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\subsection{Data handling and calculation of $\Delta$ values}  The clumped isotope Δi value of a sample is defined as:  % MathType!MTEF!2!1!+-  % faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9  % wBH5garmWu51MyVXgaruWqVvNCPvMCaerbuLwBLnharqqtubsr4rNC  % HbGeaGqiVG0de9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq-Jc9vqaq  % pepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-xfr-x  % b9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaaeaqbaaGcbaGaeuiLdq  % 0aaSbaaSqaaiaadMgaaeqaaOGaeyypa0ZaaeWaaeaadaWcaaqaaiaa  % dkfadaWgaaWcbaGaamyAaaqabaaakeaacaWGsbWaa0baaSqaaiaadM  % gaaeaacaGGQaaaaaaakiabgkHiTiaaigdaaiaawIcacaGLPaaacqGH  % xdaTcaaIXaGaaGimaiaaicdacaaIWaaaaa!4395!  \[{\Delta _i} = \left( {\frac{{{R_i}}}{{R_i^*}} - 1} \right) \times 1000\]   (1)  where Ri is the measured ratio of isotopologue i to the non-isotopically substituted isotopologue and Ri* is the expected ratio assuming a stochastic distribution of all isotopes over all possible sites in the lattice (reference). For CO2 we are largely concerned with the isotopologue 13C18O16O (i = 47) but also determine Δ values for 13C18O17O (i = 48) and 13C18O18O (i = 49).  The ratios Ri and Ri* are determined from the measured δi, δ13C and δ18O values of the sample CO2. For R47: