Introduction

If we are to have a proper understanding of the Earth’s mantle, accurate determination of the physical properties of its constituent minerals is essential. For upper-mantle minerals, many of these properties are measureable directly in the laboratory, but the conditions of pressure and temperature that obtain in the lower mantle make such in situ studies increasingly challenging with increasing depth. In particular, experimental determination of the physical properties of the “post-perovskite (or “PPV)” phase of MgSiO3, Footnote 1 which is thought to occur just above the Earth’s core-mantle boundary in the D′′ layer, presents a very severe challenge, as it is stable only at pressures in excess of about 100 GPa (see, e.g. Murakami et al. 2004; Oganov and Ono 2004; Tsuchiya et al. 2004; Shim et al. 2008; Tateno et al. 2009). It can be argued that the best way to address these difficulties is via a combination of computational and experimental studies. In this approach, experimental measurements are made on analogue phases, isostructural with the lower-mantle minerals, which are stable (or at least strongly metastable) at ambient pressure and temperature. These measurements may then be used to “ground truth” computer simulations of the analogue phases; if the agreement between experiment and simulations for the analogues proves satisfactory, and the simulations agree with experimental results on the natural system at lower pressures (if available), the computer simulations may then be applied to the natural system at high pressure with a high degree of confidence. As discussed below, two obvious examples of the need for this complementary method come from studies of the cation diffusivities in PPV-MgSiO3 (where it has been reported from computer simulations that there is a directional anisotropy of a factor of about 108, Ammann et al. 2010), and in confirming computer simulations of the possible high-pressure phases of MgSiO3 under conditions relevant to the mantles of large terrestrial exoplanets (Umemoto et al. 2006a; Tsuchiya and Tsuchiya 2011).

When examining possible lower-mantle analogue phases, fluorides can be considered to have advantages over oxides as their high-pressure phases become stable at much lower pressures. For MgSiO3 perovskite (bridgmanite), the mineral neighborite (NaMgF3) provides an obvious and attractive analogue (e.g. Umemoto et al. 2006b; Hustoft et al. 2008; Li and Weidner 2012), although other ABF3 compounds such as KCaF3 (Watson et al. 1995) and KZnF3 (Poirier et al. 1983) have also been used for this purpose. NaMgF3 may be readily synthesised at atmospheric pressure and is stable at atmospheric pressure and room temperature. At atmospheric pressure and room temperature it crystallises (e.g. Knight 2014), in common with many other compounds (e.g. Mitchell 2002), as an orthorhombically distorted perovskite, in what is commonly referred to as the gadolinium orthoferrite structure (Geller 1956); the space group is Pbnm and the system of octahedral tilting producing the distortion from the cubic perovskite aristotype, for this setting of the space group, is a−a−c+ in the notation of Glazer (1972). NaMgF3 is, therefore, isostructural with bridgmanite and the two compounds are also isoelectronic; in addition, the relative masses of the atoms in NaMgF3 are not too dissimilar from those in MgSiO3. It is well known that NaMgF3 undergoes a phase transition between the perovskite and CaIrO3 structures (Martin et al. 2006a, b) and so this compound may also be used as an analogue for PPV-MgSiO3 (e.g. Hustoft et al. 2008). In addition, on the basis of computer simulations, it has been suggested by Umemoto and Wentzcovitch (2006) and by Xu et al. (2015) that further transitions in NaMgF3 may occur at higher pressures, which may have some relevance to the mineralogy of super-earths (Umemoto et al. 2006a, b; Grocholski et al. 2010; Tackley et al. 2013). Umemoto and Wentzcovitch (2006) proposed that there would first be a transition from the CaIrO3 phase to a lower-symmetry, orthorhombic structure, of the Sb2S3 (stibnite) structure type (e.g. Lundegaard et al. 2003), as has now been found to occur by experiment in NaFeF3 (Crichton et al. 2016) and in NaCoF3 (Wood et al. 2017). At still higher pressure, Umemoto and Wentzcovitch (2006) found that a further transition occurred to a hexagonal structure with space group P63/mmc, which does not seem to correspond to any known structure type; this hexagonal structure was considered by Crichton et al. (2016) to take the Be3N2 structure (identical to that of InFeO3 or InMnO3), but although these structures are isopointal (as their space group and occupied Wyckoff positions are the same; Allmann and Hinek 2007), they cannot be considered isoconfigurational as their axial ratios differ by an amount sufficient to cause changes in the primary coordination of the atoms. It should be noted, however, that Umemoto and Wentzcovitch (2006) also concluded that both of these new high-pressure phases would be metastable with respect to dissociation of NaMgF3 into CsCl-structured NaF and cotunnite-structured MgF2 (a similar sequence of transitions was predicted for CaSnO3 by Tsuchiya and Tsuchiya 2006). In contrast, Xu et al. (2015) proposed a different high-pressure transition in NaMgF3, from the CaIrO3 phase to a structure, with space group Pnma, that they termed “ppPv”, for “post-post-Perovskite”. Xu et al. (2015) stated that this structure “has never been previously reported in any material”, but it has since been pointed out by Crichton et al. (2016) that this “ppPv” phase actually corresponds to a structure of the La2S3 typeFootnote 2 (Besançon et al. 1969). Xu et al. (2015) considered that the simulated diffraction pattern of their La2S3-type structure for NaMgF3 was consistent with the “N phase” reported to form at high pressure by Martin et al. 2006a, b, but they did not investigate its stability relative to the Sb2S3-type structure, nor did they examine its stability with respect to dissociation into NaF and MgF2.

In many respects, however, NaMgF3 is a much less convenient analogue for PPV-MgSiO3 than it is for the PV (bridgmanite) phase. The transition to the CaIrO3 structure in NaMgF3 occurs at between 28 and 30 GPa (Martin et al. 2006a), which is outside the standard range of a multi-anvil press, and although CaIrO3-type NaMgF3 can be quenched to atmospheric pressure (Hustoft et al. 2008) some amorphisation of the high-pressure phases on decompression has also been reported (Martin et al. 2006a). For this reason, other ABF3 fluorides, such as NaCoF3 and NaNiF3 (Dobson et al. 2011; Shirako et al. 2012b; Yusa et al. 2012), which may be prepared in a multi-anvil press and then recovered to atmospheric pressure, where they remain quite strongly metastable, have been suggested as more suitable analogues for PPV-MgSiO3. These compounds have already proved to be of great value in experimental studies, in revealing the relative strength of PV and PPV polymorphs (Dobson et al. 2012), in demonstrating the topotaxic nature of the PV to PPV transition (Dobson et al. 2013), in determining the degree of anisotropy in the diffusion coefficients of the PPV phase (Dobson et al. 2014) and in allowing the determination of accurate PPV crystal structures (Lindsay-Scott et al. 2014).

A major disadvantage of the ABF3 fluorides, where A = Na and B = a first-row transition element, however, is that in the majority of these compounds the d-electrons of the B cation will have unpaired spins leading to magnetic ordering at low-temperatures. Bernal et al. (2014) have reported recently that PV- and CaIrO3-type NaFeF3 show antiferromagnetic ordering at ~90 and 48 K, respectively (with weak ferromagnetism observed in PV-NaFeF3); similarly, PV- and CaIrO3-type NaNiF3 order antiferromagnetically at 156 and 22 K, respectively (Shirako et al. 2012b) and the corresponding phases of NaCoF3 have antiferromagnetic transitions at 74 K (Friedman et al. 1970) and 24 K (A S Wills, pers. comm.). Although this magnetic ordering may be of little consequence for experiments carried out at room temperature, i.e. far above the Néel temperature, it does significantly complicate quantum mechanical simulations of these materials, as the correct magnetic ground state must be determined. For this reason we have been considering whether other ABF3 compounds, where A and B are cations from groups IA and IIA of the periodic table, can also form CaIrO3-type polymorphs at high pressure. PV-structured compounds containing group IA and IIA cations are common, but to date only NaMgF3 has been found experimentally to form a phase isostructural with PPV-MgSiO3. Considering the most obvious examples, KMgF3, KCaF3 and RbCaF3, it was thought unlikely that either KMgF3 (e.g. Wood et al. 2002) or RbCaF3 (e.g. Knight et al. 2014) would transform in this way as both are cubic perovskites at ambient pressure and temperature. This supposition is confirmed by studies at high pressure. For PV-KMgF3, it was found that the cubic structure is maintained to 50 GPa (Aguado et al. 2008). In contrast, PV-RbCaF3 does undergo a lowering of symmetry with increasing pressure, but the transformation, at ~2.8 GPa, is from a cubic to a tetragonal (I4/mcm) perovskite; no further transformations were observed up to 7.9 GPa, the highest pressure reached in the experiment (Knight et al. 2014).

After NaMgF3, KCaF3 might appear to be one of the more promising candidate compounds, since at ambient pressure and temperature PV-KCaF3 is also orthorhombically distorted with space group Pbnm. Furthermore, Fujino et al. (2009) have suggested (for oxides) that for PPV formation, the t-factor (Goldschmidt 1926; defined as t = (1/√2)(R A + R F)/(R B + R F), where R X is an ionic radius) should lie in the range 0.8 ≤ t ≤ 0.9; for KCaF3, t = 0.862, almost identical to the value for NaMgF3, 0.866 (ionic radii taken from Shannon 1976). A further requirement (derived from consideration of the behaviour of some A2+B4+O3 perovskites, together with that of NaMgF3) was suggested by Tateno et al. (2010) who proposed that for the CaIrO3 structure to form, the octahedral tilt in the PV phase at atmospheric pressure and room temperature, Φ, estimated from the orthorhombic lattice parameters (in the Pbnm setting) by the relationship Φ = cos−1(√2a 2/bc) (O’Keeffe and Hyde 1977) should be greater than 13°. For these oxides, and for NaMgF3, it was found that application of pressure then led to an increase in Φ, with the transition to the PPV structure occurring when Φ exceeded 19–25°. For NaMgF3, Φ = 15.0° at atmospheric pressure and 300 K (from the cell parameters given by Mitchell et al. 2007), but for KCaF3, Φ = 9.3° (Knight et al. 2005). This, however, does not necessarily rule out the possibility of the formation of CaIrO3-type KCaF3 as: (1) the requirement is only a guide and (2) it is not yet known whether the conclusion that Φ should be greater than about 13° is appropriate for fluorides, as opposed to oxides (indeed, the present work suggests that it may well not be, see “Equations of state for KCaF3 polymorphs and phase diagram of KCaF3 ”).

It is perhaps surprising that although there have been several crystallographic analyses of PV-KCaF3 as a function of temperature (Knight et al. 2005; Mitchell et al. 2007; Knight 2009, 2011), very little has been published on the properties of KCaF3 at high pressure. Previous computational work has been confined to investigations of the effect of pressure on ionic conduction in PV-KCaF3 by molecular dynamics simulations using interatomic potentials (Watson et al. 1992, 1995) and a recent quantum mechanical study of the effect of pressure on the band structure (Mousa 2014). No experimental high-pressure structural studies have as yet been published, but a number of such experiments have been carried out as part of, or associated with, our present study of KCaF3. In the first of these, PV-KCaF3 (provided by K S Knight and synthesised as described in Knight et al. 2005) was compressed to 15 GPa and held at this pressure for 24 h at 300 K in a multi-anvil press (MAP) at UCL; on recovery of the sample to atmospheric pressure it was found that it was still composed of PV-KCaF3. In further experiments using the MAP at UCL, PV-KCaF3 was held at 5.5 GPa and 973 K for 220 min and at 14 GPa and 973 K for 8 h; on recovery to atmospheric pressure and room temperature both samples were found to have decomposed to a mixture of KF and CaF2. As mentioned briefly in a paper on RbCaF3 by Knight et al. (2014), KCaF3 has been studied by neutron powder diffraction up to pressures of 6.86 GPa (at room temperature) on the PEARL beam line at the ISIS Facility of the Rutherford Appleton Laboratory, where it was found that the Pbnm perovskite structure was maintained throughout. Similar persistence of the Pbnm PV-structure throughout the experiment was also seen in a preliminary examination of KCaF3 to ~20 GPa in a diamond-anvil cell (at room temperature) by synchrotron X-ray powder diffraction on beamline I15 at the Diamond Light Source.

In order to determine whether the failure to observe experimentally a CaIrO3-type phase of KCaF3 was due to an inherent lack of stability or was merely a consequence of the pressure and temperature range covered in these few experiments, it was decided to supplement them with ab initio computer simulations. In this way we have been able to examine the relative stabilities in KCaF3 of: (a) the four ABX3 polymorphs described by Umemoto and Wentzcovitch (2006) for NaMgF3, (b) the “ppPv” (La2S3) structure reported by Xu et al. (2015), (c) the monoclinic phase found by Shirako et al. (2012a) as an intermediate between the PV and CaIrO3 structures in CaRhO3, and also to determine the stability of these KCaF3 compounds relative to a mixture of KF and CaF2. We have found that the PV phase of KCaF3 is, as expected, the stable phase at atmospheric pressure but that at pressures in excess of about 2.6 GPa, none of the KCaF3 structures are thermodynamically stable with respect to a mixture of KF and CaF2. In addition, as the sequence of phase transitions in NaMgF3 predicted from ab initio simulations by Umemoto and Wentzcovitch (2006) and by Xu et al. (2015) differed, we also took this opportunity of carrying out a similar study of NaMgF3; we find that in NaMgF3 the PV and CaIrO3 structures are thermodynamically stable, but that none of the proposed higher-pressure phases are stable with respect to a mixture of NaF and MgF2.

Computational method

The static ab initio calculations presented here used the projector augmented wave implementation (Blöchl 1994; Kresse and Joubert 1999) of the density functional theory (Hohenberg and Kohn 1964; Kohn and Sham 1965) with the generalised gradient approximation (GGA) PBE pseudopotentials (Perdew et al. 1996) implemented in the VASP computer code (Kresse and Furthmüller 1996; Kresse et al. 2010). For KCaF3, an electronic minimization convergence criterion of 10−6 eV was used for the internal energy, with a k-point density of 4 × 4 × 4 (Monkhorst and Pack 1976) and a kinetic energy cutoff of 500 eV. For NaMgF3, the same electronic minimization convergence criterion for the internal energy was used, but (for consistency with earlier work; Lindsay-Scott 2012) the k-point density was increased to 6 × 6 × 6 (Monkhorst and Pack 1976) and the kinetic energy cutoff to 900 eV.

The procedure adopted to determine the volumetric equation of state, and hence the enthalpy of each phase, was to use VASP to calculate the internal energy (E) of the crystal at a set of chosen volumes (V), allowing the cell parameters and the fractional coordinates (where necessary) to relax in accordance with the crystal symmetry. All simulations were made along a path of decreasing volume. During the calculations the program parameters were set such that the symmetry of the crystal was maintained. Transitions to structures with lower symmetry were, therefore, forbidden but transitions to structures whose space groups are supergroups could occur, as the atoms are not prevented from moving into a more symmetrical arrangement. Since the calculations were static and effectively equivalent to T = 0 K, the pressure (P) at any point on the E vs. V curve is given by the standard thermodynamic result P = −(∂E/∂V) T = 0 (see, e.g. Pippard 1966), the actual values being determined by fitting the E (V) curve to an integrated Birch-Murnaghan 3rd-order equation of state (EoS; see, e.g. Vočadlo et al. 1999). Knowing P, V and E, the enthalpy, H, may be calculated. Since T = 0, the enthalpy is equal to the Gibbs free energy, G, and thus the most stable phase at any given pressure may be determined.

Results and discussion

General considerations

Initially, it was intended in this study to consider only the relative stabilities of the PV- and CaIrO3-phases of KCaF3. It was found, however, see Fig. 1, that when the volume of the unit cell of PV-KCaF3 was reduced below about 280 Å3 (~17 GPa) a spontaneous transition occurred in the simulations, leading to a different crystal structure. Inspection of the resulting fractional coordinates, using the crystal structure plotting package Diamond (Putz and Brandenburg 2006), revealed that this structure corresponded to the hypothetical high-pressure hexagonal phase of NaMgF3 (space group P63/mmc) previously described by Umemoto and Wentzcovitch (2006); in this structure, which does not appear yet to have been observed experimentally, the B cations are [8]-coordinated by the anions. The occurrence of this transition then led us to consider three further possible structures, as follows: (1) the Sb2S3—structured phase of NaMgF3 with space group Pmcn (Umemoto and Wentzcovitch 2006), containing B cations sevenfold coordinated by the anions (as observed recently in NaFeF3 and NaCoF3 above ~22 GPa in synchrotron X-ray diffraction experiments at room temperature by Crichton et al. 2016 and Wood et al. 2017); (2) the La2S3-structured phase found by Xu et al. (2015); (3) the monoclinic (P21/m) “intermediate phase” reported by Shirako et al. (2012a) for CaRhO3. Footnote 3 The PV, Sb2S3 and La2S3 structures are isosymmetric, but in order to more readily understand the relationship between the structures, it is convenient to use the Pbnm setting of their space group for the PV phase and the Pmcn setting for the Sb2S3 and La2S3-structured phases; the space group of the CaIrO3 structure is Cmcm. The symmetry relationships between these structures are interesting in that Cmcm is a supergroup of Pbnm and thus symmetry is increased in a PV to CaIrO3-type transition; the symmetry then reduces on transformation to either the Sb2S3 or La2S3-phases, but rises once more in the P63/mmc-NaMgF3 structure as P63/mmc is a supergroup of Cmcm (and thus also of Pmcn).

Fig. 1
figure 1

Pressure versus volume for KCaF3 obtained by compression of the Pbnm perovskite structure. The pressure values at the simulation points (symbols) are taken from the VASP output; the lines show Birch–Murnaghan 3rd-order equations of state, fitted to P (V). The transition from the PV to the P63/mmc-NaMgF3 structure can be clearly seen

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The structural relationship between the CaIrO3, Sb2S3 and P63/mmc-NaMgF3 phases in an ABF3 compound is readily understood from Fig. 2a, b (after Umemoto and Wentzcovitch 2006). Starting from the CaIrO3 structure, the Sb2S3 structure can be envisaged as forming with increasing pressure in the following way. The CaIrO3-type structure contains two-dimensional sheets of edge and corner linked BF6 octahedra lying perpendicular to the b-axis. As the b-axis (the most easily compressible of the three, see Table 2) shortens on compression, F− ions enter the primary coordination shell of B ions in adjacent sheets of octahedra, thereby linking alternating pairs of BF6 octahedra, and increasing the primary coordination of all of the B cations from 6 to 7 (Fig. 2a). The P63/mmc-NaMgF3 structure then forms by a similar mechanism, whereby further reduction of the axial ratio b/a to √3 and linkage of all octahedra (rather than just alternating pairs) by F− ions from adjacent sheets leads to a structure with hexagonal symmetry and B ions in [8]-fold coordination by F (Fig. 2b).

Fig. 2
figure 2

The structural mechanism for the transformations between a the CaIrO3 and Sb2S3 structures and b the CaIrO3 and P63/mmc-NaMgF3 structures (after Umemoto and Wentzcovitch 2006); c the La2S3 structure of NaMgF3 (coordinates from Xu et al. 2015, transformed to the Pmcn setting). The A-cations are shown in red, the B-cations in gold and the anions in green. The structures are viewed along the orthorhombic a-axis, with the orthorhombic b- and c-axes as marked; so as to enable ready comparison with the CaIrO3 and Sb2S3 structures, the hexagonal P63/mmc-NaMgF3 structure is represented here using a C-centred unit cell with an orthorhombic metric in which b = √3a. In c the coordination polyhedra are drawn as octahedra; there is a 7th Mg–F distance nearly parallel to the c-axis (shown here as a bond) which is only slightly longer; for details see text

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The transition between the CaIrO3 and La2S3-structures (Xu et al. 2015) is quite different in that the sheets of edge and corner linked BF6 octahedra found in the CaIrO3 structure are disrupted. With space group setting Pmcn for the La2S3 phase of NaMgF3, there is a direct correspondence between the axes of the unit-cell of these two structures. Xu et al. (2015) found that on forming the La2S3 structure the b-axis of CaIrO3-type NaMgF3 expanded by ~25%, whilst the c-axis contracted by a similar percentage; the a-axis shows about a 4% expansion. It is, therefore, difficult to envisage formation of the La2S3 phase by hydrostatic compression of a CaIrO3-type structure in a real material, as in all such structures for which axial incompressibilities have been measured to date (e.g. Boffa Ballaran et al. 2007; Lindsay-Scott et al. 2010; Wood et al. 2017) the b-axis has been found to be considerably more compressible than either a- or c-axes. Figure 2c shows the La2S3 structure, which can perhaps be considered to form in the following way. As the c-axis shortens, the tilt of the corner-linked octahedra (the angle Δ1 in Lindsay-Scott et al. 2011) increases until a point is reached at which the sheets of octahedra cannot be maintained and alternating double rods of edge-linked octahedra running parallel to the a-axis are formed. Whether these rods are then considered to be linked along the c-axis, depends on whether the B cation is considered to be in sixfold or sevenfold coordination; from the coordinates published by Xu et al. (2015), we find 6 Mg-F distances in the range 1.81–1.94 Å and one slightly longer distance—the Mg–F bond closely parallel to the c-axis in Fig. 2c—of 2.10 Å.

Equations of state for KCaF3 polymorphs and phase diagram of KCaF3

Table 1 shows the volumetric equation of state (EoS) parameters, derived from fitting E(V), for the PV, P21/m-CaRhO3, CaIrO3, La2S3, Sb2S3 and P63/mmc-NaMgF3 phases of KCaF3, together with those for the relevant phases of KF and CaF2. In the pressure range of interest, 0 to ~30 GPa, KF undergoes a transition from the NaCl to the CsCl structures (Weir and Piermarini 1964); similarly, the parameters for CaF2 in both the low-pressure fluorite and high-pressure cotunnite structures (Wu et al. 2006) are given. The axial EoS parameters for the non-cubic phases, derived from fitting 2nd-order Birch–Murnaghan EoS to the VASP output pressure versus volume, using the program EoSFit (Angel 2001) are given in Table 2. To enable ready comparison with the volumetric EoS, the values of K 0 listed in this table are those obtained by fitting the cubes of the lengths of the unit-cell edges; a 2nd-order Birch–Murnaghan EoS was used here as the uncertainties in some of the fitted parameters would otherwise have been very high. The axial EoS for PV-KCaF3 allows the macroscopic tilt angle, Φ, to be calculated as a function of pressure. When calculated on this basis, Φ rises from 10.1° at zero pressure to 16.1° at a pressure of 10 GPa, suggesting either that the relationship proposed by Tateno et al. (2010—see above) does not apply to the ABF3 perovskites, or that the minimum value of Φ, above which Φ will increase with pressure, is less than that for the A2+B4+O3 oxide perovskites. A second feature of the axial EoS shown in Table 2 is the extreme anisotropy of the CaIrO3-type phase of KCaF3 and the very low incompressibility (8 (2) GPa) of its b-axis. As discussed above, the likely mechanism for the formation of the Sb2S3 and P63/mmc-NaMgF3 phases requires the b-axis to be the most compressible so as to reduce the distance between the sheets of BF6 octahedra and enable them to cross-link. In KCaF3, however, the value of the incompressibility of the b-axis is so low as to suggest that the CaIrO3 structure may not be stable with respect to formation of the La2S3, or Sb2S3 polymorphs, as is indeed found to be the case (see below).

Table 1 Equation of state parameters for KCaF3, KF and CaF2 polymorphs obtained by fitting the internal energy versus volume to integrated 3rd-order Birch–Murnaghan equations of state
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Table 2 Volumetric and axial equation of state (EoS) parameters for KCaF3 and CaF2 polymorphs obtained by fitting pressure (as output by VASP) versus volume to 2nd-order Birch–Murnaghan EoS (Angel 2001); the volume range used in each case are shown
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The enthalpies derived from the EoS parameters listed in Table 1 are shown in Fig. 3, relative to that of CaIrO3-structured KCaF3. Considering first just the KCaF3 polymorphs themselves, it can be seen that a transition from the PV to the La2S3 structures might be expected at about 5.7 GPa, followed by further transitions to the Sb2S3 structure at around 10.4 GPa and to the P63/mmc-NaMgF3 structure at 28.3 GPa, confirming our original prejudice that PV-KCaF3 might be expected to transform at relatively low pressures. However, Fig. 3 also shows that the CaIrO3 structure does not appear to be the relatively most stable polymorph of KCaF3 at any pressure. Although we did not expect this result prior to the simulations, it is quite consistent with simple crystal-chemical considerations based on ionic sizes, as the large Ca2+ ion is readily able to support coordination numbers higher than six. At atmospheric pressure, the radius ratios of the B cations (in [6]—coordination; Shannon 1976) to the F− ions are 0.541 for Mg/F and 0.752 for Ca/F. The value for NaMgF3 thus lies firmly within the range expected for octahedral coordination (0.414–0.732; Pauling 1929). For KCaF3, however, the value lies slightly outside this range and just within the range for which a higher coordination number might be expected; on purely geometric grounds, radius ratios in the range 0.732–1.0 are sufficient to support [8]-fold cubic coordination. The CaF6 octahedra found in PV-KCaF3 must, therefore, be almost at the limit of stability. When pressure is applied to KCaF3 and NaMgF3 it is to be expected that these radius ratios will increase, as the F− ions will be more compressible than the cations. On compression, therefore, KCaF3 may quickly reach the point where octahedral coordination will no longer be stable and thus the La2S3 and Sb2S3 structures, in which the coordination will be sevenfold, or almost sevenfold, will form. For NaMgF3 (Fig. 4 and see below) this point is not reached until much higher pressure and so there is a stable region of the CaIrO3 phase, which has a higher density than the PV phase and in which the structure still contains MgF6 octahedra.

Fig. 3
figure 3

Enthalpies of the PV, P21/m-CaRhO3, La2S3, Sb2S3 and P63/mmc-NaMgF3 polymorphs of KCaF3 relative to that of KCaF3 in the CaIrO3 structure. The relative enthalpies of mixtures of the simple fluorides are also shown. The lines for the Sb2S3 and P63/mmc-NaMgF3 polymorphs cross at 28.3 GPa

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Fig. 4
figure 4

Enthalpies of the PV, P21/m-CaRhO3, La2S3, Sb2S3 and P63/mmc-NaMgF3 polymorphs of NaMgF3 relative to that of NaMgF3 in the CaIrO3 structure. The relative enthalpies of mixtures of the simple fluorides are also shown

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However, when the enthalpies of the KCaF3 polymorphs are considered relative to that of a mixture of KF and CaF2, Fig. 3 also shows that the PV structure is the only polymorph of KCaF3 that is ever the thermodynamically most stable phase, and that this is the case only at low pressures. Above about 2.6 GPa, none of the other polymorphs considered are predicted to be stable with respect to dissociation into KF and CaF2. The simulations presented here are static and so are effectively athermal, but it seems unlikely that consideration of temperature will significantly affect the thermodynamic stability of the KCaF3 phases relative to dissociation into the simple fluorides, since, for example, Fig. 3 shows that by ~7 GPa the enthalpy difference, which increases with increasing pressure, is already about 100 meV per formula unit. It is possible, however, that the high-pressure KCaF3 structures could be synthesised by metastable compression in the diamond-anvil cell, since the dissociation reaction requires diffusion of atoms on the multi-unit-cell length scale, which will have a large energy barrier, whereas the structural transitions between the KCaF3 phases require no long-range atomic movements and should have much smaller energy barriers. Indeed, the spontaneous transition from the PV to P63/mmc-NaMgF3 structures found in the computer simulations suggests that there is no energy barrier to this transition. Support for this view comes from our recent study of NaCoF3 by synchrotron X-ray powder diffraction in a diamond anvil cell, where we found that at 300 K the Sb2S3 phase of NaCoF3 formed at ~23 GPa and persisted to pressures of at least 85 GPa (Wood et al. 2017). In contrast, Yusa et al. (2012) found that, on heating, NaCoF3 disproportionated into Na5Co3F11 and NaCo3F7 at ~20 GPa and 1400 K, with a negative Clapeyron slope, which implies that NaCoF3 should not be stable (with respect to other complex stoichiometries) by about 28 GPa at 300 K.

The two unpublished in situ high-pressure experimental studies that have been carried out to date are insufficient to determine whether or not any metastable high-pressure phases of KCaF3 exist. In the case of the neutron diffraction study by Knight et al. (2014), it has been confirmed (K S Knight, pers. comm.) that no decomposition products or phase transitions were observed in the range of the experiment, i.e. for pressures ≤6.86 GPa. This pressure is just greater than that at which the transition from the PV to the La2S3 phase is predicted to occur in our simulations, but when the likely slope of the phase boundary is taken into consideration it may not be sufficient. For NaCoF3, the slope of the PV/PPV boundary has been determined to be 15.5 MPa K−1 (Dobson et al. 2011); if a similar slope is assumed for the PV/La2S3 boundary in KCaF3, the transition pressure at 300 K rises to ~10.5 GPa, well above that achieved in the neutron diffraction study. Persistence of PV-KCaF3 to still higher pressures has been observed ex situ at 300 K in a sample recovered from 15 GPa in our multi-anvil press and also in our in situ diamond-anvil cell study, which showed PV-KCaF3 persisting to 20 GPa. However, the fact that no transition to the La2S3 or Sb2S3 structures was seen in this experiment does not necessarily mean that the P63/mmc-NaMgF3 structure will not form at higher pressures, as it is the transition from the PV to P63/mmc-NaMgF3 structures that we have observed to occur spontaneously in our computer simulations. Taking account of the slope of the phase boundary, as above, we might expect this transition to occur at ~33 GPa, considerably higher than the maximum pressure of the experiment. We consider, therefore, that a study of KCaF3 to ~50 GPa might well be rewarding, since if the P63/mmc-NaMgF3 polymorph of KCaF3 does form on further compression we believe that this would be the first occurrence of this structure.

A further possibility would be to determine whether (K, Na)(Ca, Mg)F3 solid solutions could provide an alternative route to the various high-pressure structures. Although NaCaF3 itself does not appear to exist (at least at atmospheric pressure), doping KCaF3 with Na should lead to a substantial reduction in the average size of the A cation, which, in the case of the CaIrO3 structure, will produce a substantial reduction in the length of the b-axis but only minor changes in the a- and c-axes. An estimate of the magnitude of these changes can be obtained by considering the CaIrO3 structure parameterised in terms of the four shortest cation–anion distances and three tilt angles, as suggested by Lindsay-Scott et al. (2011; Equations 4–6). By putting the interatomic distances equal to the sum of the ionic radii (Shannon 1976), and assuming the same tilt angles as in NaNiF3 (Lindsay-Scott et al. 2014), we obtain a = 3.48 Å, b = 12.12 Å and c = 8.67 Å for CaIrO3-type KCaF3 at P = 0, in good agreement with the cell parameters obtained by fitting the axial EoS to the VASP simulations (a = 3.47 (1) Å, b = 12.1 (2) Å and c = 8.36 (1) Å). If the AF distance is now reduced to that appropriate for a composition of K0.5Na0.5CaF3, a value of b = 11.00 Å then results (a and c are unchanged in this approximation), which is equivalent to applying a pressure of ~4.1 GPa. It thus seems quite possible that a combination of applied pressure and chemical substitution might well lead to the formation of the high-pressure KCaF3 phases under relatively easily accessible experimental conditions.

Equations of state for NaMgF3 polymorphs and phase diagram of NaMgF3

The equation of state parameters for the NaMgF3, NaF and MgF2 polymorphs are listed in Table 3 and the corresponding enthalpy curves are shown in Fig. 4. Considering first just the NaMgF3 phases, it can be seen that the predicted sequence of transitions, with increasing pressure (transition pressures in brackets), is PV to CaIrO3 (~22 GPa), CaIrO3 to Sb2S3 (~55 GPa) and Sb2S3 to P63/mmc-NaMgF3 (~205 GPa). Our results are, therefore, in good accord with those of Umemoto and Wentzcovitch (2006) who found the same sequence of transitions, occurring at about 18, 44 and 223 GPa. Conversely, Xu et al. (2015) found the sequence PV to CaIrO3 (~19 GPa), followed by CaIrO3 to La2S3 (~51 GPa). All three studies agree in that they indicate a fairly wide phase field (~30 GPa) for the CaIrO3 structure, reflecting the ability of the Mg ions to remain in octahedral coordination to higher pressures (see above). Although the stable phase at pressures above ~55 GPa predicted by Xu et al. (2015) differs from the Sb2S3 structures found by Umemoto and Wentzcovitch (2006) and in the present study, it should be pointed out that the difference in enthalpy is small, being less than 10 meV/atom, and so the sequence of phases might differ at finite temperatures. There is, however, now strong experimental evidence, including crystal structure refinements, for the formation of Sb2S3 structures in some ABF3 compounds, as essentially phase-pure samples have been reported for NaFeF3 by Crichton et al. (2016) and for NaCoF3 by Wood et al. (2017). The experimental evidence for the formation of the La2S3 structure is, at present, much weaker. Xu et al. (2015) considered that their simulated powder diffraction pattern of La2S3-structured NaMgF3 resembled that of the “N phase” reported by Martin et al. (2006a), but the space group and unit-cell parameters suggested by Martin et al. are different. Martin et al. (2006a) have space group Pnnm and experimental unit-cell parameters of a = 8.353 (3) Å, b = 5.265 (2) Å, c = 5.857 (3) Å, V = 257.58 (3) Å3 at 37 (1) GPa, whereas Xu et al. (2015) have space group Pnma, with a = 5.082 Å, b = 2.829 Å, c = 10.215 Å, V = 146.9 Å3 for their simulation of the La2S3 structure at 60 GPa. Correspondence of the “N phase” of Martin et al. (2006a) with the La2S3 structure would, therefore, seem to imply that the experimental powder pattern has been mis-indexed.

Table 3 Equation of state parameters for NaMgF3, NaF and MgF2 polymorphs obtained by fitting the internal energy versus volume to integrated 3rd-order Birch–Murnaghan equations of state
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When the enthalpies of the NaMgF3 polymorphs are considered relative to that of a mixture of NaF and MgF2, however, Fig. 4 shows that only the PV and CaIrO3 structures will be the thermodynamically most stable phases. For pressures above ~48 GPa we find that all NaMgF3 polymorphs are unstable with respect to dissociation into NaF and MgF2. Once again our results are in accord with the original findings for NaMgF3 of Umemoto and Wentzcovitch (2006) who reported a dissociation pressure of ~40 GPa.

Our findings for both NaMgF3 and KCaF3, therefore, support the view expressed previously by Umemoto et al. (2006a), Grocholski et al. (2010) and Tsuchiya and Tsuchiya (2011) that in the natural silicate system as found in the deep interiors of the mantles of large terrestrial exoplanets or in the cores of gas giants, dissociation into high-pressure polymorphs of the simple oxides, MgO and SiO2, will occur before the formation of any of the high-pressure structures considered here.