱⁱⁱ	79.67 (4)
O2^xviii—Ba1—O2^xiii	60.259 (7)	Zr1—O1—Na1^iv	100.01 (6)
O2^xvii—Ba1—O2^xiii	119.741 (7)	Zr1—O1—Na1ⁱⁱⁱ	84.71 (5)
O2—Ba1—O2^xx	119.740 (7)	Zr1—O1—Ba1	92.27 (5)
O2^xiii—Ba1—O2^xx	60.260 (7)	Si1—O1—Na1ⁱⁱⁱ	79.82 (5)
O2^xiii—Ba1—O2^xix	119.740 (7)	Si1—O1—Na1^iv	98.38 (6)
O2^xviii—Ba1—O2^xix	60.259 (7)	Si1—O1—Ba1	100.86 (7)
O2—Ba1—O2^xiii	180.0	Si1—O1—Zr1	157.49 (9)
O2^xviii—Ba1—O2^xx	119.741 (7)	Na1^xxiii—O2—Ba1^xxii	87.07 (4)
O2^xvii—Ba1—O2	60.259 (7)	Na1^xxiii—O2—Ba1	87.07 (4)
O2—Ba1—O2^xix	60.260 (7)	Na1^xxiii—O2—Ba1ⁱ	87.07 (4)
O2^xviii—Ba1—O2	119.741 (7)	Ba1—O2—Ba1^xxii	119.741 (7)
O2^xvii—Ba1—O2^xviii	180.00 (8)	Ba1ⁱ—O2—Ba1	119.741 (7)
O2^xix—Ba1—O2^xx	180.00 (8)	Ba1ⁱ—O2—Ba1^xxii	119.741 (7)
Na1^iv—Zr1—Na1^xx	77.74 (2)	Si1—O2—Na1^xxiii	180.0
Na1—Zr1—Na1^xx	102.26 (2)	Si1—O2—Ba1ⁱ	92.93 (4)
Na1^xviii—Zr1—Na1	102.26 (2)	Si1—O2—Ba1^xxii	92.93 (4)
Na1^iv—Zr1—Na1^xxi	102.26 (2)	Si1—O2—Ba1	92.93 (4)
Na1^xxi—Zr1—Na1^xx	77.74 (2)

Na1ⁱⁱ—Si1—O1—Na1^iv	80.07 (11)	Ba1ⁱ—Si1—O1—Zr1	−156.5 (2)
Na1—Si1—O1—Na1^iv	100.39 (6)	Ba1—Si1—O1—Zr1	124.6 (3)
Na1ⁱⁱⁱ—Si1—O1—Na1^iv	−168.16 (9)	Ba1^xxii—Si1—O1—Zr1	65.4 (3)
Na1—Si1—O1—Na1ⁱⁱⁱ	−91.46 (4)	Ba1—Si1—O2—Ba1^xxii	−120.0
Na1^iv—Si1—O1—Na1ⁱⁱⁱ	168.16 (9)	Ba1ⁱ—Si1—O2—Ba1	−120.000 (1)
Na1ⁱⁱ—Si1—O1—Na1ⁱⁱⁱ	−111.77 (9)	Ba1^xxii—Si1—O2—Ba1	120.0
Na1—Si1—O1—Ba1	−168.75 (6)	Ba1ⁱ—Si1—O2—Ba1^xxii	120.000 (1)
Na1^iv—Si1—O1—Ba1	90.86 (7)	Ba1^xxii—Si1—O2—Ba1ⁱ	−120.0
Na1ⁱⁱⁱ—Si1—O1—Ba1	−77.30 (5)	Ba1—Si1—O2—Ba1ⁱ	120.0
Na1ⁱⁱ—Si1—O1—Ba1	170.93 (5)	O1^ix—Si1—O1—Na1ⁱⁱⁱ	−148.17 (6)
Na1ⁱⁱⁱ—Si1—O1—Zr1	47.3 (2)	O1^x—Si1—O1—Na1ⁱⁱⁱ	−34.74 (9)
Na1—Si1—O1—Zr1	−44.1 (2)	O1^ix—Si1—O1—Na1^iv	43.67 (12)
Na1ⁱⁱ—Si1—O1—Zr1	−64.5 (3)	O1^x—Si1—O1—Na1^iv	157.10 (5)
Na1^iv—Si1—O1—Zr1	−144.5 (3)	O1^ix—Si1—O1—Ba1	134.53 (8)
Na1^iv—Si1—O2—Ba1	−60.0	O1^x—Si1—O1—Ba1	−112.04 (9)
Na1ⁱⁱⁱ—Si1—O2—Ba1ⁱ	180.0	O1^ix—Si1—O1—Zr1	−100.85 (18)
Na1ⁱⁱ—Si1—O2—Ba1	180.0	O1^x—Si1—O1—Zr1	12.6 (3)
Na1ⁱⁱ—Si1—O2—Ba1^xxii	60.0	O1^ix—Si1—O2—Ba1^xxii	109.82 (5)
Na1^iv—Si1—O2—Ba1^xxii	180.000 (1)	O1^x—Si1—O2—Ba1	109.81 (5)
Na1ⁱⁱⁱ—Si1—O2—Ba1^xxii	−60.0	O1—Si1—O2—Ba1	−10.18 (5)
Na1^iv—Si1—O2—Ba1ⁱ	60.0	O1^ix—Si1—O2—Ba1ⁱ	−10.18 (5)
Na1ⁱⁱⁱ—Si1—O2—Ba1	60.0	O1^x—Si1—O2—Ba1^xxii	−10.19 (5)
Na1ⁱⁱ—Si1—O2—Ba1ⁱ	−60.000 (1)	O1^x—Si1—O2—Ba1ⁱ	−130.19 (5)
Ba1—Si1—O1—Na1^iv	−90.86 (7)	O1—Si1—O2—Ba1ⁱ	109.82 (5)
Ba1^xxii—Si1—O1—Na1ⁱⁱⁱ	18.11 (8)	O1^ix—Si1—O2—Ba1	−130.18 (5)
Ba1ⁱ—Si1—O1—Na1^iv	−12.01 (8)	O1—Si1—O2—Ba1^xxii	−130.18 (5)
Ba1—Si1—O1—Na1ⁱⁱⁱ	77.30 (5)	O2—Si1—O1—Na1^iv	−79.61 (6)
Ba1ⁱ—Si1—O1—Na1ⁱⁱⁱ	156.15 (4)	O2—Si1—O1—Na1ⁱⁱⁱ	88.54 (4)
Ba1^xxii—Si1—O1—Na1^iv	−150.05 (5)	O2—Si1—O1—Ba1	11.25 (6)
Ba1ⁱ—Si1—O1—Ba1	78.85 (6)	O2—Si1—O1—Zr1	135.9 (2)
Ba1^xxii—Si1—O1—Ba1	−59.19 (9)

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

Disodium barium hafnium bis(orthosilicate) (II) top

Crystal data top

Na₂BaHf[SiO₄]₂	D_x = 5.007 Mg m⁻³
M_r = 545.99	Mo Kα radiation, λ = 0.71073 Å
Trigonal, P3	Cell parameters from 7145 reflections
a = 5.3889 (2) Å	θ = 2.8–51.0°
c = 7.1996 (2) Å	µ = 20.19 mm⁻¹
V = 181.07 (1) Å³	T = 293 K
Z = 1	Plate, colourless
F(000) = 242	0.03 × 0.03 × 0.01 mm

Data collection top

ROD, Synergy Custom system, HyPix-Arc 150 diffractometer	538 independent reflections
Radiation source: Rotating-anode X-ray tube, Rigaku (Mo) X-ray Source	532 reflections with I > 2σ(I)
Mirror monochromator	R_int = 0.045
Detector resolution: 10.0000 pixels mm^-1	θ_max = 35.0°, θ_min = 2.8°
ω scans	h = −8→7
Absorption correction: gaussian (CrysAlisPro; Rigaku OD, 2023)	k = −8→8
T_min = 0.641, T_max = 0.883	l = −11→11
8609 measured reflections

Refinement top

Refinement on F²	0 restraints
Least-squares matrix: full	Primary atom site location: dual
R[F² > 2σ(F²)] = 0.010	w = 1/[σ²(F_o²) + (0.0035P)² + 0.1631P] where P = (F_o² + 2F_c²)/3
wR(F²) = 0.024	(Δ/σ)_max < 0.001
S = 1.11	Δρ_max = 0.61 e Å⁻³
538 reflections	Δρ_min = −0.60 e Å⁻³
24 parameters

Special details top

Refinement. Refined as a 2-component twin. 1. Twinned data refinement Scales: 0.623 (4) 0.377 (4)

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

	x	y	z	U_iso*/U_eq
Na1	0.333333	0.666667	0.7096 (2)	0.0148 (2)
Ba1	0.000000	0.000000	0.000000	0.01327 (5)
Hf1	0.000000	0.000000	0.500000	0.00516 (4)
Si1	0.333333	0.666667	0.24191 (11)	0.00532 (12)
O1	0.1238 (3)	0.3468 (3)	0.3269 (2)	0.0118 (4)
O2	0.333333	0.666667	0.0232 (3)	0.0187 (5)

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Na1	0.0142 (4)	0.0142 (4)	0.0159 (6)	0.00710 (18)	0.000	0.000
Ba1	0.01685 (8)	0.01685 (8)	0.00611 (10)	0.00842 (4)	0.000	0.000
Hf1	0.00461 (5)	0.00461 (5)	0.00626 (7)	0.00231 (2)	0.000	0.000
Si1	0.00477 (18)	0.00477 (18)	0.0064 (3)	0.00239 (9)	0.000	0.000
O1	0.0111 (9)	0.0074 (6)	0.0146 (7)	0.0030 (5)	0.0012 (5)	0.0043 (5)
O2	0.0245 (7)	0.0245 (7)	0.0073 (9)	0.0122 (4)	0.000	0.000

Geometric parameters (Å, º) top

Na1—Hf1ⁱ	3.4579 (6)	Ba1—O1^xv	2.8687 (16)
Na1—Si1	3.3670 (16)	Ba1—O1^xvi	2.8687 (16)
Na1—Si1ⁱⁱ	3.1308 (2)	Ba1—O2^xvii	3.1157 (2)
Na1—Si1ⁱⁱⁱ	3.1308 (2)	Ba1—O2^xviii	3.1158 (2)
Na1—Si1^iv	3.1308 (2)	Ba1—O2^xiv	3.1158 (2)
Na1—O1^v	2.4423 (15)	Ba1—O2^xix	3.1157 (2)
Na1—O1^iv	2.4423 (15)	Ba1—O2	3.1158 (2)
Na1—O1^vi	2.4423 (15)	Ba1—O2^xx	3.1158 (2)
Na1—O1^vii	2.9738 (15)	Hf1—O1^v	2.0600 (15)
Na1—O1^viii	2.9738 (15)	Hf1—O1^xxi	2.0600 (15)
Na1—O1ⁱⁱⁱ	2.9738 (15)	Hf1—O1^xiii	2.0600 (15)
Na1—O1	3.145 (2)	Hf1—O1	2.0600 (15)
Na1—O1^ix	3.145 (2)	Hf1—O1^vii	2.0600 (15)
Na1—O1^x	3.145 (2)	Hf1—O1^xvi	2.0600 (15)
Na1—O2^xi	2.258 (3)	Si1—O1^x	1.6354 (16)
Ba1—O1^xii	2.8687 (16)	Si1—O1	1.6354 (16)
Ba1—O1^xiii	2.8687 (16)	Si1—O1^ix	1.6354 (16)
Ba1—O1^xiv	2.8687 (16)	Si1—O2	1.575 (2)
Ba1—O1	2.8687 (16)

Si1ⁱⁱⁱ—Na1—Hf1ⁱ	160.53 (5)	Na1^xix—Hf1—Na1^iv	180.0
Si1ⁱⁱ—Na1—Hf1ⁱ	66.522 (10)	Na1^xix—Hf1—Na1^xxi	77.62 (3)
Si1^iv—Na1—Hf1ⁱ	66.522 (10)	Na1^xix—Hf1—Na1ⁱⁱⁱ	77.62 (3)
Si1—Na1—Hf1ⁱ	64.13 (2)	Na1—Hf1—Na1^xxi	180.0
Si1ⁱⁱ—Na1—Si1ⁱⁱⁱ	118.773 (12)	Na1—Hf1—Na1ⁱⁱⁱ	77.62 (3)
Si1ⁱⁱ—Na1—Si1^iv	118.773 (12)	Na1^iv—Hf1—Na1^xx	77.62 (3)
Si1^iv—Na1—Si1ⁱⁱⁱ	118.773 (12)	Na1^xxi—Hf1—Na1ⁱⁱⁱ	102.38 (3)
Si1ⁱⁱ—Na1—Si1	96.40 (3)	Na1^xxi—Hf1—Na1^xx	77.62 (3)
Si1^iv—Na1—Si1	96.40 (3)	Na1^xix—Hf1—Na1	102.38 (3)
Si1ⁱⁱⁱ—Na1—Si1	96.40 (3)	Na1^iv—Hf1—Na1	77.62 (3)
O1ⁱⁱⁱ—Na1—Hf1ⁱ	138.33 (5)	O1^vii—Hf1—Na1^xix	116.29 (5)
O1^viii—Na1—Hf1ⁱ	36.38 (3)	O1^xvi—Hf1—Na1^xx	63.71 (5)
O1^vii—Na1—Hf1ⁱ	86.70 (3)	O1^v—Hf1—Na1	44.07 (4)
O1^iv—Na1—Hf1ⁱ	35.92 (4)	O1^xiii—Hf1—Na1^xix	63.71 (5)
O1^v—Na1—Hf1ⁱ	137.65 (5)	O1^xxi—Hf1—Na1^xx	58.91 (4)
O1^vi—Na1—Hf1ⁱ	85.99 (4)	O1^xxi—Hf1—Na1^xix	44.07 (4)
O1^iv—Na1—Si1ⁱⁱ	92.15 (4)	O1^vii—Hf1—Na1^iv	63.71 (5)
O1ⁱⁱⁱ—Na1—Si1ⁱⁱⁱ	30.94 (3)	O1^xxi—Hf1—Na1	116.29 (5)
O1^iv—Na1—Si1ⁱⁱⁱ	148.71 (4)	O1^vii—Hf1—Na1^xx	44.07 (4)
O1^vi—Na1—Si1ⁱⁱ	31.12 (4)	O1^v—Hf1—Na1^iv	121.09 (4)
O1^viii—Na1—Si1	84.93 (4)	O1^xxi—Hf1—Na1^iv	135.93 (4)
O1^v—Na1—Si1ⁱⁱ	148.71 (4)	O1^xvi—Hf1—Na1ⁱⁱⁱ	116.29 (5)
O1^viii—Na1—Si1ⁱⁱⁱ	148.92 (3)	O1^vii—Hf1—Na1ⁱⁱⁱ	135.93 (4)
O1^viii—Na1—Si1ⁱⁱ	30.94 (3)	O1—Hf1—Na1ⁱⁱⁱ	58.91 (4)
O1^vi—Na1—Si1	83.83 (5)	O1^v—Hf1—Na1^xx	116.29 (5)
O1^vii—Na1—Si1ⁱⁱ	148.92 (3)	O1^xvi—Hf1—Na1^xxi	44.07 (4)
O1ⁱⁱⁱ—Na1—Si1	84.93 (4)	O1—Hf1—Na1^xx	121.09 (4)
O1ⁱⁱⁱ—Na1—Si1ⁱⁱ	91.76 (3)	O1—Hf1—Na1^xxi	116.29 (5)
O1^v—Na1—Si1ⁱⁱⁱ	31.12 (4)	O1^xiii—Hf1—Na1^xx	135.93 (4)
O1^iv—Na1—Si1^iv	31.12 (4)	O1^vii—Hf1—Na1^xxi	121.09 (4)
O1^vii—Na1—Si1ⁱⁱⁱ	91.76 (3)	O1^xxi—Hf1—Na1ⁱⁱⁱ	121.09 (4)
O1^vi—Na1—Si1^iv	148.71 (4)	O1^xiii—Hf1—Na1^xxi	58.91 (4)
O1^iv—Na1—Si1	83.83 (5)	O1—Hf1—Na1^xix	135.93 (4)
O1^v—Na1—Si1^iv	92.15 (4)	O1^xxi—Hf1—Na1^xxi	63.71 (5)
O1^v—Na1—Si1	83.83 (5)	O1^v—Hf1—Na1ⁱⁱⁱ	63.71 (5)
O1^viii—Na1—Si1^iv	91.76 (3)	O1^xvi—Hf1—Na1^iv	58.91 (4)
O1^vii—Na1—Si1	84.93 (4)	O1—Hf1—Na1^iv	44.07 (4)
O1^vii—Na1—Si1^iv	30.94 (3)	O1^v—Hf1—Na1^xxi	135.93 (4)
O1ⁱⁱⁱ—Na1—Si1^iv	148.92 (3)	O1^xiii—Hf1—Na1ⁱⁱⁱ	44.07 (4)
O1^vi—Na1—Si1ⁱⁱⁱ	92.15 (4)	O1^xvi—Hf1—Na1	135.93 (4)
O1^v—Na1—O1ⁱⁱⁱ	57.02 (7)	O1^v—Hf1—Na1^xix	58.91 (4)
O1^iv—Na1—O1^viii	62.36 (7)	O1—Hf1—Na1	63.71 (5)
O1^vi—Na1—O1^vii	168.44 (9)	O1^xvi—Hf1—Na1^xix	121.09 (4)
O1^v—Na1—O1^vii	62.36 (7)	O1^vii—Hf1—Na1	58.91 (4)
O1^v—Na1—O1^viii	168.44 (9)	O1^xiii—Hf1—Na1	121.09 (4)
O1^viii—Na1—O1ⁱⁱⁱ	119.229 (13)	O1^xiii—Hf1—Na1^iv	116.29 (5)
O1^vi—Na1—O1^v	118.859 (19)	O1^v—Hf1—O1^xxi	87.19 (7)
O1^vii—Na1—O1ⁱⁱⁱ	119.229 (13)	O1^v—Hf1—O1^vii	87.19 (7)
O1^vi—Na1—O1^viii	57.02 (7)	O1—Hf1—O1^xiii	87.19 (7)
O1^iv—Na1—O1ⁱⁱⁱ	168.44 (9)	O1^xvi—Hf1—O1^xxi	92.81 (7)
O1^iv—Na1—O1^vii	57.02 (7)	O1^v—Hf1—O1^xiii	92.81 (7)
O1^vi—Na1—O1ⁱⁱⁱ	62.36 (7)	O1—Hf1—O1^xxi	180.0
O1^iv—Na1—O1^vi	118.859 (19)	O1^xvi—Hf1—O1	87.19 (7)
O1^iv—Na1—O1^v	118.859 (19)	O1^vii—Hf1—O1^xxi	87.19 (7)
O1^viii—Na1—O1^vii	119.229 (13)	O1—Hf1—O1^vii	92.81 (7)
O2^xi—Na1—Hf1ⁱ	115.87 (2)	O1^xiii—Hf1—O1^xxi	92.81 (7)
O2^xi—Na1—Si1ⁱⁱⁱ	83.60 (3)	O1^xvi—Hf1—O1^xiii	87.19 (7)
O2^xi—Na1—Si1	180.0	O1^v—Hf1—O1^xvi	180.0
O2^xi—Na1—Si1^iv	83.60 (3)	O1^vii—Hf1—O1^xiii	180.0
O2^xi—Na1—Si1ⁱⁱ	83.60 (3)	O1^v—Hf1—O1	92.81 (7)
O2^xi—Na1—O1^vii	95.07 (4)	O1^xvi—Hf1—O1^vii	92.81 (7)
O2^xi—Na1—O1^v	96.17 (5)	Na1ⁱⁱ—Si1—Na1^iv	118.773 (12)
O2^xi—Na1—O1^viii	95.07 (4)	Na1ⁱⁱⁱ—Si1—Na1	83.60 (3)
O2^xi—Na1—O1^vi	96.17 (5)	Na1^iv—Si1—Na1	83.60 (3)
O2^xi—Na1—O1^iv	96.17 (5)	Na1^iv—Si1—Na1ⁱⁱⁱ	118.772 (12)
O2^xi—Na1—O1ⁱⁱⁱ	95.07 (4)	Na1ⁱⁱ—Si1—Na1	83.60 (3)
O1^xiii—Ba1—O1	59.36 (5)	Na1ⁱⁱ—Si1—Na1ⁱⁱⁱ	118.772 (12)
O1^xiv—Ba1—O1	180.0	Na1ⁱⁱ—Si1—Ba1ⁱ	67.724 (16)
O1^xv—Ba1—O1^xvi	120.64 (5)	Na1^iv—Si1—Ba1^xxii	157.16 (4)
O1^xii—Ba1—O1^xvi	180.00 (8)	Na1—Si1—Ba1	119.240 (11)
O1^xiv—Ba1—O1^xvi	120.64 (5)	Na1ⁱⁱⁱ—Si1—Ba1ⁱ	157.16 (4)
O1^xiv—Ba1—O1^xii	59.36 (5)	Na1—Si1—Ba1^xxii	119.240 (11)
O1^xiii—Ba1—O1^xvi	59.36 (5)	Na1—Si1—Ba1ⁱ	119.240 (11)
O1^xv—Ba1—O1^xii	59.36 (5)	Na1ⁱⁱ—Si1—Ba1^xxii	67.723 (16)
O1^xiii—Ba1—O1^xii	120.64 (5)	Na1ⁱⁱ—Si1—Ba1	157.16 (4)
O1^xv—Ba1—O1	120.64 (5)	Na1ⁱⁱⁱ—Si1—Ba1^xxii	67.723 (16)
O1^xiv—Ba1—O1^xv	59.36 (5)	Na1^iv—Si1—Ba1	67.723 (16)
O1^xii—Ba1—O1	120.64 (5)	Na1ⁱⁱⁱ—Si1—Ba1	67.723 (16)
O1^xiv—Ba1—O1^xiii	120.64 (5)	Na1^iv—Si1—Ba1ⁱ	67.724 (16)
O1^xv—Ba1—O1^xiii	180.00 (10)	Ba1—Si1—Ba1^xxii	98.169 (14)
O1^xvi—Ba1—O1	59.36 (5)	Ba1ⁱ—Si1—Ba1	98.169 (14)
O1^xii—Ba1—O2^xviii	52.61 (5)	Ba1ⁱ—Si1—Ba1^xxii	98.169 (14)
O1—Ba1—O2	52.61 (5)	O1^ix—Si1—Na1ⁱⁱ	50.52 (5)
O1^xv—Ba1—O2^xviii	108.55 (5)	O1^x—Si1—Na1	68.03 (6)
O1—Ba1—O2^xviii	80.96 (5)	O1—Si1—Na1ⁱⁱ	149.93 (7)
O1^xii—Ba1—O2^xiv	108.55 (5)	O1^x—Si1—Na1^iv	149.94 (7)
O1^xv—Ba1—O2^xx	71.45 (5)	O1^ix—Si1—Na1^iv	69.22 (5)
O1^xiii—Ba1—O2	99.04 (5)	O1^ix—Si1—Na1ⁱⁱⁱ	149.94 (7)
O1^xii—Ba1—O2^xx	127.39 (5)	O1—Si1—Na1	68.03 (6)
O1^xv—Ba1—O2^xiv	99.04 (5)	O1—Si1—Na1ⁱⁱⁱ	69.22 (5)
O1—Ba1—O2^xx	99.04 (5)	O1^x—Si1—Na1ⁱⁱ	69.22 (5)
O1—Ba1—O2^xiv	127.39 (5)	O1^x—Si1—Na1ⁱⁱⁱ	50.51 (5)
O1—Ba1—O2^xvii	71.45 (5)	O1—Si1—Na1^iv	50.51 (5)
O1^xv—Ba1—O2	80.96 (5)	O1^ix—Si1—Na1	68.03 (6)
O1^xiv—Ba1—O2^xix	71.45 (5)	O1—Si1—Ba1^xxii	134.78 (5)
O1^xii—Ba1—O2	71.45 (5)	O1^ix—Si1—Ba1ⁱ	52.13 (6)
O1^xv—Ba1—O2^xix	127.39 (5)	O1^ix—Si1—Ba1^xxii	117.23 (5)
O1^xiv—Ba1—O2^xiv	52.61 (5)	O1^x—Si1—Ba1^xxii	52.13 (6)
O1^xiii—Ba1—O2^xix	52.61 (5)	O1—Si1—Ba1ⁱ	117.23 (5)
O1^xiii—Ba1—O2^xiv	80.96 (5)	O1^x—Si1—Ba1	117.23 (5)
O1^xii—Ba1—O2^xix	80.96 (5)	O1—Si1—Ba1	52.13 (6)
O1^xvi—Ba1—O2^xiv	71.45 (5)	O1^x—Si1—Ba1ⁱ	134.78 (5)
O1^xvi—Ba1—O2^xix	99.04 (5)	O1^ix—Si1—Ba1	134.78 (6)
O1^xiv—Ba1—O2^xviii	99.04 (5)	O1^ix—Si1—O1	106.86 (7)
O1—Ba1—O2^xix	108.55 (5)	O1—Si1—O1^x	106.86 (7)
O1^xiii—Ba1—O2^xviii	71.45 (5)	O1^ix—Si1—O1^x	106.86 (7)
O1^xiv—Ba1—O2	127.39 (5)	O2—Si1—Na1	180.0
O1^xvi—Ba1—O2^xviii	127.39 (5)	O2—Si1—Na1ⁱⁱⁱ	96.40 (3)
O1^xiv—Ba1—O2^xvii	108.55 (5)	O2—Si1—Na1^iv	96.40 (3)
O1^xiv—Ba1—O2^xx	80.96 (5)	O2—Si1—Na1ⁱⁱ	96.40 (3)
O1^xv—Ba1—O2^xvii	52.61 (5)	O2—Si1—Ba1	60.760 (11)
O1^xiii—Ba1—O2^xx	108.55 (5)	O2—Si1—Ba1^xxii	60.760 (11)
O1^xiii—Ba1—O2^xvii	127.39 (5)	O2—Si1—Ba1ⁱ	60.760 (11)
O1^xvi—Ba1—O2^xx	52.61 (5)	O2—Si1—O1	111.97 (6)
O1^xii—Ba1—O2^xvii	99.04 (5)	O2—Si1—O1^x	111.97 (6)
O1^xvi—Ba1—O2^xvii	80.96 (5)	O2—Si1—O1^ix	111.97 (6)
O1^xvi—Ba1—O2	108.55 (5)	Na1^iv—O1—Na1ⁱⁱⁱ	168.44 (9)
O2^xvii—Ba1—O2^xviii	119.716 (8)	Na1^iv—O1—Ba1	89.42 (6)
O2^xvii—Ba1—O2^xiv	119.716 (8)	Ba1—O1—Na1ⁱⁱⁱ	79.80 (5)
O2^xiv—Ba1—O2^xx	60.285 (8)	Hf1—O1—Na1ⁱⁱⁱ	84.71 (5)
O2^xix—Ba1—O2^xviii	60.284 (8)	Hf1—O1—Na1^iv	100.01 (6)
O2^xix—Ba1—O2^xx	119.716 (8)	Hf1—O1—Ba1	92.35 (6)
O2—Ba1—O2^xviii	60.285 (8)	Si1—O1—Na1^iv	98.37 (7)
O2^xvii—Ba1—O2^xix	180.0	Si1—O1—Na1ⁱⁱⁱ	79.84 (6)
O2^xiv—Ba1—O2^xviii	119.715 (8)	Si1—O1—Ba1	101.12 (7)
O2^xvii—Ba1—O2^xx	60.284 (8)	Si1—O1—Hf1	157.27 (10)
O2—Ba1—O2^xiv	180.0	Na1^xxiii—O2—Ba1^xxii	86.93 (4)
O2—Ba1—O2^xx	119.715 (8)	Na1^xxiii—O2—Ba1ⁱ	86.93 (4)
O2^xvii—Ba1—O2	60.285 (8)	Na1^xxiii—O2—Ba1	86.93 (4)
O2^xix—Ba1—O2^xiv	60.284 (8)	Ba1ⁱ—O2—Ba1^xxii	119.715 (8)
O2^xix—Ba1—O2	119.715 (8)	Ba1—O2—Ba1^xxii	119.715 (8)
O2^xviii—Ba1—O2^xx	180.0	Ba1ⁱ—O2—Ba1	119.715 (8)
Na1^iv—Hf1—Na1ⁱⁱⁱ	102.38 (3)	Si1—O2—Na1^xxiii	180.0
Na1^iv—Hf1—Na1^xxi	102.38 (3)	Si1—O2—Ba1	93.07 (4)
Na1—Hf1—Na1^xx	102.38 (3)	Si1—O2—Ba1ⁱ	93.07 (4)
Na1^xx—Hf1—Na1ⁱⁱⁱ	180.0	Si1—O2—Ba1^xxii	93.07 (4)
Na1^xix—Hf1—Na1^xx	102.38 (3)

Na1ⁱⁱ—Si1—O1—Na1^iv	79.72 (11)	Ba1ⁱ—Si1—O1—Hf1	−156.1 (2)
Na1—Si1—O1—Na1ⁱⁱⁱ	−91.40 (4)	Ba1—Si1—O1—Hf1	125.3 (3)
Na1^iv—Si1—O1—Na1ⁱⁱⁱ	168.43 (9)	Ba1^xxii—Si1—O1—Hf1	66.5 (3)
Na1ⁱⁱⁱ—Si1—O1—Na1^iv	−168.43 (9)	Ba1^xxii—Si1—O2—Ba1ⁱ	−120.0
Na1ⁱⁱ—Si1—O1—Na1ⁱⁱⁱ	−111.84 (10)	Ba1^xxii—Si1—O2—Ba1	120.0
Na1—Si1—O1—Na1^iv	100.17 (6)	Ba1ⁱ—Si1—O2—Ba1	−120.000 (1)
Na1ⁱⁱ—Si1—O1—Ba1	170.78 (6)	Ba1—Si1—O2—Ba1ⁱ	120.0
Na1—Si1—O1—Ba1	−168.78 (6)	Ba1—Si1—O2—Ba1^xxii	−120.0
Na1ⁱⁱⁱ—Si1—O1—Ba1	−77.38 (5)	Ba1ⁱ—Si1—O2—Ba1^xxii	120.0
Na1^iv—Si1—O1—Ba1	91.06 (7)	O1^ix—Si1—O1—Na1ⁱⁱⁱ	−148.46 (7)
Na1^iv—Si1—O1—Hf1	−143.7 (3)	O1^ix—Si1—O1—Na1^iv	43.11 (12)
Na1ⁱⁱⁱ—Si1—O1—Hf1	47.9 (2)	O1^x—Si1—O1—Na1^iv	157.23 (5)
Na1ⁱⁱ—Si1—O1—Hf1	−63.9 (3)	O1^x—Si1—O1—Na1ⁱⁱⁱ	−34.34 (9)
Na1—Si1—O1—Hf1	−43.5 (2)	O1^ix—Si1—O1—Ba1	134.17 (9)
Na1ⁱⁱ—Si1—O2—Ba1ⁱ	−60.000 (1)	O1^x—Si1—O1—Ba1	−111.72 (10)
Na1^iv—Si1—O2—Ba1	−60.0	O1^ix—Si1—O1—Hf1	−100.56 (19)
Na1ⁱⁱ—Si1—O2—Ba1^xxii	60.0	O1^x—Si1—O1—Hf1	13.6 (3)
Na1ⁱⁱⁱ—Si1—O2—Ba1^xxii	−60.0	O1^ix—Si1—O2—Ba1ⁱ	−10.14 (6)
Na1ⁱⁱⁱ—Si1—O2—Ba1	60.0	O1^x—Si1—O2—Ba1ⁱ	−130.14 (6)
Na1ⁱⁱⁱ—Si1—O2—Ba1ⁱ	180.0	O1^ix—Si1—O2—Ba1	−130.14 (6)
Na1^iv—Si1—O2—Ba1^xxii	180.000 (1)	O1—Si1—O2—Ba1	−10.14 (6)
Na1ⁱⁱ—Si1—O2—Ba1	180.0	O1^x—Si1—O2—Ba1	109.86 (6)
Na1^iv—Si1—O2—Ba1ⁱ	60.0	O1—Si1—O2—Ba1^xxii	−130.14 (6)
Ba1—Si1—O1—Na1^iv	−91.06 (7)	O1^x—Si1—O2—Ba1^xxii	−10.14 (6)
Ba1—Si1—O1—Na1ⁱⁱⁱ	77.38 (5)	O1^ix—Si1—O2—Ba1^xxii	109.86 (6)
Ba1ⁱ—Si1—O1—Na1ⁱⁱⁱ	155.97 (4)	O1—Si1—O2—Ba1ⁱ	109.86 (6)
Ba1^xxii—Si1—O1—Na1^iv	−149.85 (6)	O2—Si1—O1—Na1^iv	−79.83 (6)
Ba1ⁱ—Si1—O1—Na1^iv	−12.46 (9)	O2—Si1—O1—Na1ⁱⁱⁱ	88.60 (4)
Ba1^xxii—Si1—O1—Na1ⁱⁱⁱ	18.58 (9)	O2—Si1—O1—Ba1	11.22 (6)
Ba1ⁱ—Si1—O1—Ba1	78.60 (6)	O2—Si1—O1—Hf1	136.5 (2)
Ba1^xxii—Si1—O1—Ba1	−58.80 (9)

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

Funding information

Funding for this research was provided by: JST, the Core Research for Evolutionary Science and Technology (grant No. JPMJCR19J2); Innovative Science and Technology Initiative for Security (grant No. JPJ004596).

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