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New mathematical models with associated equations for isotope dilution mass spectrometry (IDMS)

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Abstract

Vector models which progressively lead to a general model for isotope dilution mass spectrometry (IDMS) are presented for the case of two ‘monitor isotopes’ and one blend involved. They enable one to find the boundary conditions for performing IDMS, and cover the cases of highly enriched isotopes, radioactive isotopes and ratios that are given with different denominator. The models identify the key measurements in their simplest form as well as the conditions which minimise the measurement effort and in some cases the propagated measurement uncertainties. The equations are discussed and compared with other published IDMS equations. Combined with discussion on fundamental aspects of IDMS, this results in an even more ‘general’ but also more complex IDMS equation.

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Notes

  1. The word isotope is synonymous for nuclide here.

  2. This is the quantity which is classically understood to be measured in IDMS. From further considerations it will turn out that in IDMS, simply amounts of substance of isotopes in different materials are compared. Obtaining an element amount of substance ratio is a consequence of dividing the isotope ratio by the corresponding isotope abundances. A mixed amount of substance ratio containing an amount of substance of an element and an amount of substance of an isotope, as in the case of for example n(E,X)/n(2E,Y), where an amount of substance of the element E in the sample material X is divided by an amount of substance of one isotope 2E in the spike Y, is also possible. However, for comparison with existing equations, the quantity element amount of substance ratio is also used here.

  3. Although in principle the probability for the formation of ions is equal for isotopes (K ion(1E)≅K ion(2E)) in very good approximation (10−6 rel. or better [15]), one cannot stress enough that in experimental practice constant and strictly proportional calibration factors do not exist. The individual calibration factors are dependent on a variety of parameters with individually linear and non-linear relationships. Typical parameters influencing the K-factors are the measurement technique, the size of the isotope amount ratio concerned, bias, dead time correction, background and mass fractionation effects at various stages of the measurement process. Establishing the correct K-factors or even their ratio is crucial for the measurement process and usually difficult to achieve [16, 17, 18].

  4. Without explicit proof given here, the requirements for a vector space (i.e. closed group, existence of neutral and inverse elements, commutativity, associativity, distributivity) are fulfilled, and the theory of linear algebra applies. Note that the inverse elements have only a limited physical reality (although material can be removed).

  5. For the case of materials as discussed here, orthogonality means that different materials are concerned (e.g. copper being different from iron and 7Li being different from 6Li).

  6. We will use the in natural materials most abundant isotope in the denominator and give it the index 1, and later generalise this to the most abundant isotope in the material concerned.

  7. Not including radioactive isotopes, the vector spaces used to describe IDMS for stable elements range from dimension 2 (e.g. Li) to dimension 10 for tin.

  8. A generic set up of essential requirements (i.e. finding the necessary and sufficient requirements) is usually difficult for all types of problems. An indication for succeeding in this task is that the outcoming concept has successful applicability for the purpose, is free of logical conflicts and does not carry any unnecessary built-in limitations.

  9. See footnote 2.

  10. For IDMS considerations, it is not important whether the ions are positively, negatively, singly or multiply charged.

  11. See footnote 3.

  12. For elements, the molar mass (in kg mol−1) is numerically equal to the atomic weight A r, which is dimensionless.

  13. See footnote 3.

References

  1. De Bièvre P (1990) Fresenius J Anal Chem 337:766–771

    Google Scholar 

  2. De Bièvre P (1993) Anal Proc 30:328–333

    Google Scholar 

  3. Hintenberger H (1956) In: Smith ML (ed) Electromagnetically enriched isotopes and mass spectrometry. Butterworths, London, pp 177–189

  4. De Bièvre P, Debus GH (1965) Nucl Inst Meth 32:224–228

    Article  Google Scholar 

  5. Heumann KG (1988) In: Adams F, Gijbels R, van Grieken R (eds) Inorganic mass spectrometry. Wiley, New York, pp 301–377

  6. De Bièvre P (1994) In: Herber RFM, Stoeppler M (eds) Techniques and instrumentation in analytical chemistry, Vol 15, Trace element analysis in biological specimens. Elsevier, Amsterdam, pp 169–183

  7. Fasset JD (1989) Anal Chem 61: 643a-549a

    CAS  Google Scholar 

  8. Peiser HS, De Bièvre P (1997) Fresenius J Anal Chem 359:523–525

    Article  Google Scholar 

  9. Henrion A (1994) Fresenius J Anal Chem 350:657–658

    Google Scholar 

  10. Milton MJT, Wielgosz RI (2000) Metrologia 37:199–206

    Article  Google Scholar 

  11. Hamelin B, Manhes G, Albarede F, Allègre CJ (1985) Geochim Cosmochim Acta 49:173–182

    Article  CAS  Google Scholar 

  12. Galer SJG (1999) Chem Geol 157:255–274

    CAS  Google Scholar 

  13. Long LE (1966) Earth Planet Sci Lett 1:289–292

    Article  CAS  Google Scholar 

  14. Hofmann A (1971) Earth Planet Sci Lett 10:397–402

    Article  CAS  Google Scholar 

  15. De Bièvre P (1996) Private communication

  16. Prohaska T, Quétel CR, Hennessy C, Liesegang D, Papadakis I, Taylor PDP, Latkoczy C, Hann S, Stingeder G (2000) J Environ Monit 2:613–620

    Article  CAS  PubMed  Google Scholar 

  17. Diemer J, Quétel CR, Taylor PDP (2002) J Anal At Spectrom 17:1137–1142

    Article  CAS  Google Scholar 

  18. Kipphardt H, Valkiers S, Taylor PDP, De Bièvre P (2000) Int J Mass Spectrom 198:71-76

    Article  CAS  Google Scholar 

  19. Monna F, Loizeau JL, Thomas BA, Gueguen C, Favarger PY (1998) Spectrochim Acta 53B:1317–1333

    CAS  Google Scholar 

  20. Kipphardt H (1995) Dissertation, University of Dortmund

  21. Audi G, Wapstra AH (1993) Nucl Phys A 565:1–65

    Article  CAS  Google Scholar 

  22. Rosman KJR, Taylor PDP (1998) Pure Appl Chem 70:217–235

    CAS  Google Scholar 

  23. De Bièvre P (1999) Private communication

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Acknowledgements

The authors are indebted to the EC for granting a Training and Mobility of Researchers (TMR) fellowship to H. Kipphardt, enabling him to work at IRMM. The scientific discussions with several staff members and visiting scientists at IRMM and with colleagues from JEPPIM (Joint European Programme for Primary Isotopic Measurements), a programme co-ordinated by IRMM, are gratefully acknowledged. The authors would like to acknowledge the careful and constructive comments from the referees.

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Authors and Affiliations

  1. Bundesanstalt für Materialforschung und -prüfung (BAM), 12200, Berlin, Germany

    Heinrich Kipphardt

  2. Duineneind 9, 2460, Kasterlee, Belgium

    Paul De Bièvre

  3. European Commission, Joint Research Centre, Institute for Reference Materials and Measurements (IRMM), 2440, Geel, Belgium

    Philip D. P. Taylor

Authors
  1. Heinrich Kipphardt
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  2. Paul De Bièvre
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  3. Philip D. P. Taylor
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Correspondence to Heinrich Kipphardt.

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Kipphardt, H., De Bièvre, P. & Taylor, P.D.P. New mathematical models with associated equations for isotope dilution mass spectrometry (IDMS). Anal Bioanal Chem 378, 330–341 (2004). https://doi.org/10.1007/s00216-003-2232-3

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  • DOI: https://doi.org/10.1007/s00216-003-2232-3

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