First-principles calculations of the structure, energetics, compressibility, and elastic constants of aluminous stishovite reveal that the dissolution of aluminum in stishovite can accommodate a significant population of both oxygen vacancies and hydrogen defects. The results show that the incorporation of aluminum in stishovite account for a 1.2(±0.1)% decrease in K0 per mole percent dissolved aluminum, independent of dissolution mechanism (hydrous or anhydrous). Elastic constants, calculated to be in agreement with experimentally-derived values, indicate that the rutile to CaCl2 transition is likely to be unaffected by the dissolution of small amounts of aluminum and/or hydrogen. As a result, the dissolution of aluminum into stishovite in a subducting basaltic layer will have no more than a 0.2% effect on the density of that layer as it enters the lower mantle.
 The effect of minor elements in lower mantle phases can have a dramatic effect on material properties such as structure, compressibility and phase transitions [e.g., Yagi et al., 2004]. Aluminum is one of the most abundant minor elements in the mantle, yet its effect on lower mantle minerals is difficult to quantify due to the complicated manner in which this trivalent cation substitutes into silicates. For instance, the effect of alumina in perovskite can have dramatically different effects on the compressibility of MgSiO3 depending on whether or not Al2O3 dissolves into the structure through Tschermak-type substitutions (Mg2+ + Si4+ = 2 Al3+) or through Brownmillerite-type substitutions (2Si4+ = 2Al3+ + VO2+) [Navrotsky et al., 2003; Akber-Knutson and Bukowinski, 2004].
 Stishovite is the archetypal octahedrally coordinated silicate, and is expected to form in the basaltic layer of a subducting slab below 300 km [Irifune and Ringwood, 1993]. Despite its relatively simple structure, it is extraordinarily difficult to quantify the effects of minor elements on its structure and compressibility. Much of this is due to its stiffness (>300 GPa), low solubilities of minor components, and the difficulty in quantifying the composition of the small grains typically synthesized under high-pressure conditions.
Panero and Stixrude  calculated the effects of the Si4+ = Al3+ + H+ substitution on the structure and equation of state, via the reaction
finding a slight decrease in bulk compressibility or density. Lakshtanov et al.  find a similar relative decrease in the bulk compressibility for solid solutions containing only 1.8 wt% Al2O3 on samples with 500 ppm H2O. However, with Al/H ratios of ∼7, the relative influence of the anhydrous and hydrous substitutions remain unquantified. Presumably these samples accommodate the rest of the aluminum through the reaction
where 2Si4+ = 2Al3+ + VO2+, and the presence of oxygen vacancies would lead to a less dense, more compressible structure. Indeed, Ono et al.  measured a 10% decrease in the bulk modulus for just 2.1 wt% Al2O3 with unquantified, but presumably low, hydrogen concentrations. On samples with similar synthesis methods, Stebbins et al.  shows with high-resolution 27Al NMR that the aluminum has 6-fold coordination in stishovite and that the vacancies are not directly associated with the aluminum defects, which would otherwise lead to a significant portion of 5-fold coordinated aluminum. Measurements and calculations indicate this is the dominant mechanism low concentrations of aluminum incorporation in TiO2-rutile [Gesenhues and Rentschler, 1999; Steveson et al., 2002].
 Calculations on the anhydrous incorporation of alumina on stishovite, when paired with earlier calculations assuming just the hydrous substitution [Panero and Stixrude, 2004], shine light on the competing roles of hydrous (equation (1)) and anhydrous (equation (2)) substitutions in stishovite.
 The reaction of Al2O3 incorporation was calculated according to equation (2) where SiO2 and the solid solution are in the stishovite structure (P 42/mnm). Calculations were done at xAl = 0.0833 in a 2 × 2 × 3 supercell resulting in 22 SiO2 formula units and 1 Al2O3 unit. The arrangement of aluminum defects and oxygen vacancies were considered with corner-linked aluminum octahedral and edge-linked octahedral with a common oxygen vacancy. The effect of dispersed aluminum atoms and oxygen vacancies were also considered with identical composition with an Al—Al distance of 5.9 Å and an Al—VO distances of 4.3 Å and 6.1Å (maximum possible separation for the 2 × 2 × 3 supercell).
 An alternate reaction as suggested by Al incorporation in TiO2 [Steveson et al., 2002] is considered for a cluster of aluminum atoms according to the reaction
In this case, three Al are substituted for Si in one edge- and two corner-linked octahedra with an interstitial placed nearby the alumina octahedra. This reaction produced a substitution with an enthalpy of reaction of 1.25 eV per Al2O3 substitution at 50 GPa, far in excess of the reaction enthalpy found for the substitutions in equation (2).
 Static, density functional theory calculations (VASP) were performed with the local density approximation (LDA) and Vanderbilt-type ultrasoft pseudopotentials as described by Panero and Stixrude . The Brillion zone was sampled with a 1 × 1 × 1 mesh for solid solution calculations (71–73 atoms) and a 600 eV cutoff energy. SiO2-stishovite end member calculations sampled the Brillion zone with a 8 × 8 × 8 k-point mesh and a 900 eV cutoff. Solid solution calculations are converged to 10 meV (0.4 meV per formula unit) and the end member calculation was converged to less than 1 meV (<0.5 meV per formula unit).
 Elastic constants for pure SiO2 and solid solutions were calculated through ±0.01 strains on the tetragonal structure fitting for the best-fit elastic constants at constant pressure [Cohen, 1991; Holm et al., 1999]. Transitions determined through relative enthalpies are not reliable in this case as the CaCl2 structure relaxes to the stishovite structure below the pressure stability limit of the CaCl2 structure. The determination of the transition pressure from relative enthalpies then requires detection of a change in slope as a function of pressure. Therefore, the transition pressure to the CaCl2 structure for each composition is taken as the pressure at which c11–c12 is zero [Karki et al., 1997].
 Fully relaxed structures for anhydrous substitutions are not significantly different from the pure end member SiO2 (Figure 1). While the oxygen lattice due to the Al3+ + H+ substitution is distorted by as much as 0.2 Å [Panero and Stixrude, 2004], the oxygen positions in any of the anhydrous substitutions are distorted by no more than 0.028Å, about 10% of the maximum distortion in the hydrous substitution model. As a result, anhydrous aluminum dissolution results in little effect on c/a ratios regardless of the configuration of aluminum and oxygen vacancies.
 The density of stishovite, however, is more greatly affected by the dissolution of aluminum with the incorporation of oxygen vacancies (Table 1 and Figure 2a). The effect is greatest in the case of anhydrous aluminum incorporation with dispersed defects, with density and bulk modulus decreasing by 0.33% and 1.4(±0.1)% per mol percent dissolved aluminum, respectively. The combined effect of the decreased density and increased compressibility are such that at 50 GPa, the effect on density of aluminum incorporation in stishovite is reduced by 50%. The oxygen vacancy substitution results in a slightly greater effect on the zero-pressure volume and density when compared to the hydrous substitution of Si4+ = Al3+ + H+. At zero-pressure, the hydrous substitution results in a density and bulk modulus decrease of 0.2% and 0.9(±0.3)%, respectively (Figure 2b).
Table 1. Zero-Pressure Volumes, Lattice Parameters and Equation of State for Pure SiO2 and Al2O3-(H2O)-SiO2 Solid Solutionsa
All bulk moduli were fit using an F vs. f formalism assuming dK0/dP = 4. Values for experimental compression data reflect the K0 derived from refitting, yet uncertainties are lower bounds due to no reported uncertainties on pressure (from ruby or Pt scales).
Zero-point fluctuations and 300 K thermal expansion account for ∼0.37Å3 expansion [Panero and Stixrude, 2004], for a 300 K corrected volume of 46.148 Å3.
 As expected from the differences in zero-pressure density and bulk modulus, the substitution mechanism in these calculations has varying effects on stishovite's elastic constants. The results (Table 2) show that each anhydrous defect decreases the calculated elastic moduli by about 1% per incorporated mol percent Al3+. In the dispersed case, the c12 modulus decreases at a rate 10–50% greater than the other calculated moduli. In contrast the hydrous aluminum substitution increases the c12 and c13 elastic constants while the largest decrease is in the c33 modulus (Table 2).
Table 2. Elastic Constants for Stishovite and Aluminous Solid Solutionsa
Numbers in parenthesis indicate the percent change in the elastic constant per mole percent Al3+.
 In these calculations, c11–c12 gives the transition from rutile to CaCl2 in pure stishovite at 43.2 GPa (Figure 3), very similar to the transition pressure determined by Karki et al . The effect of aluminum depends upon the dissolution mechanism, with corner-linked defects increasing the pressure of the transition to 46.9 GPa, an increase of 0.4 GPa per mole percent dissolved aluminum (Figure 3). However, the transition pressure is indistinguishable from pure SiO2 for both the anhydrous dispersed case and the hydrous substitution. This lack of shift in the phase transition pressure is supported by Lakshtanov et al. , who reported that a sample with 1.8 wt% Al2O3 appears to undergo the rutile to CaCl2 transition at ∼44 GPa.
 The enthalpy of reaction is nearly independent of the substitution type when comparing substitutions with edge-sharing or corner-sharing octahedra (Figure 4). The dispersed substitutions have higher enthalpies of reaction compared to the edge- or corner-shared substitutions, 0.017 eV per formula unit at 0 GPa increasing to 0.05 eV at 80 GPa. However, this enthalpy difference between the three types of substitution is less than 1/3 of kBT at 2000 K. The lack of major energy cost to disassociating the oxygen vacancy from the aluminum defect is also quite analogous to previous atomistic calculations on Al2O3 in silicate perovskite [Akber-Knutson and Bukowinski, 2004]. Due to the higher configurational entropy of the dispersed substitution, the oxygen vacancy substitution may actually be favored. For low concentrations of Al2O3, the configurational entropy of the dispersed case is greater by a factor of 2.5 such that at 2000 K, 3 mol% Al2O3 has a configurational entropy of 7.3 kJ/mol (75 meV per formula unit) compared to about 2.9 kJ/mol for either the edge- or corner- sharing substitution, a difference greater than the difference in the reaction enthalpies (Figure 4).
Panero and Stixrude  showed that the solubility of aluminum via equation (1) was 2 mol% Al2O3 at 2000 K and 25 GPa, where the existence of a stable hydrous aluminum phase with a similar structure allowed for a direct determination of the aluminum solubility. Without an appropriate Al2O3 end member for comparison, it is difficult to calculate the solution enthalpy of Al2O3 in stishovite via equation (2). However, many reports indicate that stishovite formed in hydrous conditions forms with Al/H ratios significantly greater than one [e.g., Pawley et al., 1993; Chung and Kagi, 2002; D. L. Lakshtanov et al., Effect of Al3+ and H+ on the elastic properties of stishovite, submitted to American Mineralogist, 2006], indicating that much of the anhydrous aluminum solubility in stishovite is greater than the hydrous substitution.
 Direct comparison of energetics of the two substitutions is calculated via the reaction
For static calculations, the hydrous substitution is more stable (Figure 4). However, this calculation is made for static ice VIII, and therefore does not include significant entropic effects of super-critical fluid at the relevant pressures and temperature of sample synthesis. The heat of fusion for water at ambient pressure is about 0.06 eV per formula unit, about half the static enthalpy difference in equation (4) at zero pressure and 20% of the difference at 80 GPa. Assuming the Dulong-Petite limit for liquid water and stishovite, the relative enthalpy of equation (4) may be as much as 0.7 eV per formula unit higher at high pressure and 2000 K.
 The majority of aluminum incorporated in stishovite is likely incorporated anhydrously without an energetic requirement for aluminum and oxygen defects to be directly associated with one another. This inference is supported by recent high-resolution 27Al NMR data [Stebbins et al., 2006] showing that aluminum is in octahedral coordination with oxygen in stishovite without any significant amount of 5-fold coordinated alumina that would indicate a preference for the oxygen vacancy to be directly coordinated with each aluminum defect. Therefore, the Stebbins et al.  conclusion of little association of oxygen vacancies with aluminum defects appears to be a thermal effect due to sample synthesis at high temperatures (1723–1973 K).
 The experimental data on the effects of aluminum (±hydrogen) in stishovite can now be interpreted. These calculations show the same effect on compressibility per mole dissolved aluminum (∼1%), but where the anhydrous substitution decreases the density at a rate 65% more than the anhydrous substitution. Ono et al.  and Lakshtanov et al.  find a decrease in the bulk modulus relative to Weidner et al.  of 10.7(±1.3) and 4.2(±1.6)% for 2.45% and 2.1% per mole percent dissolved Al3+, respectively (Table 1). Ono et al.  find a slightly greater effect of aluminum than the calculations here would indicate, while the Lakshtanov et al.  results are within error of the relative shift found in the calculations. The sample used by Lakshtanov et al.  contains 500 ppm (wt) H2O (0.3% mol% H+) such that only about 1/7 of the aluminum atoms are compensated through equation (1) while the majority of aluminum dissolves through rxn (2).
 The elastic constants c11, c12, c13 and c33 show almost uniform decrease with anhydrous aluminum incorporation, consistent with a volumetric expansion of the structure [Lakstanov et al., 2006]. c12 shows a 50% greater decrease per mole percent dissolved aluminum than the other elastic constants, consistent with the decrease reported by Lakstanov et al. , yet a smaller decrease than would be expected from a volumetric extrapolation of c12 from compression data [Carpenter et al., 2000]. In contrast to the anhydrous case, the elastic constants for the hydrous substitution show an increase in c12 and a much greater decrease in c33. This is consistent with a previous analysis showing an increase in the ratio of the linear compressibilities [Panero and Stixrude, 2004] showing that the effect of the hydrogen incorporation is to stiffen the c-axis relative to the a-axis.
 The rapid drop in shear-wave velocities due to the stishovite to CaCl2 structural transition in SiO2 [Karki et al., 1997] may be seismically evident even for a slab where stishovite does not form as a pure substance. The incorporation of a minor component can broaden the transition making it difficult to detect through geophysical methods. Assuming c12 increases with pressure with the same slope as in pure stishovite, this would predict aluminum stabilization of stishovite relative to the CaCl2 structure. These results show, however, that the incorporation of several percent Al2O3 through either equation (1) or equation (2) will have little effect on the transition pressure to the CaCl2 structure (Figure 3) due to a comparatively rapid increase in c12 with pressure. The observation here that aluminum does not dramatically affect the transition pressure of stishovite to CaCl2 contrasts with reports that aluminum and iron can both dramatically shift the perovskite to post-perovskite transition [Mao et al., 2006; Caracas and Cohen, 2005], likely broadening the transition over several tens of kilometers [Akber-Knutson et al., 2005].
 Stishovite, accounting for about 20% of a basaltic layer under lower-mantle conditions, will have a lower density than pure stishovite under identical pressure- and temperature-conditions. Indeed, a dry basaltic layer typically contains ∼15 wt% Al2O3, of which up to 5 mol% Al dissolves into stishovite [Litasov and Ohtani, 2004] resulting in a ∼6% decrease in bulk modulus (313 GPa to 294 GPa) and a 1.1% decrease in density relative to pure stishovite at the top of the lower mantle. If aluminum content has negligible effect on the elasticity and density of associated perovskite phases [Panero et al., 2006], then the effect of stishovite will be to decrease the density of the basaltic layer at this depth by just 0.2%, an amount likely negligible when considering the buoyancy of a slab just beneath the transition zone.
 All calculations were performed on the Ohio Supercomputer Center with award PAS0238-1, and support for this work comes from NSF EAR-0537813. Useful discussions with Dmitry Lakshtanov motivated this work.