In this experiment, In₂Mn₂O₇ was synthesized by the sol–gel method combined with the high-temperature and high-pressure (HTHP) method. In₂O₃ (99.99% purity), MnO (99.9% purity), citric acid (AR: analytical reagent), and nitric acid (AR) were used as raw materials. In₂O₃ and MnO with the molar ratio of 1:2 were dissolved in nitric acid and diluted in distilled water. Citric acid was then added as a fuel to the above solution to yield a citrate/nitrate ratio of 1.2. The mixed solution was continuously stirred using a magnetic agitator. The solution was evaporated by heating at 150°C until a brown sticky gel was formed. Subsequently, the gel was dried at a temperature up to 200°C. The dried gel was calcined at 500°C in air for 10 h.

Next, the target product was synthesized by the cubic anvil HPHT apparatus. The powder was pressed into cylindrical samples with a diameter of 6 mm and a height of 2.3 mm. The cylindrical sample was put into the sample synthesis chamber with a specific combination, and the chamber was put into the cubic anvil HPHT apparatus (SPD6×600) [27, 28]. The temperature was raised to 1,300°C, and the pressure was increased to 5 GPa. The heat and pressure preservation time was 60 min. At last, we made the temperature drop rapidly to room temperature within 15 s, and the pressure needed to be reduced slowly to obtain pyrochlore In₂Mn₂O₇.

In the ambient condition, the phase purity was detected by powder X-ray diffraction (XRD) measurements using a Rigaku Rotaflex X-ray diffractometer with Cu-Kα radiation (λ = 1.54056Å) at room temperature. The test voltage was 40 kV, the current was 30 mA, the scanning range was 10–90°, and the scanning speed was 4°/min. The high-pressure angle-dispersive XRD patterns were collected at the beamline X17C of the National Synchrotron Light Source at Brookhaven for In₂Mn₂O₇ with a monochromic wavelength of 0.4112 Å. A diamond anvil cell (DAC) with an anvil culet of 300 μm diameter was utilized to generate high pressure with the T301 stainless steel as the gasket, which was pre-indented to a 50 µm thickness. One piece of the as-prepared samples, a small piece of ruby as the pressure calibrant [29], and a 16:3:1 methanol/ethanol/water mixture as the pressure-transmitting medium were loaded into the diamond anvil cells. The distance between the sample and the detector and parameters of the detector were calibrated using a CeO₂ standard. All the diffraction data were collected using an MAR CCD detector, and the diffraction profiles were obtained via the radial integration of the two-dimensional diffraction rings using FIT2D software [30]. Rietveld analyses were performed with the software GSAS [31].

The structure of In₂Mn₂O₇ was determined by Rietveld refinement, shown in Figure 1A. The red, blue, and black lines in the figure represent the experiment curve, the calculation curve, and the difference between them, respectively. The green vertical line represents the fitted peak position. From the fitted results, it was determined that the compound In₂Mn₂O₇ crystallized in the cubic system, with the space group Fd-3m (No. 227) and cell parameters a = 9.7113(5)Å, V = 915.8(6)ų, and Z = 8. In₂Mn₂O₇ can be formulated as In₂Mn₂O₆O′, with Mn ions at 16c, In ions at 16d, O at 48f, and O′ at 8b. In ions formed twisted cubes with surrounding oxygen atoms, and Mn ions formed twisted octahedrons with surrounding oxygen atoms. It was worth noting that there was only one adjustable position parameter in this structure, which was the x coordinate of the oxygen atom at the position of 48f. Figure 1B is the schematic diagram of its structure. The refinement parameters of the cubic phase In₂Mn₂O₇ structure are given in Table 1.

The in situ XRD patterns of In₂Mn₂O₇ at various pressures up to 29.0 GPa have been collected, and a few representative patterns are shown in Figure 2. The results showed that, in the whole pressure range of this experiment, there was no obvious change in the high-pressure XRD diffraction pattern, and there appeared no new diffraction peak. The diffraction peak under each pressure point moved to a higher angle with increasing pressures, accompanied by the weakening of the intensity of diffraction peak, which was usually caused by the increase of lattice disorder under pressure. This indicated that the cell volume of In₂Mn₂O₇ decreased gradually in the test pressure range of 0–29.0 GPa. After releasing the pressure, the positions of diffraction peaks were almost the same as those before pressure rise. The above two points indicated that the cubic phase In₂Mn₂O₇ exhibited excellent structural stability in the whole pressure range in this experiment. When the pressure increased gradually, the compound had no structural phase transition but only slightly strained, and this change disappeared completely after the pressure was removed. In order to further study the change of the crystal structure of In₂Mn₂O₇ with pressure, Rietveld analyses were performed with the software GSAS under various pressures. We obtained the lattice constants given in Table 1 and volumes under different pressures as shown in Figure 3.

The P–V data were fitted using the Birch–Murnaghan (B–M) equation of state [32], expressed as follows: P = 3.0 2 B o [ ( V 0 V ) 7 3 − ( V 0 V ) 5 3 ] × { 1 + 3 4 ( B o ′ − 4 ) × [ ( V 0 V ) 2 3 − 1 ] } , (1) where B ₀ represents the bulk modulus at zero pressure, B ^′ _o represents the first derivative of the bulk modulus, and V ₀ represents the volume of a single cell at zero pressure. Taking B ^′ _o = 4, the fitting results are shown as the red curve in Figure 3. It can be seen that the red curve fitted very well with the experimental data within the experimental pressure range, and the fitting results showed that B ₀ = 240(4) GPa. In addition, when B ^′ _o was not fixed as 4, we obtained B = 221(6) GPa and B ^′ _o = 5.5(1.2), shown with the fluorescent green dotted line in Figure 3. This result was compared with the bulk moduli of other A₂B₂O₇ pyrochlore oxides. The bulk moduli of A₂Ti₂O₇ (A = Ho, Y, Tb, Sm) compounds were 213(2), 204(3), 199(1), and 164.8(1.5) GPa [33, 34], and the bulk moduli of A₂Sn₂O₇ (A = La, Eu) compounds were 180(6) and 170 GPa, respectively [35]. It can be seen that the bulk modulus of In₂Mn₂O₇ was obviously larger than that of the above compounds. However, we also studied A₂Ge₂O₇ (A = In, Ho, Sc) oxides before and obtained that their bulk moduli were 279(4), 328(7), and 263(4) GPa, respectively [36–38]. The bulk modulus is one of the most important physical parameters that characterize the mechanical property of materials. Several works have suggested that the bond length should be taken into account to explain the variation of the bulk modulus of materials [39, 40]. It has been known that the bulk modulus is proportional to k/dⁿ, where d is the bond length and k, n are determined by the properties of materials [40, 41]. Changes in the electronic configuration of the A and B atoms of A₂B₂O₇ pyrochlore oxides have an effect on the length of the bond. Ionic, covalent, and metallic bonds can affect bond strength which is closely related to the bond length. So to find the laws and the physical mechanisms behind these bulk moduli, further research is needed.

In the ordered cubic pyrochlore structure, all the atoms are in special positions except O _48f . Hence, the structure is entirely described by the x positional parameter of O _48f . The X-ray diffraction patterns were refined by the Rietveld method up to 29.0 GPa. The refined x parameter for O _48f as a function of pressure is shown in the inset of Figure 3. Previous studies on pyrochlore oxides have revealed that the sudden change of the x positional parameter of the O _48f atom existed in the process of pressure-induced structural phase transition. For example, with increasing pressure, the ordered pyrochlore Gd₂Zr₂O₇ begins to transform to a disordered defect-fluorite–type cubic structure up to 15 GPa. Above 15 GPa, a high-pressure phase forms that has a distorted defect-fluorite–type structure of lower symmetry. Below 10 GPa, there is no or little change in the refined x parameter for O _48f , followed by a rapid decrease with increasing pressure [42]. Above 18 GPa, the pyrochlore Sm₂Zr₂O₇ is unstable, and a pressure-induced phase transition occurred. The x coordinate for the 48f oxygen increases dramatically after 16 GPa [43]. Within the range of our study, it can be seen that x _O48f has no sudden change, which also indicates the structural stability of pyrochlore In₂Mn₂O₇.