Crystal Structure and Some Thermodynamic Properties of Ca₇MgSi₄O₁₆-Bredigite

Abstract

Bredigite with the composition Ca₇MgSi₄O₁₆ (Ca₇MgSi₄O₁₆-Bre) has been synthesized by a solid-state reaction method at 1.2 GPa and 1373 K for 7 days, and its structure has been determined by single-crystal X-ray diffraction data. Following a relevant genealogy analysis in the literature, we have refined the structure into two space groups, Pnnm and Pnn2, and found that Ca₇MgSi₄O₁₆-Bre belongs to the space group Pnnm, which can be essentially derived from the space group Pnn2 via an atomic coordinate transformation (with an average deviation of 0.039 Å only). Furthermore, some thermodynamic properties of the Ca₇MgSi₄O₁₆-Bre have been obtained in this study. Using first-principles simulations based on density functional theory, the isothermal bulk modulus has been determined as 90.6(4) GPa with a pressure derivative of 5.7(1). Using density functional perturbation technique, the phonon dispersions and vibrational density of the states (VDoS) have been calculated. The VDoS has been combined with a quasi-harmonic approximation to compute the isobaric heat capacity (C_p) and standard vibrational entropy ($S_{298}^0$), yielding C_p = 8.22(2) × 10² − 3.76(6) × 10³T^−0.5 − 1.384(4) × 10⁷T⁻² + 1.61(8) × 10⁹T⁻³ J mol⁻¹ K⁻¹ for the T range of 298-1000 K and $S_{298}^0$ = 534.1 (22) J mol⁻¹ K⁻¹.

1. Introduction

The continental crust of the Earth is rich in CaO and SiO₂, and may form some special calcium silicates while it is subducted to certain depths of the mantle. Recently larnite-structured Ca₂SiO₄₋, walstromite-structured CaSiO₃, and titanite-structured CaSi₂O₅ were found as mineral inclusions in diamonds from the mantle [1,2]. While residing in the mantle, the continental crust material may interact with the surrounding MgO-rich mantle [3] and produce some special calcium magnesium silicates such as merwinite (Ca₃MgSi₂O₈), diopside (CaMgSi₂O₆), monticellite (CaMgSiO₄), and akermanite (Ca₂MgSi₂O₇). Among these calcium magnesium silicates, merwinite has been discovered as mineral inclusions in some diamonds from São Luiz, Brazil [4], suggesting that the interaction between the subducted CaO- and SiO₂-rich continental crust material and the MgO-rich mantle material happens indeed.

Bredigite, with a typical composition Ca₇MgSi₄O₁₆, is such a potential calcium magnesium orthosilicate, which may form by the interaction between the subducted Ca-rich continental material and the MgO-rich mantle material at appropriate P–T conditions. The word “bredigite”, abbreviated as Bre hereafter, was first used by Tilley and Vincent [5] to describe a mineral coexisting with spurrite, larnite, and gehlenite in the contact zone of Chalk and Tertiary dolerite at Scawt Hill, Northern Ireland. It caused substantial confusion in terms of chemical composition, phase stability, and crystallographic structure [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. Thanks to enormous effort along all these years, it has become generally clear that the Bre from Scawt Hill is compositionally identical to "Phase T" experimentally observed by Gutt [7,8], with mutual substitution of Ca and Mg to some small extents [17,20] but without any significant Ba in the structure [11,14,16,19]. It has also become clear that Bre is stable up to ~1372 °C at ambient P [8,9,10,12,17]. Recently Xiong [20] experimentally demonstrated that Bre is stable at ~1.2 GPa.

The structure of Bre, however, remains unclear. A large number of studies provided powder X-ray diffraction patterns, but none generated single-crystal X-ray diffraction data [7,10,17,19]. It was proposed by Tilley and Vincent [5] that Bre might be structurally identical to a synthetic Ba-bearing calcium magnesium orthosilicate phase observed in some spiegeleisen slags, on the basis of the similarities of their morphologies and optical properties (compositionally approximating Ca_6.36Ba_0.32Mg_1.24Mn_0.36Si₄O₁₆). Accordingly, two single-crystal X-ray diffraction investigations were conducted with some crystals extracted from the spiegeleisen slags [6,13]. However, the synthetic crystals used in these studies contained a trigonal phase with a volume fraction up to ~20%, which substantially reduced the quality of the X-ray data. Additionally, it was rather unclear how the incorporation of Ba into the crystals might have distorted the structure, as Ba is significantly larger than Ca (e.g., the cation radius of 12-fold Ba is 1.61 Å whereas that of 12-fold Ca is 1.34 Å [21]). Indeed, Moseley and Glasser proposed different structures for them, with the Ba-bearing phase crystallizing in the space group Pnn2 and the Ba-free phase in the space group Pmnn [17].

Along with the geological significance of Bre, Bre is not only an excellent bioactive ceramic [22] but also a high-quality phosphor candidate emitting special lights when doped with some particular rare earth elements [23]. A good understanding of the detailed structure of Bre will certainly benefit all these research fields.

Here, we synthesized the Ca₇MgSi₄O₁₆-Bre by a solid-state method at high-temperature and high-pressure conditions, and determined its structure using single-crystal X-ray diffraction method. In addition, we theoretically calculated some of its thermodynamic properties using first-principles simulations. Before this study, there were no thermodynamic data for the Ca₇MgSi₄O₁₆-Bre [24].

2. Experimental and Theoretical Methods

2.1. Synthesis

The Ca₇MgSi₄O₁₆-Bre investigated in this study was synthesized by a solid-state reaction method at high P–T conditions. The starting material was made as follows: (1) 99.9% pure chemical powders CaCO₃, MgO, and SiO₂ from Alfa Aesar(Alfa Aesar, Haverhill, MA, USA)were pretreated at 1 atm and 723 K for 72 hours to remove any absorbed moisture, immediately weighed in a mole ratio of 7:1:4, and then ground and homogenized in an agate mortar; (2) the resulting mixture was pressed into a pellet and degassed in a Pt crucible at 1 atm and 1273 K for 48 hours; (3) the degassed pellet was crushed and ground into a fine powder, which was further stored in a drying oven at 383 K and used as the starting material for a later synthesizing experiment. The starting material was loaded in a Pt capsule (10 mm in length and 2.5 mm in diameter) with both ends sealed using an arc-welding technique. The synthesizing experiment was carried out with a non-end loaded piston–cylinder apparatus installed at the High-Pressure Laboratory of Peking University (Quickpress 3.0, Depths of The Earth Company, Cave Creek, AZ, USA) [25]. The experimental assembly and high-P experimental technique were generally identical to those reported in Liu and Fleet [26]. The experimental P–T conditions were 1.2 GPa and 1373 K, and the heating time was 7 days.

2.2. Characterization

The high-P experimental product was polished with a series of diamond pastes, cleaned in alcohol using an ultrasonic washing machine, carbon-coated, and then examined for texture and composition respectively with a scanning electron microscope (Quanta 650 FEG, FEI Company, Hillsboro, OR, USA) and an electron microprobe (JXA-8100, JEOL, Tokyo, Japan). It was found out that the product contained a single crystalline phase with grain sizes ranging from 50–100 microns (Supplementary Materials Figure S1). Moreover, 13 electron microprobe analyses performed on randomly selected grains suggested the following chemical formula Ca_7.01(2)Mg_1.01(1)Si_3.99(1)O₁₆, closely matching the targeted formula Ca₇MgSi₄O₁₆. The product was eventually slightly pulverized, and some loose grains with appropriate sizes and shapes were hand-picked for a later single-crystal X-ray diffraction experiment. The rest of the sample was ground down to a fine powder and analyzed with a powder X-ray diffractometer (D/Max 2550 V/PC with graphite-monochromated Cu Kα radiation, Rigaku Corporation, Tokyo, Japan) at ambient P–T conditions. The powder XRD pattern of the Ca₇MgSi₄O₁₆-Bre, as shown in Figure 1, is consistent with the simulated XRD pattern based on the single-crystal structural analysis (more later), indicating that they are isostructural.

2.3. Single-Crystal X-Ray Diffraction

The single crystals were immersed in oil and examined under an optical microscope. A suitable crystal was then selected for the single-crystal X-ray diffraction analysis. The intensity data were collected on a Bruker Smart Apex III micro-focused diffractometer using Mo Kα radiation (λ = 0.71073 nm). The raw data were processed and corrected for the absorption effects using the programs SAINT+ and SADAB. An initial structure solution was obtained via direct methods and refined by a full-matrix least-squares method using the SHELXT software included in the SHELXTL package. All heaviest atoms were first located unambiguously in the Fourier maps, and the O atoms were later found in the subsequent difference Fourier maps. All atoms were refined with the anisotropic displacement parameters. For the atoms with I > 2sigma(I), their atomic coordinates and anisotropic thermal parameters were included in the final cycles of the least-squares refinement.

2.4. Computational Method

The first-principles simulations carried out to investigate the compressibility of the Ca₇MgSi₄O₁₆-Bre were deployed with the CASTEP code using density functional theory [27,28] and planewave pseudopotential technique [29]. The exchange–correlation interaction was treated by the generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) functional [30], and a convergence criterion of 5 × 10⁻⁷ eV/atom was used in the self-consistent field calculations. We employed a planewave basis set with a cutoff of 990 eV to expand the electronic wave functions, and a norm-conserving pseudopotential to model the ion–electron interaction [31,32]. We sampled the irreducible Brillouin zone with a 2 × 6 × 4 Monkhorst-Pack grid [33]. The effects of using larger cutoff and k point mesh on the calculated properties were found to be insignificant. The computation cell contained four Ca₇MgSi₄O₁₆-Bre molecules (in total 112 atoms), with the initial structure model from our single-crystal X-ray analysis. The equilibrium lattice parameters and internal coordinates at different pressures were optimized by minimizing the Hellmann–Feynman force on the atoms and simultaneously achieving the desired hydrostatic stress tensor. These theoretical techniques were used in our previous studies for the structures and thermodynamics of some silicate minerals [34,35].

Based on the optimized structure, the phonon dispersions and vibrational density of the states (VDoS) of the Ca₇MgSi₄O₁₆-Bre were further calculated by diagonalizing the dynamical matrix with density functional perturbation theory [36,37]. The q-vector grid spacing for interpolation was 0.05 Å⁻¹, which represented the average distance between the Monkhorst-Pack q-points used in the dynamical matrix calculations. The phonon dispersions were obtained at the high-symmetry points (G, Z, T, Y, S, X, U, R). The coordinates of these points on the surface of the Brillouin zone were G = (0 0 0), Z = (0 0 ¹/₂), T = (−¹/₂ 0 ¹/₂), Y = (−¹/₂ 0 0), S = (−¹/₂ ¹/₂ 0), X = (0 ¹/₂ 0), U = (0 ¹/₂ ¹/₂), R = (−¹/₂ ¹/₂ ¹/₂).

3. Results and Discussions

3.1. Structural Analysis

Moore and Araki [13] collected single-crystal X-ray diffraction data from the Ba-rich, Bre-like phase (Ca_6.15Ba_0.3Mg_1.2Mn_0.35Si₄O₁₆) extracted from the spiegeleisen slags and found its space group to be Pnn2. Taking into account the crystal chemistry of this phase, they performed a genealogy analysis of some phases in the CaO–MgO–SiO₂ system and proposed that the ideal Bre with the Ca₇MgSi₄O₁₆ composition might adopt the space group Pnnm.

Using our single-crystal X-ray diffraction data, we solved the structure in the space group Pnnm, with a = 18.3434(17) Å, b = 6.7313(8) Å, c = 10.8844(12) Å, α = 90°, β = 90°, γ = 90°, and V = 1344.0(3) ų (

Table 1. Crystal data for bredigite structure and bredigite-like structure.

Empirical Formula	Ca7MgSi4O16 a	Ca14Mg2Si8O32 b	Ca6.15Ba0.3Mg1.2Mn0.35Si4O16 c
Crystal system	Orthorhombic	Orthorhombic	Orthorhombic
Space group	Pnnm	Pnn2	Pnn2
Unit cell dimensions (Å)	a = 18.3434(17)	a’ = 18.343(4)	10.909(9)
	b = 6.7313(8)	b’ = 6.7313(13)	18.34(1)
	c = 10.8844(12)	c’ = 10.884(2)	6.739(9)
Volume (Å3)	1344.0(3)	1344.0(5)	1348(4)
Z	4	2	/
Crystal size (mm3)	0.06 × 0.05 × 0.05	0.06 × 0.05 × 0.05	/
Theta range for data collection (°)	2.18 to 28.37	2.18 to 28.37	/
Limiting indices	−24 ≤ h ≤ 24	−24 ≤ h ≤ 24	/
	−9 ≤ k ≤ 5	−9 ≤ k ≤ 5	/
	−13 ≤ k ≤ 14	−13 ≤ k ≤ 14	/
Independent reflections	1756 [R(int) = 0.0483]	3101 [R(int) = 0.0444]	/
Completeness	99.50%	99.50%	/
Refinement method	Full-matrix least-squares on F2	Full-matrix least-squares on F2	/
Data/restraints/parameters	1756/0/147	3101/1/255	/
Goodness-of-fit on F2	1.003	1.064	/
Final R indices [I > 2sigma(I)]	R1 = 0.0472, wR2 = 0.1132	R1 = 0.0486, wR2 = 0.1095	/
R indices (all data)	R1 = 0.0675, wR2 = 0.1259	R1 = 0.0753, wR2 = 0.1279	/
Largest diff. peak and hole (e Å−3)	0.877 and −0.720	0.758 and −0.752	/

^a Data refined in space group Pnnm. ^b Data refined in space group Pnn2. ^c Moore and Araki [13], original formula Ca_24.6Ba_1.2Mg_4.8Mn_1.4[SiO₄]₁₆.

). In this structure, each asymmetric unit contains two distinct Mg sites, three distinct Si sites, and six distinct Ca sites (for their coordinates and equivalent isotropic displacement parameters, see

Table 2. Atomic coordinates (x, y, z) and equivalent isotropic displacement parameters (U(eq)) for Ca₇MgSi₄O₁₆-Bre (Pnnm).

Atom Label	x	y	z	U(eq)
Ca(1)	0.5000	0.5000	1.0000	18(1)
Ca(2)	0.5000	0.5000	0.5000	41(1)
Ca(3)	0.3328(1)	0.8028(2)	1.0000	15(1)
Ca(4)	0.6733(1)	0.1718(2)	0.5000	14(1)
Ca(5)	0.4119(1)	0.1582(2)	0.7486(1)	15(1)
Ca(6)	0.7726(1)	0.9989(2)	0.2316(1)	10(1)
Mg(1)	0.5000	1.0000	1.0000	9(1)
Mg(2)	0.5000	1.0000	0.5000	8(1)
O(1)	0.2887(2)	0.2570(6)	1.1220(4)	14(1)
O(2)	0.6968(2)	0.8473(6)	0.3803(4)	16(1)
O(3)	0.4757(2)	0.7809(6)	0.8606(4)	16(1)
O(4)	0.3389(2)	0.8100(6)	0.7862(4)	13(1)
O(5)	0.3760(3)	0.4959(8)	1.0000	14(1)
O(6)	0.4115(2)	0.4895(6)	0.7191(4)	19(1)
O(7)	0.3978(3)	0.1045(8)	1.0000	18(1)
O(8)	0.5764(3)	0.7718(8)	0.5000	15(1)
O(9)	0.6946(3)	0.5126(8)	0.5000	18(1)
O(10)	0.4444(2)	0.8409(7)	0.6295(4)	25(1)
Si(1)	0.3382(1)	0.2828(3)	1.0000	8(1)
Si(2)	0.6638(1)	0.7355(3)	0.5000	9(1)
Si(3)	0.4192(1)	0.7194(2)	0.7518(1)	9(1)

). As shown in Figure 2a, both Mg atoms are coordinated to six bridging O atoms to form [MgO₆] octahedra. The Mg1–O bond lengths vary from 2.003(4) to 2.163(4) Å, the Mg2–O bond lengths from 2.044(4) to 2.080(6) Å, and the O–Mg–O angles from 87.33(15)° to 180.000(1)° (

Table 3. Selected bond lengths (Å) and angles (°) for Ca₇MgSi₄O₁₆ (Pnnm).

Bond a	Bond Length	Bond a	Bond Angle
Ca(1)–O(5)	2.275(5)	O(7)#7–Mg(1)–O(7)#1	180.000(1)
Ca(1)–O(5)#1	2.275(5)	O(7)#7–Mg(1)–O(3)	92.67(15)
Ca(1)–O(3)#2	2.465(4)	O(7)#1–Mg(1)–O(3)	87.33(15)
Ca(1)–O(3)#3	2.465(4)	O(7)#7–Mg(1)–O(3)#18	87.33(15)
Ca(1)–O(3)#1	2.465(4)	O(7)#1–Mg(1)–O(3)#18	92.67(15)
Ca(1)–O(3)	2.465(4)	O(3)–Mg(1)–O(3)#18	180.000(1)
Ca(2)–O(8)#5	2.305(5)	O(7)#7–Mg(1)–O(3)#3	92.67(15)
Ca(2)–O(8)	2.305(5)	O(7)#1–Mg(1)–O(3)#3	87.33(15)
Ca(2)–O(10)	2.880(4)	O(3)–Mg(1)–O(3)#3	89.1(2)
Ca(2)–O(10)#6	2.880(4)	O(3)#18–Mg(1)–O(3)#3	90.9(2)
Ca(2–O(10)#5	2.880(4)	O(7)#7–Mg(1)–O(3)#19	87.33(15)
Ca(2)–O(10)#2	2.880(4)	O(7)#1–Mg(1)–O(3)#19	92.67(15)
Ca(2)–O(6)#6	2.886(4)	O(3)–Mg(1)–O(3)#19	90.9(2)
Ca(2)–O(6)#2	2.886(4)	O(3)#18–Mg(1)–O(3)#19	89.1(2)
Ca(2)–O(6)	2.886(4)	O(3)#3–Mg(1)–O(3)#19	180.000(1)
Ca(2)–O(6)#5	2.886(4)	O(10)#6–Mg(2)–O(10)#19	180.0(2)
Ca(3)–O(5)	2.212(6)	O(10)#6–Mg(2)–O(10)	87.3(3)
Ca(3)–O(4)	2.330(4)	O(10)#19–Mg(2)–O(10)	92.7(3)
Ca(3)–O(4)#3	2.330(4)	O(10)#6–Mg(2)–O(10)#15	92.7(3)
Ca(3)–O(7)#7	2.355(6)	O(10)#19–Mg(2)–O(10)#15	87.3(3)
Ca(3)–O(9)#8	2.823(6)	O(10)–Mg(2)–O(10)#15	180
Ca(3)–O(2)#8	2.989(4)	O(10)#6–Mg(2)–O(8)	87.11(16)
Ca(3)–O(2)#9	2.989(4)	O(10)#19–Mg(2)–O(8)	92.89(16)
Ca(4)–O(9)	2.327(6)	O(10)–Mg(2)–O(8)	87.11(16)
Ca(4)–O(1)#10	2.544(4)	O(10)#15–Mg(2)–O(8)	92.89(16)
Ca(4)–O(1)#11	2.544(4)	O(10)#6–Mg(2)–O(8)#15	92.89(16)
Ca(4)–O(2)#4	2.580(4)	O(10)#19–Mg(2)–O(8)#15	87.11(16)
Ca(4)–O(2)#12	2.580(4)	O(10)–Mg(2)–O(8)#15	92.89(16)
Ca(4)–O(10)#5	2.579(5)	O(10)#15–Mg(2)–O(8)#15	87.11(16)
Ca(4)–O(10)#2	2.579(5)	O(8)–Mg(2)–O(8)#15	180.000(1)
Ca(5)–O(6)	2.253(4)	O(5)–Si(1)–O(1)#3	109.92(18)
Ca(5)–O(3)#2	2.430(4)	O(5)–Si(1)–O(1)	109.92(18)
Ca(5)–O(2)#5	2.439(4)	O(1)#3–Si(1)–O(1)	110.3(3)
Ca(5)–O(10)#4	2.568(5)	O(5)–Si(1)–O(7)	111.9(3)
Ca(5)–O(4)#4	2.730(4)	O(1)#3–Si(1)–O(7)	107.37(19)
Ca(5)–O(1)#3	2.745(4)	O(1)–Si(1)–O(7)	107.37(19)
Ca(5)–O(8)#5	2.7554(16)	O(9)–Si(2)–O(8)	119.3(3)
Ca(5)–O(7)	2.7718(15)	O(9)–Si(2)–O(2)#6	107.6(2)
Ca(5)–O(10)#2	2.938(5)	O(8)–Si(2)–O(2)#6	107.40(19)
Ca(6)–O(1)#13	2.308(4)	O(9)–Si(2)–O(2)	107.6(2)
Ca(6)–O(2)	2.365(4)	O(8)–Si(2)–O(2)	107.40(19)
Ca(6)–O(1)#14	2.377(4)	O(2)#6–Si(2)–O(2)	106.9(3)
Ca(6)–O(4)#15	2.424(4)
Ca(6)–O(4)#16	2.482(4)
Ca(6)–O(6)#16	2.553(4)
Ca(6)–O(9)#17	2.5931(17)
Ca(6)–O(2)#17	2.701(4)
Mg(1)–O(7)#7	2.003(5)
Mg(1)–O(7)#1	2.003(5)
Mg(1)–O(3)	2.163(4)
Mg(1)–O(3)#18	2.163(4)
Mg(1)–O(3)#3	2.163(4)
Mg(1)–O(3)#19	2.163(4)
Mg(2)–O(10)#6	2.044(4)
Mg(2)–O(10)#19	2.044(4)
Mg(2)–O(10)	2.044(4)
Mg(2)–O(10)#15	2.044(4)
Mg(2)–O(8)	2.080(6)
Mg(2)–O(8)#15	2.080(6)
Si(1)–O(5)	1.594(6)
Si(1)–O(1)#3	1.619(4)
Si(1)–O(1)	1.619(4)
Si(1)–O(7)	1.623(6)
Si(2)–O(9)	1.603(6)
Si(2)–O(8)	1.622(6)
Si(2)–O(2)#6	1.622(4)
Si(2)–O(2)	1.622(4)
Si(3)–O(3)	1.627(4)
Si(3)–O(4)	1.637(4)
Si(3)–O(6)	1.594(4)
Si(3)–O(10)	1.629(4)

^a Symmetry transformations used to generate equivalent atoms: #1 −x + 1, −y + 1, −z + 2; #2 −x + 1, −y + 1, z; #3 x, y, −z + 2; #4 x, y − 1, z; #5 −x + 1, −y + 1, −z + 1; #6 x, y, −z + 1; #7 x, y + 1, z; #8 x − 1/2, −y + 3/2, z + 1/2; #9 x − 1/2, −y + 3/2, −z + 3/2; #10 x + 1/2, −y + 1/2, z − 1/2; #11 x + 1/2, −y + 1/2, −z + 3/2; #12 x, y − 1, −z + 1; #13 x + 1/2, −y + 3/2, −z + 3/2; #14 −x + 1, −y + 1, z − 1; #15 −x + 1, −y + 2, −z + 1; #16 x + 1/2, −y + 3/2, z − 1/2; #17 −x + 3/2, y + 1/2, −z + 1/2; #18 −x + 1, −y + 2, −z + 2; #19 −x + 1, −y + 2, z.

). All the Si sites are in 4-fold coordination, with the Si–O bond lengths varying from 1.594(4) to 1.637(4) Å (Table 3). Further, every [SiO₄] tetrahedron shares two O corners with two neighboring [MgO₆] octahedra, leading to the formation of an [Mg–O–Si] chain, which runs along the c-axis (Figure 2b). The [Mg–O–Si] chains are the basic structural units of the Ca₇MgSi₄O₁₆-Bre. The Ca sites are much more complicated (Table 3): Ca1, 6-coordinate (Ca1–O ranging from 2.275(5) to 2.465(4) Å); Ca2, 10-coordinate (Ca2–O ranging from 2.305(5) to 2.886(4) Å); Ca3, 7-coordinate (Ca3–O ranging from 2.212(6) to 2.989(4) Å); Ca4, 7-coordinate (Ca4–O ranging from 2.327(6] to 2.579(5) Å); Ca5, 9-coordinate (Ca5–O, ranging from 2.253(4] to 2.938(5) Å); Ca6, 8-coordinate (Ca6–O ranging from 2.308(4) to 2.701(4) Å).

We also attempted to solve the structure in the space group Pnn2, with the unit-cell parameters a′, b′, and c′, respectively, equivalent to the unit-cell parameters a, b, and c of the Pnnm space group (Table 1). The resulting atomic coordinates and equivalent isotropic displacement parameters are listed in Supplementary Table S1. In the Pnn2 model, the atoms Ca1, Ca2, Mg1, and Mg2 deviate from symmetric centers in the Pnnm model (Figure 3), which splits the mirror plane and lowers the symmetry from Pnnm to Pnn2. Nevertheless, the atom arrangement in the Pnn2 model is very similar to that in the Pnnm model. We therefore used the PLATON software to exam the relations between these two structural models. The result suggests that the Pnn2 model can transform to the Pnnm model via the following matrix operation:

$$\left(\begin{array}{ccc}{x}^{\prime }& y^{\prime }& z^{\prime }\end{array}\right)=\left(\begin{array}{ccc}x& y& z\end{array}\right)\times \left[\begin{array}{ccc}-1.0000& 0.0000& 0.0000\\ 0.0000& -1.0000& 0.0000\\ 0.0000& 0.0000& 1.0000\end{array}\right]+\left(\begin{array}{ccc}0.0000& 0.0000& 0.2344\end{array}\right)$$

(1)

The average deviation of the atom positions in the two Pnnm models is just 0.039 Å. This indicates that the space group of the Ca₇MgSi₄O₁₆-Bre is Pnnm indeed.

It is thus clear that due to substantial amounts of Ba²⁺ and Mn²⁺ substitutions, respectively, for Ca²⁺ and Mg²⁺, the atoms on the Ca2, Mg1, and Mg2 sites of the Pnnm model can significantly depart from high-symmetry sites, leading to the space group Pnn2, as observed by Moore and Araki [13]. Recently, a natural Mn-free Bre with the composition Ca_7.006Na_0.015Ba_0.014Mg_0.938Si_4.000P_0.014O₁₆ was reported from the Hatrurim Complex, Negev Desert, Israel [38], and its space group was determined as Pnnm as well. This suggests that small amounts of impurities such as Na, Ba, and P incorporated into the structure do not alter the basic crystallographic features of the Ca₇MgSi₄O₁₆-Bre. As a result, it is important to investigate the physical–chemical properties of Ca₇MgSi₄O₁₆-Bre.

3.2. Isothermal Bulk Modulus, Heat Capacity, and Entropy

The unit-cell parameters of the Ca₇MgSi₄O₁₆-Bre at zero pressure obtained from the first-principles calculation are summarized in

Table 4. Comparison between energy-optimized (0 P and 0 K) and experimentally refined (1 atm and 298 K) unit-cell parameters of the Ca₇MgSi₄O₁₆-Bre (Pnnm).

Cell Parameters	Experimental	Calculated	R.D. (%) a
a	18.3434(17) Ǻ	18.5125 Ǻ	~0.92
b	6.7313(8) Ǻ	6.7833 Ǻ	~0.77
c	10.8844(12) Ǻ	11.0067 Ǻ	~1.12
V	1344.0(3) Ǻ3	1382.18 Ǻ3	~2.84

^a Relative difference.

. They compare well with the results obtained from the single-crystal X-ray diffraction data.

The E–V data from our static calculation for the Ca₇MgSi₄O₁₆-Bre are shown in Figure 4. To determine the isothermal bulk modulus, the third-order E–V Birch–Murnaghan equation of state [39] was fitted with the E–V data by a least-squares method:

$$E\left(V\right)=E_0+\frac{9B_0{V}_0}{16}{\left[{\left(\frac{V_0}{V}\right)}^{\frac{2}{3}}-1\right]}^2\times \left[\left(B_0^{\prime }-4\right)\times {\left(\frac{V_0}{V}\right)}^{\frac{2}{3}}-B_0^{\prime }+6\right]$$

(2)

where E is the free energy, B₀ is the isothermal bulk modulus, $B_0^{\prime }$ is the first pressure derivative of B₀, V₀ is the volume at zero pressure, and V are the volumes at other pressures. When $B_0^{\prime }$ is set as 4, B₀ is 93.1(22) GPa and V₀ is 1394.2(16) ų. If $B_0^{\prime }$ is not fixed, B₀ is 90.6(4) GPa, $B_0^{\prime }$ is 5.7(1), and V₀ is 1388.0(6) ų.

We calculated the phonon dispersions and VDoS. The dynamical matrices were computed at 21 wavevectors in the Brillouin zone of the cell and interpolated to obtain the bulk phonon dispersions. Figure 5 shows the dispersion curves along several symmetry directions and the VDoS. Clearly, the Ca₇MgSi₄O₁₆-Bre in the space group Pnnm is dynamically stable.

The phonon spectrum for the Ca₇MgSi₄O₁₆-Bre has been used to compute the internal energy (E) and isochoric heat capacity (C_v) as functions of temperature. The temperature dependence of the E was obtained by the following equation,

$$E\left(T\right)=E_{tot}+E_{zp}+{\displaystyle \int }\frac{h\omega }{exp\left(\frac{h\omega }{kT}\right)-1}F\left(\omega \right)d\omega$$

(3)

with E_tot representing the total electronic energy at 0 K, E_zp the zero-point vibrational energy, h the Planck’s constant, k the Boltzmann constant, and F(ω) the vibrational density of states. We evaluated the E_zp term in Equation (3) using the following equation,

$$E_{zp}={\displaystyle \int }F\left(\omega \right)h\omega d\omega$$

(4)

Furthermore, we approximated the lattice contribution to C_v with the following equation:

$$C_v\left(T\right)=k{\displaystyle \int }\frac{{\left(\frac{h\omega }{kT}\right)}^{2}exp\left(\frac{h\omega }{kT}\right)}{{\left[{\left(\frac{h\omega }{kT}\right)}^2-1\right]}^2}F\left(\omega \right)d\omega$$

(5)

The C_v result calculated in this way is shown in

Table 5. Thermal expansion coefficient (×10⁵ K⁻¹), and heat capacity (J mol⁻¹ K⁻¹) of Ca₇MgSi₄O₁₆-Bre at selected T (K).

T (K)	CV (J mol−1 K−1)	CP (J mol−1 K−1)	αT (×105 K−1)
50	77.36	78.74	0.1048
100	221.73	224.49	0.6847
150	331.46	335.61	1.4269
200	409.8	415.35	1.9949
250	467.65	474.6	2.3858
300	511.55	519.9	2.6588
350	545.31	555.07	2.8590
400	571.5	582.67	3.0141
450	592.01	604.6	3.1394
500	608.23	622.25	3.2483
550	621.19	636.64	3.3451
600	631.67	648.55	3.4340
650	640.22	658.55	3.5179
700	647.27	667.05	3.5991
750	653.15	674.37	3.6775
800	658.08	680.76	3.7579
850	662.26	686.4	3.8385
900	665.82	691.44	3.9203
950	668.89	695.97	4.0040
1000	671.54	700.1	4.0903

.

C_p was calculated by adding an anharmonic effect to C_v obtained from the above calculations, using the following equation:

$$C_p=C_v+{\alpha }_T^2{B}_T{V}_{T}T$$

(6)

where α_T, B_T, and V_T are the thermal expansion coefficient, isothermal bulk modulus, and volume at 1 atm and T (K), respectively. The thermal expansion coefficient, as shown in Figure 6 (with some values at some selected temperatures listed in Table 5), was calculated by the Gibbs software through a quasi-harmonic Debye model [40]. The temperature derivative of B_T was assumed as zero in our calculation. V_T at T K was calculated with the following equation:

$$V_T=V_{298}exp\left({{\displaystyle \int }}_{298}^T{\alpha }_{T}dT\right)$$

(7)

where V₂₉₈ = 809.1(2) cm³/mol. The calculated C_p values are listed in Table 5, and compared to the C_v values in Figure 7. The C_p values are divided into three T ranges, 10–100, 100–298, and 298–1000 K, and empirically fitted to the equation from [24], with the coefficients summarized in

Table 6. Coefficients of the C_P polynomials of Ca₇MgSi₄O₁₆-Bre (Pnnm).

T Range: 10–100 K	T Range: 100–298 K	T Range: 298–1000 K
k0 = 2.8(6)	k0 = −1.24(2) × 102	k0 = 8.22(2) × 102
k4 = −7.0(6) × 10−1	k4 = 4.50(3)	k1 = −3.76(6) × 103
k5 = 5.9(2) × 10−2	k5 = −1.16(2) × 10−2	k2 = −1.38(4) × 107
k6 = − 3.0(1) × 10−4	k6 = 1.23(3) × 10−5	k3 =1.61(8) × 109

^aCp = k₀ + k₁T^−0.5 + k₂T⁻² + k₃T⁻³ + k₄T + k₅T² + k₆T³ (J mol⁻¹ K⁻¹).

.

The C_p values have been applied to the calculation of the vibrational entropy at T K using the following equation,

$$S_T^0={{\displaystyle \int }}_0^T\frac{C_p}{T}dT$$

(8)

The vibrational entropy of the Ca₇MgSi₄O₁₆-Bre at 298 K ($S_{298}^0$) is calculated as 534.1 (22) J mol⁻¹ K⁻¹.

4. Conclusions

Ca₇MgSi₄O₁₆-Bre has been successfully synthesized at 1.2 GPa and 1373 K, and its structure has been determined by single-crystal X-ray diffraction data. We find that the Ca₇MgSi₄O₁₆-Bre has the space group Pnnm. Although this space group may be reduced to the space group Pnn2 as significant amounts of Ba and Mn enter the structure, it tolerates low levels of chemical impurities such as Na, Ba, and P.

Some first-principles calculations have been carried out as well. The results suggest that the isothermal bulk modulus of the Ca₇MgSi₄O₁₆-Bre is 90.6(4) GPa (with a pressure derivative of 5.7(1)), the isobaric heat capacity is C_p = 8.22(2) × 10² – 3.76(6) × 10³T^−0.5 – 1.384(4) × 10⁷T⁻² + 1.61(8) × 10⁹T⁻³ J mol⁻¹ K⁻¹ (298–1000 K), and the standard vibrational entropy is $S_{298}^0$ = 534.1 (22) J mol⁻¹ K⁻¹.