Experimental calibration of the reduced partition function ratios of tetrahedrally coordinated silicon from the Debye–Waller factors

Abstract

We present a new force constants approach that combines experimental and theoretical data to constrain the reduced partition function ratio ($\beta$-factor) of tetrahedrally coordinated silicon ($^{IV}$Si) in the crust and upper mantle minerals. Our approach extends the experiment-based general moment approach, which relies on nuclear resonant scattering and is only applicable to Mössbauer-active elements, to Mössbauer-inactive elements such as Si. We determine the resilience of $^{IV}$Si from the Debye–Waller factor, which is derived from the temperature dependence of single crystal X-ray diffraction data, and calculate the stiffness of $^{IV}$Si from the density-functional theory results. The relationship between the resilience the stiffness is calibrated, and we have used an experimentally measurable parameter, the effective coordination number of the $\hbox {SiO}_4$ tetrahedron, to correct the stiffness. The correction is most pronounced for pyroxenes ($\sim 2\%$). The corrected stiffness is used to calculate the equilibrium isotope fractionation $\beta$-factor of each mineral, and the $\alpha$-factors is calculated by taking the ratio of $\beta$-factors of different minerals. We calculate the ln$\alpha _{Si30/28}$ between minerals that contains $\hbox {SiO}_4$ tetrahedra, and our results are consistent with DFT calculations and mass spectrometry results. Our results suggest that the Si isotopic equilibrium temperature between cristobalite and pyroxene in lunar basalt was underestimated by $\sim$250 $^\circ$C, and the pyroxene sample in IL-14 marble is in equilibrium with $\beta$-quartz.