Abstract
 Top of page
 Abstract
 1. Introduction
 2. Computational Details
 3. Results and Discussion
 Acknowledgments
 References
 Supporting Information
[1] We use stateoftheart 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 postperovskite with formulas AlFeO_{3} and FeAlO_{3} in which Fe or Al respectively occupy only octahedral sites, for the stable magnetic configurations. The phase transition between perovskite and postperovskite is associated with a site exchange, during which Fe from the interoctahedral site in perovskite moves into the octahedral site in postperovskite. Following this transition path the elastic moduli show positive jumps, considerably larger than for MgSiO_{3}. 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 MgSiO_{3} perovskite and postperovskite depend on the crystallography of the substitution, namely the position the exchanged cations take in the structure.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. Computational Details
 3. Results and Discussion
 Acknowledgments
 References
 Supporting Information
[2] In the Earth's lower mantle, the most abundant silicate phase, both in perovskite (pv) and its highpressure polymorph, postperovskite (ppv) structures, is not pure MgSiO_{3}, 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 MgSiO_{3}FeSiO_{3}FeAlO_{3}AlFeO_{3}Al_{2}O_{3}Fe_{2}O_{3}. Realistic compositions would fall close to MgSiO_{3} but will not be pure [Anderson, 1983].
[3] Although pure AlFeO_{3} composition is highly unlikely to exist, certain amounts of Al and Fe^{3+} 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.
[4] 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: AlFeO_{3} denotes the structure where Fe occupies the octahedral site and FeAlO_{3} denotes the situation where Al occupies the octahedral space. At low pressure the stable phase is FeAlO_{3} pv and above 90 GPa the stable phase is AlFeO_{3} ppv, both in antiferromagnetic configuration. If the cation exchange fails to occur then the highspin antiferromagnetic FeAlO_{3} pv can metastably exist up to pressures beyond the Earth's mantle limit.
[5] 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; NishioHamane et al., 2008]. They showed that there is a direct correlation between the amount of Al and Fe^{3+} that can be dissolved in MgSiO_{3} 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 Fe^{3+} increases the transition pressure [Andrault et al., 2008], in contradiction with the static theoretical findings for pure Al,Fe^{3+} oxide [Caracas, 2010]. The discrepancy between the theoretical and the experimental studies arises mainly from the sluggish perovskite  postperovskite phase transition, which is associated with the Al and Fe^{3+} 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.
[6] 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 FeAlO_{3} pv and AlFeO_{3} ppv and ferromagnetic for AlFeO_{3} pv and FeAlO_{3} ppv with net residual magnetic moments of respectively 4 and 8 magneton Bohrs per primitive unit cell.
2. Computational Details
 Top of page
 Abstract
 1. Introduction
 2. Computational Details
 3. Results and Discussion
 Acknowledgments
 References
 Supporting Information
[7] We perform static (i.e., T = 0K) firstprinciples 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 highsymmetry 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 cutoff for the wavefunctions on the coarse mesh and a 36 Ha cutoff 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.
[8] 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.
[9] 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 nonhydrostatic 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 stressstrain relation in the linear limit.
[10] Both structures are orthorhombic and have nine independent, nonzero elastic constants, which in matrix (Voigt) notation are: C_{11}, C_{22}, C_{33}, C_{12}, C_{13}, C_{23}, C_{44}, C_{55}, C_{66}. The first three are purestrain 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
 Top of page
 Abstract
 1. Introduction
 2. Computational Details
 3. Results and Discussion
 Acknowledgments
 References
 Supporting Information
[11] 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, FeAlO_{3} pv and AlFeO_{3} ppv the density difference is almost 0.5 g/cm^{3}, corresponding to 8.23% positive jump.
Table 1. Density in Grams per Cubic Centimeter at Several Pressures in the Two Crystallochemical Cases^{a}P  FeAlO_{3} PV  AlFeO_{3} PV  FeAlO_{3} PPV  AlFeO_{3} PPV 


0  4.633  4.800   
30  5.158  5.406   
60  5.652  5.789  5.685  5.928 
90  6.030  6.215  6.037  6.526 
120  6.368  6.537  6.347  6.592 
150  6.675  6.828  6.626  6.870 
[12] 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. C_{12}, C_{13} and C_{23} and the pure strain elastic constants C_{11} and C_{22} are larger in AlFeO_{3} than in FeAlO_{3}. C_{33} and all the three pure shear constants are larger in FeAlO_{3}. This results in opposite trends for the bulk and the shear elastic moduli (for homogeneous aggregates) with K smaller and G larger in FeAlO_{3}. pv in AlFeO_{3} configuration shows a pronounced elastic softening at ambient pressure conditions with C_{66} only 3 GPa and C_{44} = 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 Cases^{a}P  AlFeO_{3}  FeAlO_{3} 

0  30  60  90  120  150  0  30  60  90  120  150 


C_{11}  397  532  647  753  895  983  421  513  609  698  820  883 
C_{22}  478  650  814  939  1074  1176  477  625  749  893  1020  1116 
C_{33}  256  437  586  712  825  922  314  446  590  728  871  953 
C_{12}  159  288  404  529  638  713  205  239  333  410  497  575 
C_{13}  136  239  319  407  497  577  159  227  290  347  428  490 
C_{23}  150  245  342  438  537  624  184  249  319  393  465  528 
C_{44}  54  127  171  218  255  278  126  170  204  247  284  307 
C_{55}  103  118  119  132  147  155  101  114  136  153  171  168 
C_{66}  3  39  78  95  126  142  97  142  173  201  225  236 
K  216  345  456  563  672  757  250  331  422  508  606  676 
G  89  202  265  302  349  374  216  282  336  390  442  455 
Y  236  506  666  768  893  963  503  659  796  932  1066  1116 
η  0.32  0.26  0.26  0.27  0.28  0.29  0.17  0.17  0.19  0.19  0.21  0.23 
[13] 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 C_{11} and C_{22} constants: the first one is larger in FeAlO_{3} and the second one in AlFeO_{3}. The differences are accentuated by pressure. The behavior of the pure shear constants is different: C_{44} is larger in AlFeO_{3}, C_{66} is larger in FeAlO_{3} and C_{55} 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 AlFeO_{3}. 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 PostPerovskite at Several Pressures in the Two Crystallochemical Cases^{a}P  AlFeO_{3}  FeAlO_{3} 

90  120  150  90  120  150 


C_{11}  865  955  1091  860  1010  1132 
C_{22}  865  993  1125  767  862  962 
C_{33}  891  1028  1165  868  1015  1164 
C_{12}  456  549  647  412  511  602 
C_{13}  367  439  518  375  450  534 
C_{23}  472  569  672  450  543  641 
C_{44}  209  223  245  144  170  207 
C_{55}  103  128  171  91  114  149 
C_{66}  342  387  444  291  335  370 
K  578  675  782  552  654  756 
G  407  454  522  349  403  463 
Y  988  1113  1280  865  1003  1153 
η  0.22  0.23  0.23  0.24  0.24  0.25 
[14] 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 MgSiO_{3} [Caracas and Cohen, 2005].
[15] 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 FeAlO_{3} crystallochemical arrangement are larger than those of the AlFeO_{3} 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 AlFeO_{3} by 0.12–0.27 km/s and Vs larger in AlFeO_{3} 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 AlFeO_{3} we used for the fit only the 60–120 GPa pressure range because of the lowpressure softening). This behavior stems from different equations of state, with larger compressibility and larger density for the ppv.
[16] 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 FeAlO_{3} pv to AlFeO_{3} 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.
[17] Of course these values are not meaningful by themselves for the lower mantle, but they should be used to apply corrections to MgSiO_{3} case, alongside FeSiO_{3} or Al_{2}O_{3}.
[18] Static calculations for the pure MgSiO_{3}, FeSiO_{3} and Al_{2}O_{3} 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 [2007] show that MgSiO_{3} 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 Fe^{2+} 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 Mgrich 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 (Mg_{0.9375}Fe_{0.0625}) (Si_{0.9375}Al_{0.0625})O_{3} compositions with ferric iron on the Mg site and aluminum on the Si site, in several substitutional patterns Li et al. [2005] 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.
[19] 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 Alonly and/or Fe^{2+}. 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 MgSiO_{3} in both crystallochemical cases and for both pv and ppv. Moreover it is interesting to note that the quantitative behavior of FeAlO_{3} is similar to the one of Al_{2}O_{3}. 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 AlFe substitution in pv (Fe on octahedral site; Vs = 7.31 km/s) and the AlFe substitution in ppv (Fe on octahedral site; Vs = 7.97 km/s) tend to lower or at least keep constant the Vs of MgSiO_{3} (Vs = 7.7 km/s for pv and Vs = 7.8–8.3 km/s for ppv). But the FeAl substitution (Al on octahedral site) in pv tends to increase the shear velocity, with Vs for pure FeAlO_{3} of 8.33 km/s at 120 GPa. The AlFe substitution in ppv (Al on octahedral site) also tends to increase Vs of MgSiO_{3}, with Vs for pure AlFeO_{3} ppv of 8.30 km/s at 120 GPa.
Table 4. Seismic Wave Velocities of Perovskite and PostPerovskite at Several Pressures in the Two Crystallochemical Cases^{a}P  30  60  90  120  150 


Vp FeAlO_{3} pv  11.71  12.41  13.06  13.70  
Vs FeAlO_{3} pv  7.40  7.71  8.04  8.33  
Vp AlFeO_{3} pv  10.66  11.82  12.47  13.19  
Vs AlFeO_{3} pv  6.11  6.76  6.97  7.31  
Vp FeAlO_{3} ppv   11.92  12.98  13.70  14.39 
Vs FeAlO_{3} ppv   6.94  7.61  7.97  8.36 
Vp AlFeO_{3} ppv   12.16  13.10  13.94  14.66 
Vs AlFeO_{3} ppv   7.27  7.89  8.30  8.71 
[20] Experimental Brillouin measurements on pure MgSiO_{3} perovskite and postperovskite [Murakami et al., 2007] show similar velocity jumps at the transition between the two structures and similar pressure dependencies. But pure Mgcomposition are not enough to explain the seismic discontinuity at the top of the D″ layer (at 120–130 GPa). The velocity jumps for homogeneous MgSiO_{3} aggregates are too small and various mechanisms [Wookey et al., 2005], including latticepreferred 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.
[21] Consequently, the theoretical values of this study are important as they show that (i) the thermodynamicallyfavorable 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.