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

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

 

Huiming Bao^{1 ,2, 3} and Xiaobin Cao^{1 ,2}

 

¹ International Center for Isotope Effects Research, Nanjing University, Nanjing 210023, P. R. China

² School of Earth Sciences and Engineering, Nanjing University, Nanjing 210023, PR China

³ Department of Geology and Geophysics, Louisiana State University, E235 Howe Russell Kniffen, Baton Rouge, LA 70803

 

In a triple isotope system, taking oxygen for example, the deviation of the δ¹⁷O (or δ’) from a defined δ¹⁷O-δ¹⁸O relationship is measured by the term Δ, defined as the value of δ¹⁷O - C×δ¹⁸O, in which “C” is a reference slope number. The use of Δ 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 Δ values and when we must compare Δ values among different compounds. A second, more serious problem is the lack of appreciation that a Δ value scales with its corresponding δ¹⁸O value. That means even for the same process we may get different Δ values depending on the magnitude of fractionation and/or laboratory references used.

A pair of radial-angular parameters in a polar coordinate system or a pair of δ¹⁸O and δ¹⁷O 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 Δ. 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 φ of a triple isotope composition stays the same for the same fractionation process regardless of its radial coordinate r which is determined by δ¹⁸O and δ¹⁷O values. Thus, the use of a polar coordinate (r, φ) to describe a triple isotope composition in 2D space would avoid the δ¹⁸O scaling issue for Δ values of the same process. Unfortunately, polar coordinate does not offer straightforward representation of process-specific δs or fractionation factors. Using just a pair of δ¹⁸O and δ¹⁷O values to describe a triple isotope system also eliminates additional symbols. Unfortunately, the direct use of the δ¹⁸O and δ¹⁷O 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 Δ value in case when the analytical yield is not 100%.

The limitations of a polar coordinate system or a pair of δ¹⁸O and δ¹⁷O in Cartesian space outweigh their advantages and we are left with no better alternative than the use of Δ. Therefore, when reporting small Δ values, we must report their corresponding δ¹⁸O values as well to avoid scaling bias when dealing with small Δ values.