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\[{\Delta _i} = \left( {\frac{{{R_i}}}{{R_i^*}} - 1} \right) \times 1000\]     where R_{i} $R_{i}$  is the measured ratio of isotopologue i to the non-isotopically substituted isotopologue and R_{i}^{*} $R_{i}^{*}$  is the expected ratio assuming a stochastic distribution of all isotopes over all possible sites in the lattice (reference). For CO_{2} we are largely concerned with the isotopologue ^{13}C^{18}O^{16}O (i = 47) but also determine $\Delta$ values for ^{13}C^{18}O^{17}O (i = 48) and ^{13}C^{18}O^{18}O (i = 49). The ratios $R_{i}$ and $R_{i}^{*}$ are determined from the measured $\delta^{i}$, $\delta^{13}C$ and $\delta^{18}O$ values of the sample CO_{2}. For R^{47}: $R^{47}$:  \[{R_{47}} = \left( {\frac{{\delta _{sam - wrg}^{47}}}{{1000}} - 1} \right) \times R_{47}^{wrg}\]