First-principles investigation of the concentration effect on equilibrium fractionation of Ca isotopes in forsterite

Previous theoretical studies have found that the concentration variations within a certain range have a prominent effect on inter-mineral equilibrium isotope fractionation(10~3lna).Based on the density functional theory,we investigated how the average Ca–O bond length and the reduced partition function ratios(10~3lnb)and10~3lna ofCa/Ca in forsterite(Fo)are affected by its Ca concentration.Our results show that Ca–O bond length in forsterite ranges from 2.327 to 2.267 A?with the Ca/(Ca+Mg)varying between a narrow range limited by an upper limit of 1/8 and a lower limit of 1/64.However,outside this narrow range,i.e.,Ca/(Ca+Mg)is lower than1/64 or higher than 1/8,Ca–O bond length becomes insensitive to Ca concentration and maintains to be a constant.Because the 10~3lnb is negatively correlated with Ca–O bond length,the 10~3lnb significantly increases with decreasing Ca/(Ca+Mg)when 1/64\Ca/(Ca+Mg)\2/16.As a consequence,the 10~3lna between forsterite and other minerals also strongly depend on the Ca content in forsterite.Combining previous studies with our results,the heavier Ca isotopes enrichment sequence in minerals is:forsterite[orthopyroxene[clinopyroxene[calcite&diopside[dolomite[aragonite.Olivineand pyroxenes are enriched in heavier Ca isotope compared to carbonates.The 10~3lna between forsterite with a Ca/(Ca+Mg)of 1/64 and clinopyroxene(Ca/Mg=1/1,i.e.,diopside)is up to～0.64%at 1200 K.The large 10~3lnarelative to the current analytical precision for Ca isotope measurements suggests that the dependence of10~3lnaon temperature can be used as a thermometer,similar to the one based on the 10~3lna ofCa/Ca between orthopyroxene and diopside.These two Ca isotope thermometers both have a precision approximate to that of elemental thermometers and provide independent constraints on temperature.