Phase relations of bridgmanite, the most abundant mineral in the Earth’s lower mantle

Abstract

The knowledge of phase relations of constitutive minerals is essential to investigate the structure, dynamics and evolution of the Earth and planetary interiors. This paper reviews the phase relations of bridgmanite, the most abundant mineral in the Earth’s lower mantle, with an ideal composition of MgSiO₃. Bridgmanite has an orthorhombic structure with larger dodecahedral A and smaller octahedral B cation sites. The A-sites can incorporate Mg²⁺, Fe²⁺, Fe³⁺, and Al³⁺, while the B-sites accommodate Si⁴⁺, Al³⁺ and Fe³⁺. The incorporation of hydrogen and large cations like Ca is likely limited, although these issues are still debated. Al³⁺ and Fe³⁺, respectively, can form the charge-coupled components, AlAlO₃ and Fe³⁺Fe³⁺O₃ occupying both A- and B-sites. When both Al³⁺ and Fe³⁺ are present, Al³⁺ occupies B-sites, and Fe³⁺ occupies A-sites, forming Fe³⁺AlO₃. In systems with excess MgO, Al and Fe³⁺ also form the oxygen vacancy components MgAl³⁺O_2.5□_0.5 and MgFe³⁺O_2.5□_0.5. The phase relationships of bridgmanite with coexisting phases are discussed as a function of pressure, temperature, and oxygen fugacity from the simple MgSiO₃ system to the complex MgO-Fe²⁺O-Fe³⁺₂O₃-Al₂O₃-SiO₂ system.

Introduction

Bridgmanite ((Mg, Fe)SiO₃-Al₂O₃, Bdm) is the Earth’s lower mantle’s most abundant mineral. The lower mantle extends from 660 Km to 2890 Km depth, corresponding to 23–136 GPa, and comprises 56% of Earth’s volume. The mantle’s composition is primarily peridotitic, containing 45.0% SiO₂, 37.8% MgO, 8.1% Fe_xO, 3.6% CaO, and 4.45% Al₂O₃ by weight¹. The lower mantle’s mineralogy consists of 80% Bdm, 15% ferropericlase ((Mg, Fe)O, Fper), and 10% of davemaoite (CaSiO₃, Dvm) by volume (Fig. 1A)². Below the D” discontinuity in high-velocity regions, postperovskite ((Mg, Fe)SiO₃-Al₂O₃) likely replaces Bdm³. The differentiation under mid-oceanic ridges creates oceanic plates with 18% basalt and 82% harzburgite², which subduct into the lower mantle. The resulting basaltic domain contains 32% Bdm, 25% Dvm, 26% postspinel phase (MgAl₂O₄, PS), and 18% stishovite (SiO₂, Sti) (Fig. 1B), while the harzburgite domain comprises 74% Bdm, 22% Fper, and 3% Dvm (Fig. 1C)². As the predominant mineral in all lithologies of the lower mantle, Bdm’s phase relations with secondary minerals, i.e., Fper, Dvm, PS, and Sti, are crucial for understanding mantle dynamics and evolution. Bdm is named after Percy Williams Bridgman, a pioneering American high-pressure physicist.

Fig. 1: Mineral constitution of mantle rocks.

A Pyrolite, B basalt, C harzburgite. Mineral abbreviation: Bdm (red), bridgmanite; Fper (cyan), ferropericlase; Dvm (light green), davemaoite: pPv (pink), postperovskite; Ol (blue), olivine ((Mg, Fe)₂SiO₄); Wds (blue-violet), wadsleyite ((Mg, Fe)₂SiO₄); Rwd (magenta), ringwoodite ((Mg, Fe)₂SiO₄); Gnt (orange), garnet ((Mg, Fe, Ca)SiO₃-Al₂O₃); Px, pyroxenes (dark green); Pl (yellow), plagioclase; PS (dark red): postspinel phase (CaFe₂O₄-type MgAl₂O₄), Qz (very light grey): quartz (SiO₂), Coe (light grey): coesite (SiO₂), Sti (gray): stishovite (SiO₂), pSti (dark gray): post-stishovite (CaCl₂-type SiO₂), Sft (very dark gray): seifertite (α-PbO₂ type SiO₂). Modified from Strixrude and Lithgow-Bertelloni².

This review examines Bdm’s phase relations with secondary phases under pressure (P)—temperature (T) conditions relevant to Earth’s mantle. It focuses on major mantle elements (O, Si, Mg, Fe, Al, and Ca) and H, which, although not a major constituent, could significantly alter mineral properties⁴. This study excludes phase relations in natural systems. Figure 1 illustrates Bdm’s predominance in the lower mantle but does not specify exact mineral proportions.

Bdm phase relations are primarily studied using laser-heated diamond anvil cells (LH-DAC) and multi-anvil presses (MAP). LH-DAC covers Bdm’s entire stability field but has significant temperature uncertainties (100–400 K)^{5,6,7,8,9,10,11} and pressure uncertainties (1–5 GPa or up to 7%)^{6,7,9,10,11,12}. It also risks chemical heterogeneity due to the Soret effect¹³. Sample analysis techniques have evolved from unit cell volume estimates to transmission electron microscopy (TEM) with focused ion beam (FIB), though spatial resolution remains a challenge for very small grain sizes. Notably, silicate samples prepared using FIB are typically more than 100 nm, and the spatial resolution of TEM analysis is comparable to the sample thickness¹⁴. On the other hand, MAP offers more precise P-T conditions, with temperature fluctuations of typically 5 K^15,16,17,18 and pressure uncertainties of 0.05–1 GPa^{15,16,17,18,19}. It avoids Soret effect issues. The size of recovered samples is several hundred μm, allowing various post-analysis using multiple techniques. The phases present can be identified using a microfocused powder X-ray diffractometer (MF-XRD). Sample textures can be observed using scanning electron microscopy (SEM) with backscattered electron imaging (BEI). The grain sizes are larger than 2-3 μm below 27 GPa and above 1700 K, allowing for reliable compositional analysis using an electron microprobe (EPMA), whose precision is better than 0.1 wt.%^{20,21,22,23,24,25,26}. The grain size becomes smaller with increasing pressure but usually above several 100 nm, which allows reliable analysis using an analytical TEM with an energy-dispersive X-ray spectrometer (EDS). The determination of Fe³⁺/ΣFe is also possible using Mössbauer spectroscopy with a precision of 0.02 ~ 0.05^{25,26,27,28,29,30}. MAP’s pressure range was historically limited to 26 GPa but recently extended to 52 GPa³¹. This paper reviews the experimental data obtained using MAP and LH-DAC, supplemented by ab initio calculations.

Crystal chemistry of bridgmanite

Bdm has an orthorhombic perovskite structure (Fig. 2A)^32,33 with MgSiO₃ as its principal component. In this structure, Mg²⁺ and Si⁴⁺ occupy the A- and B-sites, respectively, surrounded by eight~twelve and six O^2− 33, expressed as [Mg²⁺]_A[Si⁴⁺]_BO₃²⁻. Dvm also has a perovskite structure, but in cubic form³⁴ (Fig. 2B). This difference between Bdm and Dvm is due to the ionic radii of their A-site cations: 8-coordinated Mg²⁺ (89 pm) and Ca²⁺ (112 pm), respectively^35,36. The smaller Mg²⁺ in Bdm causes A-site distortion and SiO₆ octahedron rotation, resulting in the orthorhombic structure. In contrast, the larger Ca²⁺ in Dvm allows for the cubic perovskite structure.

Fig. 2: Perovskite structures.

A orthorhombic perovskite structure, B cubic perovskite structure. The A and B site cations and oxygen, respectively, are expressed using green, blue, and red spheres. Modified from Akaogi³⁵.

Ionic radii of cations provide insights into Bdm chemistry. The A-site, typically occupied by 8-coordinated Mg²⁺, can accommodate similarly sized cations like Fe²⁺ (92 pm) and Fe³⁺ (78 pm) in high-spin states. It doesn’t primarily accommodate larger cations like Ca²⁺. Al³⁺ (estimated 61 pm) can occupy the A-site³⁷, but its smaller size may explain why high pressure is needed for incorporation. The B-site, occupied by 6-coordinated Si⁴⁺ (40 pm), can only accommodate small cations. Al³⁺ (54 pm) fits well, while Fe³⁺ (66 pm) is accommodated in limited amounts²⁰, possibly due to its larger size. The 6-coordinated ionic radius of high-spin Fe²⁺ (78 pm) is likely too large for the B-site. Note: Effective ionic radii were extrapolated from Shannon’s³⁶ data when not explicitly given.

The valence differences between Mg²⁺ and Si⁴⁺ (2+ and 4+ valence) and Al³⁺ and Fe³⁺ (3+ valence) lead to various substitution mechanisms in Bdm. When Al or Fe³⁺ occupies the A- or B-site, it is favourable for the other site to also be occupied by a trivalent cation, forming charge-coupled (CC) components like [Al³⁺]_A[Al³⁺]_BO²⁻₃ and [Fe³⁺]_A[Fe³⁺]_BO²⁻₃^20,37. However, as the ionic radii of Al and Fe³⁺ are more similar to those of Si and Mg, respectively, the Fe³⁺ and Al cations are preferably accommodated in the A- and B-sites, respectively, forming [Fe³⁺]_A[Al³⁺]_BO²⁻₃²¹. Al and Fe³⁺ can also occupy the B-site even when divalent cations (Mg²⁺ and Fe²⁺) are in the A-site. This creates oxygen vacancies (□) to balance charges, forming oxygen vacancy (OV) components like [Mg²⁺]_A[Al³⁺]_BO²⁻_2.5□_0.5 and [Fe²⁺]_A[Al³⁺]_BO²⁻_2.5□_0.5³⁸. Conversely, when Al or Fe³⁺ occupies the A site with Si⁴⁺ in the B site, cation vacancies form to compensate for excess positive charge, resulting in A-site vacancy (AV) components like [□_1/3Al³⁺_2/3]_A[Si⁴⁺]_BO²⁻₃ and [□_1/3Fe³⁺_2/3]_A[Si⁴⁺]_BO²⁻₃^20,38.

Molar volume is crucial for understanding chemical changes with P. Table 1 shows molar volumes of various Bdm components at ambient conditions. All secondary components have larger molar volumes than MgSiO₃ (24.447(4) cm³/mol), including OV, despite their lack of oxygen.

Table 1 Molar volumes of mineral components at ambient conditions

The incorporation mechanism of H⁺ in Bdm is not yet fully understood until recently due to the limited H₂O content. Drawing parallels from Mg₂SiO₄ wadsleyite, an upper mantle mineral, where H⁺ occupies the Mg site forming the [Mg²⁺2H⁺]₂[Si⁴⁺]O²⁻₄³⁹, it was hypothesised that H⁺ might similarly occupy the Mg (A-) site in Bdm⁴⁰, creating [2H⁺]_A[Si⁴⁺]_BO²⁻₃. An alternative mechanism proposed that H⁺ could couple with Al³⁺ to substitute Si⁴⁺ in the B-site⁴¹, forming [Mg²⁺]_A[Al³⁺H⁺]_BO²⁻₃ A recent neutron scattering study has provided evidence supporting this latter substitution mechanism in Bdm⁴².

The variety of components in Bdm underscores the importance of coexisting phases in determining its chemistry. The species of coexisting phases should vary with the bulk composition. The content of different components in Bdm varies with the coexisting phases, which in turn depend on the bulk composition. In (Mg, Fe²⁺)O excess or SiO₂-deficient systems, OV likely forms, while in SiO₂ excess systems, AV forms³⁸. To define a unique defect structure at a given P and T, the Gibbs phase rule must be considered. For a system with n components, n-1 additional phases must coexist with Bdm. For instance, in a three-component system (MgO-Al₂O₃-SiO₂), two additional phases besides Bdm are necessary. Many studies have not fully considered the implication of the phase rule, often having an insufficient number of coexisting phases. While the following sections will interpret Bdm phase relations, it is important to note that this review is not exhaustive.

Phase relations of bridgmanite in various systems

MgSiO₃

The MgSiO₃ system is fundamental for understanding Bdm stability. Figure 3 summarises Bdm’s stability field, bounded by transitions to akimotoite (Aki) at low P and T ^{43,44,45,46,47,48,49,50,51}, to postperovskite (pPv) at high P^{5,52,53,54,55}, and MgSiO₃ melt at high T ^{56,57,58,59,60,61,62,63,64}. A small region of tetragonal garnet (Gnt), often referred to as majorite, exists between Aki and melt^47,56. Recent studies define these boundaries with more precision. Figure 3 shows the most recently determined boundaries with Aki⁴⁹, pPv⁵, and melt⁶⁴. The Bdm- Aki boundary has a negative slope (dP/dT), −3.2 to −8.1 MPaK⁻¹ as T decreases from 2100 to 1250 K⁴⁹ with transition P at 24.0 GPa at 2100 K and 20.5 GPa at 1250 K. The Bdm-pPv boundary exhibits a steep positive dP/dT of 13.3 MPaK⁻¹, with transition P at 107 GPa at 1500 K and 150 GPa at 4500 K³⁵. The melting curve starts at 2700 ~ 2800 K at 22 GPa^56,63, rapidly increases to 4300 K at 60 GPa, then gradually reaches 5200 K at 140 GPa⁶⁴. Extrapolation suggests a triple point (Bdm-pPv-melt) at 5200 K and 160 GPa. The Bdm-Gnt boundary remains less studied.

Fig. 3: Stability of Bdm in pure MgSiO₃.

A at 1000 to 7000 K and 0 to 200 GPa, and B at 1000–3000 K and 19–26 GPa. Colour code: Red–Bdm, Green - Aki, Purple - Gnt, Blue - pPv, Yellow - Melt. Data sources: Circles - Chanysheve et al.⁴⁹, Triangles - Tateno et al.⁵, Squares - Ito and Katsura⁵⁶, Stars - Shen and Lazor⁵⁸, Crosses - Zerr and Boehler⁵⁹, Plus sings - Akins et al.⁵⁹.

MgO-SiO₂

In the MgO-rich region of the MgO-SiO₂ system, Bdm coexists with periclase (MgO, Per). The Bdm+Per field is bounded by ringwoodite (Mg₂SiO₄, Rwd) at low-P, as shown in Fig. 4A. This phase boundary, extensively studied for its geophysical importance in relation to the 660-km seismic discontinuity, occurs at 23–24 GPa^{15,31,43,49,65,66,67}. The Rwd to Bdm + Per transition exhibits a near-zero dP/dT below 1700 K, becoming increasingly negative at higher T, reaching −0.9 MPaK⁻¹ at 2000 K⁶⁸. Above 2500 K, Rwd transforms into wadsleyite (Mg₂SiO₄, Wds) following a boundary 11.0 (GPa) + 4.5 (MPaK⁻¹) × T (K)⁶⁸, leading to a transition from Bdm + Per to Wds rather than Rwd. Both Bdm + Per and Wds become melt + Per above 2500 K at 19 to 27 GPa⁶³.

Fig. 4: Stability of Bdm in MgO-SiO₂.

A Phase relations in Mg₂SiO₄ system at 19–27 GPa and 1000–3000 K. Red: Bdm, marron: Rwd, light green: Wds, blue-green: Per, yellow: melt. Triangles and diamonds: Chanyshev et al.⁴⁹, circle: Liebske and Frost⁶³. The boundary between Rwd and Wds is the extrapolation of Tsujino et al.⁶⁸. B Melting relations in Mg₂SiO₄ at 24 and 136 GPa and 2000–8000 K. Modified from Yao et al.⁶⁹.

On the SiO₂-rich region of the MgO-SiO₂ system, Bdm coexists with SiO₂ stishovite (Sti) at a relatively lower P. At 68-78 GPa, Sti transforms to CaCl₂-type SiO₂, known as post-stishovite (pSt)⁶. At even higher P, pSt may transform to α-PbO₂-type SiO₂, called seifertite (Sft), although it is unclear if this transformation occurs within the Bdm stability field⁷. Thus, above 68-78 GPa, Bdm primarily coexists with pSt. No intermediate phases between Bdm and silica phases have been reported in the literature.

The melting relations of Bdm + Per and Bdm + Sti, though not extensively studied, exhibit congruent melting⁶⁹. Figure 4B illustrates the melting relations in the MgO-SiO₂ system proposed by Yao et al.⁶⁹ based on multi-anvil experiments and first-principle calculations. At 24 GPa, the eutectic T for Bdm + Per and Bdm + Sti, respectively, are 2650 K⁶³ and 2700 K⁶⁹. Per and Sti have significantly higher melting T (4800 and 4200 K) than Bdm (2750 K) at this P. Consequently, the compositions of Bdm + Per and Bdm + Sti melts are similar to pure Bdm with MgO/(MgO+SiO₂) of 0.43 and 0.53, respectively, compared to 0.5 for Bdm. As P increases, the melting T in the MgO-SiO₂ system rises while maintaining similar geometrical relations. At 136 GPa, the eutectic T for Bdm + Per and Bdm + Sti reaches 5500 and 5400 K, respectively. As the melting temperature differences between Bdm (5550 K) and Per (7500 K) and Sti (5900 K) decrease, the eutectic compositions of Bdm + Per and Bdm + Sti, MgO/(MgO+SiO₂) = 0.35 and 0.67, respectively, deviate more significantly from the MgSiO₃ composition.

MgO-Fe²⁺O-SiO₂

Following the MgO-SiO₂ system, the MgO-Fe²⁺O-SiO₂ system was initially anticipated to be the second most important system in geophysics due to Fe’s abundance as the fourth element in Earth’s mantle¹. While the majority of Earth’s Fe is stored in the liquid outer core, the deep mantle’s reducing conditions⁷⁰ led to the expectations that Fe would primarily exist in its 2+ state. However, recent studies have revealed that Fe³⁺ plays an even more crucial role in mantle chemistry than Fe^{2+ 27,71}, a topic that will be explored in later sections. The investigation of Bdm’s phase relations in this system began shortly after its discovery⁷², reflecting its perceived importance in understanding Earth’s lower mantle composition and dynamics.

Maximum Fe²⁺SiO₃ content

Fe²⁺ can substitute for Mg²⁺ in the A-site, forming Fe²⁺SiO₃⁷³. However, there is a limit to the maximum Fe²⁺SiO₃ content in Bdm, denoted as ${\varphi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$. When the Fe²⁺SiO₃ content in Bdm, ${{{\rm{\chi }}}}_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$, exceeds this limit, Fper + Sti forms. In planets with mantles richer in Fe²⁺O than Earth’s, Fper + Sti could be the major mantle phase rather than Bdm. Due to its geophysical significance, ${\varphi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ has been extensively studied as a function of T ^43,75,76,77 and P ^8,76,77,78. While Fe²⁺ undergoes a spin transition in certain minerals like Fper, at high P ⁷⁹, Fe²⁺ in the Bdm A-site remains in the high-spin state throughout its stability field⁷⁹. Consequently, the chemistry of Fe²⁺-bearing Bdm should change gradually or remain unaltered with P.

Figure 5A illustrates the variation of ${\varphi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ at ca. 26 GPa as a function of T determined using a MAP^43,75,76,77. The data can be divided into two groups based on the anvil material: tungsten carbide (WC)^43,74,75 and sintered diamond (SD)^76,77. Studies using WC anvils found ${\varphi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ to be 7 mol.% at 1400 K, increasing to 12 mol.% at 2000 K. In contrast, studies using SD anvils reported 16 mol.% at 1800 K, rising to to 18 mol.% at 2300 K. The SD anvils results are 1.3-2.0 times higher than WC anvil results at comparable T. The reason for this discrepancy is not fully understood. One possible explanation is that the actual T in the WC anvil experiments may have been higher than that in the SD anvil experiments, though this would require a difference of over 1000 K to account for this difference in ${\varphi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$. Another possibility is that extremely small grain sizes in the SD anvil experiments may have led to microprobe analyses of bridgmanite inadvertently including more Fe-rich ferropericlase grains.

Fig. 5: The maximum Fe²⁺SiO₃ content in Bdm, ${\varphi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$.

A With T at ca. 26 GPa. Thick red circles: Ito and Takahashi⁴³; red circles: Ito et al.⁷⁴; green square: Fei et al.⁷⁵, cyan diamond: Tange et al.⁷⁶, blue triangle: Arimoto et al.⁷⁷. B With P at various T. Triangles: Mao et al.⁷⁸; Diamond: Tange et al.⁷⁶; Circles: Arimoto et al.⁷⁷. The crosses are the average values of Ito et al.⁷⁴, Ito and Takahashi⁴³, and Fei et al.⁷⁵ at ca. 26 GPa.

Figure 5B illustrates the variation in ${\varphi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ as a function of P at various T using data from LH-DAC experiments^8,78 and MAP with SD anvils^76,77. Both data indicate that ${\varphi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ increases with P at similar rates of 0.42 to 0.60 mol.%GPa⁻¹ with no apparent effect from the Fe²⁺ spin transition in Fper⁷⁹. LH-DAC results show lower ${\varphi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ ^8,78 compared to MAP with SD anvils^76,77, but are consistent with MAP using WC anvils^43,74,75. MAP with SD anvils^76,77 yielded a ${\varphi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ of 38 mol.% at 2000 K near 60 GPa. Dorfman et al.‘s LH-DAC study on the MgSiO₃-Fe²⁺SiO₃ system⁸ showed a rapid increase in ${\varphi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ from 60 to 70 GPa, implying a possible formation of pure Fe²⁺SiO₃ Bdm. However, this drastic change is challenging to interpret thermodynamically without assuming significant changes in partial molar volumes with increasing P. The Sti-pSti transition occurring between 58 and 78 GPa has minimal effect on phase relations due to similar volumes of these minerals⁶.

Given the inconsistencies in the reported ${\varphi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$, a comprehensive reinvestigation is necessary. This study should cover a wide P-T range and employ reliable experimental and analytical techniques to establish more definitive conclusions about ${\varphi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$.

Fe²⁺-Mg exchange coefficient with ferropericlase

The Fe-Mg exchange between Bdm and Fper is geophysically important, as Fper is the only ferromagnesian mineral coexisting with Bdm in Earth’s mantle. The substitution of Mg by Fe²⁺ notably affects the physical properties of these minerals, altering their density and electrical conductivity^6,80. The exchange coefficient of Fe²⁺ and Mg of Bdm with Fper, ${K}_{{{\rm{F}}}{{{\rm{e}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$, is defined as:

$${K}_{{{\rm{F}}}{{{\rm{e}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}=\frac{\left({{{\rm{\chi }}}}_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}/{{{\rm{\chi }}}}_{{{\rm{Mg}}}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}\right)}{\left({{{\rm{\chi }}}}_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Fper}}}}/{{{\rm{\chi }}}}_{{{\rm{Mg}}}{{{\rm{SiO}}}}_{3}}^{{{\rm{Fper}}}}\right)}$$

(1)

where χ^α_i is the fraction of the component i in phase α. While the apparent Fe-Mg exchange coefficient is significantly altered by Al incorporation⁸¹, making direct application to Earth’s mantle challenging, ${K}_{{{\rm{F}}}{{{\rm{e}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$ in the MgO-Fe²⁺O-SiO₂ system remains important as a foundatioin for understanding Fe distribution in these minerals.

Numerous studies have attempted to determine ${K}_{{{\rm{F}}}{{{\rm{e}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$ at 23–25 GPa using MAP^{43,74,83,84,85}. However, achieving equilibrium has been challenging. Many workers used olivine as a starting material, but this approach led to issues with low-T compositions persisting due to slow kinetics⁸² and Fe²⁺ oxidation to Fe³⁺ during the dissociation⁸⁴.

Nakajima et al.⁸⁵ provided the best understanding of ${K}_{{{\rm{F}}}{{{\rm{e}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$ for the topmost lower mantle conditions. They measured compositions of Bdm and Fper coexisting with metallic iron at significantly high T (2400–2600 K) using various bulk compositions. This approach minimised Fe³⁺ content and allowed for chemical equilibrium and grain growth suitable for microprobe analysis. While data from Katsura and Ito⁸² and Frost and Langenhorst⁸⁴ generally align with Nakajima et al.’s⁸⁵ findings, Katsura and Ito’s⁸² ${K}_{{{\rm{F}}}{{{\rm{e}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$ was higher, especially for low-Fe samples. This discrepancy may be due to significant Fe³⁺ in Bdm’s B-site, caused by oxidation from B₂O₃ flux. The consistency improves for higher Fe samples because the B site’s Fe³⁺ capacity is limited, reducing its impact on ${K}_{{{\rm{F}}}{{{\rm{e}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$.

The Fe²⁺-Mg exchange coefficient is directly related to the chemical potential change of the Fe²⁺-Mg exchange reaction between Bdm and Fper. This reaction can be expressed as:

$$\begin{array}{c}{{{\rm{MgSiO}}}}_{3}\\ {{\rm{Bdm}}}\end{array}+\begin{array}{c}{{{\rm{Fe}}}}^{2+}{{\rm{O}}}\\ {{\rm{Fper}}}\end{array}=\begin{array}{c}{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}\\ {{\rm{Bdm}}}\end{array}+\begin{array}{c}{{\rm{MgO}}}\\ {{\rm{Fper}}}\end{array}$$

(2)

The conditions for equilibrium of Eq. (1) can be written as:

$${RT} \, {{\mathrm{ln}}}\,{K} \, _{{{\rm{F}}}{{{\rm{e}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}=-\left(\Delta {H}^{0}+P\Delta {V}^{0}{{\rm{\hbox{-}}}}T\Delta {S}^{0}\right)-{{\rm{R}}}T{\mathrm{ln}}\frac{\left({{{\rm{\gamma }}}}_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}/{{{\rm{\gamma }}}}_{{{\rm{Mg}}}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}\right)}{\left({{{\rm{\gamma }}}}_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Fper}}}}/{{{\rm{\gamma }}}}_{{{\rm{Mg}}}{{{\rm{SiO}}}}_{3}}^{{{\rm{Fper}}}}\right)}$$

(3)

where ΔH ⁰, ΔV ⁰, and ΔS ⁰, respectively, are the enthalpy, volume, and entropy changes associated with the exchange reaction in the standard state, γ^α_i is the activity coefficient of the component i in phase α, R is the gas constant, and T is the absolute temperature. Using the regular solution model, the term of the acitivty coefficient can be expressed as:

$$-{{\rm{R}}}T\,{{\mathrm{ln}}}\frac{\left({{{\rm{\gamma }}}}_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}/{{{\rm{\gamma }}}}_{{{\rm{Mg}}}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}\right)}{\left({{{\rm{\gamma }}}}_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Fper}}}}/{{{\rm{\gamma }}}}_{{{\rm{Mg}}}{{{\rm{SiO}}}}_{3}}^{{{\rm{Fper}}}}\right)}= {W}_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}-{{{\rm{Mg}}}{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}(1{{{-}}}2{\chi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}) \\ {{{-}}}{W}_{{{{\rm{Fe}}}}^{2+}{{\rm{O}}}-{{\rm{Mg}}}{{\rm{O}}}}^{{{\rm{Fper}}}}(1{{\rm{\hbox{-}}}}2{\chi }_{{{{\rm{Fe}}}}^{2+}{{\rm{O}}}}^{{{\rm{Fper}}}})$$

(4)

where ${W}_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}-{{{\rm{Mg}}}{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ and ${W}_{{{{\rm{Fe}}}}^{2+}{{\rm{O}}}-{{\rm{Mg}}}{{\rm{O}}}}^{{{\rm{Fper}}}}$, respectively, are the symmetric interaction parameters (Margules parameters) for Bdm and Fper. The parameters for this model, as determined by Nakajima et al.⁸⁵, are given in Table 2.

Table 2 Thermodynamic parameters to obtain ${K}_{{{{\rm{Fe}}}}^{2+}-{{\rm{Mg}}},{{\rm{Bdm}}}-{{\rm{Fper}}}}$ at 25 GPa

Figure 6A shows ${K}_{{{\rm{F}}}{{{\rm{e}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$ for various ${\chi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ as a function of T at 25 GPa calculated using the above thermodynamic parameters. The graph also includes experimental data by Katsura and Ito⁸², Frost and Langenhorst⁸⁴, and Nakajima et al.⁸⁵. As expected, ${K}_{{{\rm{F}}}{{{\rm{e}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$ increases with increasing T. For instance, at ${\chi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ of 8 mol.%, ${K}_{{{\rm{F}}}{{{\rm{e}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$ is 0.19 at 1800 K, whereas it is 0.30 at 2600 K. More notably, ${K}_{{{\rm{F}}}{{{\rm{e}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$ is strongly compositional dependent, decreasing with increasing Fe²⁺/(Fe²⁺ + Mg) ratio in the system. For instance, at 2000 K, ${K}_{{{\rm{F}}}{{{\rm{e}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$ is 0.32 at ${\chi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ of 4 mol.%. but drops to 0.16 at 16 mol.%. This compositional effect is significantly larger than the T effect, primarily due to the large ${W}_{{{{\rm{Fe}}}}^{2+}{{\rm{O}}}-{{\rm{Mg}}}{{\rm{O}}}}^{{{\rm{Fper}}}}$, because ${W}_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}-{{{\rm{Mg}}}{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ can be assumed to be zero⁸⁵.

Fig. 6: Fe²⁺-Mg exchange coefficient between Bdm and Fper, ${K}_{{{{\rm{Fe}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$.

A With T at 25 GPa. The symbols denote the experimental data. Circles: Nakajima et al.⁸⁵, square: Frost and Langenhorst⁸⁴, diamond: Katsura and Ito⁸². The curves are ${K}_{{{{\rm{Fe}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$ of ${\chi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ calculated using the thermodynamic parameters given by Nakajima et al.⁸⁵. The numbers denote ${\chi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ in mol.%. The colors indicate ${\chi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$: red: 4 mol.%, green: 8 mol.%, blue: 12 mol.%, and violet: 16 mol.%. The colours of the symbols also denote ${\chi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$. B With P at relatively similar T and bulk compositions. Violet: 1500–1600 K using San Carlos olivine by Kobayashi et al.⁹, cyan: 2000-2450 K using San Carlos olivine by Auzende et al.⁸⁸, 1800–2200 K using (Mg_0.89Fe_0.11)₂SiO₄ gel by Sinmyo et al.⁸⁹, green: 1900–2100 K using San Carlos olivine by Sakai et al.¹⁰. The cyan and green broken lines showing a negative P dependence are fitting to the data given by Auzende et al.⁸⁸ and Sakai et al.¹⁰.

${K}_{{{\rm{F}}}{{{\rm{e}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$ has been studied at higher P using LH-DAC^{9,78,82,86,87,88,89}. Additionally, ${K}_{{{\rm{F}}}{{{\rm{e}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$ can also be derived from data determining ${\varphi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ ^8,76,77,78 with P. However, understanding the P dependence of ${K}_{{{\rm{F}}}{{{\rm{e}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$ is challenging due to its complex relationship with T and compositions. Figure 6A shows ${K}_{{{\rm{F}}}{{{\rm{e}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$ from four studies with minimal variations in T and bulk compositions^9,82,88,89. Two of the studies, Auzende et al.⁸⁸ and Sakai et al.⁸², showed a negative P dependence of ${K}_{{{\rm{F}}}{{{\rm{e}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$, which they attributed to the spin transition of Fper. However, Nakajima et al.⁸⁵ suggested that this P dependence could be interpreted without the spin transition. Given these conflicting interpretations, a comprehensive study is required to determine ${K}_{{{\rm{F}}}{{{\rm{e}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$ as a function of P, T, ${f}_{{{{\rm{O}}}}_{2}}$, and composition precisely.

MgO-CaO-SiO₂

Calcium is the sixth most abundant element in Earth’s mantle¹. In the lower mantle, CaO forms Dvm³⁴. Bdm and Dvm are the two main phases in the MgO-CaO-SiO₂ system. Since non-stoichiometry is not Known in Dvm, the coexistence of Sti, Per, or other Ca-bearing phases should not affect the phase relations between Bdm and Dvm. Consequently, the binary MgSiO₃-CaSiO₃ system is crucial for understanding the ternary MgO-CaO-SiO₂ system. As Bdm and Dvm melt congruently⁹⁰, the eutectic melting T limits the stability of Bdm at high T in this system. Furthermore, no phase transition of Dvm is known within Bdm’s stability field at mantle temperatures.

The maximum CaSiO₃ content in Bdm, ${\varphi }_{{{\rm{Ca}}}{{{\rm{SiO}}}}_{3}\,}^{{{\rm{Bdm}}}}$, below the eutectic point is geochemically significant due to Ca’s large ionic radius. This property allows Dvm to incorporate much larger amounts of trace elements, such as REE, U, and Th, than Bdm⁹¹. Consequently, the presence of Dvm significantly impacts trace element profiles in the mantle. Given that CaSiO₃ is relatively small compared to MgSiO₃ in Earth’s mantle, it is crucial to determine whether ${\varphi }_{{{\rm{Ca}}}{{{\rm{SiO}}}}_{3}\,}^{{{\rm{Bdm}}}}$ exceeds the mantle’s CaSiO₃ content. If ${\varphi }_{{{\rm{Ca}}}{{{\rm{SiO}}}}_{3}\,}^{{{\rm{Bdm}}}}$ surpasses 8 mol.%, it would imply that all Ca is contained in Bdm, and Dvm does not exist in the lower mantle. This determination is essential for understanding the distribution of calcium and associated trace elements in the lower mantle.

The phase relations of Bdm and Dvm in the MgSiO₃-CaSiO₃ system under topmost lower-mantle conditions (24 GPa) have been studied using MAP^90,92,93. Eutectic melting occurs between 2620 and 2700 K at this P ⁹⁰ (Fig. 7). ${\varphi }_{{{\rm{Ca}}}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ is very low, limited to 2 mol.% even at the eutectic T, suggesting Dvm’s presence in the top of the lower mantle.

Fig. 7: Phase relations in MgSiO₃-CaSiO₃, i.e., ${\varphi }_{{{\rm{Ca}}}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ and ${\varphi }_{{{\rm{Mg}}}{{{\rm{SiO}}}}_{3},}^{{{\rm{Dvm}}}}$, with T at 25 GPa^90,92,93,115.

The red and violet circles indicate T and ${\varphi }_{{{\rm{Ca}}}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ of coexisting Bdm and Dvm, respectively, taken from Irifune et al.⁹² and Irifune et al.⁹³. The yellow and cyan circles, respectively, indicate T and ${\varphi }_{{{\rm{Ca}}}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ of Bdm and Dvm coexisting with eutectic melt, from Nomura et al.⁹⁰. The open red and violet diamonds indicate T and ${\varphi }_{{{\rm{Ca}}}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ of coexisting Bdm and Dvm, respectively, from Nomura et al.⁹⁰.

It remains unknown whether ${\varphi }_{{{\rm{Ca}}}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ reaches 8 mol.% at higher P. As the eutectic T should increase with pressure, the maximum T of Bdm stability may increase, potentially resulting in higher ${\varphi }_{{{\rm{Ca}}}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$. The miscibility gap between Bdm and Dvm could narrow with increasing P if the partial molar volume of CaSiO₃ in Bdm is smaller than Dvm’s molar volume. In situ X-ray studies with MAP observed metastable Ca-rich Bdm formation at various P and T, but it transformed to Bdm + Dvm at higher T ^19,93. The volumes of Ca-rich Bdm were reported to be larger than Bdm + Dvm¹⁹, suggesting the miscibility gap is unlikely to narrow with increasing P. Recent laser-heated diamond anvil cells (LH-DAC) studies reported the formation of Ca-rich Bdm at high P and T in Fe- and Al-bearing systems^11,94. Reference ¹² suggested that the secondary components of Fe and Al promoted ${\varphi }_{{{\rm{Ca}}}{{{\rm{SiO}}}}_{3}\,}^{{{\rm{Bdm}}}}$. It is noted that, however, Raoult’s law implies that the presence of minor components does not significantly alter the thermodynamic properties of the major component. Further investigation is necessary to reach conclusive results about ${\varphi }_{{{\rm{Ca}}}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}$ at higher P.

MgO-Al₂O₃-SiO₂

Al is the sixth most abundant element in the Earth’s mantle¹. Significant quantities of Al₂O₃ can be present in Bdm, depending on prevailing conditions. Extensive research has been conducted on the phase relations in the MgO-Al₂O₃-SiO₂ system^{16,22,23,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110}. Figure 8A illustrates the phase assemblage in this system at 27 GPa and 2000 K³⁸. As the Mg/Si ratio decreases from infinite to zero, the phases coexisting with Bdm change from Per, Per + PS^111,112, PS, PS + corundum (Crn), Crn, Crn + Sti, and Sti. This PS has a CaFe₂O₄-structure¹¹¹, with MgAl₂O₄ as its primary component, but can contain Mg₂SiO₄¹¹³. Similarly, Crn’s primary component is Al₂O₃, but it can contain MgSiO₃^{37,98,100,114}. At these conditions, the Mg₂SiO₄ and MgSiO₃ contents in CaFe₂O₄-type PS and Crn coexisting with Bdm are 29-34 and 19-21 mol.%, respectively³⁸. The Al₂O₃ contents in Per and Sti coexisting with Bdm are limited to 0.5 and 5 mol.%, respectively. Notably, the Al₂O₃ content in Sti may be attributed to H₂O potential incorporation through the substitution Si⁴⁺ ↔ Al³⁺ + H^{+ 42}.

Fig. 8: Bdm’s phase relations in MgO-Al2O₃-SiO₂.

A Phase assemblage at 27 GPa and 2000 K. Modified from Liu et al.³⁷. B Phase relations in MgSiO₃-Al₂O₃ at 1700, 2000, and 2300 K and up to 50 GPa. The phase relations of Bdm, Gnt, and Aki were only studied near 2000 K. Modified from Liu et al.^37,114. C Gnt to Bdm + Crn transition. Modified from Ishii et al.¹⁶. D ${\varphi }_{{{{\rm{MgAlO}}}}_{2.5}}^{{{\rm{Bdm}}}}\,$ and ${\varphi }_{{{{{\rm{Mg}}}{{\rm{Fe}}}}^{3+}{{\rm{O}}}}_{2.5}}^{{{\rm{Bdm}}}}$ in Bdm with P at 2000 K. Based on Liu et al.¹¹⁰ and Fei et al.¹²³. E ${\varphi }_{{{{\rm{OV}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ with T at 27 GPa. Violet: coexisting with PS and Crn, blue: coexisting with Per and PS, blue-green: coexisting with Per, light green: no coexisting phase. The coloured broken lines indicate the T dependence of ${\varphi }_{{{{\rm{OV}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ with various coexisting phases. F ${\chi }_{{{{\rm{OV}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ and ${\chi }_{{{{\rm{CC}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ coexisting with Per with ${\chi }_{{{{\rm{Al}}}}_{2}{{{\rm{O}}}}_{3}}^{{{\rm{Bdm}}}}$ at 2000 K and 27 GPa. Modified from Huang et al.⁸¹.

Al is primarily incorporated into Bdm as CC, Al₂O₃, especially when the Mg/Si ratio is unity, where the coexisting phase is Crn^{22,37,98,100,102,106,114,115}. The maximum Al₂O₃ CC content in Bdm, ${\varphi }_{{{{\rm{CC}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$, increases with P ^{37,98,100,102,114,115} (Fig. 8B). For example, ${\varphi }_{{{{\rm{CC}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ increases from 12 to 22 mol.% as P rises from 27 to 42 GPa at 2000 K¹¹⁴. Notably, when the ${\varphi }_{{{{\rm{CC}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ is high (at least 25 mol.%), Bdm cannot be recovered to ambient conditions but transforms to the LiNbO₃ structure upon decompression^109,114. The increase in ${\varphi }_{{{{\rm{CC}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ with P is associated with a decrease in the maximum MgSiO₃ content in Crn, ${\varphi }_{{{\rm{MgSi}}}{{{\rm{O}}}}_{3}}^{{{\rm{Crn}}}}$. For instance, ${\varphi }_{{{\rm{MgSi}}}{{{\rm{O}}}}_{3}}^{{{\rm{Crn}}}}$ decreases from 32 to 22 mol.% as P rises from 27 to 35 GPa at 2300 K. Similarly, ${\varphi }_{{{{\rm{CC}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ increases with T ^22,37,114. For instance, ${\varphi }_{{{{\rm{CC}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ increases from 16 to 30 mol.% as T rises from 1700 to 3000 K at 27 GPa. This increase is accompanied by a corresponding increase in ${\varphi }_{{{\rm{MgSi}}}{{{\rm{O}}}}_{3}}^{{{\rm{Crn}}}}$.

The stability field of Bdm + Crn in the MgSiO₃-Al₂O₃ system is constrained on the lower P side by the formation of pyrope Gnt^{16,98,100,101} at 26 GPa (Fig. 8C). The boundary in the P-T space is curved. The dP/dT is negative, with a value of −1.5 MPaK⁻¹ at 1400–1800 K. At higher T up to 1900–2100 K, dP/dT becomes positive, reaching a value of +2.5 MPaK⁻¹. At high P, ca. 100 GPa, Crn first transforms to the Rh₂O₃(II) structure¹¹⁶. The melting relations in the Bdm + Crn system have not yet been investigated.

In MgO-rich or SiO₂-poor systems, Bdm contains OV, MgAlO_2.5, in addition to CC^{103,107,110,117,118,119}. The maximum OV content, ${\varphi }_{{{{\rm{OV}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$, is considerably more limited than ${\varphi }_{{{{\rm{CC}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$^{103,110,118,119}. ${\varphi }_{{{{\rm{OV}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ rapidly decreases with P¹¹⁰, as shown in Fig. 8D. For instance, ${\varphi }_{{{{\rm{OV}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ decreases from 6 to 1 mol.% as P rises from 27 to 40 GPa at 2000 K when coexisting with Per. The T dependence of ${\varphi }_{{{{\rm{OV}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ varies based on the coexisting phases (Fig. 8E)¹¹⁹. When Bdm coexists with Per and Per + PS, respectively, ${\varphi }_{{{{\rm{OV}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ increases from 3 to 6 mol.% and from 3 to 4 mol.%, respectively, as T rises from 1700 to 2300 K. However, when coexisting with PS + Crn, ${\varphi }_{{{{\rm{OV}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ decreases from 4 to 3 mol.% as T rises from 2000 to 2300 K²³. Generally, the T dependence of ${\varphi }_{{{{\rm{OV}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ decreases with increasing the Al₂O₃ content, i.e., in the order of Per → Per + PS → PS + Crn.

The local chemical environment of Al³⁺ in OV is complex. Nuclear magnetic resonance (NMR) studies have indicated that Al³⁺ in Bdm can have coordination numbers of 4, 5, 6, and 8^29,120,121. Among these, 6-coordinated Al³⁺ are predominant, suggesting a random distribution of Al³⁺ and O²⁻ vacancies. However, the presence of 4-coordinated Al³⁺ indicates the formation of clusters comprising both Al³⁺ and O²⁻ vacancies.

The variation in the OV content, ${\varphi }_{{{{\rm{OV}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$, is more complex than ${\varphi }_{{{{\rm{CC}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$. When the bulk Al₂O₃ content in Bdm, ${\chi }_{{{{\rm{Al}}}}_{2}{{{\rm{O}}}}_{3}}^{{{\rm{Bdm}}}}={\chi }_{{{{\rm{CC}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}+0.5{\chi }_{{{{\rm{OV}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$, increases to approximately 10 mol.% in the presence of Per, the ${\chi }_{{{{\rm{OV}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ initially increases and then decreases at ${\chi }_{{{{\rm{Al}}}}_{2}{{{\rm{O}}}}_{3}}^{{{\rm{Bdm}}}}$. In contrast, ${\chi }_{{{{\rm{CC}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ increases monotonically (Fig. 8F).

Huang et al.⁸¹ employed a thermodynamic approach to express the equilibrium of Bdm and Per in the MgO-Al₂O₃-SiO₂ system. To address the complexity of OV, they introduced the ${{{\rm{MgSi}}}}_{\frac{7}{8}}{{{\rm{Al}}}}_{\frac{1}{8}}{{{\rm{O}}}}_{\frac{15}{16}}{{{\rm{\square }}}}_{\frac{1}{16}}{{{\rm{O}}}}_{2}$ component instead of MgAlO_2.5. This equilibrium was represented by the following equation:

$$\begin{array}{c}{{{\rm{MgSi}}}}_{\frac{7}{8}}{{{\rm{Al}}}}_{\frac{1}{8}}{{{\rm{O}}}}_{\frac{15}{16}}{{{\square }}}_{\frac{1}{16}}{{{\rm{O}}}}_{2}\\ {{\rm{Brg}}}\end{array}=\frac{1}{16}\begin{array}{c}{{{\rm{AlAlO}}}}_{3}\\ {{\rm{Brg}}}\end{array}+\frac{7}{8}\begin{array}{c}{{{\rm{MgSiO}}}}_{3}\\ {{\rm{Brg}}}\end{array}+\frac{1}{8}\begin{array}{c}{{\rm{MgO}}}\\ {{\rm{Per}}}\end{array}$$

(5)

The standard Gibbs energy of reaction (5) at equilibrium, ΔG₍₅₎⁰, was expressed using the following equation:

$$\Delta {G}_{\left(5\right)}^{0}=-{{\rm{R}}}T\,{{\mathrm{ln}}}\frac{{\left({a}_{{{{\rm{AlAlO}}}}_{3}}^{{{\rm{Bdm}}}}\right)}^{\frac{1}{16}}{\left({a}_{{{\rm{MgSi}}}{{{\rm{O}}}}_{3}}^{{{\rm{Bdm}}}}\right)}^{\frac{7}{8}}{\left({a}_{{{\rm{MgO}}}}^{{{\rm{Per}}}}\right)}^{\frac{1}{8}}}{{a}_{{{{\rm{MgSi}}}}_{\frac{7}{8}}{{{\rm{Al}}}}_{\frac{1}{8}}{{{\rm{O}}}}_{\frac{15}{16}}{{{\square }}}_{\frac{1}{16}}{{{\rm{O}}}}_{2}}^{{{\rm{Bdm}}}}}$$

(6)

where a^A_i is the activity of component i in phase A. To consider the fraction of ${{{\rm{MgSi}}}}_{\frac{7}{8}}{{{\rm{Al}}}}_{\frac{1}{8}}{{{\rm{O}}}}_{\frac{15}{16}}{{{\square }}}_{\frac{1}{16}}{{{\rm{O}}}}_{2}$, ${\chi }_{{{{\rm{MgSi}}}}_{\frac{7}{8}}{{{\rm{Al}}}}_{\frac{1}{8}}{{{\rm{O}}}}_{\frac{15}{16}}{{{\rm{\square }}}}_{\frac{1}{16}}{{{\rm{O}}}}_{2}}^{{{\rm{Bdm}}}}$, they assumed that O²⁻ vacancies occur on the O1 site (multiplicity of unity) and not on the O2 site (multiplicity of two). This results in:

$${\chi }_{{{{\rm{MgSi}}}}_{\frac{7}{8}}{{{\rm{Al}}}}_{\frac{1}{8}}{{{\rm{O}}}}_{\frac{15}{16}}{{{\square }}}_{\frac{1}{16}}{{{\rm{O}}}}_{2}}^{{{\rm{Bdm}}}}=1.841{x}_{{{{\rm{Mg}}}}_{{{\rm{A}}}}}^{{{\rm{Bdm}}}}{\left({x}_{{{{\rm{Al}}}}_{{{\rm{B}}}}}^{{{\rm{Bdm}}}}\right)}^{\frac{1}{8}}{\left({x}_{{{{\rm{Si}}}}_{{{\rm{B}}}}}^{{{\rm{Bdm}}}}\right)}^{\frac{7}{8}}{\left({x}_{{{{\square }}}_{{{\rm{O}}}1}}^{{{\rm{Bdm}}}}\right)}^{\frac{1}{16}}{\left({x}_{{{{\rm{O}}}}_{{{\rm{O}}}1}}^{{{\rm{Bdm}}}}\right)}^{\frac{15}{16}}$$

(7)

where ${x}_{{{{\rm{Mg}}}}_{{{\rm{A}}}}}^{{{\rm{Bdm}}}}$, ${x}_{{{{\rm{Al}}}}_{{{\rm{B}}}}}^{{{\rm{Bdm}}}}$, and ${x}_{{{{\rm{Si}}}}_{{{\rm{B}}}}}^{{{\rm{Bdm}}}}$ are the fractions of Mg on the A site, Al and Si on the B site, respectively, and ${x}_{{{{\rm{\square }}}}_{{{\rm{O}}}1}}^{{{\rm{Bdm}}}}$ and ${x}_{{{{\rm{O}}}}_{{{\rm{O}}}1}}^{{{\rm{Bdm}}}}$ are the fractions of vacancies and oxygen on the O1 site, respectively. The coefficient 1.841 ensures that the activity of the endmember MgSi_7/8Al_1/8O_15/16□_1/16O₂ equals unity and is driven from $1/{\left(\frac{1}{8}\right)}^{\frac{1}{8}}{\left(\frac{7}{8}\right)}^{\frac{7}{8}}{\left(\frac{1}{16}\right)}^{\frac{1}{16}}{\left(\frac{15}{16}\right)}^{\frac{15}{16}}$.

The MgSiO₃ fraction in Bdm, ${\chi }_{{{\rm{MgSi}}}{{{\rm{O}}}}_{3}}^{{{\rm{Bdm}}}}$, was expressed as follows:

$${\chi }_{{{\rm{MgSi}}}{{{\rm{O}}}}_{3}}^{{{\rm{Bdm}}}}={x}_{{{\rm{Mg}}},{{\rm{A}}}}^{{{\rm{Bdm}}}}{x}_{{{\rm{Si}}},{{\rm{B}}}}^{{{\rm{Bdm}}}}{x}_{{{\rm{O}}},{{\rm{O}}}1}^{{{\rm{Bdm}}}}$$

(8)

The regular symmetric solution model was used to express the activity coefficient of MgSi_7/8Al_1/8O_15/16□_1/16O₂, ${\gamma }_{{{{\rm{MgSi}}}}_{\frac{7}{8}}{{{\rm{Al}}}}_{\frac{1}{8}}{{{\rm{O}}}}_{\frac{15}{16}}{{{\square }}}_{\frac{1}{16}}{{{\rm{O}}}}_{2}}^{{{\rm{Bdm}}}}$:

$${{\rm{R}}}T\,{{\mathrm{ln}}}{\gamma }_{{{{\rm{MgSi}}}}_{\frac{7}{8}}{{{\rm{Al}}}}_{\frac{1}{8}}{{{\rm{O}}}}_{\frac{15}{16}}{{{\square }}}_{\frac{1}{16}}{{{\rm{O}}}}_{2}}^{{{\rm{Bdm}}}}= {W}_{{{\rm{Mg}}}-{{\rm{Al}}},{{\rm{A}}}}^{{{\rm{Bdm}}}}{\left(1-{x}_{{{\rm{Mg}}},{{\rm{A}}}}^{{{\rm{Bdm}}}}\right)}^{2}+{W}_{{{\rm{Al}}}-{{\rm{Si}}},{{\rm{B}}}}{\left(1-{x}_{{{\rm{Al}}},{{\rm{B}}}}^{{{\rm{Bdm}}}}\right)}^{2} \\ +{W}_{{{\rm{O}}}-{{\square }},{{\rm{O}}}1}^{{{\rm{Bdm}}}}{\left(1-{x}_{{{\square }},{{\rm{O}}}1}^{{{\rm{Bdm}}}}\right)}^{2}$$

(9)

where W^Bdm_i-j,α are the interaction parameters between components i and j in the α site in Bdm. Similar expressions were provided for the activity coefficients of MgSiO₃ and AlAlO₃:

$${{\rm{R}}}T\,{{\mathrm{ln}}}{\gamma }_{{{\rm{MgSi}}}{{{\rm{O}}}}_{3}}^{{{\rm{Bdm}}}}={W}_{{{\rm{Al}}}-{{\rm{Si}}},{{\rm{B}}}}^{{{\rm{Bdm}}}}{\left(1-{x}_{{{\rm{Si}}}{,}_{{{\rm{B}}}}}^{{{\rm{Bdm}}}}\right)}^{2}$$

(10)

$${{\rm{R}}}T\,{{\mathrm{ln}}}{\gamma }_{{{{\rm{Al}}}}_{2}{{{\rm{O}}}}_{3}}^{{{\rm{Bdm}}}}={W}_{{{\rm{Al}}}-{{\rm{Si}}},{{\rm{B}}}}^{{{\rm{Bdm}}}}{\left(1-{x}_{{{{\rm{Al}}}}_{{{\rm{B}}}}}^{{{\rm{Bdm}}}}\right)}^{2}$$

(11)

The parameters evaluated by Huang et al.⁸¹ are presented in Table 3. This formulation and the estimated parameters provide a comprehensive description of ${\chi }_{{{{\rm{OV}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ and ${\chi }_{{{{\rm{CC}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ in relation to ${\chi }_{{{{\rm{Al}}}}_{2}{{{\rm{O}}}}_{3}{{\rm{m}}}}^{{{\rm{Bdm}}}}$, as illustrated in Fig. 8F.

Table 3 Thermodynamic parameters to describe Bdm and Fper chemistry in MgO-Al₂O₃-SiO₂ at 27 GPa and 2000 K from Huang et al.⁸¹

MgO-Fe³⁺ ₂O₃-SiO₂

Despite Fe being the fourth most abundant element in Earth’s mantle¹, Fe³⁺ was previously thought to play a minor role in the lower mantle. This assumption was based on the presumed reducing conditions of the lower mantle, inferred from its contact with the outer core. As a result, phase relations in the MgO-Fe³⁺₂O₃-SiO₂ system have not been extensively studied. Another factor contributing to the limited study on this system is the difficulty in maintaining and accurately measuring Fe valence during and after high P-T experiments, making it challenging to obtain reliable data on Fe³⁺ behaviour under lower-mantle conditions.

The phase relations in the MgO-Fe³⁺₂O₃-SiO₂ system are similar to those in the MgO-Al₂O₃-SiO₂ system^{20,30,122,123}. Fe³⁺ can occupy the A- and B-sites, like Al³⁺. However, the coexisting phases with Bdm have not been precisely determined as a function of the Mg/Fe³⁺/Si ratio. As the Mg/Fe³⁺ ratio decreases at Mg/Si ratios less than 1:1, the coexisting phases change from Per, Per + PS, and PS,^20,122,123 (Fig. 9). This PS has a MgFe₂O₄ composition and a CaMn₂O₄ structure. At Mg/Si ratios greater than 1:1, Bdm coexists with Sti and an unidentified phase with a LiNbO₃ structure after recovery³⁰. This phase has a composition of approximately Mg/Si = 1:1 and MgSiO₃/Fe₂O₃ = 1:2 at 27 GPa and 1700 to 2000 K. It is possible that this LiNbO₃-structured phase may also have a perovskite structure at high pressure, suggesting a potential miscibility gap in perovskite phases within the MgSiO₃-Fe₂O₃ system. At Mg/Si ≈ 1, Bdm or the LiNbO₃-structured phase should coexist with a Fe₂O₃ polymorph. Although numerous Fe₂O₃ polymorphs exist¹²⁴, it is unclear which one coexists with Bdm under specific P-T conditions. Bykova et al.¹²⁴ proposed a series of phase transitions for Fe₂O₃ with P: α-phase (hematite, Hem) → ι-phase (Rh₂O₃(II) structure) at 25-45 GPa → ζphase (distorted perovskite structure) at 45-55 GPa → η-phase (CaIrO₃ structure).

Fig. 9: Phase relations in MgO-Fe³⁺₂O₃-SiO₂.

A Ternary diagram at 27 GPa and 2000~2300 K. B ${\varphi }_{{{\rm{CC}}},{{\rm{OV}}}}^{{{\rm{Bdm}}}}$ for Fe³⁺ and Al coexisting with Per and PS with T at 27 GPa. Green: Fe³⁺; Violet: Al; Closed symbols: OV; Open symbols: CC.

The incorporation of Fe³⁺ is more limited than that of Al³⁺. While the maximum Fe³⁺ CC content in Bdm, ${\varphi }_{{{{\rm{CC}}}}_{{{{\rm{Fe}}}}^{3+}}}^{{{\rm{Bdm}}}}$, coexisting with Fe₂O₃ phases has not been studied, research on ${\varphi }_{{{{\rm{CC}}}}_{{{{\rm{Fe}}}}^{3+}}}^{{{\rm{Bdm}}}}$ in other systems suggests its comparable significance at top lower mantle P. For instance, ${\varphi }_{{{{\rm{CC}}}}_{{{{\rm{Fe}}}}^{3+}}}^{{{\rm{Bdm}}}}$ increases from 2 to 8 mol.% as T rises from 1700 to 2300 K when coexisting with Per and PS at 27 GPa²⁰, which is a similar increases in ${\varphi }_{{{{\rm{CC}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ from 4 to 7 mol.% under the same conditions¹¹⁹ (Fig. 9B).

P appears to have contrasting effects on ${\varphi }_{{{{\rm{CC}}}}_{{{{\rm{Fe}}}}^{3+}}}^{{{\rm{Bdm}}}}$ and ${\varphi }_{{{{\rm{CC}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$. As P rises from 27 to 40 GPa at 2300 K, ${\varphi }_{{{{\rm{CC}}}}_{{{{\rm{Fe}}}}^{3+}}}^{{{\rm{Bdm}}}}$ decreases from 8 to 5 mol.% when coexisting with Per and PS¹²³. Conversely, ${\varphi }_{{{{\rm{CC}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ increases from 12 to 22 mol.% as P increases from 27 to 42 GPa at 2000 K when coexisting with Crn¹¹⁴. It should be noted, however, that these two data sets cannot be directly compared due to the difference in the coexisting phase.

Unlike Fe³⁺ CC, the Fe³⁺ OV component, MgFe³⁺O_2.5□_0.5, is more limited compared to its Al counterpart, MgAlO_2.5□_0.5. When coexisting with Per and PS, ${\varphi }_{{{{\rm{OV}}}}_{{{{\rm{Fe}}}}^{3+}}}^{{{\rm{Bdm}}}}$ remains around 2 mol.% between 1700 and 2300 K at 27 GPa²⁰ (Fig. 9B). Contrastly, ${\varphi }_{{{{\rm{OV}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ increases from 3 to 4 mol.% under similar conditions¹¹⁹. The formation of both Fe³⁺ and Al OV is inhibited by P, which is more profound for Fe³⁺. As P rises from 27 to 40 GPa at 2300 K, ${\varphi }_{{{{\rm{OV}}}}_{{{{\rm{Fe}}}}^{3+}}}^{{{\rm{Bdm}}}}$ decreases from 3 to 0 mol.%, showing a more significant reduction compared to ${\varphi }_{{{{\rm{OV}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ (Fig. 8D).

Huang et al.⁸¹ extended their thermodynamic approach from the Mg-Al-Si-O system to investigate the phase equilibrium of Bdm coexisting with Fper in the Mg-Fe³⁺-Si-O system. They proposed a reaction similar to Eq. (5):

$$\begin{array}{c}{{{\rm{MgSi}}}}_{\frac{7}{8}}{{{\rm{Fe}}}}_{\frac{1}{8}}^{3+}{{{\rm{O}}}}_{\frac{15}{16}}{{{\square }}}_{\frac{1}{16}}{{{\rm{O}}}}_{2}\\ {{\rm{Bdm}}}\end{array}=\frac{1}{16}\begin{array}{c}{{{{\rm{Fe}}}}^{3+}{{{\rm{Fe}}}}^{3+}{{\rm{O}}}}_{3}\\ {{\rm{Bdm}}}\end{array}+\frac{7}{8}\begin{array}{c}{{{\rm{MgSiO}}}}_{3}\\ {{\rm{Bdm}}}\end{array}+\frac{1}{8}\begin{array}{c}{{\rm{MgO}}}\\ {{\rm{Per}}}\end{array}$$

(12)

Thermodynamic parameters for this model were suggested (as listed in Table 4). However, the model’s accuracy in reproducing experimental data is unsatisfactory. This imprecision likely stems from the limited number of experiments conducted on this system and the potential for Fe³⁺ to be reduced to Fe²⁺ during experiments.

Table 4 Thermodynamic parameters to describe Bdm and Fper chemistry in MgO-Fe³⁺-SiO₂ at 27 GPa and 2000 K

MgO-FeAlO₃-SiO₂

Due to the similarity in ionic radii between Fe³⁺ and Al and between Mg and Si, it is expected that the Fe³⁺ and Al would predominantly occupy the A- and B-sites, respectively¹²⁵. This expectation has been confirmed by single-crystal X-ray diffraction¹²⁵. Notably, even under reduced conditions, Fe in Fe, Al-bearing Bdm remains predominantly Fe³⁺ through coupling with Al^27,71.

This coupling is thought to promote the formation of CC of FeAlO₃ while inhibiting OV. Consequently, the maximum CC fraction of FeAlO₃, ${\varphi }_{{{{\rm{CC}}}}_{{{\rm{FeAl}}}}}^{{{\rm{Bdm}}}}$, is much larger than ${\varphi }_{{{{\rm{CC}}}}_{{{\rm{Al}}}}}^{{{\rm{Bdm}}}}$ and ${\varphi }_{{{{\rm{CC}}}}_{{{{\rm{Fe}}}}^{3+}}}^{{{\rm{Bdm}}}}$ ^21,126,127. At 27 GPa and 2000 K, ${\varphi }_{{{{\rm{CC}}}}_{{{\rm{FeAl}}}}}^{{{\rm{Bdm}}}}$ reaches 67 mol.%, when Bdm coexists with Crn and Hem. Notably, Bdm with ${\chi }_{{{{\rm{CC}}}}_{{{\rm{FeAl}}}}}^{{{\rm{Bdm}}}}$ exceeding 40 mol.% transforms to the LiNbO₃ structure upon decompression^21,24, similar to Al₂O₃-rich Bdm^109,114. When the Fe and Al contents are equal, the OV content is virtually zero^25,125. A small quantity of MgAlO_2.5□_0.5 is observed when the Al content exceeds the Fe content¹²⁵.

While the phase relations in the MgO-FeAlO₃-SiO₂ remain largely unexplored, those in the MgO-FeAlO₃-MgSiO₃ system were studied at 1700–2300 K at 27 GPa²⁵. Under these conditions, Bdm coexisted with PS, primarily composed of MgAl₂O₄ and MgFe₂O₄ (likely with a CaTi₂O₄-type structure), and minor Crn. Bdm’s main composition was 28–32 mol.% MgSiO₃, 65–68 mol.% FeAlO₃ with minor Fe₂O₃. The formation of Crn and the incorporation of Fe₂O₃ in Bdm suggest that Bdm favours Fe₂O₃ over Al₂O₃. Despite a slight increase in FeAlO₃ with rising T, no significant compositional changes were observed across the T range.

MgO-FeO-Fe₂O₃-Al₂O₃-SiO₂

This system is most relevant for understanding Bdm chemistry in the lower mantle. However, its complexity, with five components, makes comprehending phase relations extremely challenging. The difficulty in directly controlling the Fe³⁺/ΣFe ratio further complicates investigations. Some studies assume that the initial Fe³⁺/ΣFe in starting materials remains unchanged during a high P-T experiment in LH-DAC experiments¹². This assumption is quite questionable, especially at high T due to diffusion¹³. Indeed, measurements of Fe³⁺/ΣFe in Bdm have yielded inconsistent results across various studies^{12,13,128,129,130,131}, suggesting that bulk Fe³⁺/ΣFe likely changes during high P-T experiments.

In contrast to the conventional approach of maintaining constant bulk Fe³⁺/ΣFe, some studies have estimated ${f}_{{{{\rm{O}}}}_{2}}$ based on the equilibrium between Fe-bearing phases with different valence states in the run products^79,81. Specifically, ${f}_{{{{\rm{O}}}}_{2}}$ is estimated using the reaction between Fe-bearing alloy and ferropericlase:

$$2 \begin{array}{c}{{\rm{Fe}}} \\ {{\rm{alloy}}}\end{array}+{{{\rm{O}}}}_{2} = 2 \begin{array}{c}{{\rm{FeO}}} \\ {{\rm{Fper}}}\end{array}$$

(13)

To produce an alloy, a platinum group metal, typically Ir, is added, as it is not incorporated into the silicate or oxide phase. Using reaction (13), ${f}_{{{{\rm{O}}}}_{2}}$ is calculated with the following equation:

$$\log {f}_{\!\!{{\mbox{O}}}_{2}}= \frac{2{\mu }_{{{{\rm{Fe}}}}^{2+}{{\rm{O}}}}^{{{\rm{Fper}}}}-\,2{\mu }_{{{\rm{Fe}}}}^{{{\rm{alloy}}}}}{{{\mathrm{ln}}}10{{\rm{R}}}T}=\frac{\Delta {G}_{P,T(13)}^{0}}{{{\mathrm{ln}}}10{{\rm{R}}}T}+2\log \left({\gamma }_{{{{\rm{Fe}}}}^{2+}{{\rm{O}}}}^{{{\rm{Fper}}}}{\chi }_{{{{\rm{Fe}}}}^{2+}{{\rm{O}}}}^{{{\rm{Fper}}}}\right) \\ -2\log \left({\gamma }_{{{\rm{Fe}}}}^{{{\rm{alloy}}}}{\chi }_{{{\rm{Fe}}}}^{{{\rm{alloy}}}}\right)$$

(14)

The phase relations of Bdm and Fper in the MgO-FeO-Fe₂O₃-Al₂O₃-SiO₂ are described based on Wang et al.²⁶.

The two exchange coefficients are defined as follows:

$${K}_{{{{\rm{Fe}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}=\frac{\frac{{\chi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}}}{{\chi }_{{{{\rm{MgSiO}}}}_{3}}^{{{\rm{Bdm}}}} \, + \, {\chi }_{{{{\rm{Mg}}}{{\rm{Al}}}{{\rm{O}}}}_{2.5}{{{\square }}}_{0.5}}^{{{\rm{Bdm}}}}}}{\frac{{\chi }_{{{{\rm{Fe}}}}^{2+}{{\rm{O}}}}^{{{\rm{Fper}}}}}{{\chi }_{{{\rm{Mg}}}{{\rm{O}}}}^{{{\rm{Fper}}}}}}$$

(15)

$${K}_{\Sigma {{\rm{Fe}}}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}=\frac{\frac{{\chi }_{{{{\rm{Fe}}}}^{2+}{{{\rm{SiO}}}}_{3}}^{{{\rm{Bdm}}}} \, + \, 2{\chi }_{{{{{\rm{Fe}}}}^{3+}{{{\rm{Fe}}}}^{3+}{{\rm{O}}}}_{3}}^{{{\rm{Bdm}}}} \, + \, {\chi }_{{{{{\rm{Fe}}}}^{3+}{{\rm{Al}}}{{\rm{O}}}}_{3}}^{{{\rm{Bdm}}}}}{{\chi }_{{{{\rm{MgSiO}}}}_{3}}^{{{\rm{Bdm}}}} \, + \, {\chi }_{{{{\rm{Mg}}}{{\rm{Al}}}{{\rm{O}}}}_{2.5}{{{\square }}}_{0.5}}^{{{\rm{Bdm}}}}}}{\frac{{\chi }_{{{{\rm{Fe}}}}^{2+}{{\rm{O}}}}^{{{\rm{Fper}}}}}{{\chi }_{{{\rm{Mg}}}{{\rm{O}}}}^{{{\rm{Fper}}}}}}$$

(16)

Unlike the previous definition of ${K}_{{{{\rm{Fe}}}}^{2+}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$, Eq. (15) includes ${\chi }_{{{{\rm{Mg}}}{{\rm{Al}}}{{\rm{O}}}}_{2.5}{{{\square }}}_{0.5},{{\rm{Bdm}}}}$ to account for all Mg. ${K}_{\Sigma {{\rm{Fe}}}-{{\rm{Mg}}}}^{{{\rm{Bdm}}}-{{\rm{Fper}}}}$ is referred to as the “apparent” exchange coefficient, incorporating both Fe²⁺ and Fe^{3+27,81,84,85}. This value is obtained by electron microprobe analysis without distinguishing Fe²⁺ and Fe³⁺.

Wang et al.²⁶ described the equilibrium of Bdm and Fper in the current system using the following reactions in addition to reaction (2):

$$2\begin{array}{c}{{\rm{Mg}}}{{\rm{Al}}}\left[{{{\rm{O}}}}_{0.5}{{{\square }}}_{0.5}\right]{{{\rm{O}}}}_{2}\\ {{\rm{Bdm}}}\end{array}=\begin{array}{c}{{\rm{AlAl}}}\left[{{\rm{O}}}\right]{{{\rm{O}}}}_{2}\\ {{\rm{Bdm}}}\end{array}+2\begin{array}{c}{{\rm{MgO}}}\\ {{\rm{FPer}}}\end{array}$$

(17)

$$2\begin{array}{c}{{{\rm{Fe}}}}^{2+}{{\rm{O}}}\\ {{\rm{FPer}}}\end{array}+0.5{{{\rm{O}}}}_{2}=\begin{array}{c}{{{\rm{Fe}}}}^{3+}{{{\rm{Fe}}}}^{3+}{{{\rm{O}}}}_{3}\\ {{\rm{Bdm}}}\end{array}$$

(18)

$$\begin{array}{c}{{{\rm{Fe}}}}^{3+}{{{\rm{Fe}}}}^{3+}{{{\rm{O}}}}_{3}\\ {{\rm{Bdm}}}\end{array}+\begin{array}{c}{{\rm{AlAl}}}{{{\rm{O}}}}_{3}\\ {{\rm{Bdm}}}\end{array}=2\begin{array}{c}{{\rm{Al}}}{{{\rm{Fe}}}}^{3+}{{{\rm{O}}}}_{3}\\ {{\rm{Bdm}}}\end{array}$$

(19)

The thermodynamics framework, based on the work of Stixrude and Lithgow-Bertelloni^132,133, begins with the following expression for the Helmholtz energy, F, of the endmembers:

$$F\left(V,{T}_{1}\right)= \, F\left({V}_{0},{T}_{0}\right)+\frac{9{\!V}_{\!\!0}{K}_{T0}}{2}{f}_{{{\rm{E}}}}^{2}+\frac{9{V}_{\!\!0}{K}_{T0}\left({K}_{T0}^{{\prime} }-4\right)}{2}{f}_{{{\rm{E}}}}^{3} \\ +9n{{{\rm{k}}}}_{{{\rm{B}}}}\left[{T}_{1}D\left(\frac{\theta }{{T}_{1}}\right)-{T}_{0}D\left(\frac{\theta }{{T}_{0}}\right)\right] \\ +\left[{F}_{{{\rm{mag}}}}\left({V}_{\!0},{T}_{1}\right)-{F}_{{{\rm{mag}}}}\left({V}_{\!0},{T}_{0}\right)\right]$$

(20)

where K_T0 and K_T0’ are the bulk modulus and its pressure derivative, f_E is the Murnaghan’s finite stran¹³⁴, k_B is the Boltzmann constant, θ is the Debye temperature, D is the Debye function¹³⁵, and F_mag is the magnetic contribution of iron to F. Wang et al.²⁶ applied a symmetric regular solution model to determine Bdm and Fper compositions. The parameters provided are presented in Tables 5 and 6.

Table 5 Parameters for calculating the Gibbs free energy of the end members of Bdm and Fper

Table 6 Interaction parameters for the excess enthalpy in the solution mixing model

We can obtain various insights about Brm and Fper’s chemistry from the above formula and parameters. To show some examples, I have simulated the various contents in Bdm and Fper and the exchange coefficients in a bulk composition similar to pyrolite by varying, ${f}_{{{{\rm{O}}}}_{2}}$, T, P, Fe/Mg, and Al/Si, whose results are given in the supplementary information. The most important insight from these investigations is that Bdm Fe³⁺/ΣFe is non-zero but is several-tenths even at ${f}_{{{{\rm{O}}}}_{2}}$ below the IW buffer (Fig. S1A). Bdm Fe³⁺/ΣFe particularly increases from less than 0.1 to about 0.6 as rising Al/Si from 0 to 0.3, indicating that the Fe³⁺ content is due to the coupling of Fe³⁺ and Al, especially the high stability of [Fe³⁺]_A[Al³⁺]_BO²⁻₃ (Fig. S1I).

From an Earth science perspective, this property of Bdm results in the disproportionation of Fe²⁺ into Fe⁰ and Fe³⁺¹³⁶.

$$3{{\rm{FeO}}}={{\rm{Fe}}}+{{{\rm{Fe}}}}_{2}{{{\rm{O}}}}_{3}$$

(21)

In peridotite systems, this disproportionation involves Fper:

$$3\begin{array}{c}{{{\rm{Fe}}}}^{2+}{{\rm{O}}}\\ {{\rm{Fper}}}\end{array}+\begin{array}{c}{{\rm{AlAl}}}{{{\rm{O}}}}_{3}\\ {{\rm{Bdm}}}\end{array}={{\rm{Fe}}}+2\begin{array}{c}{{\rm{Al}}}{{{\rm{Fe}}}}^{3+}{{{\rm{O}}}}_{3}\\ {{\rm{Bdm}}}\end{array}$$

(22)

While Fe³⁺/ΣFe observed in upper-mantle rocks is only about 0.04¹³⁷, Bdm constitutes 80% of pyrolite with Bdm Fe³⁺/ΣFe of several tenths at various P, T and f_O2 (Fig. S1A, C, E). Consequently, free metallic iron is likely present in the lower mantle^136,137.

H₂O bearing systems

Many studies have been conducted to determine the maximum H₂O content in Bdm, ${\varphi }_{{{{\rm{H}}}}_{2}{{\rm{O}}}}^{{{\rm{Bdm}}}}$^{138,139,140,141,142,143,144,145,146,147,148}. As excess H₂O coexisting with silicate forms a hydrous melt under mantle conditions, the coexistence with hydrous melt is essential for determining ${\varphi }_{{{{\rm{H}}}}_{2}{{\rm{O}}}}^{{{\rm{Bdm}}}}$. A challenge in determining ${\varphi }_{{{{\rm{H}}}}_{2}{{\rm{O}}}}^{{{\rm{Bdm}}}}$ is the potential presence of hydrous mineral inclusions¹⁴⁰. As ${\varphi }_{{{{\rm{H}}}}_{2}{{\rm{O}}}}^{{{\rm{Bdm}}}}$ is typically minimal, such inclusions can lead to erroneous interpretation of ${\varphi }_{{{{\rm{H}}}}_{2}{{\rm{O}}}}^{{{\rm{Bdm}}}}$.

${\varphi }_{{{{\rm{H}}}}_{2}{{\rm{O}}}}^{{{\rm{Bdm}}}}$ in MgSiO₃ Bdm was first reported by Meade et al.¹³⁸ as 60–70 wt. ppm, using a 200 μm single crystal synthesised by Ito and Weidner¹⁴⁹. Litasov et al.¹⁴¹ later reported 100 wt. ppm H₂O, but their smaller crystal size (100 μm) may have led to the broadband signals from the grain boundary rather than the crystal interior. The sharp peaks in their study indicated only 40 wt. ppm H₂O. In contrast, Bolfan-Casanova et al.¹⁴⁰ detected only 1 ~ 2 wt. ppm H₂O, suggesting that higher reported ${\varphi }_{{{{\rm{H}}}}_{2}{{\rm{O}}}}^{{{\rm{Bdm}}}}$ might be due to inclusions. Recently, Liu et al.¹⁴⁵ measured ${\varphi }_{{{{\rm{H}}}}_{2}{{\rm{O}}}}^{{{\rm{Bdm}}}}$ using a large (300 μm) single crystal and reported that it is less than 50 wt. ppm. Considering these studies, ${\varphi }_{{{{\rm{H}}}}_{2}{{\rm{O}}}}^{{{\rm{Bdm}}}}$ in MgSiO₃ Bdm is likely low, at most 50 wt. ppm.

Several studies showed ${\varphi }_{{{{\rm{H}}}}_{2}{{\rm{O}}}}^{{{\rm{Bdm}}}}$ in Fe, Al-bearing Bdm is significantly higher than in pure MgSiO₃ Bdm^{140,141,144,146}. This is attributed to the coupling of H⁺ with Al³⁺ to substitute Si⁴⁺ in the B-site, forming [Mg²⁺]_A[Al³⁺H⁺]_BO²⁻⁴². The majority of studies^{42,141,142,144} reported relatively high values, such as 1000 wt. ppm H₂O in Fe, Al-bearing Bdm, although Liu et al. reported only 10~30 wt. ppm H₂O.

Outlook

Determining Bdm chemistry with high reliability is crucial, as it is the most abundant mineral in Earth’s interior. Two high P-T apparatuses are commonly used for studying Bdm chemistry: LH-DAC and MAP. Each has its limitations: LH-DAC can cover the entire P-T range of the lower mantle but produces less reliable results, while MAP yields more reliable results but has limited P-T coverage. To address these issues, efforts should focus on improving the reliability of LH-DAC experiments and extending the P-T range of MAP experiments. It is recommended that future studies investigate the compositions of Bdm coexisting with Fper in the MgO-FeO-Fe₂O₃-Al₂O₃-SiO₂ system as a function of pressure at Known ${f}_{{{{\rm{O}}}}_{2}}$ using these advanced experimental techniques.