Isotopic Constraints on Earth System Processes. Группа авторов

      Applying the exponential function to both sides of this equation it becomes , which dividing by and rearranging gives . Writing the isotope ratios as as R_{i, j} and as R_o one arrives at the Rayleigh fractionation equation for the evaporation residue

(1.15)

where . This Rayleigh equation states that the isotopic ratio of a continuously homogenized condensed phase with initial ratio R_o becomes R_{i, j}when a fraction f of the initial amount of isotope j remains in the residue (or when a fraction 1–f had evaporated). In most practical application one can replace by the fraction of the parent element remaining in the residue. Equation 1.15 shows that given an experimentally determined value of α _ij and the degree of isotopic fractionation of an evaporation residue one can calculate how much of the parent element evaporated (i.e., 1–f).

      1.6.3. High‐Temperature Vacuum Evaporation Experiments

      The experiments involving the evaporation of CAI‐like liquids described in this section were run at the University of Chicago in a high‐temperature vacuum furnace (pressure < 10^–6 Torr) that was designed and constructed by Akihiko Hashimoto (see Hashimoto, 1990, for a description of the furnace). The experimental methods for evaporating molten samples in this furnace are described in the first paper documenting high‐temperature isotopic fractionations of evaporation residues from a silicate liquid (molten fayalite) by Davis et al. (1990). For CAI evaporation experiments powders of CAI‐like CMAS composition (CaO+MgO+SiO₂+Al₂O₃) were loaded onto small iridium wire loops (1–6 mm in diameter) that were then placed for different lengths of time in the hot spot of the vacuum furnace at temperatures between 1600°C and 1900°C and pressure less than 10^–6 Torr. The objective of the evaporation experiments was to determine the evaporation coefficients γ _Mg and γ _Si for calculating the fluxes using equation 1.10 and the isotopic fractionation coefficients α _Mg and α _Si for calculating the isotopic fractionation of evaporation residues using equation 1.15. When calculating the bulk composition of evaporation residues, one assumes CaO and Al₂O₃ are effectively conserved, being much too refractory to significantly evaporate while there is still any MgO left in the melt.

The evaporation coefficients γ _Mg and γ _Si are determined by the amount of initial magnesium and silicon lost by evaporation from a molten bead of a given surface area as a function of temperature and run duration. The evaporation coefficients of magnesium and silicon are very similar to each other and decrease with temperature from about 0.2 at 1900°C to about 0.05 at 1500°C (see Fig. 3 in Richter et al., 2007). The kinetic isotope fractionation factors α _Mg and α _Si are determined by fitting the isotopic fractionation of evaporation residues as a function of the fraction of the initial amount of the parent element remaining in the residue with calculations using the Rayleigh fractionation equation (equation 1.15). Fig. 1.17 shows the isotopic fractionation of magnesium and silicon of evaporation residues of a CAI‐like melt from vacuum evaporation experiments run for different lengths of time by Richter et al. (2007) and Knight (2009). The curved profiles are fits to the data calculated assuming perfect Rayleigh fractionation (equation 1.15) with α _Mg = 0.9855 for ²⁵Mg/²⁴Mg and α _Si = 0.0990 for ²⁹Si/³⁰Si. The fits are sufficiently good to confirm not only that the vacuum evaporation was a Rayleigh process but also that α _Mg and are α _Si do not significantly change as the composition of the evaporating material evolves by evaporation of its volatile components MgO and SiO₂. The experimentally determined value α _Mg = 0.9855 is significantly larger (i.e., significantly less fractionation) than the inverse square root of the atomic mass of ²⁴Mg/²⁵Mg (0.9797). In the case of silicon, α _Si = 0.9901 is only slightly larger than the inverse square root the mass ratio ²⁹Si¹⁶O/³⁰Si¹⁶O (0.9890). A more detailed analysis of the magnesium isotope fractionation data in Fig. 1.17 showed that when the data from experiments run at different temperatures are fit separately with Rayleigh fractionation curves, the α _Mg increased (i.e., became less fractionating) as the temperature declined from 1900°C to 1600°C (see figure 10 in Richter et al., 2007). This temperature dependence of α _Mg must be extrapolated from the high temperature of the evaporation experiments down to temperatures in the range of 1450°C to 1300°C, at which the Type B CAIs were molten. If one were to use α _Mg as given by the inverse square root law in the Rayleigh fractionation equation to invert the isotopic fractionation of a CAI for the amount of the parent element evaporated, it would result in an estimate of about a factor of two more magnesium evaporated compared to that calculated with α _Mg derived by extrapolating the experimental values down to 1400°C. Using the experimentally determined α _Mg and α _Si is particularly important when using the magnesium isotopic fractionation of a CAI to determine the bulk composition of its pre‐evaporation precursor.

Figure 1.17 The unfilled circles and squares are the isotopic fractionation of magnesium and silicon in vacuum evaporation residues of a CAI‐like