Mantle oxidation state
Mantle oxidation state (redox state) applies the concept of oxidation state in chemistry to the study of the Earth's mantle. The chemical concept of oxidation state mainly refers to the valence state of one element, while mantle oxidation state provides the degree of decreasing of increasing valence states of all polyvalent elements in mantle materials confined in a closed system. The mantle oxidation state is controlled by oxygen fugacity and can be benchmarked by specific groups of redox buffers. Mantle oxidation state changes because of the existence of polyvalent elements (elements with more than one valence  state, e.g. Fe, Cr, V, Ti, Ce, Eu, C and others). Among them, Fe is the most abundant (~8 wt% of the mantle) and its oxidation state largely reflects the oxidation state of mantle. Examining the valence state of other polyvalent elements could also provide the information of mantle oxidation state. It is well known that the oxidation state can influence the partitioning behavior of elements and liquid water between melts and minerals, the speciation of C-O-H-bearing fluids and melts, as well as transport properties like electrical conductivity and creep. The formation of diamond requires both reaching  high pressures and high temperatures and a carbon source. The most common carbon source in deep Earth is not elemental carbon and redox reactions need to be involved in diamond formation. Examining the oxidation state can help us predict the P-T conditions of diamond formation and elucidate the origin of deep diamonds. Thermodynamic description of oxidation state. Mantle oxidation state can be quantified as the oxygen fugacity (formula_1) of the system within the framework of thermodynamics. A higher oxygen fugacity implies a more oxygen-rich and more oxidized environment. At each given pressure-temperature conditions, for any compound or  element M that bears the potential to be oxidized by oxygen, we can write formula_2 For example, if M is Fe, the redox equilibrium reaction can be Fe+1/2O=FeO; if M is FeO, the redox equilibrium reaction can be 2FeO+1/2O=FeO. Gibbs energy change associated with this reaction is therefore formula_3 Along each isotherm, the partial derivation of "ΔG" with respect to "P" is "ΔV", formula_4 Under deeper upper mantle conditions, Olivine-Orthopyroxene-Garnet oxygen barometer is the redox reaction that is used to calibrate oxygen fugacity. formula_5 In this reaction, 4 mole of ferrous ions were oxidized  to ferric ions and the other 2 mole of ferrous ions remain unchanged. Transition zone. Garnet-Garnet reaction can be used to estimate the redox state of transition zone. formula_6 A recent study showed that the oxygen fugacity of transition referred from Garnet-Garnet reaction is -0.26 formula_7 to +3 formula_7 relative to the Fe-FeO (IW, iron- wütstite) oxygen buffer. Lower mantle. Disproportionation of ferrous iron at lower mantle conditions also affect the mantle oxidation state. This reaction is different from the reactions mentioned above as it does not incorporate the participation of free oxygen. formula_9, FeO resides in  the form of ferropericlase ("Fp") and FeO resides in the form of bridgmanite ("Bdg"). There is no oxygen fugacity change associated with the reaction. However, as the reaction products differ in density significantly, the metallic iron phase could descend downwards to the Earth's core and get separated from the mantle. In this case, the mantle loses metallic iron and becomes more oxidized. Implications for Diamond Formation. The equilibrium reaction involving diamond is formula_10. Examining the oxygen fugacity of the upper mantle and transition enables us to compare it with the conditions (equilibrium reaction shown above) required for diamond formation . The results show that the formula_7 is usually 2 units lower than the carbonate-carbon reaction which means favoring the formation of diamond at transition zone conditions. It has also been reported that pH decrease would also facilitate the formation of diamond in Mantle conditions. formula_12 formula_13 where the subscript "aq" means 'aqueous', implying H is dissolved in the solution. Deep diamonds have become important windows to look into the mineralogy of the Earth's interior. Minerals not stable at the surface could possibly be found within inclusions of superdeep diamonds-- implying they were stable where these diamond crystallized. Because  of the hardness of diamonds, the high pressure environment is retained even after transporting to the surface. So far, these superdeep minerals brought by diamonds include ringwoodite, ice-VII, cubic δ-N and Ca-perovskite.