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\subsection{Data handling and calculation of $\Delta$ values}  The clumped isotope Δi Δ_{i}  value of a sample is defined as: \[{\Delta _i} = \left( {\frac{{{R_i}}}{{R_i^*}} - 1} \right) \times 1000\]  (1)  where Ri R_{i}  is the measured ratio of isotopologue i to the non-isotopically substituted isotopologue and Ri* R_{i}^{*}  is the expected ratio assuming a stochastic distribution of all isotopes over all possible sites in the lattice (reference). For CO2 CO_{2}  we are largely concerned with the isotopologue 13C18O16O ^{13}C^{18}O^{16}O  (i = 47) but also determine Δ values for 13C18O17O ^{13}C^{18}O^{17}O  (i = 48) and 13C18O18O ^{13}C^{18}O^{18}O  (i = 49). The ratios Ri and Ri* are determined from the measured δi, δ13C and δ18O values of the sample CO2. For R47:   (2)  where R47wrg is the 47/44 ratio of the working reference gas (wrg) and is determined as: