Can we not use the &#916; value to measure a triple isotope system?
Authors: Bao Huiming , Cao Xiaobin 
Published: March 2020

&lt;p&gt;&lt;strong&gt;Can we not use the &lt;/strong&gt;&lt;strong&gt;&amp;#916;&lt;/strong&gt;&lt;strong&gt; value to measure a triple isotope system?&lt;/strong&gt;&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;&lt;p&gt;Huiming Bao&lt;sup&gt;1 ,2, 3&lt;/sup&gt; and Xiaobin Cao&lt;sup&gt;1 ,2&lt;/sup&gt;&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;&lt;p&gt;&lt;sup&gt;1&lt;/sup&gt; International Center for Isotope Effects Research, Nanjing University, Nanjing 210023, P. R. China&lt;/p&gt;&lt;p&gt;&lt;sup&gt;2&lt;/sup&gt; School of Earth Sciences and Engineering, Nanjing University, Nanjing 210023, PR China&lt;/p&gt;&lt;p&gt;&lt;sup&gt;3 &lt;/sup&gt;Department of Geology and Geophysics, Louisiana State University, E235 Howe Russell Kniffen, Baton Rouge, LA 70803&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;&lt;p&gt;In a triple isotope system, taking oxygen for example, the deviation of the &amp;#948;&lt;sup&gt;17&lt;/sup&gt;O (or &amp;#948;&amp;#8217;) from a defined &amp;#948;&lt;sup&gt;17&lt;/sup&gt;O-&amp;#948;&lt;sup&gt;18&lt;/sup&gt;O relationship is measured by the term &amp;#916;, defined as the value of &amp;#948;&lt;sup&gt;17&lt;/sup&gt;O - C&amp;#215;&amp;#948;&lt;sup&gt;18&lt;/sup&gt;O, in which &amp;#8220;C&amp;#8221; is a reference slope number. The use of &amp;#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 &amp;#916; values and when we must compare &amp;#916; values among different compounds. A second, more serious problem is the lack of appreciation that a &amp;#916; value scales with its corresponding &amp;#948;&lt;sup&gt;18&lt;/sup&gt;O value. That means even for the same process we may get different &amp;#916; values depending on the magnitude of fractionation and/or laboratory references used.&lt;/p&gt;&lt;p&gt;A pair of radial-angular parameters in a polar coordinate system or a pair of &amp;#948;&lt;sup&gt;18&lt;/sup&gt;O and &amp;#948;&lt;sup&gt;17&lt;/sup&gt;O 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 &amp;#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 &amp;#966; of a triple isotope composition stays the same for the same fractionation process regardless of its radial coordinate r which is determined by &amp;#948;&lt;sup&gt;18&lt;/sup&gt;O and &amp;#948;&lt;sup&gt;17&lt;/sup&gt;O values. Thus, the use of a polar coordinate (r, &amp;#966;) to describe a triple isotope composition in 2D space would avoid the &amp;#948;&lt;sup&gt;18&lt;/sup&gt;O scaling issue for &amp;#916; values of the same process. Unfortunately, polar coordinate does not offer straightforward representation of process-specific &amp;#948;s or fractionation factors. Using just a pair of &amp;#948;&lt;sup&gt;18&lt;/sup&gt;O and &amp;#948;&lt;sup&gt;17&lt;/sup&gt;O values to describe a triple isotope system also eliminates additional symbols. Unfortunately, the direct use of the &amp;#948;&lt;sup&gt;18&lt;/sup&gt;O and &amp;#948;&lt;sup&gt;17&lt;/sup&gt;O 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 &amp;#916; value in case when the analytical yield is not 100%.&lt;/p&gt;&lt;p&gt;The limitations of a polar coordinate system or a pair of &amp;#948;&lt;sup&gt;18&lt;/sup&gt;O and &amp;#948;&lt;sup&gt;17&lt;/sup&gt;O in Cartesian space outweigh their advantages and we are left with no better alternative than the use of &amp;#916;. Therefore, when reporting small &amp;#916; values, we must report their corresponding &amp;#948;&lt;sup&gt;18&lt;/sup&gt;O values as well to avoid scaling bias when dealing with small &amp;#916; values.&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;