Low reactivity of stoichiometric FeS with hydrogen at high-pressure and high-temperature conditions

Abstract

Hydrogenation of iron sulfide (FeS) under high-pressure and high-temperature conditions has attracted attention because hydrogen and sulfur are promising candidates as light elements in the cores of the Earth and other terrestrial planets. In earlier reports describing the hydrogenation of FeS, the chemical compositions of starting materials were not fully clarified. This study reports in-situ neutron and X-ray diffraction measurements under high-pressure and high-temperature conditions of an Fe-S-H system using a stochiometric Fe_1.000S (troilite) as a starting material. The site occupancies of hydrogen atoms in FeS, estimated by Rietveld refinement of neutron diffraction patterns collected at about 5 GPa, were 0.014(2) at 700 K and 0.024(2) at 1000 K. The hydrogen occupancy at 900 K and 18.2 GPa was estimated as 0.067(6) from the unit-cell volume determined by X-ray diffraction using the hydrogen-induced volume expansion calculated from first-principles calculations. These occupancies were significantly lower than those reported from earlier studies, indicating that the hydrogenation of FeS can be affected strongly by the stoichiometry of iron sulfide.

INTRODUCTION

Iron sulfide, Fe_1−δS, has a very complex phase diagram depending on temperature and the amount of iron in a system under ambient pressure. Stoichiometric or near stoichiometric Fe_1−δS (0 ≤ δ ≤ 0.05), called troilite, has a NiAs superstructure and belongs to the P62c space group. With iron deficiency (δ) greater than 0.05, iron sulfide called pyrrhotite becomes stable. In general, pyrrhotite has a hexagonal or monoclinic structure. Both structures of pyrrhotite are ferromagnetic, whereas troilite is anti-ferromagnetic (Wang and Salveson, 2005). In the region where the atomic concentration of Fe is close to 50% at room temperature, hexagonal pyrrhotite can coexist with troilite, but monoclinic pyrrhotite cannot (e.g., Nakazawa and Morimoto, 1970; Kissin and Scott, 1982).

FeS, which is found universally in meteorites with Fe-Ni alloy, is one of the plausible components of terrestrial planet cores. The phase relations of FeS under high-pressure and high-temperature conditions were first reported by Fei et al. (1995), after which phase diagrams were modified several times (Kusaba et al., 1998; Kavner et al., 2001; Urakawa et al., 2002; Urakawa et al., 2004). The phase diagram of FeS up to 22 GPa and 1600 K is now well defined. Also, the phase stability of FeS at high pressure was investigated using first-principles calculations (Martin et al., 2001; Terranova and de Leeuw, 2017). In addition, FeS VI (non-magnetic MnP phase) was found at pressures higher than 40 GPa (Ono et al., 2008); B2 phase of FeS was found above 180 GPa (Sata et al., 2008). The phase relations of FeS in planetary core conditions are now well known.

Several light elements are believed to be dissolved in the core of the Earth and other terrestrial planets (e.g., Birch, 1964). Hydrogen is a plausible candidate as a light element dissolved in the Earth’s and other planets’ cores. Hydrogen, the most abundant element in the solar system, decreases the density of iron considerably by forming iron hydrides FeH_x under high-pressure and high-temperature conditions (Stevenson, 1977). In-situ X-ray diffraction studies of iron hydrides have been conducted to determine the hydrogen solubility, compressibility and phase diagram of FeH_x (e.g., Fukai and Akimoto, 1983; Sakamaki et al., 2009; Tagawa et al., 2022). Crystal structures of iron hydrides (deuterides) including site occupancies and atomic positions of hydrogen (deuterium) were reported from in-situ neutron diffraction measurements at high pressure and high temperature (Machida et al., 2014; Ikuta et al., 2019; Machida et al., 2019; Saitoh et al., 2020b). In addition, recent neutron diffraction measurements have revealed that the substitution of Ni and Si for metallic Fe notably changes hydrogen-induced volume expansion (Mori et al., 2021; Shito et al., 2023; Mori et al., 2024).

Because FeS contains 50 atomic percent of sulfur, the volume expansion effect of hydrogenation, the decrease in melting temperature, and the hydrogen position in hydrogenated FeS might be markedly different from that in pure Fe. Shibazaki et al. (2011) first reported the hydrogenation of FeS from volume expansion and decreased melting temperature. Recently, Abeykoon et al. (2023) reported the crystal structure of hydrogenated phase V of FeS (hereinafter FeS V) by in-situ neutron diffraction measurements. However, these studies did not clearly show the chemical compositions of the starting materials. The Fe deficit in troilite or the coexistence of hexagonal pyrrhotite can decrease the unit cell volume of the starting materials and might lead to uncertainty in hydrogen-induced volume expansion at high pressure and high temperature. We therefore conducted high-pressure and high-temperature experiments using stoichiometric FeS (troilite) as the starting material.

METHODS

Sample preparation

Powdered FeS (troilite) was synthesized by referring to earlier studies (e.g., Morimoto, 1976; King and Prewitt, 1982). Pellets of a mixture of Fe (99.9% purity, Wako Pure Chemical Industries Co., Ltd.) and S (99.99% purity; Kojundo Chemical Lab. Co., Ltd.) powders with an atomic ratio of 103:100 were loaded in evacuated quartz tubes. The quartz tubes were heated at 690 °C in an electric furnace for two weeks and were then annealed at 150 °C for two weeks. After the recovered products were crushed, magnetic components (pyrrhotite and unreacted Fe) were removed from the products using a magnet. Figure 1 shows the X-ray diffraction pattern of the obtained sample, presenting the final product as the single phase of troilite, with no additional phase such as metallic iron, pyrrhotite, or magnetite. The unit-cell parameters of the obtained troilite were a = 5.9654(6) Å and c =11.760(3) Å, which are comparable to earlier studies (e.g., King and Prewitt, 1982). Yund and Hall (1969) reported the Fe concentration of pyrrhotite as a function of the d-spacing of (102) planes (d₁₀₂ value). Kusaba et al. (1997) applied this relation to troilite and estimated the Fe concentration using the d₁₁₄ value. Using this relation, our troilite sample was found to be Fe_1.000S.

Figure 1. Angle-dispersive X-ray diffraction pattern of starting material (troilite) at room temperature and ambient pressure.

First-principles calculations

Static (T = 0 K) first-principles calculations based on density functional theory (DFT) (Hohenberg and Kohn, 1964; Kohn and Sham, 1965) were conducted using the Quantum-ESPRESSO package (Giannozzi et al., 2009). Plane-wave basis and ultrasoft pseudopotentials generated by Vanderbilt’s scheme (Vanderbilt, 1990) were used to calculate the electronic structure and force acting on each atom. The same pseudopotentials have been used in earlier works; their quality has already been well tested (Ichikawa et al., 2014; Ichikawa and Tsuchiya, 2015, 2020). In structural relaxation calculations, the unit-cell parameters and all atomic coordinates were fully optimized until residual forces became less than 1.0 × 10⁻⁵ Ry/a.u. within the generalized gradient approximation for the exchange-correlation potential (Perdew et al., 1996). The kinetic energy cutoff for the plane-wave expansion was set to 50 Ry.

Non-spin polarized calculations were performed using FeS V and FeS V hydride structures. The FeS V structure with hydrogen atoms occupying all octahedral sites (FeSH₂ in stoichiometry) was used for the initial structure of the calculations of FeS V hydride. The first-Brillouin zone of the crystal was sampled on the 16 × 16 × 12 k-point grid with the Fermi-Dirac smearing of the electron occupation at the Fermi level. These calculation conditions ensure convergence with respect to energy and stress, respectively within 1 mRy/atom and 0.1 GPa. All computations were performed using the GRC parallel computing system installed at Ehime University.

Neutron diffraction measurements

Neutron diffraction measurements under high-pressure and high-temperature conditions were performed at BL11 (PLANET), MLF, J-PARC (Hattori et al., 2015). High pressure was generated using a six-axis multi-anvil press ‘ATSUHIME’ (Sano-Furukawa et al., 2014). The multi-anvil 6-6 (MA6-6) assembly was used. The second-stage anvils were made of tungsten carbide (WC) with a truncation edge length of 10 mm (Nishiyama et al., 2008).

In the cell assembly, a pellet of FeS was sandwiched by two deuterium sources (ND₃BD₃). Thin hexagonal BN (hBN) disks were placed in between as a separator (Fig. 2a). Deuterated ammonia borane (ND₃BD₃) was used as the hydrogen (deuterium) source. Deuteration avoids the incoherent neutron-scattering of H atoms, which produces high background. No marked isotopic effect on site occupancies of hydrogen atoms was found for the Fe-H(D) system (Ikuta et al., 2019; Machida et al., 2019). We therefore assumed that no isotopic effects exist in the FeS-H(D) system. The sample and deuterium sources were loaded into a NaCl container to prevent deuterium leakage from the assembly. A NaCl container was also used to determine the generated pressure during the experiment using the third-order Birch-Murnaghan equation of state reported by Matsui et al. (2012). The container was surrounded by the cylindrical graphite heater and was packed in 5% Cr₂O₃-doped MgO pressure medium (OMCR; Mino Ceramic Co., Ltd.). During the experiment, the sample temperature was estimated from the relation between temperature and electrical power applied to the heater, which was obtained in a separate experiment.

Figure 2. Cell assembly for (a) neutron diffraction measurements and (b) X-ray diffraction measurements.

For the experiment, the pressure was increased to 4.86 GPa. Then the temperature was increased to 1000 K to decompose ND₃BD₃ and to serve deuterium into the system. Diffraction patterns for structure refinement were obtained for 5 h at 1000 and 700 K. Structure refinement using the Rietveld method (Rietveld, 1969) was conducted using GSAS software (Larson and Von Dreele, 2004) with an EXPGUI interface (Toby, 2001). Also, D atoms were assumed to occupy only the octahedral site, which is the same structure model proposed by Shibazaki et al. (2011). Structural parameters of FeS V, including the site occupancy and the isotropic atomic displacement parameter (ADP) of the D atoms, were varied through the refinement, whereas all atomic coordinates were fixed. The preferred orientation of the c-axis of FeS was also optimized.

Synchrotron X-ray diffraction measurements

Energy-dispersive X-ray diffraction measurements were conducted at BL04B1 at SPring-8 (Utsumi et al., 1998). The diffraction angle (2θ) was fixed at 6°. High pressure was generated using a uniaxial multi-anvil press ‘SPEED Mk.II’ (Katsura et al., 2004) with DIA-type guide block system (Ohtani et al., 1989). A set of cubic anvils made of WC with 4 mm truncation edge length was used.

Figure 2b depicts the cell assembly for the X-ray diffraction measurements. Pellets of Fe + hBN (2:3 in volume), FeS + hBN (2:3 in volume), and NH₃BH₃ were packed in the NaCl capsule. Each pellet was separated by hBN disks with thickness of approximately 0.1 mm. The molar ratio of H/(Fe + FeS) was 4.0; that of FeS/Fe was 0.62. A pressure marker composed of NaCl and MgO was placed under the container. High temperature was generated using a cylindrical TiB₂ + BN heater (EBN; Denka Co. Ltd.). The sample temperature was measured by W97%Re3%-W75%Re25% thermocouple. The pressure effect on emf of the thermocouple was ignored.

First, the assembly was compressed to 19.1 GPa. Then the temperature was increased to 900 K to decompose NH₃BH₃. X-ray diffraction patterns were obtained every 2 min to observe time-dependent changes in the unit cell volume of Fe and FeS. Hydrogen content (x) can be estimated as:

  

\begin{equation*} x = \frac{V_{\text{MH}} - V_{\text{M}}}{N\Delta v_{\text{H}}}, \end{equation*}

where V_MH, V_M, Δv_H, and N respectively represent the unit-cell volume of hydrides, the unit-cell volume of non-hydrides, the hydrogen-induced volume expansion coefficient, and the number of atoms in a unit cell (e.g., Fukai, 2005). The EoS determined in earlier studies (Urakawa et al., 2004 for FeS IV; Tsujino et al., 2013 for fcc-Fe) were used to calculate V_M. The value of Δv_H for fcc-Fe is 2.21(4) (Machida et al., 2014). For our study, and for that of Shibazaki et al. (2011), the Δv_H and hydrogen solubility of FeS IV and FeS V under each pressure condition were assumed to be the same because both FeS IV (space group: P6₃mc) and FeS V (space group: P6₃/mmc) have a NiAs-type crystal structure and because the phase transition between those two phases is a second-order transition (Kusaba et al., 2000). The hydrogen contents of FeS IV and FeS V were estimated from the Δv_H obtained from first-principles calculations of hydrogenated FeS V.

RESULTS AND DISCUSSION

First-principles calculations

Figure 3 shows the hydrogen-induced volume expansion coefficient Δv_H of FeS V. The Δv_H values of FeS V calculated for this study (yellow solid circles) are shown every 5 GPa up to 30 GPa. The values of Δv_H of hcp-Fe from Caracas (2015) (blue solid line) and Machida et al. (2019) (blue triangle) are also shown for comparison. The value of Δv_H of FeS V was smaller than the Δv_H of hcp-Fe calculated by Caracas (2015). This result can be derived from the difference in chemical bonds between FeS and metallic Fe: Fe-S bonds in FeS are more covalent and can be stronger than the metallic bonds in the Fe lattice (Shibazaki et al., 2011).

Figure 3. Hydrogen-induced volume expansion Δv_H of FeS V (yellow circles) obtained from our first-principles calculations. The solid blue curve represents the Δv_H of hcp-Fe obtained from first-principles calculations (Caracas, 2015). The blue triangle denotes the Δv_H of hcp-Fe obtained by neutron diffraction experiments (Machida et al., 2019).

As described herein, the isotopic effect on the hydrogen (deuterium)-induced volume expansion coefficient was assumed to be negligible (Δv_H = Δv_D). The Δv_H calculated for this study was used to estimate the deuterium (hydrogen) content in the following discussion of our neutron (X-ray) diffraction experiments.

Neutron diffraction measurements

After the sample temperature reached 1000 K, neutron diffraction patterns of the sample were obtained at 5.35 GPa and 1000 K to track the time-dependent volume change (Fig. 4). The sample was a single phase of FeS V, which is thermodynamically stable in this P-T condition. During measurements, gradual volume expansion of FeS V was observed; completion of the volume expansion took almost 2 h. After the volume expansion stopped, long-time measurements for 5 h were conducted respectively at 5.35 GPa and 1000 K and at 4.68 GPa and 700 K.

Figure 4. Time-dependent change of the unit-cell volume of FeS V obtained by neutron diffraction measurements.

The obtained neutron diffraction patterns and simulated profiles calculated using the refinement are displayed in Figures 5a and 5b. Table 1 presents refinement results. The obtained site occupancies of deuterium were 0.024(2) at 1000 K and 0.014(2) at 700 K. The obtained deuterium occupancies were markedly smaller than those reported from earlier studies (Shibazaki et al., 2011; Abeykoon et al., 2023). Shibazaki et al. (2011) observed the hydrogen-induced volume expansion of FeS IV and FeS V using X-ray diffraction measurements. They estimated the hydrogen content of FeS IV and FeS V using the Δv_H of dhcp-FeH_x (Badding et al., 1991; Fukai, 1992) and obtained the hydrogen content x as 0.2 immediately before melting at about 5 GPa and 1500 K. Abeykoon et al. (2023) conducted neutron diffraction experiments and reported the hydrogen content x of FeS V as 0.74(9) at 6.9 GPa and 960 K.

Figure 5. Neutron diffraction patterns (blue crosses), and simulated patterns by Rietveld refinement (solid red line) (a) at 5.35 GPa and 1000 K and (b) at 4.68 GPa and 700 K. Residues are shown below.

Table 1. P-T conditions, atomic coordinates, site occupancies, and atomic displacement parameters of FeS deuterides

Phase	P (GPa)	T (K)	Atom	Site	x	y	z	Occupancy	Biso	Rwp	χ2
FeSD0.028(4) V (This study)	4.68	700	Fe	2a	0	0	0	1.0 (fixed)	2.13(2)	3.99%	3.08
a = 3.42492(4) Å			S	2c	2/3	1	1/4	1.0 (fixed)	1.07(5)
c = 5.66605(11) Å			D	4f	1/3	2/3	0.125	0.014(2)	3.3(15)
Pref. orient [001] = 0.94(2)
FeSD0.048(4) V (This study)	5.35	1000	Fe	2a	0	0	0	1.0 (fixed)	2.46(1)	2.81%	3.34
a = 3.44290(4) Å			S	2c	2/3	1	1/4	1.0 (fixed)	1.24(4)
c = 5.6668(1) Å			D	4f	1/3	2/3	0.125	0.024(2)	3.9(8)
Pref. orient [001] = 0.949(17)

The ADP values of the D atoms (B_D) obtained from our refinements were comparable to those of metal deuterides reported earlier (Machida et al., 2014; Machida et al., 2019; Saitoh et al., 2020a, 2020b). However, the obtained B_D values have large errors because of the extremely small deuterium occupancies obtained through our refinements (Fig. 6). To avoid this error, Rietveld refinements were also performed using fixed B_D values. The fixed values of B_D were 2.45 Ų at 700 K and 3.5 Ų at 1000 K, as obtained using linear interpolation from the B_D values reported from earlier in-situ neutron diffraction measurements of FeD_x and NiD_x taken at temperatures higher than 500 K, as depicted in Figure 6. Through refinements using the fixed B_D values, the site occupancies of deuterium and the other varied parameters (e.g., ADP of Fe and S atoms) changed only negligibly. In these refinements, the ADP values of Fe and S atoms were comparable to those reported by Fei et al. (1998) for non-deuterated FeS at 5.2 GPa and 530 K.

Figure 6. Atomic displacement parameter of deuterium atoms (B_D) obtained by Rietveld refinement. Circles denote the B_D in FeS V lattice determined in this study. Triangles represent B_D in Fe lattice reported by earlier studies. Squares represent the B_D in Ni lattice.

Using the Δv_D (Δv_H) obtained from the first-principles calculations, the deuterium content in FeS can be estimated from our experimentally obtained results. The Δv_D (Δv_H) at 5 GPa was 1.55 (Fig. 3). The unit-cell volumes of FeS V obtained in the first 15 min and in the last 15 min were used respectively as V_M and V_MD (V_MH). The estimated hydrogen (deuterium) content x was 0.01(2), which was comparable to the result obtained from Rietveld refinement, 0.048(4), for the data obtained at 5.35 GPa and 1000 K. Shibazaki et al. (2011) used the third-order Vinet equation of state (V₀ = 2.7 ų, K₀ = 99.7 GPa, and K₀’ = 3.98) of the Δv_H of dhcp-FeH_x and adopted it to estimate the hydrogen content of FeS IV and FeS V under each pressure condition. Using the equation of state, the Δv_D (Δv_H) of FeS V at 5.35 GPa was calculated to be 2.57, which is larger than that of our calculation. Using Δv_H (Δv_D) = 2.57, the deuterium content of FeS V was calculated to be 0.006(20) at 5.35 GPa and 1000 K. The change in the hydrogen contents [0.01(2) and 0.006(20)] was negligibly small because the observed expansion of the unit-cell volume of FeS V was very small. Consequently, the difference of the Δv_D (Δv_H) when calculating the hydrogen content of FeS V does not affect the discussion above.

Synchrotron X-ray diffraction measurements

The extremely small hydrogen (deuterium)-induced increase in the unit-cell volume of FeS V obtained from our neutron diffraction measurements might indicate that the maximum hydrogen solubility of FeS V is much smaller than the values reported in the previous studies. The maximum hydrogen content of FeS under higher pressure conditions is expected to be greater than that obtained in our neutron diffraction measurements because the chemical potential of hydrogen rapidly increases concomitantly with increasing pressure (Sugimoto and Fukai, 1992). To ascertain whether the small volume expansion observed in our neutron diffraction experiments reflected the low reactivity between Fe_1.000S and hydrogen (deuterium), X-ray diffraction patterns of FeS and Fe with the excess amount of H₂ fluid were obtained at 19.3 GPa and 900 K: conditions under which the maximum hydrogen content of FeS is expected to be much greater than at 5 GPa.

The observed phases were fcc-Fe and FeS IV. As described in the METHODS section, the Δv_H and hydrogen solubilities of FeS IV and FeS V were assumed to be equal. During the time-sliced observations, the pressure was dropped from 19.3 to 18.2 GPa. The pressure drop was caused mainly by decomposition of NH₃BH₃ and deformation of the pressure medium and gaskets. As shown in Figure 7, the observed unit-cell volume of fcc-Fe was much greater (approximately 20%) than the unit-cell volume at 18.2 GPa and 900 K calculated using the EoS reported by Tsujino et al. (2013). Consequently, volume expansion is attributable to the hydrogenation of iron. The hydrogen content of Fe was estimated using the Δv_H = 2.21(4), as reported by Machida et al. (2014). The hydrogen content (x) of fcc-Fe was 0.94(1).

Figure 7. Time evolution of the unit-cell volume of hydrogenated FeS IV (yellow circles) and fcc-Fe (gray triangles). The volume values were normalized by the EoS of their non-hydrogenated counterparts (Urakawa et al., 2004 for FeS; Tsujino et al., 2013 for fcc-Fe).

The amount of hydrogen released from NH₃BH₃ was greater than the amount of Fe and FeS in the system. The maximum hydrogen content (x) of Fe is 1. Therefore, hydrogen was not leaked from the NaCl capsule because hydrogenation of Fe occurred and hydrogen remained in the system. By contrast, the volume expansion rate of FeS IV was only 0.8% (Fig. 7): significantly lower than that of Fe. The hydrogen content (x) estimated using the Δv_H = 1.32 obtained using first-principles calculations was 0.067(6). The hydrogen content obtained in our X-ray diffraction measurements was much smaller than that estimated by Shibazaki et al. (2011) at 16.5 GPa and 1723 K (x = 0.4). The difference in Δv_H values between our estimates and the earlier study was again negligible.

As discussed above, the maximum hydrogen content of FeS V around 5 GPa is expected to be less than that around 18 GPa because of the pressure dependence of the chemical potential of hydrogen. Consequently, the maximum hydrogen content of FeS V around 5 GPa, at which our neutron diffraction experiments were performed, can be much less than earlier reported values. It can be comparable to the hydrogen (deuterium) contents obtained in our neutron diffraction measurements.

Comparison with earlier studies

Although Shibazaki et al. (2011) and Abeykoon et al. (2023) observed the volume expansion of FeS caused by hydrogenation (deuteration) under high pressure and high temperature, we were unable to observe the hydrogen (deuterium)-induced volume expansion of FeS clearly. Abeykoon et al. (2023) optimized structural parameters for a model structure of FeS deuteride, in which deuterium atoms were assumed to occupy the high-multiplicity (6h) sites. Although the deuterium occupancies seemed reasonable, their estimated values of the atomic displacement parameters of deuterium (B_D) were considerably larger than those reported earlier for iron deuteride presented in Figure 6. For example, Abeykoon et al. (2023) reported the B_D of deuterium in deuterated FeS V as 30(4) Ų at 6.9 GPa and 960 K, which is about 10 times greater than the values shown in Figure 6. The large occupancies might arise from the larger displacement parameter because of the strong positive correlation between these two parameters.

Shibazaki et al. (2011) reported the unit-cell volumes of FeS IV and FeS V (both non-hydride), which were clearly smaller than the volume estimated from the EoS of stoichiometric FeS by Urakawa et al. (2004). For example, the unit-cell volume of FeS IV at 16.5 GPa and 573 K was reported as about 188 ų, as shown in Figure 3 of Shibazaki et al. (2011). However, the unit-cell volume of FeS IV calculated using EoS from Urakawa et al. (2004) is 192.5 ų. The difference in the unit-cell volume of FeS IV can be attributed to the difference in Fe/S ratio. Under ambient conditions, the unit-cell volume and the d₁₁₄ value of troilite decrease, the atomic ratio of Fe in the starting materials estimated from the equation established by Yund and Hall (1969) decreases. Therefore, the Fe/S ratio of the iron sulfides used by Shibazaki et al. (2011) can be less than 1. In addition, Abeykoon et al. (2023) reported that the stoichiometry of the starting materials of their experiments was Fe_0.953(5)S by chemical analysis of the recovered sample using EPMA. Changes in the compositional ratio of Fe and S, which can occur because of the presence of hexagonal pyrrhotite in the starting material or the iron deficiency of troilite, can strongly affect the hydrogenation reactions of iron sulfide under high-pressure and high-temperature conditions.

ACKNOWLEDGMENTS

We are grateful to Dr. Hirotada Gotou (ISSP, The University of Tokyo) for his technical assistance in the preparation of X-ray diffraction experiments under the ISSP Joint Research Program. We would like to thank Dr. Ken-ichi Funakoshi (CROSS), Dr. Jun Abe (CROSS), and Dr. Takanori Hattori (JAEA) for technical support for neutron diffraction experiments. We appreciate Mr. Hiroki Kobayashi’s technical advice on Rietveld refinement using GSAS. We are grateful to Prof. Takaya Nagai, who handled our manuscript, and to two anonymous reviewers for their insightful comments greatly improving the manuscript. We acknowledge the support of the PRIUS project (Nos. 2022-A43 and 2023-A31) by the Joint Usage/Research Center PRIUS, Ehime University, Japan. This research was supported by JSPS KAKENHI Grant Numbers JP18H05224 and JP23H00140. Neutron diffraction measurements were conducted using BL11(PLANET), MLF, J-PARC (Proposal Number 2022B0208). X-ray diffraction experiments were performed at BL04B1, SPring-8 (Proposal Numbers 2023A1410 and 2023B1524).

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