We use state-of-the-art ab initio calculations based on the generalized gradient approximation of the density functional theory in the planar augmented wavefunction formalism to determine the elastic constants tensor of perovskite and post-perovskite with formulas AlFeO3 and FeAlO3 in which Fe or Al respectively occupy only octahedral sites, for the stable magnetic configurations. The phase transition between perovskite and post-perovskite is associated with a site exchange, during which Fe from the inter-octahedral site in perovskite moves into the octahedral site in post-perovskite. Following this transition path the elastic moduli show positive jumps, considerably larger than for MgSiO3. The phase transition is marked by a positive jump of 0.04 km/s (0.33%) in the velocity of the compressional waves and by a negative jump of −0.15 km/s (−1.87%) in shear wave velocity. We find that the effects of the Mg + Si <=> Al + Fe substitution on the seismic properties of MgSiO3 perovskite and post-perovskite depend on the crystallography of the substitution, namely the position the exchanged cations take in the structure.
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 In the Earth's lower mantle, the most abundant silicate phase, both in perovskite (pv) and its high-pressure polymorph, post-perovskite (ppv) structures, is not pure MgSiO3, but contains a certain amount of ferrous iron, ferric iron and aluminum as major substituents for Mg and Si. The exact proportion of each is still a matter of debate and obviously depends on a variety of factors, like local mantle conditions, temperature and pressure, mineralogical and petrological history, etc. An improved description should be obtained in a multidimensional system whose major components are MgSiO3-FeSiO3-FeAlO3-AlFeO3-Al2O3-Fe2O3. Realistic compositions would fall close to MgSiO3 but will not be pure [Anderson, 1983].
 Although pure AlFeO3 composition is highly unlikely to exist, certain amounts of Al and Fe3+ are most certainly present in both pv and ppv in double substitution to Mg and Si. Consequently this substitution can change the thermodynamic conditions of the phase transition, modify the seismic properties, or affect the partitioning of elements or the spin state of iron.
 In a previous computational study [Caracas, 2010] we analyzed the relative static stability of pv and ppv in this system. We treated only ordered stoichiometric structures and showed that the transition from pv to ppv involves an exchange of cation sites. We adopted a notation based on crystallochemistry: AlFeO3 denotes the structure where Fe occupies the octahedral site and FeAlO3 denotes the situation where Al occupies the octahedral space. At low pressure the stable phase is FeAlO3 pv and above 90 GPa the stable phase is AlFeO3 ppv, both in antiferromagnetic configuration. If the cation exchange fails to occur then the high-spin antiferromagnetic FeAlO3 pv can metastably exist up to pressures beyond the Earth's mantle limit.
 Most of the previous experimental studies on this system focused on the mechanism of incorporation of ferric iron in pv and on the possible role played by aluminum [e.g., Andrault et al., 2001; Daniel et al., 2004; Nishio-Hamane et al., 2008]. They showed that there is a direct correlation between the amount of Al and Fe3+ that can be dissolved in MgSiO3 pv, and that Al occupies preferentially the octahedral sites. More recent experiments addressing the transition from pv to ppv showed that the addition of Al and Fe3+ increases the transition pressure [Andrault et al., 2008], in contradiction with the static theoretical findings for pure Al,Fe3+ oxide [Caracas, 2010]. The discrepancy between the theoretical and the experimental studies arises mainly from the sluggish perovskite - post-perovskite phase transition, which is associated with the Al and Fe3+ exchanging crystallographic sites. This involves breaking strong interatomic bonds that are hard to achieve, which in turn results in strong kinetic effects that delay the experimental transition pressure with respect to the theoretical one.
 In this study we focus on the elastic and seismic properties. We determine the full elastic constants tensor and derive the corresponding bulk seismic properties for both crystallochemical cases for both structures. We consider only the stable magnetic configurations in each case: antiferromagnetic for FeAlO3 pv and AlFeO3 ppv and ferromagnetic for AlFeO3 pv and FeAlO3 ppv with net residual magnetic moments of respectively 4 and 8 magneton Bohrs per primitive unit cell.
2. Computational Details
 We perform static (i.e., T = 0K) first-principles calculations based on the planar augmented wavefunctions (PAW) formalism [Blochl, 1994] within the generalized gradient approximation [Perdew et al., 1996] of the density functional theory, as implemented in the ABINIT code [Torrent et al., 2008; Gonze et al., 2002]. We sample the electronic density in the reciprocal space (in the first Brillouin zone) using 6 × 6 × 6 and 6 × 6 × 4 grids of special high-symmetry k points [Monkhorst and Pack, 1976] for the pv and ppv structures, respectively. We employ a 16 Ha (1 Ha = 27.2116 eV) kinetic energy cut-off for the wavefunctions on the coarse mesh and a 36 Ha cut-off for the wavefunctions on the finer grid inside the PAW spheres. This set of parameters ensures an accuracy of the calculation better than 1 GPa in pressure and 1 mHa/unit cell in energy.
 A drawback of our calculations might be their static character. But many of the physical properties of geophysical interest, like elasticity and compressibility, depend in a first approximation on the specific volume [Wentzcovitch et al., 2004], which is shifted in static calculations by a few percent relative to the experimental values at higher temperature; but the shift arises in a consistent manner. This makes the static (0K) elastic data extremely useful. As GGA tends to overestimate the experimental ambient volume and LDA to underestimate it, results in GGA are comparable, though not similar, to thermal LDA [Payne et al., 1992]. Moreover for this particular system the primary effect of temperature would be in affecting the Fe/Al site occupancy. A second limitation, the use of standard GGA rather than more sophisticated ways of treating strongly correlated electrons, like the GGA + U, is less important since the +U formalism is expected to have little effect on mechanical properties. This arises from the shift of the total energy in GGA + U relative to GGA, which is commensurable to the U parameter and weakly dependent on volume. Consequently derivatives of the GGA + U energy would not drastically differ from the derivatives of the GGA energy at a given density.
 First we fully relax the crystal structure in the desired magnetic configuration at a given target pressure, i.e. under respective symmetry constraints we allow the atoms to move to minimize the residual forces and the unit cell to distort to eliminate the non-hydrostatic stresses. Then we apply positive and negative strains of 1% and 2%. For each case we allow only the atoms to relax and we measure the residual stresses. Then we obtain the elastic constants tensor using the stress-strain relation in the linear limit.
 Both structures are orthorhombic and have nine independent, non-zero elastic constants, which in matrix (Voigt) notation are: C11, C22, C33, C12, C13, C23, C44, C55, C66. The first three are pure-strain and the last three pure shear. We consider homogenous aggregates and use standard expressions for the bulk (K) and shear (G) elastic moduli, and for the compressional (Vp) and shear (Vs) seismic wave velocities.
3. Results and Discussion
Table 1 summarizes the densities of the pv and ppv structures in the two crystallochemical cases. For both structures the arrangement with Fe occupying the octahedral site yields larger densities, partly because of the lower value of the magnetic spin. At the transition between the stable thermodynamical phases, FeAlO3 pv and AlFeO3 ppv the density difference is almost 0.5 g/cm3, corresponding to 8.23% positive jump.
Table 1. Density in Grams per Cubic Centimeter at Several Pressures in the Two Crystallochemical Casesa
Pressure values are in GPa. PV and PPV denote respectively the perovskite and post-perovskite structures. The jump at the phase transition pressure (90 GPa) between FeAlO3 PV and AlFeO3 PPV is almost 0.5 g/cm3.
 The elastic constants of pv at several pressures are listed in Table 2. The compressibility of the two crystallochemical cases is different and in general the pressure enhances these differences. C12, C13 and C23 and the pure strain elastic constants C11 and C22 are larger in AlFeO3 than in FeAlO3. C33 and all the three pure shear constants are larger in FeAlO3. This results in opposite trends for the bulk and the shear elastic moduli (for homogeneous aggregates) with K smaller and G larger in FeAlO3. pv in AlFeO3 configuration shows a pronounced elastic softening at ambient pressure conditions with C66 only 3 GPa and C44 = 54 GPa. This yields a smaller than expected shear modulus. The poisson ratio also shows an anomaly with its largest value at 0 GPa. However at ambient conditions pv is not thermodynamically stable.
Table 2. Elastic Constants of Perovskite at Several Pressures in the Two Crystallochemical Casesa
All values are in GPa, except for ν, which is adimensional.
 The pressure variation of the elastic constants of ppv is listed in Table 3. The elastic properties of the two crystallochemical cases are similar. The major differences are in the C11 and C22 constants: the first one is larger in FeAlO3 and the second one in AlFeO3. The differences are accentuated by pressure. The behavior of the pure shear constants is different: C44 is larger in AlFeO3, C66 is larger in FeAlO3 and C55 is roughly the same in the two cases. The bulk moduli differ by about 20 GPa and the shear moduli by about 50 GPa both larger for AlFeO3. Both the Young modulus and the Poisson ratio are smaller than most of other perovskites [Caracas and Cohen, 2007; Caracas et al., 2010].
Table 3. Elastic Constants of Post-Perovskite at Several Pressures in the Two Crystallochemical Casesa
All values are in GPa, except for ν, which is adimensional.
 At the thermodynamic transition accompanied by the cationic site exchange at 90 GPa [Caracas, 2010] the bulk modulus has a positive jump of 70 GPa (13.8%) and the shear modulus a positive jump of 16 GPa (4.2%) as shown in Figure 1. These values are larger than at the same transition in pure MgSiO3 [Caracas and Cohen, 2005].
Figure 2 shows the variation of the seismic wave velocities with respect to pressure for all the cases considered here. In pv the velocities of the FeAlO3 crystallochemical arrangement are larger than those of the AlFeO3 one by almost 0.6 km/s for Vp and about 1 km/s for Vs in the 60–120 GPa pressure range. In ppv the trend is reversed with Vp larger in AlFeO3 by 0.12–0.27 km/s and Vs larger in AlFeO3 by 0.29–0.36 km/s in the 60–150 GPa pressure range. The velocities of the pv and ppv structures have different slopes under pressure, regardless of the crystallochemistry. The slope of Vp changes from 0.022 km/s/GPa for pv to 0.027 km/s/GPa in ppv and the slope of Vs changes from 0.010 km/s/GPa for pv to 0.016 km/s/GPa in ppv (For Vp of AlFeO3 we used for the fit only the 60–120 GPa pressure range because of the low-pressure softening). This behavior stems from different equations of state, with larger compressibility and larger density for the ppv.
 In Figures 1 and 2 we outline with arrows the path followed by the system at thermodynamic equilibrium, namely during a phase transition at 90 GPa from the FeAlO3 pv to AlFeO3 ppv. The change in Vp is small: a positive jump of 0.043 km/s corresponding to 0.33% of the velocity in pv at the transition pressure. The change in Vs is larger and of opposite sign: a negative jump of −0.15 km/s corresponding to −1.87% of the shear velocity in pv. At larger pressure, namely at 120 GPa, the difference in Vp between the two structures and crystallochemical cases increases to 0.24 km/s, corresponding to 1.75% relative difference, while the difference in Vs vanishes.
 Of course these values are not meaningful by themselves for the lower mantle, but they should be used to apply corrections to MgSiO3 case, alongside FeSiO3 or Al2O3.
 Static calculations for the pure MgSiO3, FeSiO3 and Al2O3 compositions [e.g., Oganov and Ono, 2004; Caracas and Cohen, 2005; Tsuchiya and Tsuchiya, 2006; Stackhouse et al., 2005, 2006] and summarized by Caracas and Cohen  show that MgSiO3 has the largest seismic wave velocities, both for compressional waves and for shear waves. Any addition of ferrous iron or aluminum or a combination thereof decrease both Vp and Vs for both pv and ppv. Seismic anisotropy instead does not present clear trends and adding Fe2+ or Al [Caracas and Cohen, 2007] or allowing for the iron spin to change [Caracas et al., 2010] can yield different anisotropy patterns. For a Mg-rich silicate composition [Caracas and Cohen, 2007] compatible with an average pyrolitic mantle composition [Kesson et al., 1998; Murakami et al., 2005] and assuming that all Fe in the silicate is ferrous, the differences between seismic velocities of postperovskite and perovskite structures are positive for Vp at pressures above 112 GPa and for Vs at pressures above 86 GPa: at 100 GPa the differences are −0.2% in Vp and +0.7% in Vs; at 130 GPa the differences are +0.5% in Vp and +2% in Vs. The only static elastic calculations so far on perovskite (Mg0.9375Fe0.0625) (Si0.9375Al0.0625)O3 compositions with ferric iron on the Mg site and aluminum on the Si site, in several substitutional patterns Li et al.  showed a decrease of the velocities. Moreover the velocities varied within narrow ranges for both Vp, 14.06–14.09 km/s, and Vs, 7.53–7.55 km/s, regardless of the geometric arrangement of the substitution.
 In case of mixed addition of both Al and ferric Fe, our results summarized in Figure 2 and Table 4 shows that the trends are different than in the case of the substitutions with Al-only and/or Fe2+. If we refer in particular to the values at 120 GPa (summarized by Caracas and Cohen [2007, Table 2]) the Mg + Si < = > Al + Al substitution decreases Vp of pure MgSiO3 in both crystallochemical cases and for both pv and ppv. Moreover it is interesting to note that the quantitative behavior of FeAlO3 is similar to the one of Al2O3. One could cautiously speculate that the presence of Al in the octahedral framework is determinant to the compressibility pattern and that the cations on the interoctahedral site play a minor role. At 120 GPa, for Vs the Al-Fe substitution in pv (Fe on octahedral site; Vs = 7.31 km/s) and the Al-Fe substitution in ppv (Fe on octahedral site; Vs = 7.97 km/s) tend to lower or at least keep constant the Vs of MgSiO3 (Vs = 7.7 km/s for pv and Vs = 7.8–8.3 km/s for ppv). But the Fe-Al substitution (Al on octahedral site) in pv tends to increase the shear velocity, with Vs for pure FeAlO3 of 8.33 km/s at 120 GPa. The Al-Fe substitution in ppv (Al on octahedral site) also tends to increase Vs of MgSiO3, with Vs for pure AlFeO3 ppv of 8.30 km/s at 120 GPa.
Table 4. Seismic Wave Velocities of Perovskite and Post-Perovskite at Several Pressures in the Two Crystallochemical Casesa
Pressure (P) is in GPa and velocities in km/s; pv, perovskite; ppv, post-perovskite.
Vp FeAlO3 pv
Vs FeAlO3 pv
Vp AlFeO3 pv
Vs AlFeO3 pv
Vp FeAlO3 ppv
Vs FeAlO3 ppv
Vp AlFeO3 ppv
Vs AlFeO3 ppv
 Experimental Brillouin measurements on pure MgSiO3 perovskite and post-perovskite [Murakami et al., 2007] show similar velocity jumps at the transition between the two structures and similar pressure dependencies. But pure Mg-composition are not enough to explain the seismic discontinuity at the top of the D″ layer (at 120–130 GPa). The velocity jumps for homogeneous MgSiO3 aggregates are too small and various mechanisms [Wookey et al., 2005], including lattice-preferred orientation (LPO) [Merkel et al., 2006] have been proposed to match mineral physics with seismology. However a viable alternative to LPO or at least a partial alternative that would reduce the constraints on LPO would be the change in chemistry, namely the presence of either ferrous [Caracas and Cohen, 2007] or ferric (this study) iron.
 Consequently, the theoretical values of this study are important as they show that (i) the thermodynamically-favorable substitution tends to decrease Vp and increase Vs and (ii) the influence of the Mg + Si < = > Al + Fe substitution on the seismic properties of both pv and ppv is strongly dependent on the substitution mechanism and the ordering of the cations on the crystallographic sites. Thus these results are part of the greater puzzle represented by the pv - ppv transition and represent critical information necessary to model the lowermost part of the Earth's mantle at realistic chemical compositions. A full integration of all the chemical information into a general seismic model of pv and ppv will be the subject of a future study.
 All the calculations were performed on the jade machine at CINES under stl2816 computational grant.