Short communication: On the potential use of materials with heterogeneously distributed parent and daughter isotopes as primary standards for nonUPb geochronological applications of laser ablation inductively coupled mass spectrometry (LAICPMS)
 School of Environment, Earth and Ecosystem Sciences, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK
 School of Environment, Earth and Ecosystem Sciences, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK
Abstract. Many new geochronological applications of laser ablation inductively coupled mass spectrometry (LAICPMS) have been proposed in recent years. One of the problems associated with this rapid growth is the lack of chemically and isotopically homogeneous matrixmatched primary standards to control elemental fractionation during LAICPMS analysis. In UPb geochronological applications of LAICPMS this problem is often addressed by utilising matrixmatched primary standards with variable chemical and isotopic compositions. Here I derive a set of equations to adopt this approach for nonUPb geochronological applications of LAICPMS.
Daniil Popov
Status: final response (author comments only)

RC1: 'Comment on gchron202137', Anonymous Referee #1, 07 Jan 2022
This short communication is a techniquebased manuscript, useful for those performing LAICPMS dating for systems other than UPb—that is, those with only one parent/daughter —that also have variable parent and daughter concentrations. It includes a standardization technique for correcting raw parent/daughter ratios, subject to elemental fractionation by laser ablation, transport, ionization efficiency, etc.. The general idea, as follows, is no different than correction of LAICPMS UPb data, which has been explored by many of the authors referenced within: 1) correct for mass bias of the daughter ratio (can be done a number of ways, including the use of a nonmatrixed matched RM (reference material), via solution, or internal standardization of a nonUPb system) and correct all RMs and unknowns accordingly; 2) assume concordance for the RM and correct the parent/daughter ratio, such that the age matches it's accepted value. This is a relatively straightforward correction that has been explained many times over, primarily for UPb. As such, this communication seems a touch superfluous, as a single isotopic geochronometer is simpler than the UPb system, but nevertheless is rarely mentioned and therefore warrants more discussion, especially in the light of recent developments in LAICP dating techniques (e.g., Zack and Hogmalm, 2016 and Simpson et al., 2021).
In my experience, the best example of standardization of elemental fractionation of commondaughterbearing minerals is that in Chew et al., 2014, and I shall thus refer to it often below; though the Chew et al. study discusses the UPb system, it does so on a systembysystem basis, that is, it corrects 206Pb/238U and 207Pb/235U ratios using any of the other isotopes of the daughter product of the system (i.e., 204¬Pb, 207¬Pb, 208Pb for 206Pb/238U and 204¬Pb, 206¬Pb, 208Pb for 207Pb/235U). As an example, one can look at Fig. 2E, in which each parent/daughter ratio has been corrected using a nonradiogenic daughter (204¬Pb); the math by which to do this should be identical to the math by which to correct any spot analysis for any radioisotopic system  that is, it is should be identical to Equation 21 in this manuscript. Nevertheless, it is not spelled out in this paper at least, that the calculation for UPb applies the same way for other isotopic systems such as RbSr, SmNd, LuHf etc., which is presumably why the author has endeavored to write this short communication.
What the Chew et al. study doesn't explain as well is how to correct the mass bias for the ratio of the daughter isotopes (e.g., 207Pb/206¬Pb, 207Pb/204¬Pb, 87Sr/86¬Sr, etc.). Unfortunately, that is also mostly missing from this manuscript, which should be revised to state how this can/should be done in a clear and concise manner; for nonUPb LAICPMS geochronology—Rb/Sr, Sm/Nd, Lu/Hf—the mass fractionation (Yaxis value) can be calculated internally, unlike for UPb, which has no two nonradiogenic isotopes (however this internal standardization is rarely done  this needs discussion). The analytical uncertainty in this correction is likely to be in the 10's low 100's of ppm (<<1%) and for intents and purposes, can be considered negligible when calculating age uncertainties, however, the actual uncertainty of the measurement—because of interferences and matrix effects, for example—is likely to be much larger.
On this note, these excess uncertainties are not included in the equations herein, as far as I can tell, and in many cases, these types of uncertainties are likely to be the biggest cause of the actual uncertainty of the measurement. One of the seminal papers in uncertainty propagation for LAICPMS dating is that of Horstwood et al., 2016, in which they explain how the reproducibility of measurements can easily overwhelm the instrument analytical uncertainty. In that paper, without equations, they give their best practices for data reduction workflow, which include propagating excess uncertainty (different than external uncertainty). This is a critical step in reporting ages and uncertainties in all LAICPMS derived data and cannot be ignored in the current manuscript.
The main aspect of this paper that is relevant, and has not been discussed in great detail, is the correction of parent/daughter ratios and consequent age calculation using a standard isochron method, that is, a graph in which both axes have a nonradiogenic, nonradioactive daughter isotope as the denominator (or numerator on the Yaxis in an inverse diagram; this is opposed to a TeraWasserburg diagram, for example, which uses radiogenic daughters on both axes). Again, the correction of the ratios for each axis (ratio) of this diagram have been described in numerous publications (primarily for UPb, but see Zack and Hogmalm, 2016 and Simpson et al., 2021, and furthermore there is no difference in the correction method between that and nonUPb geochronometers), but few 1) demonstrate visually the uncorrected vs. corrected data, or 2) give the equations for uncertainties for each parameter. Point 1) is easy enough to do on one's own to get a visual representation of the 2step correction for each ratio, and is analogous to the correction of UPb on a TW diagram as shown in Chew et al., 2014, Fig. A1. As noted above, this figure is missing the daughterratio correction, and would be more appropriate shown below, but this time in a singlesystem isochron diagram (analogous to Fig 1b in the submitted manuscript):Note that the figures in the current manuscript are either misleading or wrong. Given that there is little discussion about the correction of the yaxis, my impression is that it is the latter; the plots do not accurately represent theoretical data, as data of the same age, whether real or synthetic, should be isochronous, whether corrected for elemental fractionation or not. Given that the math for generating such apparent and corrected isochrons is trivial, it is worrisome that the plots in Figure 1 are incorrectly represented.
In conclusion, for this manuscript to merit publication, it must first contain a broader background of previous work, and a better description of the workflow to correcting measured ratios, both for elemental fractionation (including differences fractionation downhole which is completely missing). Second, it needs a better description of all possible sources of uncertainty and how and when they should be properly propagated. Third, any figure must accurately represent realworld data.
 AC2: 'Reply on RC1', Daniil Popov, 10 Feb 2022

RC2: 'Comment on gchron202137', Pieter Vermeesch, 07 Jan 2022

AC1: 'Reply on RC2', Daniil Popov, 20 Jan 2022
Pieter Vermeesch noted that I got stuck with finding rigorous formulas for a in Equation 14 and suggested to use an alternative approach to error propagation, which is the maximum likelihood method. However, I have now derived appropriate formulas for a, which can be found in the attached file (not sure why I did not see the solution before). Therefore, there is no need to use the maximum likelihood method to estimate k and its uncertainty. This is particularly useful for those who use Excel spreadsheets to reduce mass spectrometry data, since using the maximum likelihood method would require some numerical optimisation, which would be cumbersome to implement in Excel.
Additionally, I spotted an error in Equation 12. The revised equation is provided in the attached file.
It may be important to highlight that the factors k in my manuscript and Pieter Vermeesch’s review are the inverse of each other.

AC1: 'Reply on RC2', Daniil Popov, 20 Jan 2022
Daniil Popov
Daniil Popov
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