Rb2Ca2Si2O7: a new alkali alkaline-earth silicate based on [Si2O7]6− anions

Kahlenberg, V.

doi:10.1107/S2053229625001196

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ISSN: 2053-2296

Volume 81| Part 3| March 2025|

https://doi.org/10.1107/S2053229625001196

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Rb₂Ca₂Si₂O₇: a new alkali alkaline-earth silicate based on [Si₂O₇]⁶⁻ anions

Volker Kahlenberg ^a ^*

^aInstitute of Mineralogy and Petrography, University of Innsbruck, Innrain 52, Innsbruck, Tyrol, A-6020, Austria
^*Correspondence e-mail: volker.kahlenberg@uibk.ac.at

Edited by A. Lemmerer, University of the Witwatersrand, South Africa (Received 7 January 2025; accepted 10 February 2025; online 17 February 2025)

Synthesis experiments were conducted in the ternary Rb₂O–CaO–SiO₂ system, resulting in the formation of a hitherto unknown compound with the composition Rb₂Ca₂Si₂O₇, i.e. dirubidium dicalcium pyrosilicate. Single crystals of sufficient size and quality were recovered from a starting mixture with an Rb₂O:CaO:SiO₂ molar ratio of 2:1:3. The educts were confined in a lid-covered platinum crucible and gradually cooled from 1050 °C at a rate of 0.3 °C min⁻¹ to 800 °C before being finally quenched in air to ambient conditions. The crystal structure was investigated at −80 and 15 °C from single-crystal X-ray diffraction data, with structure determination performed using direct methods. The compound was found to be of orthorhombic symmetry, belonging to the space group Pmmn (No. 59), with a = 5.7363 (6), b = 13.8532 (12), c = 9.9330 (10) Å, V = 789.34 (13) Å³ and Z = 4 (at 15 °C). The final refinement calculations at ambient temperature converged at R1 = 0.030 and wR2 = 0.076 for 773 observed reflections with I > 2σ(I). The silicate anion is based on pyrosilicate units of composition [Si₂O₇]⁶⁻ with point-group symmetry m (C_s). Charge compensation is achieved by the incorporation of rubidium and calcium cations distributed among a total of five independent sites within the asymmetric unit. Two of the nontetrahedrally coordinated cation sites (M4 and M5) are exclusively occupied by calcium cations, which are surrounded by six O atoms in the form of octahedra or trigonal prisms, respectively. The rubidium cations on the M1–M3 sites show more complex coordination environments. The M2 site, for example, is characterized by a tricapped trigonal prism polyhedron. Notably, the M3 site exhibits a 50% population of Ca²⁺ and Rb⁺, respectively. The compound shows closer structural resemblances with K₂Ca₂Si₂O₇ and can be derived from a hexagonal aristotype with space-group symmetry P6₃/mmc by displacements of the atoms. The corresponding distortion modes can be classified by certain irreducible representations of the high-symmetry parent phase. Structural investigations were completed by determining the thermal expansion tensor for the temperature interval between −80 and 15 °C.

Keywords: crystal structure; Rb₂Ca₂Si₂O₇; pyrosilicate; sorosilicate; rubidium; calcium.

CCDC references: 2422698; 2422697

1. Introduction

The ternary A₂O–CaO–SiO₂ systems, where A represents a chemical element in Group 1 of the Periodic Table have been the subject of many investigations in the past. However, the extent to which these systems were studied through synthesis experiments and/or thermodynamic modelling varies considerably depending on the specific alkali metal in question. This observation is directly correlated with the importance of a particular system for certain disciplines within the fields of inorganic chemistry and technical mineralogy. To date, the Na₂O–CaO–SiO₂ system has undoubtedly attracted the greatest attention (Morey & Bowen, 1925 ; Segnit, 1953 ; Williamson & Glasser, 1965 ; Shahid & Glasser, 1971 ; Zhang et al., 2011 ; Santoso et al., 2022 ). The corresponding melts have been of fundamental importance to the glass industry, facilitating the production of flat and hollow soda-lime silicate glass products that are ubiquitous in everyday life (Varshneya, 1994 ; Shelby, 2009 ). The crystalline counterparts cannot only occur as glass defects due to devitrification problems (Holland & Preston, 1938 ; Kahlenberg et al., 2010 ), but have also been studied as optical diffusers (Butt et al., 2014 ), constituents of bioactive ceramics (Reddy et al., 2014 ; Zandi Karimi et al., 2018 ) or as host materials for rare-earth-element-based silicate phosphors (Liu et al., 2014 ; Parauha et al., 2022 ). It is noteworthy that some sodium calcium silicates, such as combeite (Na₂Ca₂Si₃O₉), are also present in nature as exotic species in rather unusual petrological environments (Mitchell & Dawson, 2012 ; Weidendorfer et al., 2016 ; Kahlenberg, 2023 ).

Conversely, the K₂O–CaO–SiO₂ system has only recently experienced a resurgence of interest, because several of the ternary phases can occur in ashes from biomass combustion and gasification (Olanders & Steenari, 1995 ; Chen & Zhao, 2016 ; Santoso et al., 2020 ). During the last 15 years, eight potassium calcium silicates have been structurally characterized in detail, most of them for the first time. A recent summary of these phases, encompassing a vast array of connectivities of the [SiO₄] tetrahedra silicate anions and new structure types, can be found in the article by Liu et al. (2021 ).

According to West (1978 ), four thermodynamically stable ternary phases occur in the Li₂O–CaO–SiO₂ system. A major subject of interest for lithium calcium silicates is their applications in glass ceramics (Al-Harbi, 2007 ) or for the synthesis of phosphors after doping with rare earth elements (Kim et al., 2012 ; Wu et al., 2020 ).

Because the highly radioactive and extremely unstable alkali metal francium is not a suitable subject for phase analytical studies, only two other systems remain for consideration, Rb₂O–CaO–SiO₂ and Cs₂O–CaO–SiO₂, both of which have been relegated to the backwaters of silicate research. One potential explanation for this fact is that these ternary silicate phases are expected to be more sensitive to water and humidity in comparison to the corresponding compounds containing alkali elements with lower atomic numbers. Naturally, this aspect presents a clear disadvantage from an applicational perspective. For the rubidium-containing system, the existence of only one crystalline compound has been reported so far. Single crystals of Rb₂Ca₂Si₃O₉ were obtained using a polycrystalline precursor and an RbCl₂ flux (Kahlenberg et al., 2016 ). Its crystal structure is based on silicate anions forming sechser single chains.

In the course of a systematic investigation of the ternary Rb₂O–CaO–SiO₂ system, the crystal structure of a previously unknown compound, which was crystallized from the melt without the application of a mineralizer, is reported.

2. Experimental

2.1. Synthesis

The synthesis experiment for 1 g of a sample with an Rb₂O:CaO:SiO₂ molar ratio of 2:1:3 (or Rb₄CaSi₃O₉) was based on a stoichiometric mixture of the following educts: Rb₂CO₃ (Aldrich, 99.8%), CaCO₃ (calcite, Merck, >99.9%) and SiO₂ (quartz, AlfaAesar, 99.995%). Prior to weighing on an analytical balance, the reagents were dried at 400 °C for a period of 24 h. In addition to removing physically adsorbed water, this step is important because rubidium carbonate is known to be very hygroscopic. Homogenization was performed with an agate mortar and a pestle for a duration of 15 min in a glove-bag under argon. The mixture was immediately transferred to a 50 ml platinum crucible, which was covered with a platinum lid. The container was heated in a box furnace in air from room temperature to 1050 °C at a rate of 6 °C min⁻¹. The sample was annealed at the maximum temperature for 60 min and subsequently cooled to 800 °C at a rate of 0.3 °C min⁻¹ before final quenching to ambient conditions. Weight losses were determined from weight differences before and after heating. The observed difference was 0.6% higher than the predicted value (based on CO₂ release from the disintegration of the carbonates), indicating that losses due to Rb₂O evaporation were small. A preliminary visual inspection of the product following the removal of the lid indicated that the sample had melted. The crucible was then stored in an evacuated desiccator for further analysis.

2.2. Single-crystal diffraction

The solidified melt cake was mechanically separated from the crucible and further crushed in an agate mortar. Portions of the sample were immediately transferred to a glass slide into a drop of Paratone-N oil (Hampton Research) and investigated under a polarizing binocular, which revealed the existence of transparent colourless birefringent single crystals (showing sharp extinction between crossed polarizers) up to 350 µm in size. The majority of the crystals were found to be at least partially embedded in an optically isotropic glassy matrix.

Notably, fresh transparent crystals when exposed to air at 38% relative humidity and 21 °C (laboratory conditions) on a glass slide began to become slightly opaque after 3 d. After 6 d, the samples were completely opaque with a milky white colour, indicating an ongoing hydration reaction. However, the hydration product was not analysed further.

Several crystals displaying prismatic to plate-like morphology were isolated from the oil and affixed to glass fibers using fingernail hardener. They were subsequently studied using single-crystal diffraction performed on an Oxford Diffraction Gemini R Ultra diffractometer, which was equipped with a four-circle kappa goniometer and a Ruby CCD detector. Preliminary diffraction experiments were conducted with the objective of determining the unit-cell parameters. The screening process was performed in a dried air gas stream of −80 (2) °C generated by an Oxford Cryosystems Desktop Cooler to protect the samples from potential hydration. All crystals were found to belong to the same phase and exhibited an orthorhombic primitive metric that did not correspond to any entries of the Rb₂O–CaO–SiO₂ system or one of the relevant silicate subsystems currently available in the Inorganic Crystal Structure Database (Hellenbrandt, 2004 ). The sample with the best overall diffraction quality was selected for further structural analysis. A full sphere of reciprocal space up to 29.50° θ was obtained with Mo Kα radiation (see Table 1). The data were processed using the CrysAlis PRO software package (Rigaku OD, 2020 $[Rigaku OD (2020). CrysAlis PRO. Rigaku Oxford Diffraction Ltd, Yarnton, Oxfordshire, England.]$ ). Following indexing, the diffraction pattern was integrated. The data reduction process involved Lorentz and polarization corrections. Finally, the temperature was raised to 15 (2) °C and a second data collection was started using the identical run list employed for the low-temperature study, while keeping the crystal immersed in a dry environment (see Table 1). There was no evidence of a phase transition upon heating to ambient conditions. Once the correct chemical formula had been established on the basis of structure determination (see below), an analytical numeric absorption correction was applied to both data sets using a multifaceted crystal model.

Table 1
Experimental details

For both determinations: Rb₂Ca₂Si₂O₇, M_r = 419.28, orthorhombic, Pmmn, Z = 4. Experiments were carried out with Mo Kα radiation using a Rigaku Gemini R Ultra diffractometer equipped with a four-circle kappa goniometer and a Ruby CCD detector. The absorption correction was analytical [CrysAlis PRO (Rigaku OD, 2020 $[Rigaku OD (2020). CrysAlis PRO. Rigaku Oxford Diffraction Ltd, Yarnton, Oxfordshire, England.]$ ), based on expressions derived by Clark & Reid (1995 )]. Refinement was on 78 parameters.

	Data at 15 °C (RT)	Data at −80 °C (LT)
Crystal data
Temperature (K)	288	193
a, b, c (Å)	5.7363 (6), 13.8532 (12), 9.933 (1)	5.7281 (6), 13.8361 (13), 9.9233 (11)
V (Å³)	789.34 (13)	786.47 (14)
μ (mm⁻¹)	14	14.05
Crystal size (mm)	0.32 × 0.1 × 0.04	0.32 × 0.1 × 0.04

T_min, T_max	0.121, 0.678	0.109, 0.675
No. of measured, independent and observed [I > 2σ(I)] reflections	10834, 931, 773	10870, 929, 777
R_int	0.058	0.059
(sin θ/λ)_max (Å⁻¹)	0.625	0.625

Refinement
R[F² > 2σ(F²)], wR(F²), S	0.030, 0.076, 1.06	0.029, 0.067, 1.07
No. of reflections	931	929
Δρ_max, Δρ_min (e Å⁻³)	0.86, −0.70	0.93, −0.75
Computer programs: CrysAlis PRO (Rigaku OD, 2020 $[Rigaku OD (2020). CrysAlis PRO. Rigaku Oxford Diffraction Ltd, Yarnton, Oxfordshire, England.]$ ), SIR2002 (Burla et al., 2003) and SHELXL97 (Sheldrick, 2008).

The intensity statistics clearly indicated the presence of a centre of symmetry. Merging the two data sets in the orthorhombic Laue group 2/m 2/m 2/m resulted in reasonable internal R values (see Table 1). Based on the observed reflection conditions (hk0): h+k = 2n, only the space groups P2₁mn, Pm2₁n and Pmmn remained. The structure solution for the low-temperature set was successfully initiated in the centrosymmetric space group using direct methods (SIR2002; Burla et al., 2003 ), which provided a crystal-chemically sound starting model. One missing O atom was found from a difference Fourier map (SHEXL97; Sheldrick, 2008 ). The same software was also employed for subsequent full-matrix least-squares refinements. The scattering curves and anomalous dispersion coefficients were obtained from the International Tables for Crystallography (Vol. C; Prince, 2004 ). The final structure model obtained from the low-temperature data collection was then used as a starting point for the refinement of the structure under ambient conditions. The calculations with anisotropic displacement parameters for all atoms resulted in R1 residuals of 0.029 (at −80 °C) and 0.030 (at 15 °C) The largest shift/e.s.d. in the final cycles was < 0.001. Section 2.3 provides a detailed analysis of the site populations of the five nontetrahedrally coordinated cation sites in the asymmetric unit. The resulting chemical composition from the structure analysis was Rb₂Ca₂Si₂O₇. Table 2 lists the final coordinates, site occupancies and equivalent isotropic displacement parameters, while Table 3 provides the anisotropic displacement parameters. Table 4 summarizes the selected interatomic distances. Structural features were illustrated using the VESTA3 program (Momma & Izumi, 2011 ). Bond valence sum (BVS) calculations have been performed with the program VaList (Wills, 2010 ) using the parameter sets of Brown & Altermatt (1985 ) for Ca—O and Rb—O interactions, as well as Brese & O'Keeffe (1991 ) for the Si—O bonds. For the illustration of the three-dimensional representation surface of the thermal expansion tensor, the program WinTensor was employed (Kaminsky, 2014 ).

Table 2
Atomic coordinates (×10⁴, origin choice 2 of space group Pmmn) and equivalent isotropic displacement parameters (Å² × 10³) for Rb₂Ca₂Si₂O₇

First line 15 °C and second line −80 °C. U_eq is defined as one third of the trace of the orthogonalized U_ij tensor. M1 and M2 are exclusively occupied by rubidium, while M4 and M5 represent pure Ca sites. M3 is a mixed Rb–Ca position, with a population of 50% rubidium and 50% calcium.

Atom	Wyckoff site	Site symmetry	x	y	z	U_eq
M1	4e	m..	7500	1049 (1)	854 (1)	18 (1)
			7500	1048 (1)	853 (1)	14 (1)
M2	2a	mm2	2500	2500	2585 (1)	16 (1)
			2500	2500	2588 (1)	12 (1)
M3	4e	m..	2500	980 (1)	5869 (1)	21 (1)
			2500	980 (1)	5869 (1)	21 (1)
M4	2b	mm2	7500	2500	7494 (2)	10 (1)
			7500	2500	7492 (2)	8 (1)
M5	4e	m..	2500	5106 (1)	2489 (1)	10 (1)
			2500	5108 (1)	2489 (1)	9 (1)
Si1	4e	m..	7500	1362 (1)	4187 (2)	15 (1)
			7500	1362 (1)	4186 (2)	14 (1)
Si2	4e	m..	2500	1396 (1)	9193 (2)	9 (1)
			2500	1396 (1)	9192 (2)	8 (1)
O1	8g	1	181 (5)	1274 (2)	8280 (3)	23 (1)
			183 (5)	1274 (2)	8280 (3)	21 (1)
O2	4e	m..	2500	734 (3)	10515 (4)	19 (1)
			2500	732 (3)	10520 (4)	16 (1)
O3	2a	mm2	2500	2500	9820 (7)	26 (2)
			2500	2500	9825 (6)	22 (2)
O4	8g	1	5246 (7)	822 (3)	3605 (4)	44 (1)
			5242 (7)	821 (3)	3604 (4)	43 (1)
O5	2b	mm2	7500	2500	3599 (6)	14 (1)
			7500	2500	3598 (6)	12 (1)
O6	4e	m..	7500	1355 (3)	5789 (4)	35 (1)
			7500	1361 (3)	5793 (5)	32 (1)

Table 3
Anisotropic displacement parameters (Å² × 10³) for Rb₂Ca₂Si₂O₇

The anisotropic displacement factor exponent takes the form: −2π²[h²a*²U₁₁ + ⋯ + 2hka*b*U₁₂]. First line 15 °C and second line −80 °C.

	U₁₁	U₂₂	U₃₃	U₂₃	U₁₃	U₁₂
M1	14 (1)	26 (1)	13 (1)	0 (1)	0	0
	11 (1)	20 (1)	10 (1)	0 (1)	0	0
M2	18 (1)	14 (1)	16 (1)	0	0	0
	14 (1)	10 (1)	13 (1)	0	0	0
M3	18 (1)	41 (1)	13 (1)	−7 (1)	0	0
	16 (1)	35 (1)	11 (1)	−7 (1)	0	0
M4	13 (1)	7 (1)	11 (1)	0	0	0
	10 (1)	5 (1)	9 (1)	0	0	0
M5	13 (1)	7 (1)	10 (1)	0 (1)	0	0
	12 (1)	6 (1)	9 (1)	0 (1)	0	0
Si1	22 (1)	11 (1)	12 (1)	0 (1)	0	0
	21 (1)	10 (1)	11 (1)	0 (1)	0	0
Si2	9 (1)	7 (1)	12 (1)	2(1)	0	0
	7 (1)	6 (1)	11 (1)	2(1)	0	0
O1	17 (2)	21 (2)	32 (2)	7 (1)	−11 (1)	−7 (1)
	14 (2)	19 (2)	31 (2)	8(1)	−11 (1)	−7 (1)
O2	20 (2)	18 (2)	19 (2)	7(2)	0	0
	17 (2)	14 (2)	19 (2)	6(2)	0	0
O3	43 (4)	15 (3)	21 (3)	0	0	0
	40 (4)	10 (3)	15 (3)	0	0	0
O4	53 (3)	32 (2)	46 (2)	5 (2)	−25 (2)	−18 (2)
	50 (3)	28 (2)	50 (2)	5 (2)	−25 (2)	−16 (2)
O5	21 (3)	13 (3)	9 (3)	0	0	0
	20 (3)	10 (3)	6 (3)	0	0	0
O6	76 (4)	17 (2)	11 (2)	0(2)	0	0
	70 (4)	12 (2)	15 (2)	−1 (2)	0	0

Table 4
Selected bond lengths up to 3.2 Å and bond angles (°) for Rb₂Ca₂Si₂O₇

For the tetrahedra and octahedra, the distortion parameters QE (quadratic elongation) and AV (angle variance) have been calculated.

15 °C		−80 °C
M1—O2ⁱ	2.819 (4)	M1—O2ⁱ	2.815 (4)
M1—O2ⁱⁱ	2.9206 (9)	M1—O2ⁱⁱ	2.9160 (9)
M1—O2ⁱⁱⁱ	2.9206 (9)	M1—O2ⁱⁱⁱ	2.9160 (9)
M1—O1^iv	2.999 (3)	M1—O1^iv	2.997 (3)
M1—O1ⁱⁱⁱ	2.999 (3)	M1—O1ⁱⁱⁱ	2.997 (3)
M1—O4^v	3.039 (4)	M1—O4	3.037 (4)
M1—O4	3.039 (4)	M1—O4^v	3.037 (4)
<M1—O>	2.962	<M1—O>	2.959
M2—O3ⁱⁱ	2.746 (7)	M2—O3ⁱⁱ	2.742 (6)
M2—O4^vi	2.985 (4)	M2—O4^vi	2.980 (4)
M2—O4^vii	2.985 (4)	M2—O4^vii	2.980 (4)
M2—O4^viii	2.985 (4)	M2—O4	2.980 (4)
M2—O4	2.985 (4)	M2—O4^viii	2.980 (4)
M2—O5	3.0401 (19)	M2—O5	3.0343 (19)
M2—O5^ix	3.0401 (19)	M2—O5^ix	3.0343 (19)
M2—O2ⁱⁱ	3.196 (4)	M2—O2ⁱⁱ	3.193 (4)
M2—O2^x	3.196 (4)	M2—O2^x	3.193 (4)
<M2—O>	3.018	<M2—O>	3.013
M3—O4^viii	2.754 (5)	M3—O4^vii	2.752 (5)
M3—O4	2.754 (5)	M3—O4	2.752 (5)
M3—O1^viii	2.770 (4)	M3—O1^vii	2.765 (4)
M3—O1	2.770 (4)	M3—O1	2.765 (4)
M3—O4^xii	2.860 (4)	M3—O4^xii	2.855 (4)
M3—O4ⁱ	2.860 (4)	M3—O4ⁱ	2.855 (4)
M3—O6	2.9160 (9)	M3—O6	2.9131 (9)
M3—O6^ix	2.9160 (9)	M3—O6^ix	2.9131 (9)
<M3—O>	2.825	<M3—O>	2.821
M4—O6	2.320 (5)	M4—O6	2.308 (5)
M4—O6^xi	2.320 (5)	M4—O6^xi	2.308 (5)
M4—O1^xiii	2.421 (3)	M4—O1^xiii	2.419 (3)
M4—O1^vii	2.421 (3)	M4—O1^vii	2.419 (3)
M4—O1^viii	2.421 (3)	M4—O1^xiv	2.419 (3)
M4—O1^xiv	2.421 (3)	M4—O1^viii	2.419 (3)
<M4—O>	2.387	<M4—O>	2.382
M5—O2^x	2.279 (4)	M5—O2^x	2.274 (4)
M5—O4^vii	2.317 (4)	M5—O4^viii	2.312 (4)
M5—O4^vi	2.317 (4)	M5—O4^vi	2.312 (4)
M5—O1^xix	2.359 (3)	M5—O1^xix	2.355 (3)
M5—O1^xx	2.359 (3)	M5—O1^xx	2.355 (3)
M5—O6^xxi	2.435 (5)	M5—O6^xxi	2.431 (5)
<M5—O>	2.345	<M5—O>	2.340
QE = 1.020	AV = 69.94	QE = 1.020	AV = 70.19
Si1—O6	1.591 (5)	Si1—O6	1.595 (5)
Si1—O4	1.601 (4)	Si1—O4	1.602 (4)
Si1—O4^v	1.601 (4)	Si1—O4^v	1.602 (4)
Si1—O5	1.681 (2)	Si1—O5	1.679 (2)
<Si1—O>	1.618	<Si1—O>	1.620
QE = 1.001	AV = 2.51	QE = 1.001	AV = 2.48
Si2—O2	1.602 (4)	Si2—O2	1.606 (4)
Si2—O1	1.619 (3)	Si2—O1	1.615 (3)
Si2—O1^viii	1.619 (3)	Si2—O1^vii	1.615 (3)
Si2—O3	1.651 (3)	Si2—O3	1.652 (3)
<Si2—O>	1.622	<Si2—O>	1.622
QE = 1.004	AV = 16.51	QE = 1.004	AV = 17.44

O—Ca—O angles
O6—M4—O6^xi	86.2 (2)	O6—M4—O6^xi	86.2 (2)
O6—M4—O1^xiii	135.66 (10)	O6—M4—O1^xiii	135.64 (10)
O6^xi—M4—O1^xiii	75.89 (12)	O6^xi—M4—O1^xiii	75.93 (12)
O6—M4—O1^vii	135.66 (10)	O6—M4—O1^vii	75.93 (12)
O6^xi—M4—O1^vii	75.89 (12)	O6^xi—M4—O1^vii	135.64 (10)
O1^xiii—M4—O1^vii	78.88 (14)	O1^xiii—M4—O1^vii	142.31 (18)
O6—M4—O1^viii	75.89 (12)	O6—M4—O1^xiv	75.93 (12)
O6^xi—M4—O1^viii	135.66 (10)	O6^xi—M4—O1^xiv	135.64 (10)
O1^xiii—M4—O1^viii	142.33 (18)	O1^xiii—M4—O1^xiv	89.08 (14)
O1^vii—M4—O1^viii	89.11 (14)	O1^vii—M4—O1^xiv	78.90 (14)
O6—M4—O1^xiv	75.89 (12)	O6—M4—O1^viii	135.64 (10)
O6^xi—M4—O1^xiv	135.66 (10)	O6^xi—M4—O1^viii	75.93 (12)
O1^xiii—M4—O1^xiv	89.11 (14)	O1^xiii—M4—O1^viii	78.90 (14)
O1^vii—M4—O1^xiv	142.33 (18)	O1^vii—M4—O1^viii	89.08 (14)
O1^viii—M4—O1^xiv	78.88 (14)	O1^xiv—M4—O1^viii	142.31 (18)

O2^x—M5—O4^vii	97.40 (13)	O2^x—M5—O4^viii	97.30 (13)
O2^x—M5—O4^vi	97.40 (13)	O2^x—M5—O4^vi	97.30 (13)
O4^vii—M5—O4^vi	85.7 (2)	O4^viii—M5—O4^vi	85.6 (2)
O2^x—M5—O1^xix	94.14 (12)	O2^x—M5—O1^xix	94.14 (12)
O4^vii—M5—O1^xix	168.20 (13)	O4^viii—M5—O1^xix	168.32 (13)
O4^vi—M5—O1^xix	95.30 (14)	O4^vi—M5—O1^xix	95.29 (14)
O2^x—M5—O1^xx	94.14 (12)	O2^x—M5—O1^xx	94.14 (12)
O4^vii—M5—O1^xx	95.30 (14)	O4^viii—M5—O1^xx	95.29 (14)
O4^vi—M5—O1^xx	168.20 (13)	O4^vi—M5—O1^xx	168.32 (13)
O1^xix—M5—O1^xx	81.37 (16)	O1^xix—M5—O1^xx	81.50 (16)
O2^x—M5—O6^xxi	165.38 (16)	O2^x—M5—O6^xxi	165.29 (16)
O4^vii—M5—O6^xxi	93.30 (13)	O4^viii—M5—O6^xxi	93.48 (13)
O4^vi—M5—O6^xxi	93.30 (13)	O4^vi—M5—O6^xxi	93.48 (13)
O1^xix—M5—O6^xxi	74.91 (11)	O1^xix—M5—O6^xxi	74.85 (11)
O1^xx—M5—O6^xxi	74.91 (11)	O1^xx—M5—O6^xxi	74.85 (11)

O—Si—O angles
O6—Si1—O4	110.99 (19)	O6—Si1—O4	111.10 (19)
O6—Si1—O4^v	110.99 (19)	O6—Si1—O4^v	111.10 (19)
O4—Si1—O4^v	107.7 (3)	O4—Si1—O4^v	107.6 (3)
O6—Si1—O5	110.6 (3)	O6—Si1—O5	110.4 (3)
O4—Si1—O5	108.21 (18)	O4—Si1—O5	108.26 (18)
O4^v—Si1—O5	108.21 (18)	O4^v—Si1—O5	108.26 (18)
<O—Si1—O>	109.45	<O—Si1—O>	109.45

O2—Si2—O1	113.52 (14)	O2—Si2—O1	113.58 (14)
O2—Si2—O1^viii	113.52 (14)	O2—Si2—O1^vii	113.58 (14)
O1—Si2—O1^viii	110.5 (3)	O1—Si2—O1^vii	110.5 (3)
O2—Si2—O3	102.8 (3)	O2—Si2—O3	102.5 (3)
O1—Si2—O3	107.96 (17)	O1—Si2—O3	108.05 (16)
O1^viii—Si2—O3	107.96 (17)	O1^vii—Si2—O3	108.05 (17)
<O—Si2—O>	109.38	<O—Si2—O>	109.38

Si—O—Si angles
Si2^vii—O3—Si2	135.7 (4)	Si2—O3—Si2^viii	135.3 (4)
Si1^xi—O5—Si1	139.4 (4)	Si1^xi—O5—Si1	139.3 (4)
Symmetry codes: (i) −x + 1, −y, −z + 1; (ii) x, y, z − 1; (iii) x + 1, y, z − 1; (iv) −x + $[{1\over 2}]$ , y, z − 1; (v) −x + $[{3\over 2}]$ , y, z; (vi) x, −y + $[{1\over 2}]$ , z; (vii) −x + $[{1\over 2}]$ , −y + $[{1\over 2}]$ , z; (viii) −x + $[{1\over 2}]$ , y, z; (ix) x − 1, y, z; (x) −x + $[{1\over 2}]$ , −y + $[{1\over 2}]$ , z − 1; (xi) −x + $[{3\over 2}]$ , −y + $[{1\over 2}]$ , z; (xii) x − $[{1\over 2}]$ , −y, −z + 1; (xiii) x + 1, −y + $[{1\over 2}]$ , z; (xiv) x + 1, y, z; (xix) x + $[{1\over 2}]$ , y + $[{1\over 2}]$ , −z + 1; (xx) −x, y + $[{1\over 2}]$ , −z + 1; (xxi) x − $[{1\over 2}]$ , y + $[{1\over 2}]$ , −z + 1.

2.3. Crystal structure

The crystal structure is based on [Si₂O₇]⁶⁻ anions and can be classified as a sorosilicate (Liebau, 1985 ). The unit cell contains a total of two symmetrically independent bitetrahedral units. Both silicate anions exhibit point-group symmetry m or C_s (see Fig. 1). Charge compensation within the structure is achieved by monovalent rubidium and divalent calcium cations, which are distributed among a total of five different positions (M1–M5). Bond distance considerations indicated that M1 and M2 are pure rubidium sites, while M4 and M5 are occupied by calcium ions. Indeed, the results of the site-population refinements pointed to full occupancy with the respective two cation species. Conversely, M3 was identified as a mixed cation position, with a population of 51 (2)% Rb and 49 (2)% Ca. Therefore, we finally assumed a 1:1 ratio of rubidium and calcium on the M3 site, leading to a charge-neutral chemical composition of Rb₂Ca₂Si₂O₇. Taking into account the initial composition of the starting material, it can be concluded that the glass phase is enriched in Rb₂O compared to the crystalline samples.

Figure 1
Side view of a single [Si₂O₇]⁶⁻ unit. Displacement ellipsoids are shown at the 80% probability level. Colour key: O atoms are red and Si atoms are blue.

It is important to note that all geometrical parameters presented in this paragraph have been derived from the refinement of the data set collected at 15 °C. The Si—O bond distances within the two silicate dimers cover a considerable range, spanning from 1.591 (5) to 1.681 (2) Å. However, this variation aligns with the anticipated trends for [Si₂O₇]⁶⁻ groups comprising one bridging and three terminal O atoms. The distances between the Si atoms and the terminal O atoms are notably shorter (average of both dimers = 1.599 Å) than the corresponding bond lengths to the bridging O atoms (average of the dimers = 1.680 Å). The observed shortening of the mean Si—O(terminal) bond distance by 0.081 Å can be attributed to the stronger attraction between the O and Si atoms than between the O atoms and the rubidium/calcium cations present in the structure. The distortion of the tetrahedra is also reflected in the O—Si—O angles, which range from 102.8 (3) to 113.52 (14)°, respectively. Nevertheless, the mean O—Si—O angles are in close proximity to the ideal values for an ideal tetrahedron (see Table 4). The degree of tetrahedral distortion can be quantified using the following two parameters: quadratic elongation (QE) and angle variance (AV) (Robinson et al., 1971 ). The numerical values for these parameters are also provided in Table 4. In fact, the distortions of both the crystallographically independent [Si₂O₇]⁶⁻ units are relatively minor, with the tetrahedra around Si1 showing the least strain. Moreover, due to the point-group symmetry m, the silicate dimers display an eclipsed conformation. The Si—O—Si bond angles deviate from linearity, exhibiting significantly smaller values of 135.7 (4) and 139.4 (4)°. In particular, the second value is close to 140°, which is postulated to correspond to an unstrained Si—O—Si angle (Liebau, 1985).

The Ca positions (M4 and M5) are coordinated by six oxygen ligands that form distorted octahedra (around M5) and trigonal prisms (around M4). Each trigonal prism shares opposing faces with two adjacent octahedra (see Fig. 2). The resulting tripolyhedral cluster has point-group symmetry m. Neighbouring clusters are linked by rubidium cations located at the M2 position. They are coordinated by nine O atoms, forming a highly elongated tricapped trigonal prism. Again, linkage is provided by shared faces, but this time between M5O₆ and M2O₉ polyhedra. Consequently, linear rod-like building blocks are obtained, that run parallel to the b axis (see Fig. 3). The [Si₂O₇]⁶⁻ groups serve to connect adjacent rods. Each dimer shares common oxygen anions with (i) a single trigonal prism around M4 and (ii) several surrounding M2O₉ groups. Finally, the Rb and/or Ca atoms on the M1 and M3 positions complete the structure, occupying cavities above and below the silicate dimers. They are bonded to seven and eight O atoms, respectively, forming more irregular coordination polyhedra. A projection of the whole structure parallel to [100] is given in Fig. 4.

Figure 2
Side view of a single trimer containing two octahedra and a central trigonal prism. Displacement ellipsoids are drawn at the 80% probability level. Colour key: O atoms are red and Ca atoms are orange.

Figure 3
Side view of a single rod-like building element. Rb atoms occupying M2 are shown in green and are coordinated by nine O atoms in the form of a tricapped trigonal prism. O atoms are presented in red.

Figure 4
Projection of the whole crystal structure of Rb₂Ca₂Si₂O₇ along [100]. Potassium and calcium cations are illustrated in green and orange, respectively. O atoms are shown in red. The sizes of the two-coloured segments of the M3 site are the percentages determined from the site-occupancy refinements.

With the exception of the M3 site, the BVS calculations for the various cation positions yielded values that were close to the formal charges of the ions, assuming full occupancy with a single cation species: M1 1.129, M2 1.243, M4 1.942, M5 2.182, Si1 4.078 and Si2 4.019 (all data in valence units, v.u.). In the case of M3, however, a pronounced overbonding (1.778 v.u. for Rb) or underbonding (0.800 v.u. for Ca) was observed. This outcome provides further evidence for a mixed Rb/Ca occupancy. Notably, BVS calculations can permit an independent, though typically rather rough estimation, of the contents of two distinct atom types sharing the same position (Brown, 2016 ). The concentrations obtained using the corresponding bond-valence parameters in combination with the M3—O bond distances determined at 15 °C are as follows: 61% Rb and 39% Ca. This result is deemed to be in sufficiently good agreement with the percentages determined from the site-population refinements.

2.4. Thermal expansion

The two sets of lattice parameters determined at −80 and 15 °C were employed to calculate the average thermal expansion tensor α_ij for the specified temperature interval from the thermal strain tensor ɛ_ij and the relationship $[ \alpha_{ij} = {{\varepsilon_{ij}}\over{\Delta T}}]$ . Due to the orthorhombic symmetry restrictions, the off-diagonal terms of the symmetric second-rank tensor ɛ_ij with i ≠ j must be strictly zero. The remaining three components can be obtained from the following expressions: ɛ₁₁ = $[ {{a}\over{a_0}} - 1]$ , ɛ₂₂ = $[ {{b}\over{b_0}} - 1]$ and ɛ₃₃ = $[ {{c}\over{c_0}} - 1]$ . Notably, the lattice parameters with the suffix `zero' pertain to the low-temperature data. In consequence, the thermal expansion tensor has the following form:

$[ \alpha_{ij} = \left( \matrix{ {15 \ (1)} & {0} & {0} \cr {0} & {13 \ (1)} & {0} \cr {0} & {0} & {10\ (1)} \cr } \right) \times 10^{-6}]$

From the comparison of the numerical values it is obvious that the thermal expansion is not extremely anisotropic. The largest (α₁₁) and the smallest (α₃₃) value differ by only a factor of 1.5. The expansion along [010], that is, along the rod-like building blocks of the crystal structure, is observed to have an intermediate value which is equivalent to the average of α₁₁ and α₃₃ within one standard deviation. By plotting the values of the thermal expansion tensor as a function of all directions one obtains a convenient geometric representation of the anisotropic behaviour of the tensor in the form of a surface in three-dimensional space (Fig. 5). The corresponding two-dimensional sections (a–b, b–c and a–c) are presented in Fig. S1 of the supporting information.

Figure 5
Three-dimensional representation surface of the average thermal expansion tensor of Rb₂Ca₂Si₂O₇ in the temperature interval between −80 and 15 °C.

3. Discussion

It is somewhat unexpected to find that the M3 site is occupied by both rubidium and calcium. Indeed, the two cations differ considerably concerning their ionic radii: r(Rb^+,[8]) = 1.61 Å and r(Ca^2+,[8]) = 1.12 Å (Shannon, 1976 ). In the only other structurally characterized rubidium calcium silicate, Rb₂Ca₂Si₃O₉, the two nontetrahedrally coordinated cation species are well ordered (Kahlenberg et al., 2016). To exclude the possibility that the observation of a mixed population is an artifact due to an incorrect unit cell and/or symmetry, several additional tests were performed.

First, the frequency distributions of the experimentally determined Bragg peak positions when projected onto the a, b and c axes were computed and the corresponding maxima along these lines were visualized. Secondly, precession-type reconstructions of reciprocal space were calculated for the zero, first and second layers of reciprocal space for each of the three symmetry directions of the orthorhombic crystal system. Neither method yielded any indication of additional reflections requiring a larger cell. In other words, it can be excluded that our model corresponds to an average structure.

Finally, it was tested whether the unit cell was correct, but the structure was described in a symmetry that was too high. Indeed, a reduction in symmetry could allow for the possibility of cation ordering among the four symmetry-equivalent positions belonging to the Wyckoff position 4e of the M3 site. In the light of the aforementioned observed systematic absences, which clearly indicated the presence of an n-glide plane perpendicular to [001], the following translationengleiche subgroups of Pmmn were considered for a potential symmetry reduction: P2₁mn, P112/n, Pm2₁n and P11n. Notably, only a description in one of the latter two space groups involves a Wyckoff splitting of the 4e position, which is a prerequisite for cation ordering. Therefore, the model in Pmmn was transformed for each of the two relevant acentric subgroups and the refinement calculations were repeated. In both instances, instabilities of the refinements were recognized, which can be attributed to the presence of significant correlations between the coordinates and displacement parameters of those atom pairs, which were previously coupled by the centres of inversion present in Pmmn. Consequently, the refinements were restarted using adapted models in the low-symmetry space groups, wherein the atomic coordinates of all atoms except those of the former M3 site were constrained manually to conform to centrosymmetric structures. Despite the expected successful resolution of the correlation issue, the Rb and Ca ions demonstrated no tendency to order among the new sets of sites obtained from the Wyckoff splitting of the former M3 position. In conclusion, we found no evidence to suggest that the distribution of the rubidium and calcium on M3 is not statistically random.

The c/a ratio of the orthorhombic lattice parameters had a value close to $[\sqrt{3}]$ , which is characteristic of an orthohexagonal cell. Although there is no doubt that the actual symmetry of Rb₂Ca₂Si₂O₇ is only orthorhombic, this observation prompted us to check for potential pseudosymmetry using the program PSEUDO, implemented in the Bilbao Crystallographic Server (Capillas et al., 2011 ). Relative coordinates of all atoms were used, without distingushing between the various cation species present on the M sites. The search involving minimal supergroups was successful and indicated that the structure can be derived from an aristotype or the parent structure in G = P6₃/mmc with a′ = 5.7352 and c′ = 13.8532 Å, provided that an intermediate step to Z = Cmcm is introduced and atomic shifts up to 1 Å are permitted. The 4 × 4 transformation matrix leading directly from parent structure to the structure in H = Pmmn is as follows:

$[ \left( \matrix{ {1} & {\ \ 0} & {-1} & {0} \cr {1} & {\ \ 0} & {\ \ 1} & {-0.5} \cr {0} & {-1} & {\ \ 0} & {0} \cr {0} & {\ \ 0} & {\ \ 0} & {1} \cr } \right)]$

In the hexagonal aristotype, the sites M1 and M3, as well as M2 and M4, are symmetrically equivalent (Wyckoff positions 4f and 2b of P6₃/mmc, respectively). M5 (in 2a) corresponds to a third nontetrahedrally coordinated cation position. Moreover, the asymmetric unit of the P6₃/mmc structure contains a single Si atom (in 4f) and two O atoms (in 12k and 2d), that is, all [Si₂O₇] dimers in the unit cell are coupled by symmetry. Notably, the aristotype is isostructural with K₃ScSi₂O₇ (Napper et al., 2004 ). In this pyrosilicate, the Sc atoms occupy the octahedrally coordinated positions, while the potassium ions are located in the centres of the trigonal prisms.

With respect to the atomic coordinates, the symmetry break from the parent structure can be explained with the onset of several distortion modes, whose symmetry properties are given by the irreducible representations (irreps) of the space group P6₃/mmc of the aristotype. The corresponding mode analysis was performed using the program AMPLIMODES (Orobengoa et al., 2009 ). The observed symmetry reduction requires the irrep M₄⁻ associated with the point M ( $[1 \over 2]$ 0 0) of the first Brillouin zone, as well as the zone-center irrep Γ₅⁺. Furthermore, the fully symmetrical Γ₁⁺ distortion is also allowed that retains the symmetry of the aristotype.

Using the crystallographic data, a reference structure was calculated in a first step representing the aristotype, but expressed in the low-symmetry space group Pmmn. From the comparison of the reference structure with the actual model in Pmmn the resulting displacement field can be obtained, which is defined by the individual displacement vectors u for the atoms in the asymmetric unit of the reference structure. It defines the displacive distortions relating both structures. A detailed analysis of the shifts revealed that the O atoms are distinctly more affected. Their absolute values for the shifts vary between 0.345 and 0.649 Å. The Si atoms and most of the M sites show considerably smaller displacements. An exception, however, is the M5 position, which is displaced by about 0.147 Å. The average shift of all corresponding atom pairs in both structures has a value of 0.249 Å.

In the next step of mode analysis, the absolute amplitudes for the three individual components of the global distortions were calculated. Notably, the amplitudes were normalized with respect to the primitive unit cell of the high-symmetry structure. The relevant values are 1.61 (8) (for M₄⁻), 0.147 (6) (for Γ₅⁺) and 0.02 (8) Å (for Γ₁⁺), indicating that the onset of the M₄⁻ distortion triggers the symmetry break. Finally, for each involved irrep the corresponding polarization vector was obtained. The actual distortion of a specific mode can then be obtained by multiplying the components of the polarization vector with the amplitudes mentioned above. In order to obtain a concise graphical overview of the distortion fields, individual displacements of the most affected atoms O1 to O5 belonging to the [Si₂O₇]⁶⁻ groups calculated for the dominant M₄⁻ representation only have been visualized using the program VESTA3 (see Fig. S2 in the supporting information).

To date, four alkali alkaline-earth silicates with the general formula A₂Ca₂Si₂O₇ have been the subject of structural investigations. With the exception of Na₂Ca₂Si₂O₇, which is a mixed anion silicate containing insular [SiO₄] tetrahedra and [Si₃O₁₀] trimers in a 1:1 ratio (Kahlenberg & Hösch 2002 ), the corresponding lithium, potassium and rubidium compounds are characterized by [Si₂O₇]⁶⁻ units. Li₂Ca₂Si₂O₇ represents a unique structure type (Kahlenberg et al., 2015 ) and can be rationalized as a three-dimensional framework structure based on corner-sharing [LiO₄] and [SiO₄] tetrahedra, including an O^[3]-type bridging oxygen node linking three adjacent tetrahedra. While the lithium calcium disilicate and the compound under investigation exhibit no closer structural relationship, K₂Ca₂Si₂O₇ (space group P6₃/m) and Rb₂Ca₂Si₂O₇ can be derived from the condensation of the same type of rod-like elements that have been introduced in Section 2.3. In more detail, both compounds can be considered as derivative structures of the same aristotype. However, the distortion patterns resulting in the hexagonal potassium and the orthorhombic rubidium phase are different. Fig. 6 illustrates the differences between the parent structure, the two above-mentioned less-symmetric alkali alkaline-earth silicates, as well as related Na₃ScSi₂O₇ (Skrzat et al., 1969 ), by showing a 6.5 Å wide slab for each structure that contains a sequence of four consecutive rods linked by [Si₂O₇]⁶⁻ units. For the sake of clarity, the structures have been simplified slightly. In fact, some of the coordination polyhedra are shown as distorted trigonal prisms, although in some cases there are additional O atoms capping two or three of the prismatic faces that actually do belong to the coordination sphere.

Figure 6
Structural differences within a 6.5 Å wide slab containing a sequence of four rods between (a) the aristotype, view along [120] (P6₃/mmc, a = 5.6065 and c = 13.6420 Å), (b) K₂Ca₂Si₂O₇, view along [110] (P6₃/m, a = 9.8020 and c = 13.8781 Å), (c) Rb₂Ca₂Si₂O₇, view along [001] (Pmmn, a = 5.7363, b = 13.8532 and c = 9.9330 Å), and (d) Na₃ScSi₂O₇, view along [1 $[\overline 1]$ 0] (Pbnm, a = 5.3540, b = 9.3470 and c = 13.0890 Å). Different colours simply highlight the various coordination polyhedra, but do not provide any information about their occupation with different cation species.

4. Conclusion

The current crystal structure is an interesting example of the substitution of two cation species with very different ionic radii. For several potassium–calcium silicates, including K₂Ca₂Si₂O₇, the replacement of calcium by the substantially larger potassium ion has been reported previously. However, the evidence in these cases was based solely on bond-valence calculations, as the two cation types are isoelectronic with 19 electrons each and thus cannot be discriminated from each other using direct site-population refinements by X-ray diffraction data. Conversely, in the case of Rb₂Ca₂Si₂O₇, the ion types differ significantly from each other in terms of the number of electrons, thereby rendering diffraction methods a viable additional evidence for a substitution of Rb with Ca. It is noteworthy that the discrepancy between the respective values of the radii for eight-coordinated potassium and rubidium is a mere 6% (Shannon, 1976).

As mentioned in the Introduction, the crystalline compounds of the rubidium–calcium and caesium–calcium silicate groups have been studied only rudimentarily, if at all. This opens up new possibilities for systematic crystal chemical investigations of the influence of the size of the alkali cations on the formation of certain structure types. As Liebau (1985) demonstrated in his seminal book on oxosilicates, correlations between the radii of the nontetrahedrally coordinated cations and various structural aspects, including, among others, ring sizes or chain periodicities in phyllo- and inosilicates, could be deciphered. However, this requires that the data set under consideration contains a sufficiently large number of representatives. Of particular interest, of course, are compounds with equal proportions of the various monovalent, divalent and tetravalent cation types in the formula unit, such as the family of A₂Ca₂Si₂O₇ compounds. In order to obtain a comprehensive crystallographic understanding of the compounds belonging to this general composition, it is necessary to ascertain the structure of the missing Cs phase. This objective is currently being pursued through synthesis experiments.

The present work is a first contribution to a project that will investigate structural relationships systematically and at the same time provide new information on phase equilibria in the Rb₂O–CaO–SiO₂ and Cs₂O–CaO–SiO₂ systems.

Supporting information

CCDC references: 2422698; 2422697

Crystal structure: contains datablocks RT, LT, global. DOI: https://doi.org/10.1107/S2053229625001196/ef3065sup1.cif

Structure factors: contains datablock RT. DOI: https://doi.org/10.1107/S2053229625001196/ef3065RTsup2.hkl

Structure factors: contains datablock LT. DOI: https://doi.org/10.1107/S2053229625001196/ef3065LTsup3.hkl

Supporting information file. DOI: https://doi.org/10.1107/S2053229625001196/ef3065sup4.pdf

Computing details top

Dirubidium dicalcium pyrosilicate (RT) top

Crystal data top

Rb₂Ca₂Si₂O₇	D_x = 3.528 Mg m⁻³
M_r = 419.28	Mo Kα radiation, λ = 0.71073 Å
Orthorhombic, Pmmn	Cell parameters from 2256 reflections
a = 5.7363 (6) Å	θ = 5.0–29.5°
b = 13.8532 (12) Å	µ = 14 mm⁻¹
c = 9.933 (1) Å	T = 288 K
V = 789.34 (13) Å³	Thin plate, colourless
Z = 4	0.32 × 0.1 × 0.04 mm
F(000) = 792

Data collection top

Xcalibur, Ruby, Gemini ultra diffractometer	931 independent reflections
Radiation source: fine-focus sealed X-ray tube, Enhance (Mo) X-ray Source	773 reflections with I > 2σ(I)
Graphite monochromator	R_int = 0.058
Detector resolution: 10.3575 pixels mm^-1	θ_max = 26.4°, θ_min = 3.6°
ω scans	h = −7→7
Absorption correction: analytical [CrysAlis PRO (Rigaku OD, 2020), based on expressions derived by Clark & Reid (1995)]	k = −17→17
T_min = 0.121, T_max = 0.678	l = −12→12
10834 measured reflections

Refinement top

Refinement on F²	Primary atom site location: structure-invariant direct methods
Least-squares matrix: full	Secondary atom site location: difference Fourier map
R[F² > 2σ(F²)] = 0.030	w = 1/[σ²(F_o²) + (0.0295P)² + 3.6732P] where P = (F_o² + 2F_c²)/3
wR(F²) = 0.076	(Δ/σ)_max < 0.001
S = 1.06	Δρ_max = 0.86 e Å⁻³
931 reflections	Δρ_min = −0.70 e Å⁻³
78 parameters	Extinction correction: SHELXL97 (Sheldrick, 2008), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
0 restraints	Extinction coefficient: 0.0100 (5)

Special details top

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

Refinement. Refinement of F² against ALL reflections. The weighted R-factor wR and goodness of fit S are based on F², conventional R-factors R are based on F, with F set to zero for negative F². The threshold expression of F² > 2σ(F²) is used only for calculating R-factors(gt) etc. and is not relevant to the choice of reflections for refinement. R-factors based on F² are statistically about twice as large as those based on F, and R- factors based on ALL data will be even larger.

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

	x	y	z	U_iso*/U_eq	Occ. (<1)
Rb1	0.75	0.10486 (5)	0.08538 (6)	0.0178 (2)
Rb2	0.25	0.25	0.25846 (8)	0.0159 (2)
Rb3	0.25	0.09802 (7)	0.58685 (8)	0.0243 (3)	0.5
Ca3	0.25	0.09802 (7)	0.58685 (8)	0.0243 (3)	0.5
Ca4	0.75	0.25	0.74935 (15)	0.0101 (4)
Ca5	0.25	0.51058 (8)	0.24873 (11)	0.0102 (3)
Si1	0.75	0.13621 (12)	0.41868 (17)	0.0149 (4)
Si2	0.25	0.13963 (11)	0.91931 (16)	0.0093 (3)
O1	0.0181 (5)	0.1274 (2)	0.8280 (3)	0.0233 (8)
O2	0.25	0.0734 (3)	1.0515 (4)	0.0188 (10)
O3	0.25	0.25	0.9820 (7)	0.0262 (15)
O4	0.5246 (7)	0.0822 (3)	0.3605 (4)	0.0439 (11)
O5	0.75	0.25	0.3599 (6)	0.0143 (12)
O6	0.75	0.1355 (3)	0.5789 (4)	0.0347 (14)

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Rb1	0.0143 (3)	0.0261 (4)	0.0128 (3)	0	0	0.0002 (2)
Rb2	0.0178 (5)	0.0137 (4)	0.0162 (4)	0	0	0
Rb3	0.0181 (4)	0.0413 (6)	0.0134 (4)	0	0	−0.0074 (4)
Ca3	0.0181 (4)	0.0413 (6)	0.0134 (4)	0	0	−0.0074 (4)
Ca4	0.0125 (8)	0.0069 (7)	0.0109 (8)	0	0	0
Ca5	0.0133 (6)	0.0073 (5)	0.0099 (6)	0	0	−0.0001 (4)
Si1	0.0224 (9)	0.0107 (8)	0.0116 (8)	0	0	−0.0001 (6)
Si2	0.0085 (7)	0.0075 (7)	0.0118 (8)	0	0	0.0024 (6)
O1	0.0171 (16)	0.0207 (16)	0.0320 (19)	−0.0073 (13)	−0.0110 (14)	0.0074 (13)
O2	0.020 (2)	0.018 (2)	0.019 (2)	0	0	0.0074 (18)
O3	0.043 (4)	0.015 (3)	0.021 (3)	0	0	0
O4	0.053 (3)	0.032 (2)	0.046 (2)	−0.0184 (19)	−0.025 (2)	0.0047 (17)
O5	0.021 (3)	0.013 (3)	0.009 (3)	0	0	0
O6	0.076 (4)	0.017 (2)	0.011 (2)	0	0	0.0002 (18)

Geometric parameters (Å, º) top

Rb1—O2ⁱ	2.819 (4)	Si1—O4	1.601 (4)
Rb1—O2ⁱⁱ	2.9206 (9)	Si1—O4^v	1.601 (4)
Rb1—O2ⁱⁱⁱ	2.9206 (9)	Si1—O5	1.681 (2)
Rb1—O1^iv	2.999 (3)	Si2—O2	1.602 (4)
Rb1—O1ⁱⁱⁱ	2.999 (3)	Si2—O1	1.619 (3)
Rb1—O4^v	3.039 (4)	Si2—O1^viii	1.619 (3)
Rb1—O4	3.039 (4)	Si2—O3	1.651 (3)
Rb1—O5	3.388 (4)	O1—Si2	1.619 (3)
Rb2—O3ⁱⁱ	2.746 (7)	O1—Ca5^xviii	2.359 (3)
Rb2—O4^vi	2.985 (4)	O1—Ca4^ix	2.421 (3)
Rb2—O4^vii	2.985 (4)	O1—Rb3	2.770 (4)
Rb2—O4^viii	2.985 (4)	O1—Rb1^xix	2.999 (3)
Rb2—O4	2.985 (4)	O2—Si2	1.602 (4)
Rb2—O5	3.0401 (19)	O2—Ca5^xx	2.279 (4)
Rb2—O5^ix	3.0401 (19)	O2—Rb1ⁱ	2.819 (4)
Rb2—O2ⁱⁱ	3.196 (4)	O2—Rb1^xxi	2.9206 (9)
Rb2—O2^x	3.196 (4)	O2—Rb1^xix	2.9206 (9)
Rb3—O4^viii	2.754 (5)	O2—Rb2^xxi	3.196 (4)
Rb3—O4	2.754 (5)	O3—Si2^vii	1.651 (3)
Rb3—O1^viii	2.770 (4)	O3—Si2	1.651 (3)
Rb3—O1	2.770 (4)	O3—Rb2^xxi	2.746 (7)
Rb3—O4^xi	2.860 (4)	O4—Si1	1.601 (4)
Rb3—O4ⁱ	2.860 (4)	O4—Ca5^vii	2.317 (4)
Rb3—O6	2.9160 (9)	O4—Rb3	2.754 (5)
Rb3—O6^ix	2.9160 (9)	O4—Rb3ⁱ	2.860 (4)
Ca4—O6	2.320 (5)	O4—Rb2	2.985 (4)
Ca4—O6^xii	2.320 (5)	O4—Rb1	3.039 (4)
Ca4—O1^xiii	2.421 (3)	O5—Si1^xii	1.681 (2)
Ca4—O1^vii	2.421 (3)	O5—Si1	1.681 (2)
Ca4—O1^viii	2.421 (3)	O5—Rb2^xiv	3.0401 (19)
Ca4—O1^xiv	2.421 (3)	O5—Rb2	3.0401 (19)
Ca5—O2^x	2.279 (4)	O5—Rb1^xii	3.388 (4)
Ca5—O4^vii	2.317 (4)	O5—Rb1	3.388 (4)
Ca5—O4^vi	2.317 (4)	O6—Si1	1.591 (4)
Ca5—O1^xv	2.359 (3)	O6—Ca4	2.320 (5)
Ca5—O1^xvi	2.359 (3)	O6—Ca5^xxii	2.435 (5)
Ca5—O6^xvii	2.435 (5)	O6—Rb3^xiv	2.9160 (9)
Si1—O6	1.591 (5)	O6—Rb3	2.9160 (9)

O2ⁱ—Rb1—O2ⁱⁱ	79.25 (8)	O1^viii—Rb3—O4^xi	100.79 (11)
O2ⁱ—Rb1—O2ⁱⁱⁱ	79.25 (8)	O1—Rb3—O4^xi	75.71 (10)
O2ⁱⁱ—Rb1—O2ⁱⁱⁱ	158.25 (17)	O4^viii—Rb3—O4ⁱ	109.79 (8)
O2ⁱ—Rb1—O1^iv	71.35 (9)	O4—Rb3—O4ⁱ	79.71 (12)
O2ⁱⁱ—Rb1—O1^iv	54.11 (10)	O1^viii—Rb3—O4ⁱ	75.71 (10)
O2ⁱⁱⁱ—Rb1—O1^iv	114.88 (10)	O1—Rb3—O4ⁱ	100.79 (11)
O2ⁱ—Rb1—O1ⁱⁱⁱ	71.35 (9)	O4^xi—Rb3—O4ⁱ	53.74 (16)
O2ⁱⁱ—Rb1—O1ⁱⁱⁱ	114.88 (10)	O4^viii—Rb3—O6	123.68 (12)
O2ⁱⁱⁱ—Rb1—O1ⁱⁱⁱ	54.11 (10)	O4—Rb3—O6	55.20 (12)
O1^iv—Rb1—O1ⁱⁱⁱ	61.69 (11)	O1^viii—Rb3—O6	61.64 (11)
O2ⁱ—Rb1—O4^v	110.08 (10)	O1—Rb3—O6	118.01 (11)
O2ⁱⁱ—Rb1—O4^v	120.40 (11)	O4^xi—Rb3—O6	127.25 (12)
O2ⁱⁱⁱ—Rb1—O4^v	70.77 (11)	O4ⁱ—Rb3—O6	73.50 (12)
O1^iv—Rb1—O4^v	174.31 (9)	O4^viii—Rb3—O6^ix	55.20 (12)
O1ⁱⁱⁱ—Rb1—O4^v	123.98 (9)	O4—Rb3—O6^ix	123.68 (12)
O2ⁱ—Rb1—O4	110.08 (10)	O1^viii—Rb3—O6^ix	118.01 (11)
O2ⁱⁱ—Rb1—O4	70.77 (11)	O1—Rb3—O6^ix	61.64 (11)
O2ⁱⁱⁱ—Rb1—O4	120.40 (11)	O4^xi—Rb3—O6^ix	73.50 (12)
O1^iv—Rb1—O4	123.98 (9)	O4ⁱ—Rb3—O6^ix	127.25 (12)
O1ⁱⁱⁱ—Rb1—O4	174.31 (9)	O6—Rb3—O6^ix	159.22 (18)
O4^v—Rb1—O4	50.35 (14)	O6—Ca4—O6^xii	86.2 (2)
O3ⁱⁱ—Rb2—O4^vi	109.85 (8)	O6—Ca4—O1^xiii	135.66 (10)
O3ⁱⁱ—Rb2—O4^vii	109.85 (8)	O6^xii—Ca4—O1^xiii	75.89 (12)
O4^vi—Rb2—O4^vii	63.71 (15)	O6—Ca4—O1^vii	135.66 (10)
O3ⁱⁱ—Rb2—O4^viii	109.85 (8)	O6^xii—Ca4—O1^vii	75.89 (12)
O4^vi—Rb2—O4^viii	140.29 (17)	O1^xiii—Ca4—O1^vii	78.88 (14)
O4^vii—Rb2—O4^viii	102.26 (13)	O6—Ca4—O1^viii	75.89 (12)
O3ⁱⁱ—Rb2—O4	109.85 (8)	O6^xii—Ca4—O1^viii	135.66 (10)
O4^vi—Rb2—O4	102.26 (13)	O1^xiii—Ca4—O1^viii	142.33 (18)
O4^vii—Rb2—O4	140.29 (17)	O1^vii—Ca4—O1^viii	89.11 (14)
O4^viii—Rb2—O4	63.71 (15)	O6—Ca4—O1^xiv	75.89 (12)
O3ⁱⁱ—Rb2—O5	109.36 (10)	O6^xii—Ca4—O1^xiv	135.66 (10)
O4^vi—Rb2—O5	52.37 (7)	O1^xiii—Ca4—O1^xiv	89.11 (14)
O4^vii—Rb2—O5	112.66 (10)	O1^vii—Ca4—O1^xiv	142.33 (18)
O4^viii—Rb2—O5	112.66 (10)	O1^viii—Ca4—O1^xiv	78.88 (14)
O4—Rb2—O5	52.37 (7)	O2^x—Ca5—O4^vii	97.40 (13)
O3ⁱⁱ—Rb2—O5^ix	109.36 (10)	O2^x—Ca5—O4^vi	97.40 (13)
O4^vi—Rb2—O5^ix	112.66 (10)	O4^vii—Ca5—O4^vi	85.7 (2)
O4^vii—Rb2—O5^ix	52.37 (7)	O2^x—Ca5—O1^xv	94.14 (12)
O4^viii—Rb2—O5^ix	52.37 (7)	O4^vii—Ca5—O1^xv	168.20 (13)
O4—Rb2—O5^ix	112.66 (10)	O4^vi—Ca5—O1^xv	95.30 (14)
O5—Rb2—O5^ix	141.3 (2)	O2^x—Ca5—O1^xvi	94.14 (12)
O3ⁱⁱ—Rb2—O2ⁱⁱ	49.96 (7)	O4^vii—Ca5—O1^xvi	95.30 (14)
O4^vi—Rb2—O2ⁱⁱ	144.54 (9)	O4^vi—Ca5—O1^xvi	168.20 (13)
O4^vii—Rb2—O2ⁱⁱ	144.54 (9)	O1^xv—Ca5—O1^xvi	81.37 (16)
O4^viii—Rb2—O2ⁱⁱ	67.82 (10)	O2^x—Ca5—O6^xvii	165.38 (16)
O4—Rb2—O2ⁱⁱ	67.82 (10)	O4^vii—Ca5—O6^xvii	93.30 (13)
O5—Rb2—O2ⁱⁱ	102.31 (6)	O4^vi—Ca5—O6^xvii	93.30 (13)
O5^ix—Rb2—O2ⁱⁱ	102.31 (6)	O1^xv—Ca5—O6^xvii	74.91 (11)
O3ⁱⁱ—Rb2—O2^x	49.96 (7)	O1^xvi—Ca5—O6^xvii	74.91 (11)
O4^vi—Rb2—O2^x	67.82 (10)	O6—Si1—O4	110.99 (19)
O4^vii—Rb2—O2^x	67.82 (10)	O6—Si1—O4^v	110.99 (19)
O4^viii—Rb2—O2^x	144.54 (9)	O4—Si1—O4^v	107.7 (3)
O4—Rb2—O2^x	144.54 (9)	O6—Si1—O5	110.6 (3)
O5—Rb2—O2^x	102.31 (6)	O4—Si1—O5	108.21 (18)
O5^ix—Rb2—O2^x	102.31 (6)	O4^v—Si1—O5	108.21 (18)
O2ⁱⁱ—Rb2—O2^x	99.92 (15)	O2—Si2—O1	113.52 (14)
O4^viii—Rb3—O4	69.79 (16)	O2—Si2—O1^viii	113.52 (14)
O4^viii—Rb3—O1^viii	172.90 (10)	O1—Si2—O1^viii	110.5 (3)
O4—Rb3—O1^viii	116.29 (9)	O2—Si2—O3	102.8 (3)
O4^viii—Rb3—O1	116.29 (9)	O1—Si2—O3	107.96 (17)
O4—Rb3—O1	172.90 (10)	O1^viii—Si2—O3	107.96 (17)
O1^viii—Rb3—O1	57.39 (12)	Si2^vii—O3—Si2	135.7 (4)
O4^viii—Rb3—O4^xi	79.71 (12)	Si1^xii—O5—Si1	139.4 (4)
O4—Rb3—O4^xi	109.79 (8)

Symmetry codes: (i) −x+1, −y, −z+1; (ii) x, y, z−1; (iii) x+1, y, z−1; (iv) −x+1/2, y, z−1; (v) −x+3/2, y, z; (vi) x, −y+1/2, z; (vii) −x+1/2, −y+1/2, z; (viii) −x+1/2, y, z; (ix) x−1, y, z; (x) −x+1/2, −y+1/2, z−1; (xi) x−1/2, −y, −z+1; (xii) −x+3/2, −y+1/2, z; (xiii) x+1, −y+1/2, z; (xiv) x+1, y, z; (xv) x+1/2, y+1/2, −z+1; (xvi) −x, y+1/2, −z+1; (xvii) x−1/2, y+1/2, −z+1; (xviii) x−1/2, y−1/2, −z+1; (xix) x−1, y, z+1; (xx) −x+1/2, −y+1/2, z+1; (xxi) x, y, z+1; (xxii) x+1/2, y−1/2, −z+1.

Dirubidium dicalcium pyrosilicate (LT) top

Crystal data top

Rb₂Ca₂Si₂O₇	D_x = 3.541 Mg m⁻³
M_r = 419.28	Mo Kα radiation, λ = 0.71073 Å
Orthorhombic, Pmmn	Cell parameters from 2430 reflections
a = 5.7281 (6) Å	θ = 5.0–30.4°
b = 13.8361 (13) Å	µ = 14.05 mm⁻¹
c = 9.9233 (11) Å	T = 193 K
V = 786.47 (14) Å³	Thin plate, colourless
Z = 4	0.32 × 0.1 × 0.04 mm
F(000) = 792

Data collection top

Rigaku Xcalibur Ruby Gemini ultra diffractometer	929 independent reflections
Radiation source: fine-focus sealed X-ray tube, Enhance (Mo) X-ray Source	777 reflections with I > 2σ(I)
Graphite monochromator	R_int = 0.059
Detector resolution: 10.3575 pixels mm^-1	θ_max = 26.4°, θ_min = 3.6°
ω scans	h = −7→7
Absorption correction: analytical [CrysAlis PRO (Rigaku OD, 2020), based on expressions derived by Clark & Reid (1995)]	k = −17→17
T_min = 0.109, T_max = 0.675	l = −12→12
10870 measured reflections

Refinement top

Refinement on F²	Primary atom site location: structure-invariant direct methods
Least-squares matrix: full	Secondary atom site location: difference Fourier map
R[F² > 2σ(F²)] = 0.029	w = 1/[σ²(F_o²) + (0.018P)² + 4.8532P] where P = (F_o² + 2F_c²)/3
wR(F²) = 0.067	(Δ/σ)_max < 0.001
S = 1.07	Δρ_max = 0.93 e Å⁻³
929 reflections	Δρ_min = −0.75 e Å⁻³
78 parameters	Extinction correction: SHELXL97 (Sheldrick, 2008), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
0 restraints	Extinction coefficient: 0.0075 (4)

Special details top

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

	x	y	z	U_iso*/U_eq	Occ. (<1)
Rb1	0.75	0.10480 (4)	0.08531 (6)	0.0138 (2)
Rb2	0.25	0.25	0.25884 (8)	0.0124 (2)
Rb3	0.25	0.09799 (7)	0.58693 (8)	0.0206 (2)	0.5
Ca3	0.25	0.09799 (7)	0.58693 (8)	0.0206 (2)	0.5
Ca4	0.75	0.25	0.74921 (15)	0.0080 (4)
Ca5	0.25	0.51081 (8)	0.24885 (11)	0.0088 (3)
Si1	0.75	0.13623 (12)	0.41858 (17)	0.0137 (4)
Si2	0.25	0.13959 (11)	0.91918 (16)	0.0078 (3)
O1	0.0183 (5)	0.1274 (2)	0.8280 (3)	0.0211 (7)
O2	0.25	0.0732 (3)	1.0520 (4)	0.0165 (9)
O3	0.25	0.25	0.9825 (6)	0.0220 (15)
O4	0.5242 (7)	0.0821 (3)	0.3604 (4)	0.0425 (11)
O5	0.75	0.25	0.3598 (6)	0.0119 (12)
O6	0.75	0.1361 (3)	0.5793 (5)	0.0323 (13)

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Rb1	0.0107 (3)	0.0204 (3)	0.0104 (3)	0	0	0.0001 (2)
Rb2	0.0141 (4)	0.0100 (4)	0.0130 (4)	0	0	0
Rb3	0.0161 (4)	0.0345 (5)	0.0113 (4)	0	0	−0.0071 (4)
Ca3	0.0161 (4)	0.0345 (5)	0.0113 (4)	0	0	−0.0071 (4)
Ca4	0.0099 (8)	0.0046 (7)	0.0094 (8)	0	0	0
Ca5	0.0117 (6)	0.0059 (5)	0.0088 (6)	0	0	0.0001 (4)
Si1	0.0208 (9)	0.0096 (8)	0.0107 (8)	0	0	−0.0003 (6)
Si2	0.0070 (7)	0.0055 (7)	0.0110 (8)	0	0	0.0022 (6)
O1	0.0137 (15)	0.0189 (15)	0.0307 (19)	−0.0071 (13)	−0.0112 (14)	0.0077 (14)
O2	0.017 (2)	0.014 (2)	0.019 (2)	0	0	0.0059 (18)
O3	0.040 (4)	0.010 (3)	0.015 (3)	0	0	0
O4	0.050 (3)	0.0280 (19)	0.050 (2)	−0.0163 (19)	−0.025 (2)	0.0048 (18)
O5	0.020 (3)	0.010 (3)	0.006 (3)	0	0	0
O6	0.070 (4)	0.012 (2)	0.015 (2)	0	0	−0.0008 (18)

Geometric parameters (Å, º) top

Rb1—O2ⁱ	2.815 (4)	Si1—O4^v	1.602 (4)
Rb1—O2ⁱⁱ	2.9160 (9)	Si1—O5	1.679 (2)
Rb1—O2ⁱⁱⁱ	2.9160 (9)	Si2—O2	1.606 (4)
Rb1—O1^iv	2.997 (3)	Si2—O1	1.615 (3)
Rb1—O1ⁱⁱⁱ	2.997 (3)	Si2—O1^vii	1.615 (3)
Rb1—O4	3.037 (4)	Si2—O3	1.652 (3)
Rb1—O4^v	3.037 (4)	O1—Si2	1.615 (3)
Rb1—O5	3.385 (4)	O1—Ca5^xviii	2.355 (3)
Rb2—O3ⁱⁱ	2.742 (6)	O1—Ca4^ix	2.419 (3)
Rb2—O4^vi	2.980 (4)	O1—Rb3	2.765 (4)
Rb2—O4^vii	2.980 (4)	O1—Rb1^xix	2.997 (3)
Rb2—O4	2.980 (4)	O2—Si2	1.606 (4)
Rb2—O4^viii	2.980 (4)	O2—Ca5^xx	2.274 (4)
Rb2—O5	3.0343 (19)	O2—Rb1ⁱ	2.815 (4)
Rb2—O5^ix	3.0343 (19)	O2—Rb1^xxi	2.9160 (9)
Rb2—O2ⁱⁱ	3.193 (4)	O2—Rb1^xix	2.9160 (9)
Rb2—O2^x	3.193 (4)	O2—Rb2^xxi	3.193 (4)
Rb3—O4^vii	2.752 (5)	O3—Si2^viii	1.652 (3)
Rb3—O4	2.752 (5)	O3—Si2	1.652 (3)
Rb3—O1^vii	2.765 (4)	O3—Rb2^xxi	2.742 (6)
Rb3—O1	2.765 (4)	O4—Si1	1.603 (4)
Rb3—O4^xi	2.855 (4)	O4—Ca5^viii	2.312 (4)
Rb3—O4ⁱ	2.855 (4)	O4—Ca3ⁱ	2.855 (4)
Rb3—O6	2.9131 (9)	O4—Rb3	2.752 (5)
Rb3—O6^ix	2.9131 (9)	O4—Rb3ⁱ	2.855 (4)
Ca4—O6	2.308 (5)	O4—Rb2	2.980 (4)
Ca4—O6^xii	2.308 (5)	O4—Rb1	3.037 (5)
Ca4—O1^xiii	2.419 (3)	O5—Si1^xii	1.679 (2)
Ca4—O1^vii	2.419 (3)	O5—Si1	1.679 (2)
Ca4—O1^xiv	2.419 (3)	O5—Rb2^xiv	3.0343 (19)
Ca4—O1^viii	2.419 (3)	O5—Rb2	3.0343 (19)
Ca5—O2^x	2.274 (4)	O5—Rb1^xii	3.385 (4)
Ca5—O4^viii	2.312 (4)	O5—Rb1	3.385 (4)
Ca5—O4^vi	2.312 (4)	O6—Si1	1.595 (5)
Ca5—O1^xv	2.355 (3)	O6—Ca4	2.308 (4)
Ca5—O1^xvi	2.355 (3)	O6—Ca5^xxii	2.431 (5)
Ca5—O6^xvii	2.431 (5)	O6—Rb3^xiv	2.9131 (9)
Si1—O6	1.595 (5)	O6—Rb3	2.9131 (9)
Si1—O4	1.602 (4)

O2ⁱ—Rb1—O2ⁱⁱ	79.29 (8)	O1^vii—Rb3—O4^xi	100.79 (11)
O2ⁱ—Rb1—O2ⁱⁱⁱ	79.29 (8)	O1—Rb3—O4^xi	75.67 (10)
O2ⁱⁱ—Rb1—O2ⁱⁱⁱ	158.35 (17)	O4^vii—Rb3—O4ⁱ	109.78 (8)
O2ⁱ—Rb1—O1^iv	71.27 (9)	O4—Rb3—O4ⁱ	79.70 (12)
O2ⁱⁱ—Rb1—O1^iv	54.21 (10)	O1^vii—Rb3—O4ⁱ	75.67 (10)
O2ⁱⁱⁱ—Rb1—O1^iv	115.00 (10)	O1—Rb3—O4ⁱ	100.79 (11)
O2ⁱ—Rb1—O1ⁱⁱⁱ	71.27 (9)	O4^xi—Rb3—O4ⁱ	53.86 (16)
O2ⁱⁱ—Rb1—O1ⁱⁱⁱ	115.00 (10)	O4^vii—Rb3—O6	123.69 (12)
O2ⁱⁱⁱ—Rb1—O1ⁱⁱⁱ	54.21 (10)	O4—Rb3—O6	55.40 (12)
O1^iv—Rb1—O1ⁱⁱⁱ	61.71 (11)	O1^vii—Rb3—O6	61.57 (11)
O2ⁱ—Rb1—O4	110.14 (10)	O1—Rb3—O6	117.88 (11)
O2ⁱⁱ—Rb1—O4	70.62 (11)	O4^xi—Rb3—O6	127.43 (12)
O2ⁱⁱⁱ—Rb1—O4	120.31 (11)	O4ⁱ—Rb3—O6	73.58 (12)
O1^iv—Rb1—O4	123.94 (9)	O4^vii—Rb3—O6^ix	55.39 (12)
O1ⁱⁱⁱ—Rb1—O4	174.34 (10)	O4—Rb3—O6^ix	123.69 (12)
O2ⁱ—Rb1—O4^v	110.14 (10)	O1^vii—Rb3—O6^ix	117.88 (11)
O2ⁱⁱ—Rb1—O4^v	120.31 (11)	O1—Rb3—O6^ix	61.57 (11)
O2ⁱⁱⁱ—Rb1—O4^v	70.62 (11)	O4^xi—Rb3—O6^ix	73.58 (12)
O1^iv—Rb1—O4^v	174.34 (10)	O4ⁱ—Rb3—O6^ix	127.43 (12)
O1ⁱⁱⁱ—Rb1—O4^v	123.94 (9)	O6—Rb3—O6^ix	158.94 (18)
O4—Rb1—O4^v	50.41 (15)	O6—Ca4—O6^xii	86.2 (2)
O3ⁱⁱ—Rb2—O4^vi	109.76 (9)	O6—Ca4—O1^xiii	135.64 (10)
O3ⁱⁱ—Rb2—O4^vii	109.76 (9)	O6^xii—Ca4—O1^xiii	75.93 (12)
O4^vi—Rb2—O4^vii	140.49 (17)	O6—Ca4—O1^vii	75.93 (12)
O3ⁱⁱ—Rb2—O4	109.76 (9)	O6^xii—Ca4—O1^vii	135.64 (10)
O4^vi—Rb2—O4	102.46 (14)	O1^xiii—Ca4—O1^vii	142.31 (18)
O4^vii—Rb2—O4	63.62 (15)	O6—Ca4—O1^xiv	75.93 (12)
O3ⁱⁱ—Rb2—O4^viii	109.76 (8)	O6^xii—Ca4—O1^xiv	135.64 (10)
O4^vi—Rb2—O4^viii	63.62 (15)	O1^xiii—Ca4—O1^xiv	89.08 (14)
O4^vii—Rb2—O4^viii	102.46 (14)	O1^vii—Ca4—O1^xiv	78.90 (14)
O4—Rb2—O4^viii	140.49 (17)	O6—Ca4—O1^viii	135.64 (10)
O3ⁱⁱ—Rb2—O5	109.28 (10)	O6^xii—Ca4—O1^viii	75.93 (12)
O4^vi—Rb2—O5	52.47 (7)	O1^xiii—Ca4—O1^viii	78.90 (14)
O4^vii—Rb2—O5	112.70 (10)	O1^vii—Ca4—O1^viii	89.08 (14)
O4—Rb2—O5	52.47 (7)	O1^xiv—Ca4—O1^viii	142.31 (18)
O4^viii—Rb2—O5	112.70 (10)	O2^x—Ca5—O4^viii	97.30 (13)
O3ⁱⁱ—Rb2—O5^ix	109.28 (10)	O2^x—Ca5—O4^vi	97.30 (13)
O4^vi—Rb2—O5^ix	112.70 (10)	O4^viii—Ca5—O4^vi	85.6 (2)
O4^vii—Rb2—O5^ix	52.47 (7)	O2^x—Ca5—O1^xv	94.14 (12)
O4—Rb2—O5^ix	112.70 (10)	O4^viii—Ca5—O1^xv	168.32 (13)
O4^viii—Rb2—O5^ix	52.47 (7)	O4^vi—Ca5—O1^xv	95.29 (14)
O5—Rb2—O5^ix	141.4 (2)	O2^x—Ca5—O1^xvi	94.14 (12)
O3ⁱⁱ—Rb2—O2ⁱⁱ	49.99 (7)	O4^viii—Ca5—O1^xvi	95.29 (14)
O4^vi—Rb2—O2ⁱⁱ	144.54 (9)	O4^vi—Ca5—O1^xvi	168.32 (13)
O4^vii—Rb2—O2ⁱⁱ	67.67 (10)	O1^xv—Ca5—O1^xvi	81.50 (16)
O4—Rb2—O2ⁱⁱ	67.67 (10)	O2^x—Ca5—O6^xvii	165.29 (16)
O4^viii—Rb2—O2ⁱⁱ	144.54 (9)	O4^viii—Ca5—O6^xvii	93.48 (13)
O5—Rb2—O2ⁱⁱ	102.26 (6)	O4^vi—Ca5—O6^xvii	93.48 (13)
O5^ix—Rb2—O2ⁱⁱ	102.26 (6)	O1^xv—Ca5—O6^xvii	74.85 (11)
O3ⁱⁱ—Rb2—O2^x	49.99 (7)	O1^xvi—Ca5—O6^xvii	74.85 (11)
O4^vi—Rb2—O2^x	67.67 (10)	O6—Si1—O4	111.10 (19)
O4^vii—Rb2—O2^x	144.54 (9)	O6—Si1—O4^v	111.10 (19)
O4—Rb2—O2^x	144.54 (9)	O4—Si1—O4^v	107.6 (3)
O4^viii—Rb2—O2^x	67.67 (10)	O6—Si1—O5	110.4 (3)
O5—Rb2—O2^x	102.26 (6)	O4—Si1—O5	108.26 (18)
O5^ix—Rb2—O2^x	102.26 (6)	O4^v—Si1—O5	108.26 (18)
O2ⁱⁱ—Rb2—O2^x	99.98 (15)	O2—Si2—O1	113.58 (14)
O4^vii—Rb3—O4	69.62 (16)	O2—Si2—O1^vii	113.58 (14)
O4^vii—Rb3—O1^vii	172.97 (10)	O1—Si2—O1^vii	110.5 (3)
O4—Rb3—O1^vii	116.39 (10)	O2—Si2—O3	102.5 (3)
O4^vii—Rb3—O1	116.39 (10)	O1—Si2—O3	108.05 (16)
O4—Rb3—O1	172.97 (10)	O1^vii—Si2—O3	108.05 (17)
O1^vii—Rb3—O1	57.36 (12)	Si2—O3—Si2^viii	135.3 (4)
O4^vii—Rb3—O4^xi	79.70 (12)	Si1^xii—O5—Si1	139.3 (4)
O4—Rb3—O4^xi	109.78 (8)

Symmetry codes: (i) −x+1, −y, −z+1; (ii) x, y, z−1; (iii) x+1, y, z−1; (iv) −x+1/2, y, z−1; (v) −x+3/2, y, z; (vi) x, −y+1/2, z; (vii) −x+1/2, y, z; (viii) −x+1/2, −y+1/2, z; (ix) x−1, y, z; (x) −x+1/2, −y+1/2, z−1; (xi) x−1/2, −y, −z+1; (xii) −x+3/2, −y+1/2, z; (xiii) x+1, −y+1/2, z; (xiv) x+1, y, z; (xv) x+1/2, y+1/2, −z+1; (xvi) −x, y+1/2, −z+1; (xvii) x−1/2, y+1/2, −z+1; (xviii) x−1/2, y−1/2, −z+1; (xix) x−1, y, z+1; (xx) −x+1/2, −y+1/2, z+1; (xxi) x, y, z+1; (xxii) x+1/2, y−1/2, −z+1.

Atomic coordinates (× 104, origin choice 2 of space group Pmmn) and equivalent isotropic displacement parameters (Å² × 10³) for Rb₂Ca₂Si₂O₇. First line: 15 °C, second line: -80 °C. U_eq is defined as one third of the trace of the orthogonalised U_ij tensor. M1 and M2 are exclusively occupied by rubidium, while M4 and M5 represent pure Ca sites. M3 is a mixed Rb–Ca position, with a population of 50% rubidium and 50% calcium top

Atom	Wyckoff site	Site symmetry	x	y	z	U_eq
M1	4e	m..	7500	1049 (1)	854 (1)	18 (1)
			7500	1048 (1)	853 (1)	14 (1)
M2	2a	mm2	2500	2500	2585 (1)	16 (1)
			2500	2500	2588 (1)	12 (1)
M3	4e	m..	2500	980 (1)	5869 (1)	21 (1)
			2500	980 (1)	5869 (1)	21 (1)
M4	2b	mm2	7500	2500	7494 (2)	10 (1)
			7500	2500	7492 (2)	8(1)
M5	4e	m..	2500	5106 (1)	2489 (1)	10 (1)
			2500	5108 (1)	2489 (1)	9(1)
Si1	4e	m..	7500	1362 (1)	4187 (2)	15 (1)
			7500	1362 (1)	4186 (2)	14 (1)
Si2	4e	m..	2500	1396 (1)	9193 (2)	9(1)
			2500	1396 (1)	9192 (2)	8(1)
O1	8g	1	181 (5)	1274 (2)	8280 (3)	23 (1)
			183 (5)	1274 (2)	8280 (3)	21 (1)
O2	4e	m..	2500	734 (3)	10515 (4)	19 (1)
			2500	732 (3)	10520 (4)	16 (1)
O3	2a	mm2	2500	2500	9820 (7)	26 (2)
			2500	2500	9825 (6)	22 (2)
O4	8g	1	5246 (7)	822 (3)	3605 (4)	44 (1)
			5242 (7)	821 (3)	3604 (4)	43 (1)
O5	2b	mm2	7500	2500	3599 (6)	14 (1)
			7500	2500	3598 (6)	12 (1)
O6	4e	m..	7500	1355 (3)	5789 (4)	35 (1)
			7500	1361 (3)	5793 (5)	32 (1)

Selected bond lengths up to 3.2 Å and bond angles (°) for Rb₂Ca₂Si₂O₇. For the tetrahedra and octahedra, the distortion parameters QE (quadratic elongation) and AV (angle variance) have been calculated. top

15 °C		-80 °C
M1—O2ⁱ	2.819 (4)	M1—O2ⁱ	2.815 (4)
M1—O2ⁱⁱ	2.9206 (9)	M1—O2ⁱⁱ	2.9160 (9)
M1—O2ⁱⁱⁱ	2.9206 (9)	M1—O2ⁱⁱⁱ	2.9160 (9)
M1—O1^iv	2.999 (3)	M1—O1^iv	2.997 (3)
M1—O1ⁱⁱⁱ	2.999 (3)	M1—O1ⁱⁱⁱ	2.997 (3)
M1—O4^v	3.039 (4)	M1—O4	3.037 (4)
M1—O4	3.039 (4)	M1—O4^v	3.037 (4)
<M1—O> = 2.962		<M1—O> = 2.959
M2—O3ⁱⁱ	2.746 (7)	M2—O3ⁱⁱ	2.742 (6)
M2—O4^vi	2.985 (4)	M2—O4^vi	2.980 (4)
M2—O4^vii	2.985 (4)	M2—O4^vii	2.980 (4)
M2—O4^viii	2.985 (4)	M2—O4	2.980 (4)
M2—O4	2.985 (4)	M2—O4^viii	2.980 (4)
M2—O5	3.0401 (19)	M2—O5	3.0343 (19)
M2—O5^ix	3.0401 (19)	M2—O5^ix	3.0343 (19)
M2—O2ⁱⁱ	3.196 (4)	M2—O2ⁱⁱ	3.193 (4)
M2—O2^x	3.196 (4)	M2—O2^x	3.193 (4)
<M2—O> = 3.018		<M2—O> = 3.013
M3—O4^viii	2.754 (5)	M3—O4^vii	2.752 (5)
M3—O4	2.754 (5)	M3—O4	2.752 (5)
M3—O1^viii	2.770 (4)	M3—O1^vii	2.765 (4)
M3—O1	2.770 (4)	M3—O1	2.765 (4)
M3—O4^xii	2.860 (4)	M3—O4^xii	2.855 (4)
M3—O4ⁱ	2.860 (4)	M3—O4ⁱ	2.855 (4)
M3—O6	2.9160 (9)	M3—O6	2.9131 (9)
M3—O6^ix	2.9160 (9)	M3—O6^ix	2.9131 (9)
<M3—O> = 2.825		<M3—O> = 2.821
M4—O6	2.320 (5)	M4—O6	2.308 (5)
M4—O6^xi	2.320 (5)	M4—O6^xi	2.308 (5)
M4—O1^xiii	2.421 (3)	M4—O1^xiii	2.419 (3)
M4—O1^vii	2.421 (3)	M4—O1^vii	2.419 (3)
M4—O1^viii	2.421 (3)	M4—O1^xiv	2.419 (3)
M4—O1^xiv	2.421 (3)	M4—O1^viii	2.419 (3)
<M4—O> 2.387		<M4—O> 2.382
M5—O2^x	2.279 (4)	M5—O2^x	2.274 (4)
M5—O4^vii	2.317 (4)	M5—O4^viii	2.312 (4)
M5—O4^vi	2.317 (4)	M5—O4^vi	2.312 (4)
M5—O1^xix	2.359 (3)	M5—O1^xix	2.355 (3)
M5—O1^xx	2.359 (3)	M5—O1^xx	2.355 (3)
M5—O6^xxi	2.435 (5)	M5—O6^xxi	2.431 (5)
<M5—O> = 2.345		<M5—O> = 2.340
QE = 1.020	AV = 69.94	QE = 1.020	AV = 70.19
Si1—O6	1.591 (5)	Si1—O6	1.595 (5)
Si1—O4	1.601 (4)	Si1—O4	1.602 (4)
Si1—O4^v	1.601 (4)	Si1—O4^v	1.602 (4)
Si1—O5	1.681 (2)	Si1—O5	1.679 (2)
<Si1—O> = 1.618		<Si1—O> = 1.620
QE = 1.001	AV = 2.51	QE = 1.001	AV = 2.48
Si2—O2	1.602 (4)	Si2—O2	1.606 (4)
Si2—O1	1.619 (3)	Si2—O1	1.615 (3)
Si2—O1^viii	1.619 (3)	Si2—O1^vii	1.615 (3)
Si2—O3	1.651 (3)	Si2—O3	1.652 (3)
<Si2—O> = 1.622		<Si2—O> = 1.622
QE = 1.004	AV = 16.51	QE = 1.004	AV = 17.44

O—Ca—O angles
O6—M4—O6^xi	86.2 (2)	O6—M4—O6^xi	86.2 (2)
O6—M4—O1^xiii	135.66 (10)	O6—M4—O1^xiii	135.64 (10)
O6^xi—M4—O1^xiii	75.89 (12)	O6^xi—M4—O1^xiii	75.93 (12)
O6—M4—O1^vii	135.66 (10)	O6—M4—O1^vii	75.93 (12)
O6^xi—M4—O1^vii	75.89 (12)	O6^xi—M4—O1^vii	135.64 (10)
O1^xiii—M4—O1^vii	78.88 (14)	O1^xiii—M4—O1^vii	142.31 (18)
O6—M4—O1^viii	75.89 (12)	O6—M4—O1^xiv	75.93 (12)
O6^xi—M4—O1^viii	135.66 (10)	O6^xi—M4—O1^xiv	135.64 (10)
O1^xiii—M4—O1^viii	142.33 (18)	O1^xiii—M4—O1^xiv	89.08 (14)
O1^vii—M4—O1^viii	89.11 (14)	O1^vii—M4—O1^xiv	78.90 (14)
O6—M4—O1^xiv	75.89 (12)	O6—M4—O1^viii	135.64 (10)
O6^xi—M4—O1^xiv	135.66 (10)	O6^xi—M4—O1^viii	75.93 (12)
O1^xiii—M4—O1^xiv	89.11 (14)	O1^xiii—M4—O1^viii	78.90 (14)
O1^vii—M4—O1^xiv	142.33 (18)	O1^vii—M4—O1^viii	89.08 (14)
O1^viii—M4—O1^xiv	78.88 (14)	O1^xiv—M4—O1^viii	142.31 (18)

O2^x—M5—O4^vii	97.40 (13)	O2^x—M5—O4^viii	97.30 (13)
O2^x—M5—O4^vi	97.40 (13)	O2^x—M5—O4^vi	97.30 (13)
O4^vii—M5—O4^vi	85.7 (2)	O4^viii—M5—O4^vi	85.6 (2)
O2^x—M5—O1^xix	94.14 (12)	O2^x—M5—O1^xix	94.14 (12)
O4^vii—M5—O1^xix	168.20 (13)	O4^viii—M5—O1^xix	168.32 (13)
O4^vi—M5—O1^xix	95.30 (14)	O4^vi—M5—O1^xix	95.29 (14)
O2^x—M5—O1^xx	94.14 (12)	O2^x—M5—O1^xx	94.14 (12)
O4^vii—M5—O1^xx	95.30 (14)	O4^viii—M5—O1^xx	95.29 (14)
O4^vi—M5—O1^xx	168.20 (13)	O4^vi—M5—O1^xx	168.32 (13)
O1^xix—M5—O1^xx	81.37 (16)	O1^xix—M5—O1^xx	81.50 (16)
O2^x—M5—O6^xxi	165.38 (16)	O2^x—M5—O6^xxi	165.29 (16)
O4^vii—M5—O6^xxi	93.30 (13)	O4^viii—M5—O6^xxi	93.48 (13)
O4^vi—M5—O6^xxi	93.30 (13)	O4^vi—M5—O6^xxi	93.48 (13)
O1^xix—M5—O6^xxi	74.91 (11)	O1^xix—M5—O6^xxi	74.85 (11)
O1^xx—M5—O6^xxi	74.91 (11)	O1^xx—M5—O6^xxi	74.85 (11)

O—Si—O angles
O6—Si1—O4	110.99 (19)	O6—Si1—O4	111.10 (19)
O6—Si1—O4^v	110.99 (19)	O6—Si1—O4^v	111.10 (19)
O4—Si1—O4^v	107.7 (3)	O4—Si1—O4^v	107.6 (3)
O6—Si1—O5	110.6 (3)	O6—Si1—O5	110.4 (3)
O4—Si1—O5	108.21 (18)	O4—Si1—O5	108.26 (18)
O4^v—Si1—O5	108.21 (18)	O4^v—Si1—O5	108.26 (18)
<O—Si1—O>	109.45	<O—Si1—O>	109.45

O2—Si2—O1	113.52 (14)	O2—Si2—O1	113.58 (14)
O2—Si2—O1^viii	113.52 (14)	O2—Si2—O1^vii	113.58 (14)
O1—Si2—O1^viii	110.5 (3)	O1—Si2—O1^vii	110.5 (3)
O2—Si2—O3	102.8 (3)	O2—Si2—O3	102.5 (3)
O1—Si2—O3	107.96 (17)	O1—Si2—O3	108.05 (16)
O1^viii—Si2—O3	107.96 (17)	O1^vii—Si2—O3	108.05 (17)
<O—Si2—O>	109.38	<O—Si2—O>	109.38

Si—O—Si angles
Si2^vii—O3—Si2	135.7 (4)	Si2—O3—Si2^viii	135.3 (4)
Si1^xi—O5—Si1	139.4 (4)	Si1^xi—O5—Si1	139.3 (4)

Symmetry codes: (i) -x+1, -y, -z+1; (ii) x, y, z-1; (iii) x+1, y, z-1; (iv) -x+1/2, y, z-1; (v) -x+3/2, y, z; (vi) x, -y+1/2, z; (vii) -x+1/2, -y+1/2, z; (viii) -x+1/2, y, z; (ix) x-1, y, z; (x) -x+1/2, -y+1/2, z-1; (xi) -x+3/2, -y+1/2, z; (xii) x-1/2, -y, -z+1; (xiii) x+1, -y+1/2, z; (xiv) x+1, y, z; (xix) x+1/2, y+1/2, -z+1; (xx) -x, y+1/2, -z+1; (xxi) x-1/2, y+1/2, -z+1.

Anisotropic displacement parameters (Å² × 10³) for Rb₂Ca₂Si₂O₇. The anisotropic displacement factor exponent takes the form: -2π²[h²a*²U₁₁ + ··· + 2hka*b*U₁₂]. First line 15 °C and second line -80 °C. top

	U₁₁	U₂₂	U₃₃	U₂₃	U₁₃	U₁₂
M1	14 (1)	26 (1)	13 (1)	0(1)	0	0
	11 (1)	20 (1)	10 (1)	0(1)	0	0
M2	18 (1)	14 (1)	16 (1)	0	0	0
	14 (1)	10 (1)	13 (1)	0	0	0
M3	18 (1)	41 (1)	13 (1)	-7(1)	0	0
	16 (1)	35 (1)	11 (1)	-7(1)	0	0
M4	13 (1)	7(1)	11 (1)	0	0	0
	10 (1)	5(1)	9(1)	0	0	0
M5	13 (1)	7(1)	10 (1)	0(1)	0	0
	12 (1)	6(1)	9(1)	0(1)	0	0
Si1	22 (1)	11 (1)	12 (1)	0(1)	0	0
	21 (1)	10 (1)	11 (1)	0(1)	0	0
Si2	9(1)	7(1)	12 (1)	2(1)	0	0
	7(1)	6(1)	11 (1)	2(1)	0	0
O1	17 (2)	21 (2)	32 (2)	7(1)	-11 (1)	-7(1)
	14 (2)	19 (2)	31 (2)	8(1)	-11 (1)	-7(1)
O2	20 (2)	18 (2)	19 (2)	7(2)	0	0
	17 (2)	14 (2)	19 (2)	6(2)	0	0
O3	43 (4)	15 (3)	21 (3)	0	0	0
	40 (4)	10 (3)	15 (3)	0	0	0
O4	53 (3)	32 (2)	46 (2)	5(2)	-25 (2)	-18 (2)
	50 (3)	28 (2)	50 (2)	5(2)	-25 (2)	-16 (2)
O5	21 (3)	13 (3)	9(3)	0	0	0
	20 (3)	10 (3)	6(3)	0	0	0
O6	76 (4)	17 (2)	11 (2)	0(2)	0	0
	70 (4)	12 (2)	15 (2)	-1(2)	0	0

References

Al-Harbi, O. A. (2007). Eur. J. Glass. Sci. Technol. A48, 35–40. Google Scholar
Brese, N. E. & O'Keeffe, M. (1991). Acta Cryst. B47, 192–197. CrossRef CAS Web of Science IUCr Journals Google Scholar
Brown, I. D. (2016). The Chemical bond in Inorganic Chemistry: The Bond Valence Model, 2nd ed., p. 315. Oxford University Press. Google Scholar
Brown, I. D. & Altermatt, D. (1985). Acta Cryst. B41, 244–247. CrossRef CAS Web of Science IUCr Journals Google Scholar
Burla, M. C., Camalli, M., Carrozzini, B., Cascarano, G. L., Giacovazzo, C., Polidori, G. & Spagna, R. (2003). J. Appl. Cryst. 36, 1103. CrossRef IUCr Journals Google Scholar
Butt, H., Knowles, K. M., Montelongo, Y., Amaratunga, G. A. J. & Wilkinson, T. D. (2014). ACS Nano, 8, 2929–2935. CrossRef CAS PubMed Google Scholar
Capillas, C., Tasci, E. S., de la Flor, G., Orobengoa, D., Perez-Mato, J. M. & Aroyo, M. I. (2011). Z. Kristallogr. 226, 186–196. Web of Science CrossRef CAS Google Scholar
Chen, M. & Zhao, B. (2016). Fuel, 180, 638–644. CrossRef CAS Google Scholar
Clark, R. C. & Reid, J. S. (1995). Acta Cryst. A51, 887–897. CrossRef CAS Web of Science IUCr Journals Google Scholar
Hellenbrandt, M. (2004). Crystallogr. Rev. 10, 17–22. CrossRef CAS Google Scholar
Holland, A. J. & Preston, E. (1938). J. Soc. Glass Technol. 21, 395–408. Google Scholar
Kahlenberg, V. (2023). Miner. Petrol. 117, 293–306. Web of Science CrossRef ICSD CAS Google Scholar
Kahlenberg, V., Brunello, E., Hejny, C., Krüger, H., Schmidmair, D., Tribus, M. & Többens, D. (2015). J. Solid State Chem. 225, 155–167. CrossRef ICSD CAS Google Scholar
Kahlenberg, V., Girtler, D., Arroyabe, E., Kaindl, R. & Többens, D. (2010). Miner. Petrol. 100, 1–9. CrossRef ICSD CAS Google Scholar
Kahlenberg, V. & Hösch, A. (2002). Z. Kristallogr. 217, 155–163. Web of Science CrossRef ICSD CAS Google Scholar
Kahlenberg, V., Müllner, M., Schmidmair, D., Perfler, L. & Többens, D. (2016). Z. Kristallogr. 231, 209–217. CrossRef ICSD CAS Google Scholar
Kaminsky, W. (2014). WinTensor. Version 1.5 for Windows. http://cad4.cpac.washington.edu/WinTensorhome/WinTensor.htm. Google Scholar
Kim, J. S., Song, H. J., Roh, H. S., Yim, D. K., Noh, J. H. & Hong, K. S. (2012). Mater. Lett. 79, 112–115. CrossRef CAS Google Scholar
Liebau, F. (1985). Structural Chemistry of Silicates, p. 347. Berlin, Heidelberg, New York, Tokyo: Springer. Google Scholar
Liu, H., Hildebrandt, E., Krammer, H., Kahlenberg, V., Krüger, H. & Schottenberger, H. (2021). J. Am. Ceram. Soc. 104, 6678–6695. Web of Science CrossRef CAS Google Scholar
Liu, Q., Liu, Y., Ding, Y., Peng, Z., Yu, Q., Tian, X. & Dong, G. (2014). J. Sol-Gel Sci. Technol. 71, 276–282. CrossRef CAS Google Scholar
Mitchell, R. H. & Dawson, J. B. (2012). Lithos, 152, 40–46. CrossRef CAS Google Scholar
Momma, K. & Izumi, F. (2011). J. Appl. Cryst. 44, 1272–1276. Web of Science CrossRef CAS IUCr Journals Google Scholar
Morey, G. W. & Bowen, N. L. (1925). J. Glass Technol. Soc. 9, 226–264. CAS Google Scholar
Napper, J. D., Layland, R. C., Smith, M. D. & Loye, H. (2004). J. Chem. Crystallogr. 34, 347–351. CrossRef ICSD CAS Google Scholar
Olanders, B. & Steenari, B. M. (1995). Biomass Bioenergy, 8, 105–115. CrossRef CAS Google Scholar
Orobengoa, D., Capillas, C., Aroyo, M. I. & Perez-Mato, J. M. (2009). J. Appl. Cryst. 42, 820–833. Web of Science CrossRef CAS IUCr Journals Google Scholar
Parauha, Y. R., Halwar, D. K. & Dhoble, S. J. (2022). Displays, 75, 102304. CrossRef Google Scholar
Prince, E. (2004). Editor. International Tables for X-ray Crystallography, Vol. C, Mathematical, Physical and Chemical Tables, 3rd ed. Dordrecht: Springer. Google Scholar
Reddy, P. M., Lakshmi, R., Dass, F. P. & Sasikumar, S. (2014). Sci. Eng. Compos. Mater. 23, 375–380. CrossRef Google Scholar
Rigaku OD (2020). CrysAlis PRO. Rigaku Oxford Diffraction Ltd, Yarnton, Oxfordshire, England. Google Scholar
Robinson, K., Gibbs, G. V. & Ribbe, P. H. (1971). Science, 172, 567–570. CrossRef PubMed CAS Web of Science Google Scholar
Santoso, I., Riihimäki, M., Sibarani, D., Taskinen, P., Hupa, L., Paek, M. K. & Lindberg, D. (2022). J. Eur. Ceram. Soc. 42, 2449–2463. CrossRef CAS Google Scholar
Santoso, I., Taskinen, P., Jokilaakso, A., Paek, M.-K. & Lindberg, D. (2020). Fuel, 265, 116894. CrossRef Google Scholar
Segnit, E. R. (1953). Am. J. Sci. 251, 586–601. CrossRef CAS Google Scholar
Shahid, K. A. & Glasser, F. P. (1971). Phys. Chem. Glasses, 12, 50–57. CAS Google Scholar
Shannon, R. D. (1976). Acta Cryst. A32, 751–767. CrossRef CAS IUCr Journals Web of Science Google Scholar
Shelby, J. E. (2009). Glass Science and Technology, 2nd ed., p. 291. Cambridge: The Royal Society of Chemistry. Google Scholar
Sheldrick, G. M. (2008). Acta Cryst. A64, 112–122. Web of Science CrossRef CAS IUCr Journals Google Scholar
Skrzat, Z. M., Simonov, V. I. & Belov, N. V. (1969). Dokl. Akad. Nauk SSSR, 184, 337–340. CAS Google Scholar
Varshneya, A. K. (1994). Fundamentals of Inorganic Glasses, p. 570. London: Academic Press. Google Scholar
Weidendorfer, D., Schmidt, M. W. & Mattsson, H. B. (2016). Contrib. Mineral. Petrol. 171, 43. CrossRef Google Scholar
West, A. R. (1978). J. Am. Ceram. Soc. 61, 152–155. CrossRef CAS Google Scholar
Williamson, J. & Glasser, F. P. (1965). Science, 148, 1589–1591. CrossRef PubMed CAS Web of Science Google Scholar
Wills, A. (2010). VaList. http://fermat.chem.ucl.ac.uk/spaces/willsgroup/software/. Google Scholar
Wu, Q., Zhao, Q., Zheng, P., Chen, W., Xiang, D., He, Z., Huang, Q., Ding, J. & Zhou, J. (2020). Ceram. Int. 46, 2845–2852. CrossRef CAS Google Scholar
Zandi Karimi, A., Rezabeigi, E. & Drew, R. A. L. (2018). J. Non-Cryst. Solids, 502, 176–183. CrossRef CAS Google Scholar
Zhang, Z., Xiao, Y., Voncken, J., Yang, Y., Boom, R., Wang, N. & Zou, Z. (2011). J. Am. Ceram. Soc. 94, 3088–3093. CrossRef CAS Google Scholar