Can we not use the Δ value to measure a triple isotope system?

<p><strong>Can we not use the </strong><strong>&#916;</strong><strong> value to measure a triple isotope system?</strong></p><p>&#160;</p><p>Huiming Bao^{1 ,2, 3} and Xiaobin Cao^{1 ,2}</p><p>&#160;</p><p>¹ International Center for Isotope Effects Research, Nanjing University, Nanjing 210023, P. R. China</p><p>² School of Earth Sciences and Engineering, Nanjing University, Nanjing 210023, PR China</p><p>³Department of Geology and Geophysics, Louisiana State University, E235 Howe Russell Kniffen, Baton Rouge, LA 70803</p><p>&#160;</p><p>In a triple isotope system, taking oxygen for example, the deviation of the &#948;¹⁷O (or &#948;&#8217;) from a defined &#948;¹⁷O-&#948;¹⁸O relationship is measured by the term &#916;, defined as the value of &#948;¹⁷O - C&#215;&#948;¹⁸O, in which &#8220;C&#8221; is a reference slope number. The use of &#916; has generated two problems. First, there is a spectrum of C values currently being adopted in the community, for reasons of end-member cases (e.g. 0.5305 at high-temperature limit), legacy (0.52), or compound-specificity (e.g. 0.528 for water cycle or 0.524 for silicates). These practices have brought confusions especially when we deal with small &#916; values and when we must compare &#916; values among different compounds. A second, more serious problem is the lack of appreciation that a &#916; value scales with its corresponding &#948;¹⁸O value. That means even for the same process we may get different &#916; values depending on the magnitude of fractionation and/or laboratory references used.</p><p>A pair of radial-angular parameters in a polar coordinate system or a pair of &#948;¹⁸O and &#948;¹⁷O in Cartesian space uniquely describe a triple isotope data point in 2D space. Either of the two ways would thaw any debates on the choice of reference slope value C necessary for calculating the &#916;. In addition, a polar coordinate system is usually preferred when studying behaviors centering around an origin, in this case, isotope composition deviating from a reference point (0, 0). The angular coordinate &#966; of a triple isotope composition stays the same for the same fractionation process regardless of its radial coordinate r which is determined by &#948;¹⁸O and &#948;¹⁷O values. Thus, the use of a polar coordinate (r, &#966;) to describe a triple isotope composition in 2D space would avoid the &#948;¹⁸O scaling issue for &#916; values of the same process. Unfortunately, polar coordinate does not offer straightforward representation of process-specific &#948;s or fractionation factors. Using just a pair of &#948;¹⁸O and &#948;¹⁷O values to describe a triple isotope system also eliminates additional symbols. Unfortunately, the direct use of the &#948;¹⁸O and &#948;¹⁷O presents an apparently larger uncertainty for a data point than the other approaches. And it could not take advantage of the use of an accurate &#916; value in case when the analytical yield is not 100%.</p><p>The limitations of a polar coordinate system or a pair of &#948;¹⁸O and &#948;¹⁷O in Cartesian space outweigh their advantages and we are left with no better alternative than the use of &#916;. Therefore, when reporting small &#916; values, we must report their corresponding &#948;¹⁸O values as well to avoid scaling bias when dealing with small &#916; values.</p><p>&#160;</p>