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\[{\Delta _i} = \left( {\frac{{{R_i}}}{{R_i^*}} - 1} \right) \times 1000\]     where R_{i} is the measured ratio of isotopologue i to the non-isotopically substituted isotopologue and 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 Δ values for ^{13}C^{18}O^{17}O (i = 48) and ^{13}C^{18}O^{18}O (i = 49).  The ratios Ri R_{i}  and Ri* R_{i}^{*}  are determined from the measured δi, δ13C δ^{i}, δ^{13}C  and δ18O δ^{18}O  values of the sample CO2. CO_{2}.  For R47: R^{47}:  \[{R_{47}} = \left( {\frac{{\delta _{sam - wrg}^{47}}}{{1000}} - 1} \right) \times R_{47}^{wrg}\]  (2)  where R47wrg R^{47}_{wrg}  is the 47/44 ratio of the working reference gas (wrg) and is determined as:  (3)  Note that implicit in this treatment is an assumption that the mass spectrometer working reference gas has a stochastic distribution of isotopes. It is self evident that this is incorrect since the working reference gas has been equilibrated with water at the laboratory temperature. However, since the Δ values are <1‰ we can make this assumption and carry out a later linear transformation of the data to take account of the actual reference gas R47 value without introducing any significant errors.  The ratios R13, R17 and R18 are determined from the δ13CVPDB, δ17OVSMOW and δ18OVSMOW values of the working reference gas: