[1] Using first principles computations we show that NaMgF_{3} perovskite undergoes the same series of phase transitions as MgSiO_{3} perovskite, that is, the post-perovskite transition and the dissociation into CsCl-type NaF and cotunnite-type MgF_{2}. These fluorides also undergo the same series of phase transitions as MgO and SiO_{2}, the dissociation products of MgSiO_{3}. Since the phase transformations in NaMgF_{3} are not accompanied by any soft mode, we compute quasi-harmonic free energies and the respective phase boundaries. They have positive and negative Clapeyron slopes, respectively, like MgSiO_{3}. However, the transition pressures in NaMgF_{3} are much lower and could be easily achieved in diamond anvil experiments. Therefore NaMgF_{3} should be a good low-pressure analog of MgSiO_{3}.

[2] Since the discovery of the transition from perovskite (PV) to CaIrO_{3}-type post-perovskite (PPV) phase in MgSiO_{3} near the core-mantle boundary of the Earth (at ∼125 GPa and 2500 K) [Murakami et al., 2004; Tsuchiya et al., 2004; Oganov and Ono, 2004], this transition has been attracting great interest. Very recently we have predicted the dissociation of MgSiO_{3} PPV into CsCl-type MgO and cotunnite-type SiO_{2} at ultrahigh pressure and temperature (PT) typical of solar giants, Jupiter and Saturn, and newly-found extrasolar planets [Umemoto et al., 2006]. Unfortunately, the predicted dissociation pressure is too high (∼1 TPa) to be achieved experimentally today. Therefore low-pressure analogs of MgSiO_{3} are highly desired for experimental investigation of properties of the CaIrO_{3}-type structure. NaMgF_{3} PV, neighborite, is one of the best candidates. It is a stable Pbnm perovskite phase at ambient condition and undergoes a transition to a cubic phase by elevating temperature. The temperature dependence of structural parameters has been studied in detail [Zhao et al., 1993a, 1993b, 1994; Zhou et al., 1997; Smith et al., 2000]. The PPV transition occurs also in NaMgF_{3} under pressure [Parise et al., 2004; Liu et al., 2005]. However, the pressure dependence of NaMgF_{3}'s structural properties has been reported in detail only at low pressure [Zhao et al., 1994], not near the PPV transition. If NaMgF_{3} is a good low-pressure analog of MgSiO_{3}, then the ternary system of Na, Mg, and F could be useful for comparison at higher pressures as well. It would be particularly important for investigations of the CaIrO_{3}-type structure and its dissociation into the elementary fluorides NaF and MgF_{2}, which are also low-pressure analogs of MgO and SiO_{2} [Yagi et al., 1983; Haines et al., 2001].

[3] In this paper, we investigate NaMgF_{3} under pressure by first-principles and demonstrate that NaMgF_{3} should undergo the same types of pressure-induced phase transitions as MgSiO_{3}: the transition from the PV to PPV phase and the dissociation of the PPV phase into CsCl-type NaF and cotunnite-type MgF_{2}.

2. Computational Method

[4] Calculations were performed using the local-density approximation (LDA) and the generalized-gradient approximation (GGA) [Perdew and Zunger, 1981; Perdew et al., 1996]. The valence electronic configurations used for the generation of Vanderbilt ultrasoft pseudopotentials [Vanderbilt, 1990] are 2s^{2} 2p^{6}3s^{1} 3p^{0} 3d^{0}, 2s^{2} 2p^{6} 3s^{2} 3p^{0} 3d^{0}, and 2s^{2} 2p^{5} 3d^{0}, for Na, Mg, and F, respectively. Their cutoff radii are 1.6 a.u. for all quantum numbers l in each atom. The plane-wave cutoff energy is 60 Ry. We used variable-cell-shape molecular dynamics [Wentzcovitch, 1991; Wentzcovitch et al., 1993] for structural optimization under arbitrary pressures. Dynamical matrices were computed at wave vectors q using density-functional perturbation theory [Giannozzi et al., 1991; Baroni et al., 2001]. The numbers of formula units in the unit cells, k points in the irreducible wedge, and q points are (4, 4, 8) for NaMgF_{3} PV, (2, 6, 6) for NaMgF_{3} PPV, (1, 10, 8) for NaCl-type NaF, (1, 20, 10) for CsCl-type NaF, (2, 6, 6) for rutile-type MgF_{2}, (2, 8, 8) for CaCl_{2}-type MgF_{2}, (4, 1, 4) for pyrite-type MgF_{2}, and (4, 2, 8) for cotunnite-type MgF_{2}. Force constants are extracted to build dynamical matrices at arbitrary phonon q vectors. Vibrational contributions to the free energy due to the zero-point motion (ZPM) and finite temperature are taken into account by the quasi-harmonic approximation (QHA) [Wallace, 1972].

[5] Comparisons between our calculated structural parameters (Table 1) and transition pressures (Table 2) with experimental values in Tables 1 and 2 confirm the reliability of our computational method. Notice that experimental data appear bracketed between LDA and GGA results, except the transition pressure from NaCl-type to CsCl-type NaF, where the experimental value is slightly larger than the GGA value but very close to it. Hence, the two series of calculations, one using the LDA and other the GGA should provide good predictions for the behavior of NaMgF_{3} under pressure.

Table 1. Calculated Parameters of the Third-Order Birch-Murnaghan Equations of States by LDA and GGA for Perovskite NaMgF_{3}, Post-Perovskite NaMgF_{3}, NaCl-type NaF, and Rutile-type MgF_{2}^{a}

[6] Static LDA (GGA) enthalpy calculation shows NaMgF_{3} PV transforms to PPV at 17.5 (22.5) GPa (blue line in Figure 1). Around the PPV transition pressure no phonon instability is found both in the PV and PPV phases, indicating that the PPV transition is enthalpically driven (Figure 2). At 0 GPa, all phonon frequencies in NaMgF_{3} PPV are real and there is no dynamical instability. This is consistent with the experimental result that the PPV phase can be recovered at ambient pressure and room temperature [Liu et al., 2005]. But imaginary phonon frequencies (soft mode) occur at negative pressure, ∼−7 GPa (∼−4 GPa) by LDA(GGA) (Figure 2). The lowest acoustic phonon branch along the Δ line goes soft entirely. Since this is a typical sign of amorphization, NaMgF_{3} PPV should amorphize upon decompression at high temperatures (annealing), which is somewhat analogous to decompressing to negative pressures at 0 K.

[7] Next we investigate the dissociation of NaMgF_{3} PPV into NaF and MgF_{2}. NaCl-type and CsCl-type NaF and pyrite-type and cotunnite-type MgF_{2} are the relevant phases for the dissociation, because they are the stable phases beyond the PPV transition pressure (Table 2). Figure 1 shows that NaMgF_{3} PPV dissociates into CsCl-type NaF and cotunnite-type MgF_{2} at 40 GPa (LDA). The GGA dissociation pressure is 48 GPa. This dissociation pressure is much smaller than that of MgSiO_{3} (∼1 TPa) [Umemoto et al., 2006] and can be easily achieved by diamond anvil experiments. There is no soft mode in CsCl-type NaF and cotunnite-type MgF_{2} beyond the dissociation pressure and these phases are dynamically stable. NaMgF_{3} PPV is also dynamically stable around the dissociation pressure; the dissociation as well as the PPV transition are enthalpically driven. Table 3 gives calculated structural parameters of NaMgF_{3} PPV at 30 GPa and CsCl-type NaF and cotunnite-type MgF_{2} at 50 GPa. They should be useful for the experimental identification of these phases.

Table 3. Calculated Lattice Constants and Atomic Wyckoff Positions of NaMgF_{3} PPV at Static 30 GPa and CsCl-type NaF and Cotunnite-type MgF_{2} at Static 50 GPa

LDA

GGA

NaMgF_{3}PPV (Space Group: Cmcm)

(a, b, c), a.u.

(5.238, 16.407, 12.995)

(5.343, 16.907, 13.240)

Na (4c)

(0, 0.252, 0.75)

(0, 0.252, 0.75)

Mg (4a)

(0, 0, 0)

(0, 0, 0)

F_{1} (4c)

(0, 0.071, 0.25)

(0, 0.071, 0.25)

F_{2} (8f)

(0, 0.360, 0.060)

(0, 0.362, 0.060)

CsCl-type NaF (Space Group: Pm3m)

a, a.u.

4.626

4.729

Cotunnite-type MgF_{2}(Space Group: Pbnm)

(a, b, c), a.u.

(11.016, 9.377, 5.527)

(11.195, 9.547, 5.648)

Mg (4c)

(0.118, 0.248, 0.25)

(0.116, 0.249, 0.25)

F_{1} (4c)

(0.429, 0.353, 0.25)

(0.428, 0.354, 0.25)

F_{2} (4c)

(0.667, 0.978, 0.25)

(0.667, 0.979, 0.25)

[8] We have also investigated possible structures of post-PPV NaMgF_{3}: LiSbO_{3}-type BaNiO_{3}-type, hexagonal BaTiO_{3}-type, and P6_{3}/mmc NaMgF_{3}. The former three structures are described by Wyckoff [1965] and Hyde and Andersson [1989]. The LiSbO_{3}-type structure consists of MgF_{6} octahedra interconnected in an α-PbO_{2}-like network; this is a likely higher connectivity. In BaNiO_{3}-type structure, MgF_{6} octahedra share their faces and form separate columns. In the hexagonal BaTiO_{3}-type structure, MgF_{6} octahedra share their faces and apices. In these two structures, MgF_{6} octahedra have a higher degree of connectivity than the PPV structure. The last P6_{3}/mmc structure, to our knowledge, has not been observed experimentally so far. Its space group is a supergroup of Cmcm, the space group of the PPV structure. This structure has been found by a static compression of NaMgF_{3} PPV to 150 GPa. Magnesium is 8-fold coordinated and the MgF_{8} polyhedra share edges and apices. The MgF_{8} network is related to Ni_{2}In-type structure. We found that all four phases have higher enthalpies than NaMgF_{3} PPV around the dissociation pressure. At 50 GPa, LDA calculations show that ΔH (Ry/f.u.) of these four structures with respect to NaMgF_{3} PPV are 0.092, 0.173, 0.128, and 0.033, respectively.

[9] Using the QHA in conjunction with the computed phonon density of states (Figure 3), we obtain phase boundaries of the PPV transition and the dissociation (Figure 4). Empirically, the QHA works well below the temperature at which the thermal expansivity α (P, T) starts to deviate from linear behavior [Karki et al., 1999]. Dashed lines in Figure 4 represent P-T conditions on which d^{2} α/dT^{2} = 0 for the investigated phases; the QHA should be valid at least up to these lines. The experimental PPV transition pressure at room temperature (19.7 GPa) [Liu et al., 2005] is bracketed by LDA (19.4 GPa) and GGA (24.0 GPa) values at 300 K. The unit cell volume decreases through the PPV transition by ∼2.5%. The shortest covalent Mg-F bond length also decreases through the transition by ∼1%. These decreases induce higher phonon frequencies (Figure 3a) and consequently lower vibrational entropy in the PPV phase than those in the PV phase. Hence, ZPM energy increases the PPV transition pressure from the static value. The Clapeyron slope (dP/dT = ΔS/ΔV) of the PPV transition is positive, 6 (5) MPa/K at 500 K by LDA (GGA).

[10] Throughout the dissociation, the volume decreases by ∼5–6%, while the covalent Mg-F bond length increases accompanied by an increase in the Mg coordination number (6 to 9). These longer covalent bonds give rise to smaller high (optical) phonon frequencies in general (Figure 3b). Hence, ZPM energy decreases the dissociation pressure from the static value. The Clapeyron slope is negative above ∼300 K because of the overall increase in entropy: −1.7 (−1.8) MPa/K at 500 K and −3.3(−3.6) MPa/K at 1000 K by LDA (GGA). On the other hand, the volume shrinkage increases the lowest (acoustic) phonon frequencies (<∼200–300 cm^{−1} ∼300–400 K in Figure 3b). Since this effect compensates that of the longer bond lengths, the Clapeylon slope below ∼300 K is almost independent on temperature. The same type of Clapeyron slope occurs in the transition of NaCl-type to CsCl-type NaF and pyrite-type to cotunnite-type MgF_{2}. In both cases there are increases in covalent bond lengths and cation coordination numbers and decrease of volumes.

[11] The present results show that NaMgF_{3} PV undergoes the same sequence of phase transitions as MgSiO_{3} PV: the PPV transition and the dissociation into NaF and MgF_{2}. The Clapeyron slopes of these transitions are positive and negative respectively, like the case of MgSiO_{3}. Transition pressures in NaMgF_{3} are much smaller than those in MgSiO_{3}. Therefore, NaMgF_{3} should be a good low-pressure analog of MgSiO_{3} and serve as an alternative route to investigate properties of the PPV structure, including its dissociation.

Acknowledgments

[12] Calculations have been carried out using the Quantum-ESPRESSO package (http://www.pwscf.org). Research was supported by NSF grants EAR-0135533, EAR-0230319, EAR-0510501, and ITR-0428774 (VLab).