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. 2025 Aug 25;129(35):15865–15877. doi: 10.1021/acs.jpcc.5c04262

High-Pressure X‑ray Diffraction Study of Scheelite-Type Perrhenates

Neha Bura †, Pablo Botella †, Catalin Popescu ‡, Frederico Alabarse §, Ganapathy Vaitheeswaran ∥, Alfonso Muñoz ⊥, Brendan J Kennedy #, Jose Luis Rodrigo Ramon †, Josu Sanchez-Martin †, Daniel Errandonea †,*
PMCID: PMC12415949  PMID: 40927502

Abstract

The effects of pressure on the crystal structure of scheelite-type perrhenates were studied using synchrotron powder X-ray diffraction and density-functional theory. At ambient conditions, the studied materials AgReO4, KReO4, and RbReO4, exhibit a tetragonal scheelite-type crystal structure described by space group I41/a. Under compression, a transition from scheelite-to-M′-fergusonite (space group P21/c) was observed at 1.6 and 7.4 GPa for RbReO4 and KReO4, respectively. The transition involves a relative volume decrease. On the other hand, AgReO4 underwent a phase transition to the M-fergusonite structure (space group I2/a) at 13.6 GPa. In this case there is no appreciable volume discontinuity. The room-temperature pressure–volume equation of state for the three studied perrhenates was estimated using a second-order Birch–Murnaghan equation of state. The results for the low-pressure phase are confirmed by density-functional theory calculations. The analysis of the bulk modulus shows that the compressibility of the compounds decreases following the sequence RbReO4 > KReO4 > AgReO4, which is related to the compressibility of the RbO8, KO8, and AgO8 bidisphenoid units. Density-functional theory also offers valuable insights into the elastic constants. Despite giving a good description for the low-pressure phase in the three compounds, density-functional theory cannot catch the structural phase transition observed in experiments. Reasons for it are discussed in the manuscript.

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1. Introduction

Bimetal oxides form a large family of compounds, encompassing a wide range of materials. One of such group of compounds is formed by AMO4 oxides which has received significant attention due to their multiple technological applications. In this work, we focus on three specific members of the perrhenate family, KReO4, RbReO4, and AgReO4. These compounds exhibit, at ambient conditions, a tetragonal scheelite-type structure with space group I41/a which is schematically represented in Figure . This structure has two types of polyhedra, AO8 dodecahedra (with A = K, Rb, Ag) and ReO4 tetrahedra. In this structure, the A, Re, and O atoms occupy the Wyckoff positions 4a, 4b, and 16f, respectively.

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Different projections of the tetragonal scheelite-type crystal structure, described by space group I41/a. Spheres drawn with pink and silver colors represent A (K, Rb, or Ag) atoms and Re atoms, respectively. Oxygen atoms are represented by red spheres. The ReO4 tetrahedral units are shown.

Perrhenates have garnered significant interest in recent years due to their unique structural and electronic properties and potential applications across various fields, which include calorimeters for neutrino mass measurements. Their fascinating electronic and optical behaviors make them suitable for use in optoelectronic devices, sensors, photoluminescence, and catalysts. , Despite their potential, the study of AReO4 compounds under variable temperature and pressure conditions is relatively limited compared to related materials such as vanadates and tungstates, highlighting the need for further exploration of the high-pressure properties of AReO4 perrhenates to fully understand and harness their functional potential.

The room-temperature pressure dependent Raman studies for KReO4 of Jayaraman et al. showed a sequence of transitions at 7.5, 10, and 14 GPa. Based on the changes in Raman spectra, the transitions were proposed to gradually reduce the symmetry of the crystal from tetragonal to orthorhombic to monoclinic to triclinic. Similar changes in the Raman spectra were detected at 1.6, 5.5, and 15 GPa for RbReO4. Chay et al. studied the temperature dependent behavior of RbReO4 and observed a transition from the scheelite structure (I41/a) to another tetragonal structure described by space group I41/amd near 650 K. On the other hand, using energy-dispersive X-ray diffraction (XRD), as well as Raman spectroscopy, Otto et al. observed a phase transition in AgReO4 to an unknown structure at 13 GPa. , More recently, Mukherjee et al. studied the high-pressure (HP) properties of AgReO4 using density-functional theory (DFT) simulations. These authors suggested, based on the pressure dependence of the volume, that the previous XRD results were hindered by nonhydrostatic effects. Critically, the crystal structure of the high-pressure phases of AgReO4, KReO4, and RbReO4 have not yet been determined. Evidently it is timely to perform HP XRD studies of these compounds to determine the crystal structure of the HP phases using state-of-the-art methods.

To fully characterize the behavior of these materials under extreme conditions, in this paper we concentrate on their crystal structures. We performed HP XRD experiments and focused on the determination of the crystal structure of different phases, as well as the pressure dependence of unit-cell parameters and the accurate determination of the room-temperature pressure–volume equation of state (EoS). We also performed DFT studies using different functionals to find the most appropriate functionals to describe the properties of the low-pressure phase in the different studied compounds.

2. Materials and Method

2.1. Sample Synthesis and Characterization

Polycrystalline samples of KReO4, RbReO4, and AgReO4 were prepared as described by Chay et al. A HReO4 solution (100 mL, 53.7 mmol) was prepared by mixing rhenium metal (Aldrich, 99.9%,) 11.59 mmol, with 100 mL of fresh H2O2 (Sigma-Aldrich, 30% v/v) at room temperature. The solution was stirred overnight until the Re metal had completely dissolved. The target compound was then prepared by adding A 2CO3, A = K, Rb, Ag, (30 mL, 1.223 M) to the HReO4 solution (100 mL, 0.0537 M) and allowing the product to precipitate. This was collected by filtration, washed with cold water and air-dried.

Powder XRD studies were carried out to identify the phase of the samples along with their crystal structure. These measurements were performed with Cu Kα1 radiation on a Rigaku Ultima IV diffractometer and Rietveld refinements were carried out using FULLPROF. The three compounds were found to exhibit the tetragonal scheelite-type structure. The refined lattice parameters at ambient conditions are summarized in Table and are in good agreement with the values previously reported by Chay et al.

1. Lattice Parameters and Volume of Perrhenates Obtained from XRD Measurements at Ambient Conditions and Determined from DFT Calculations .

    experiments
 DFT
material space group I41/a this work Chay et al. MetaSCAN PBEsol PBEsol + D3 + BJ
KReO4 a (Å) 5.6738(1) 5.67625(8) 5.6907 (0.30%) 5.74597 (1.26%) 5.74527 (1.24%)
c (Å) 12.6961(3) 12.6994(4) 12.5989 (−0.77%) 12.5628 (−1.06%) 12.78474 (0.69%)
V (Å3) 408.71(1) 409.17(1) 408.00 (−0.17%) 414.77 (1.46%) 422.00 (3.15%)
RbReO4 a (Å) 5.8327(3) 5.8329(1) 5.8592 (0.45%) 5.90404 (1.21%) 5.91768 (1.44%)
c (Å) 13.2578(8) 13.2543(3) 13.0497 (−1.59%) 13.04719 (−1.61%) 13.24997 (−0.06%)
V (Å3) 451.04(4) 450.94(2) 448.00 (−0.68%) 454.80 (0.83%) 464.00 (2.79%)
AgReO4 a (Å) 5.3664(6) 5.37674(7) 5.3375 (−0.54%) 5.33738 (−0.54%) 5.369 (0.05%)
c (Å) 11.8450(31) 11.8006(2) 11.5208 (−2.81%) 11.32421 (−4.60%) 11.795 (−0.42%)
V (Å3) 341.11(9) 341.15(1) 328.00 (−4.0%) 322.60 (−5.74%) 340.01 (−0.33%)
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a

For DFT calculations, we show in brackets the relative difference with the present experiments. Results from previous experiments are shown for comparison.

2.2. High-Pressure Studies

Angle dispersive XRD (AD-XRD) studies on KReO4 and RbReO4, to pressures of 10 GPa, were carried out at the Xpress beamline of the Elettra Synchrotron Radiatiom Facility (Elettra). AD-XRD studies on AgReO4 were carried out up to 18.5 GPa at the MSPD beamline of the ALBA synchrotron. The monochromatic wavelength for the experiments at Elettra was tuned to 0.4956 Å, and for experiments at ALBA it was 0.4246 Å. All the experiments were carried out using a membrane-type diamond-anvil cell (DAC) with a culet size of 500 μm. The samples were loaded in stainless steel gaskets, into which a 150 μm hole had been drilled, along with a small amount of Cu or Ag powder (used for pressure determination) and a 4:1 methanol–ethanol mixture, which acts as the pressure-transmitting medium (PTM). At each pressure, we collected two XRD patterns, one with Cu (or Ag) and sample used to determine the pressure, and one where we maximized the sample signal which was used for structural analysis. Pressure was determined with the equation of state of Cu (or Ag) reported by Dewaele et al. with a 0.5% accuracy. The selected PTM remains quasi-hydrostatic up to 10 GPa and is commonly used to study oxides in the pressure range covered by this study. This medium is the same as that used in previous experiments, − which allows a direct comparison of results. A PILATUS3 S 6M (Rayonix SX165 CCD) detector was used to collect the diffraction patterns in Elettra (ALBA). The detectors were calibrated using LaB6 (ALBA) and CeO2 (Elettra) as standards. To obtain a conventional 1-D diffraction pattern, the intensity was integrated as a function of 2θ using Dioptas.

2.3. Density-Functional Theory Calculations

First-principles calculations were performed utilizing the well-established plane-wave pseudopotential method within the context of DFT, as implemented in the Vienna Ab initio Simulation Package (VASP). Calculations for the low-pressure phase of AgReO4 were already published. It that work we have shown that the generalized-gradient approximation (GGA), using PBEsol for the exchange-correlation functional including van der Waals correction using the D3 method proposed by Grimme, incorporating the Becke–Johnson (BJ) damping variant (D3+BJ) was the most accurate method to describe the scheelite phase of AgReO4. Here, for KReO4 and RbReO4, we compare calculations using PBEsol and PBEsol including D3+BJ. We also compared these functionals for the three compounds with the newly developed strongly constrained and appropriately normed family of meta-GGA density functionals (MetaSCAN). MetaSCAN was also used in this work to study AgReO4 for the sake of completeness. Calculations with the HSE06 hybrid functionals gave very similar results to MetaSCAN.

In all calculations the projector augmented wave (PAW) pseudopotentials from the VASP database were utilized. The valence electron configurations were defined as follows: for K, [Ar] 4s1; Rb, [Kr] 5s1; Ag, [Kr] 4d10 5s1; Re, [Xe] 4f14 5d5 6s2; and O, [He] 2s2 2p4. To enhance computational precision, a dense Monkhorst–Pack sampling of a 9 × 9 × 9 k-point mesh was employed for Brillouin zone integration. A plane-wave basis set with an energy cutoff of 600 eV was selected to ensure accurate and well-converged structural results. The criteria for self-consistency in energy convergence were established at 1 × 10–8 eV per atom, while the maximum interatomic force was limited to 0.002 eV/Å. Stability criteria were established to optimize structural parameters across different volumes and pressures, ensuring that the deviation of the stress tensor from a diagonal hydrostatic form remained below 0.1 GPa. From these calculations, a data set comprising the lattice parameters of the ground state, energies (E) and volumes (V) at varying pressures (P) derived from the stress tensor was generated. This data set was fitted using the third-order Birch–Murnaghan EoS to extract the bulk modulus and its pressure derivative. As shown in Table , we found that the PBEsol+D3+BJ functional best describes the ground states of AgReO4, but MetaSCAN works better for KReO4 and RbReO4. We also computed the mechanical properties of the three compounds. Evaluation of the mechanical properties involved the calculation of elastic constants. These constants were obtained from the stress tensor, which was determined by applying strain to the relaxed structure through alterations in its lattice vectors, encompassing both magnitude and angle, using the stress–strain approach implemented in VASP. The elastic moduli were derived from the elastic constants.

3. Results and Discussion

3.1. Phase Transitions

3.1.1. Potassium Perrhenate (KReO4)

High-pressure powder XRD studies on KReO4 were carried out to up to 10 GPa. A selection of XRD patterns at different pressures is shown in Figure including Rietveld refinements. As previously mentioned, the material has a tetragonal scheelite-type structure (I41/a) at ambient conditions. This structure persists to around 6.4 GPa. The XRD pattern measured at 7.1 GPa contains a few additional peaks, most obvious around 2θ ∼ 6°, and scrutiny of this suggests the coexistence of two phases. At higher pressures no peaks diagnostic of the tetragonal scheelite phase could be observed, rather the diffraction patterns could be fit to a monoclinic phase. Analysis suggested this high-pressure phase has a primitive monoclinic cell and is isomorphic to the so-called M′-fergusonite structure described in space group P21 /c. This structure has previously been identified as a HP phase in HoNbO4 and in double molybdates. , A two phase (I41/a and P21/c) model was generated and this provided a good fit to the profile measured at 7.1 GPa. The diffraction patterns measured at and above 7.4 GPa only displayed peaks corresponding to the HP monoclinic M′-fergusonite phase.

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Rietveld refinements of XRD pattern of KReO4 at selected pressures, λ = 0.4956 Å. Pink (Purple) ticks identify the positions of Bragg reflections of the low-pressure tetragonal (high-pressure monoclinic) phase. The black circle represents the experimental data, the calculated profile is given by the red lines, and the blue lines represent the difference between the calculated and measured profiles. R-values of the refinements are included in the figure.

The HP structure is represented in Figure . The lattice parameters at 7.4 GPa are summarized in Table . The HP monoclinic phase is found to be stable up to 10.1 GPa. Given the coexistence of the low pressure tetragonal and high-pressure monoclinic phases at 7.1 GPa and the volume discontinuity associated with the transition (see section ) the phase transition is evidently first-order. The phase transition is found to be reversible, as the peaks observed at low pressure, during the decompression cycle were found to belong to the scheelite phase (see Figure ). The M′-fergusonite structure is a distorted and compressed form of the scheelite structure. This structural change is due to small deformations of the cation matrix and significant displacements of the anions. The most noticeable changes after the transition are the discontinuous decrease of the unit-cell volume and the distortion of KO8 and ReO4 polyhedra, and the way that polyhedra are interconnected. At the phase transition the discontinuity of the unit-cell volume is approximately 1.2%. On the other hand, the distortion index defined by Baur changes from 0.0174 in scheelite at 7.14 GPa to 0.0552 in M′-fergusonite at 7.44 GPa. The ReO4 tetrahedra is regular in the low-pressure phase with four identical bond distances, and distorted in the M′-fergusonite with a distortion index of 0.0078.

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Representation of the high-pressure monoclinic crystal structure of perrhenates in space group P21/c. The pink spheres represent A (K or Rb) atoms. The gray spheres represent the Re atoms and the oxygen atoms are represented by red spheres. The ReO4 polyhedra are shown.

2. Refined Lattice Parameters and Atomic Positions of the M′-Fergusonite Phase of KReO4 at 7.4 GPa.
parameters values atoms x y z Wyckoff position
space group P21/c K 0.7508(3) 0.1265(3) 0.0013(3) 4e
a (Å) 5.1461(5) Re 0.7498(3) 0.6255(3) 0.0017(3) 4e
b (Å) 12.1067(9) O1 0.9714(7) 0.7037(4) 0.1224(6) 4e
c (Å) 5.3689(4) O2 0.5283(6) 0.7040(7) 0.8761(7) 4e
β (°) 90.09(1) O3 0.1271(5) 0.4535(6) 0.2204 (6) 4e
volume (Å 3 ) 334.49(5) O4 0.3723(6) 0.4543(8) 0.7781(6) 4e
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The phase transition observed in KReO4 is in keeping with the trends described by Jayaraman et al., who reported phase transitions at 7.5 and 10 GPa. The monoclinic HP phase we are proposing involves an increase of the number of Raman modes compared to low-pressure scheelite structure. This is consistent with the changes observed in the Raman spectrum by Jayaraman et al. at the phase transition. DFT calculations were performed taking as starting models the HP monoclinic phase obtained from the current experiments. Invariably, when optimizing the HP monoclinic structure, it always reverted to the tetragonal scheelite structure, which is facilitated by the fact that both structures are related by group-subgroup relations (P21/c ⊂ C2/c ⊂ I41/a). Surprisingly irrespective of the functional employed in our calculations, PBEsol, PBEsol+D3+BJ, HSE06, MetaScan, PBE, and AM05, the DFT calculations failed to reproduce the observed transition. This discrepancy could be related to the influence of nonhydrostatic effects in experiments, which could favor the formation of metastable phases at pressures where they are not thermodynamically stable. However, we are confident that this is not the case in our study for two reasons. The transition found in our XRD measurements happens at a similar pressure to that reported in previous experiments. When methanol–ethanol is used as a pressure medium, as in ours and previous studies, nonhydrostaticity only becomes noticeable beyond 10 GPa, i.e., at pressures higher than the transition pressure. We consider, therefore, that the discrepancy might be related to the fact that DFT is not capturing specific features of the phase transition, such as a possible pressure-induced delocalization of the f-electrons of Re, which could strongly affect the HP behavior of materials. It is remarkable that a similar discrepancy was also found in this study for RbReO4 and AgReO4, see below. It is hypothesized that this might be related to a poor description of the Re f-electrons under HP. Under compression, the interatomic distances decrease, further increasing the strength of f-electrons correlation, a phenomenon that often DFT does not accurately capture. Another example of the problems of DFT for describing rhenium-based compounds under HP is ReO3, for which DFT underestimates the pressure induced changes in the unit-cell volume below 10 GPa.

3.1.2. Rubidium Perrhenate (RbReO4)

Powder XRD studies on RbReO4 were also carried out up to a pressure of 10 GPa. A selection of XRD patterns, including Rietveld refinements, is shown in Figure . A similar tetragonal to monoclinic transition to that described for KReO4 was observed around 1.0 GPa. At this pressure, extra peaks started appearing in the diffraction pattern, most obvious near 2θ ∼ 5.5 and 8.0° Although Jayaraman et al., based on Raman spectroscopy, suggested that the first HP structure in RbReO4 has the same orthorhombic structure observed in CsReO4 and TlTcO4. , The current diffraction data shows that it actually has a monoclinic structure and is isostructural to the HP monoclinic P21/c phase that forms in KReO4 above 7 GPa.

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Rietveld refinements of XRD pattern of RbReO4 at 0.3, 2.0, and 4.5 GPa, λ = 0.4956 Å. The pink and purple ticks identify the position of the Bragg peaks of the low-pressure tetragonal and HP monoclinic phase, respectively. The green peaks show the position of the Cu peaks. The black circles represent the measured data, the calculated profiles are given by the red lines, and the blue lines represent the difference between them. Cu peaks are identified. The figure also includes the XRD patterns measured at 10.6 GPa showing evidence of a second phase transition to an unidentified phase and at 4.8 GPa under pressure release. The changes in the intensity of Cu peaks at these two pressures indicate a shift in the preferred orientation of crystal planes of Cu. R-values of the refinements are included in the figure.

Table gives the lattice parameters of RbReO4 at 2.0 GPa. This HP primitive monoclinic phase of RbReO4 remains stable to pressures of up to 4.5 GPa. In this compound, the discontinuity of the unit-cell volume at the transition is approximately 2.1%. On the other hand, the distortion index of the RbO8 polyhedron changes from 0.0144 to 0.0418 and the ReO4 tetrahedron changes from regular to being irregular with a distortion index of 0.0068. Beyond 4.5 GPa a second transition occurs, however, as is obvious from Figure the diffraction peaks from this second HP phase are severely broadened. That this is not a result of a loss of hydrostatic conditions is evident from the well resolved peaks from the Cu pressure standard. The peaks corresponding to the second HP phase have very low intensity and as it is evident from Figure , this is not the primitive monoclinic M′-fergusonite phase. Due to the relatively poor quality of the XRD patterns measured at pressure above 4.5 GPa, that further decreased as the pressure increased, it was not possible to solve the structure of this phase.

3. Refined Lattice Parameters and Atomic Positions of the M′-Fergusonite Phase of RbReO4 at 2.0 GPa.
parameters values atoms x y z Wyckoff position
space group P21 /c Rb 0.7503(4) 0.1246(4) 0.0007(4) 4e
a (Å) 5.2854(5) Re 0.7505(4) 0.6252(4) 0.0011(4) 4e
b (Å) 13.2462(9) O1 0.9712(8) 0.7039(8) 0.1227(6) 4e
c (Å) 5.7556(4) O2 0.5288(7) 0.7039(8) 0.8773(9) 4e
β (°) 90.15(1) O3 0.1273(6) 0.4539(7) 0.2212(5) 4e
volume (Å 3 ) 402.96(5) O4 0.3727(7) 0.4539(7) 0.7780(7) 4e
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Our findings are consistent with the results reported by Jayaraman et al., who reported transitions in RbReO4 at 1.5 and 5.5 GPa, based on Raman studies. The changes in the Raman spectra at the first transition, included an increase in the number of Raman modes that, as mentioned above, is consistent with the tetragonal-monoclinic phase transition proposed here. Unlike in KReO4, in RbReO4 the observed phase transitions are not reversible, at least when pressure is reduced to 4.8 GPa. Figure shows that the XRD pattern measured at 4.8 GPa in the decompression cycle resembles that measured at 10.6 GPa during compression. Unfortunately, we could not collect XRD at lower pressures during decompression because of friction between the piston and cylinder of the DAC; the lowest pressure obtainable removing all the force applied to the DAC was 4.8 GPa. Since this pressure is higher than the transition pressure of the second phase transition, we cannot extract from present XRD experiments any conclusion of the reversibility of the phase transitions of RbReO4. As found for KReO4, DFT cannot capture the transition from scheelite to M′-fergusonite, in RbReO4. That the transition happens around 1 GPa effectively rules out the possibility that the observed results are impacted by nonhydrostatic effects.

3.1.3. Silver Perrhenate (AgReO4)

A selection of XRD patterns for AgReO4 measured at different pressures is given in Figure . The XRD studies revealed a systematic shift of peaks to higher 2θ upon application of pressure, which can be attributed to compression of the lattice. At 13.63 GPa, the most intense peak around 2θ ∼ 7.7° begins to split into two distinct peaks. Similar splitting occurs for other peaks. This splitting of peaks is enhanced at higher pressure. Splitting of the scheelite 101, 112/203, and 200 peaks is clearly seen in the XRD patterns measured 13.9 and 18.4 GPa as illustrated in Figure . This suggests a monoclinic distortion in the (001) plane of scheelite. It should be stressed that no new peaks emerged in the diffraction patterns until the highest studied pressure.

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Rietveld refinements of XRD patterns of AgReO4 at selected pressures, λ = 0.4246 Å. The black circles represent the measured data, the calculated profiles are given by the red lines, and the blue lines represent the difference between them. The pink and purple tick markers identify the position of peaks of the low-pressure tetragonal scheelite and high-pressure M-fergusonite phases, respectively. The green ticks indicate the position of Ag peaks. The Miller index relevant for the discussion are labeled, and Ag peaks are identified. The splitting of the 101, 112/103, and 200 peaks of scheelite is an indication of the phase transition. R-values of the refinements are included in the figure.

A key observation here is the lack of additional reflections that shows the cell remains I-centered, and a satisfactory fit to the data measured at 13.6 GPa was obtained to the monoclinic fergusonite structure. The fergusonite is best described in the nonstandard I2/b setting of space group C2/c to facilitate comparison with the tetragonal I41/a scheelite structure, but in the present case I2/a is preferred as it allows comparison with the primitive monoclinic (P21/c) M′- fergusonite structure seen for KReO4 and RbReO4 at high pressure. The HP phase remains stable up to the highest pressure covered by the present study, 18.5 GPa. These results agree with the earlier study by Otto et al. , who reported a phase transition to occur at 13(1) GPa. The scheelite to M-fergusonite transition is a typical pressure-driven transition of scheelite-structured oxides. , The two crystal structures are related via a group-subgroup relationship I2/a ⊂ I41/a. M-fergusonite can be obtained from scheelite via a shear deformation of the xy plane of the tetragonal structure and a slight displacement of the atoms, with no dramatic reconstruction of the lattice. This transition is driven by a Γ-point soft optical Bg phonon. , As shown below, the phase transition does not involve any discontinuity in the pressure dependence of the volume.

Table reports the structural information on the crystal structure of the HP phase of AgReO4. The structure of the M-fergusonite phase is shown in Figure . The HP M-fergusonite phase proposed here has 18 Raman-active modes, compared to 13 Raman active modes in the tetragonal scheelite. Otto et al. reported that the number of Raman modes increases at the tetragonal to monoclinic transition. The M-fergusonite structure is also consistent with the splitting of the high-frequency stretching modes reported previously, which is a typical fingerprint of the scheelite to M-fergusonite transition. As observed for the other perrhenates studied here, the DFT calculations did not capture the phase transition. Using the M-fergusonite structure as a starting model for the calculations, the structure invariably relaxed to the tetragonal scheelite upon optimization. As discussed above, understanding this issue is beyond the scope of the present work. DFT did however accurately describe the pressure dependence of unit-cell parameters of the scheelite phase.

4. Lattice Parameters of the M-Fergusonite of AgReO4 at 13.9 GPa.
parameters values atoms x y z Wyckoff position
space group I2/a Ag 0.25 0.1282(6) 0 4e
a (Å) 5.110(5) Re 0.25 0.6131(6) 0 4e
b (Å) 10.95(1) O1 0.3822(23) 0.0419(34) 0.2306(29) 8f
c (Å) 5.762(5) O2 0.4806(32) 0.2919(27) 0.3678(33) 8f
β (°) 90.81(9)          
volume (Å 3 ) 282.1(4)          
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Representation of the high-pressure monoclinic crystal structure of perrhenates in space group I2/a. The pink spheres represent Ag atoms. The gray spheres represent the Re atoms, and the oxygen atoms are represented by red spheres. The ReO4 polyhedra are shown.

3.2. Pressure Dependence of Lattice Parameters and Unit-Cell Volume

Structural refinements against the XRD patterns provided accurate lattice parameters and unit-cell volume, which were used to calculate the bulk modulus (K 0) using a second-order Birch–Murnaghan EoS. Figure illustrates the pressure dependence of the lattice parameters for the three studied oxides. As evident from this figure the pressure induced compression is anisotropic, reflecting the layered nature of the scheelite structure, with the compression along the c-axis being greater than in the ab-plane.

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Pressure dependence of the lattice parameters for AReO4 (A = K, Rb, Ag) oxides. The error bars are smaller than the size of the symbols. The inset of the graphs shows the change in the β-angle with pressure for the HP monoclinic phases. The results from the DFT calculations are shown as continuous lines. For AgReO4, results from previous XRD experiments by Otto et al. are included as blue symbols. The D3+BJ results for AgReO4 are taken from ref . The vertical lines show the transition pressures. Due to space limitations in this, and the subsequent figures, we use “SCAN” to describe the MetaSCAN functional.

Figure also includes the compression of the three oxides calculated using DFT. For AgReO4 and RbReO4 PBEsol+D3+BJ gives the best agreement whereas for KReO4, the MetaSCAN calculations provided the best agreement. In this figure the results for AgReO4 are compared with previous studies. Our results for AgReO4 compare better with previous DFT calculations than with the previous XRD experiments. The previous XRD experiments underestimate the compressibility of both axes possibly due to the influence of nonhydrostatic stresses caused by the sample bridging between diamonds.

In the three compounds, the scheelite phase is more compressible along the c-axis than along the a-axis. This anisotropic nature of compressibility can be seen clearly in the axial ratio vs pressure plot for the three perrhenates as shown in Figure . At the phase transition, there is no discontinuity in the lattice parameters for AgReO4, however, there is a clear discontinuity in KReO4 and RbReO4. In the HP monoclinic phases, the greatest compression is in the b-axis which corresponds to the c-axis of the tetragonal scheelite structure. We also observed that the monoclinic β angle increases with pressure in the three compounds and that the anisotropy of the basal ac plane becomes greater as pressure increases, showing that the monoclinic distortion is enhanced by compression.

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Pressure dependence of the axial ratio c/a ratio for perrhenates. The error bars are smaller than the size of the symbols. The results for the DFT calculations are also shown as solid lines. For AgReO4 the black symbols are from this study and blue symbols from the experiments performed by Otto et al.

Figure presents the volume versus pressure curves for the three studied perrhenates and the fits to the second-order Birch–Murnaghan EoS. Both KReO4 and RbReO4 exhibit a first-order phase transition, characterized by a ∼2.6% decrease in volume as they transform from the tetragonal (I41/a) to the monoclinic (P21/c) phase. In contrast there is no obvious volume discontinuity at the I41/a to I2/a phase transition in AgReO4. The scheelite to fergusonite transition observed for AgReO4 is allowed to be continuous, reflecting the group-subgroup relationship between the two structures, although there is evidence that it is often not second order. , However, a direct scheelite to M′-fergusonite transition cannot be continuous. The difference in behavior can, in part, be attributed to the differences in the ionic radii of the A-type cations. According to the phenomenological model of ABX 4 structures proposed by Bastide the fergusonite structure, that is derived from scheelite by a ferroelastic displacement of the cations coupled with rotation of the unconnected BO4 tetrahedra, is favored by smaller A-site cations such as Ag (1.28 Å). We propose that increasing the size of the A-site cation, as occurs for K (1.51 Å) and Rb (1.61 Å) restricts rotation of the tetrahedral units and the steric stresses induced by pressure are accommodated by deformation of the outer shell of the cation rather than simple rotation of the tetrahedra resulting in a reconstructive phase transition. A similar trend to that seen here for the perrhenates has been observed in tungstates where, under pressure, CaWO4 and SrWO4 undergo the same I41/a to I2/a transition observed for AgReO4 whereas BaWO4 and PbWO4 show a I41/a to P21/n transition, which was attributed to larger ionic radii of Ba and Pb than Sr and Ca. ,

9.

9

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Pressure vs Volume data for perrhenates along with corresponding second-order Birch–Murnaghan fits, illustrating the volume discontinuity indicative of a 1st order phase transition in KReO4 and RbReO4. The error bars are smaller than the size of the symbols. The absence of such a discontinuity at the phase transition in AgReO4 is indicative of a continuous transition. The figure for AgReO4 includes results from an earlier XRD experiment (Otto et al.). In all cases the closed symbols are for the tetragonal structures and the open symbols for the high-pressure monoclinic structures. The black lines represent the EoS fits described in the text, and colored lines represent the DFT calculations for the low-pressure phase. The D3+BJ calculations for AgReO4 were previously published. The vertical lines show the transition pressures.

The arguments presented in the previous paragraph combined with high-temperature studies allow us to obtain a systematic overview of the behavior of AReO4 perrhenates. It should be noted here that increasing temperature leads to an increase of the volume and increasing pressure leads to a decrease of the volume. So, both thermodynamic variables are expected to have the opposite behavior in the structural sequence. It is known that CsReO4, which has an orthorhombic structure related to scheelite (space group Pnma) and scheelite-type RbReO4 (space group I41/a) transform at high-temperature to a crystal structure described by space group I41/amd, which is derived from scheelite by removal of the tetrahedral rotation. On the other hand, a transition from a structure described by space I41/amd to scheelite has been reported in many compounds under compression. , Therefore, based on the comparable behavior of oxides with similar structures under both variable temperature and pressure conditions and the arguments proposed by Bastide, we would expect that under compression CsReO4 would transform to a M′-fergusonite structure as observed for RbReO4. This hypothesis is consistent with the fact that the Raman spectra of the HP phases of CeReO4 and RbReO4 are similar. On the other hand, given Na has an ionic radius more similar to that of Ag than that of K and Rb, we would expect that NaReO4 would transform to the M-fergusonite structure under high-pressure. Both hypotheses should be confirmed by future HP XRD studies.

The results obtained for the pressure dependence of the volume of AgReO4 are in good agreement with the present and previous calculations. As noted elsewhere, the present experimental results diverge from the previously reported values around 2 GPa. It is postulated that the previous XRD studies were hindered by nonhydrostatic stresses. The values of bulk modulus obtained for each compound in this work are summarized in Table . This table also includes the values estimated for the high-pressure phases. In all cases, a second-order EoS was used (pressure derivative of the bulk modulus K′ 0 = 4) to facilitate a direct comparison between results; in a third-order EoS K 0 and K′ 0 are correlated, so the bulk moduli cannot be directly compared. The second-order EoS fits well the experimental results.

5. Comparison of Bulk Moduli (K 0) between Various Theoretical Approaches and Experiments .

  low pressure phase
high pressure phase
  expt. SCAN PBEsol PBESol + D3+BJ expt.
KReO4 K 0 = 28.8(6) K 0 = 36.2(15) K 0 = 29.6(7) K 0 = 30.4(5) K 0 = 34.1(8)
V 0 = 404.1(5) V 0 = 408.0(1) V 0= 414.8(1) V 0 = 422.0(1) V 0 = 398.9(9)
RbReO4 K 0 = 19.5(7) K 0 = 30.5(6) K 0 = 22.9(2) K 0 = 24.3(2) K 0 = 29.4(4)
V 0 = 451.1(5) V 0 = 448.0(1) V 0 = 454.8(1) V 0 = 464.0(1) V 0 = 441.1(5)
AgReO4 K 0 = 56.2(9) K 0 = 68.1(11) K 0 = 61.9(10) K 0 = 48.7(14) K 0 = 56.2(9)
V 0 = 341.1(5) V 0 = 328.0(5) V 0 = 322.6(1) V 0 = 340.0(1) V 0 = 341.1(5)
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a

K 0 is given in GPa. The volume at zero pressure (V 0) is included to provide the complete information of the different EoS. V 0 is given in Å3.

AgReO4 (K 0 = 56.2(9) GPa) has the highest bulk modulus followed by KReO4 (K 0 = 28.8(6) GPa) and RbReO4 (K 0 = 19.5(7) GPa); this order follows that of the ionic radii of the A-site cation. Notice that the bulk modulus inversely correlates to the unit-cell volume which is correlated to the volume of the AO8 polyhedron as observed in most AMO4 oxides. ,,, The unit-cell volume and the AO8 volume decreases in going from Rb to K and to Ag while the bulk modulus increases following the Rb, K, Ag sequence. The largest bulk modulus for AgReO4 is supported by the charge density analysis by Mukherjee et al. that indicates ionic bonding in AgReO4. The sequence obtained for the bulk modulus of the three compounds is also consistent with the empirical equation for the bulk modulus proposed by Errandonea and Manjón. This equation depends on the average A–O bond distances and the effective valence of the A-site cation. The bulk modulus values calculated using the model proposed by Errandonea and Manjon are 38 GPa for AgReO4 > 27 GPa for KReO4 > 23 GPa for RbReO4. These results qualitatively agree with our experimental and computational results. The pressure dependence of the volume of the HP phase of AgReO4 can be described by the same EoS found for the low-pressure phase. The bulk modulus for the monoclinic structures of KReO4 and RbReO4, 34.1(8) and 29.4(4) respectively, are slightly larger than the values obtained for the low-pressure phase, which is consistent with the increased density of the HP phase resulting from the discontinuous volume reduction. These results are also supported by the DFT calculations where the three studied approximations show similar trends (see Table ). Previous DFT calculations performed for CsReO4, RbReO4, KReO4, and NaReO4 also support our conclusions, with the bulk modulus following the sequence CsReO4 < RbReO4 < KReO4 < NaReO4 as expected from the arguments presented in this section.

3.3. Elastic Constants

To gain a deeper insight into the mechanical behavior of scheelite-type perrhenates we have computed their elastic constants Cij. The tetragonal scheelite structure has seven independent elastic constants C11, C12, C13, C16, C33, C44, and C66. The values obtained for the three studied compounds, using PBEsol+D3+BJ, are summarized in Table . They satisfy the Born stability criteria for the stability of a tetragonal system: C11 - C12 > 0; 2 C13 2 < C33(C11 + C12); C44 > 0; C66 > 0; 2C16 2 < C66(C11–C12). From the elastic constants we obtained the bulk (B), shear (G), and Young (E) modulus, and the Poisson ratio (ν) using the Hill approximation.

6. Calculated Elastic Constants and Mechanical Moduli of AgReO4, KReO4, and RbReO4 at 0 GPa .

AgReO4 KReO4 RbReO4
C11 = 62.95 B = 42.11 C11 = 34.25 B = 21.28 C11 = 30.56 B = 17.98
C12 = 29.17 G = 13.40 C12 = 15.92 G = 11.86 C12 = 17.65 G = 8.96
C13 = 34.73 E = 36.34 C13 = 14.32 E = 30.00 C13 = 12.10 E = 23.06
C33 = 55.78 ν = 0.36 C33 = 33.93 ν = 0.26 C33 = 21.76 ν = 0.28
C44 = 12.44   C44 = 13.83   C44 = 10.64  
C66 = 14.21   C66 = 14.70   C66 = 12.41  
C16 = 0.03   C16 = −3.55   C16 = −2.56  
Open in a new tab
a

All magnitudes are given in GPa except for ν which is dimensionless.

In the three compounds we found that C11 > C33 which means that the lattice is more rigid along the a-axis compared to the c-axis, aligning with the reduction in the c/a ratio under compression shown in Figure . The bulk moduli obtained from the elastic constants follows the same trend (AgReO4 > KReO4 > RbReO4) to the values obtained from the pressure dependence of the volume (see Table ). However, the values obtained from the elastic constants, 42.11 GPa for AgReO4, 21.28 GPa for KReO4, and 17.98 GPa for RbReO4 are slightly smaller than the values obtained from the Birch–Murnaghan fit to the corresponding DFT calculations, 48.7 GPa, 30,4 GPa, and 24.3 GPa, respectively, and to the experimentally determined values of 56.2, 28.8, and 19.5 GPa. The Young’s modulus for the three compounds is of the same order as the bulk modulus, with B > E in AgReO4 and E > B in the other two compounds. This means that in scheelite-type perrhenates, the tensile or compressive stiffness of the material when subjected to a longitudinal force, is similar to its ability to withstand bulk compression. The value of the shear modulus, which is smaller than E and B in the three compounds, implies that they are more prone to shear deformations than to volume reduction, rendering it highly sensitive to nonhydrostatic stress. This is possibly the origin of the anomalous results obtained by Otto et al. The Poisson ratio of the three compounds indicates that they exhibit ductile characteristics.

4. Conclusions

By means of high-pressure powder XRD, we found that the three studied perrhenates undergo a phase transition from tetragonal to monoclinic symmetry. In KReO4 and RbReO4 this involves a first-order phase transition from scheelite (I41/a) to M′-fergusonite (space group P21/c) and in AgReO4 it occurs by a continuous phase transition to M-fergusonite (space group I2/a). The reported transitions provide an explanation for changes observed in the Raman spectra reported in previous studies. , From the pressure dependence of the volume, a Birch–Murnaghan equation of state was determined, and the anisotropic compressibility of the different phases described. RbReO4 has the smallest value of bulk modulus, and AgReO4 the highest among the three compounds. Density-functional theory calculations were also performed. The calculations correctly describe the pressure dependence of the lattice parameters of the scheelite phase. Information on the elastic constants was also obtained. Despite correctly describing the structures of the low-pressure phase, including the pressure dependence of the unit-cell parameters, DFT could not capture the pressure-driven phase transitions. It is thought that this may be a result of pressure-induced delocalization of the f-electrons of Re, a phenomenon usually not capture properly by DFT. Further studies are needed to clarify why DFT can correctly describe the properties of the low-pressure scheelite phase but not capture the phase transition induced by pressure.

Acknowledgments

Part of this research is supported by the Spanish Ministry of Science, Innovation, and Universities (MCIN/AEI/10.13039/501100011033) under grant number PID2022-138076NB-C41/44, and by Generalitat Valenciana under grants CIPROM/2021/075 and MFA/2022/007. This study is part of the Advanced Materials program supported with funding from the European Union NextGenerationEU (PRTR-C17.I1). The authors thank Elettra Sincrotrone Trieste (Proposal 20230054) and ALBA synchrotron (Proposal 2024028151) for providing beam time for the HP XRD experiments. The research leading to this result has been cofunded by the project NEPHEWS under Grant Agreement No 101131414 from the EU Framework Programme for Research and Innovation Horizon Europe. D.E. and P.B. thank the financial support of Generalitat Valenciana through grant CIAPOS/2023/406. G.V. acknowledges Institute of Eminence, University of Hyderabad (UOH-IOE-RC3-21-046) for the financial support and CMSD, University of Hyderabad for providing the computational facility.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

N.B. formal analysis, methodology, data acquisition, writing review and editing. P.B., C.P., F.A., J.L.R.R., and J.S.-M. methodology, data acquisition, writing review and editing. G.V. and A.M. computational calculations, writing review and editing. B.J.K: methodology, writing review and editing. D.E. conceptualization, formal analysis, writing original draft, funding acquisition, formal analysis, writing review and editing. All authors participated in the writing and editing of the manuscript. All authors have given approval to the final version of the manuscript.

The authors declare no competing financial interest.

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Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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