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Methodological Problems in the Replacement of Discrete Mass Spectrometric Models by Continuum Models

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Two problems of mass spectrometry, namely, involving the model of chemical kinetics of cluster formation and the model of the pseudopotential of stroboscopic samples of coordinates and ion velocities in quadrupole radio frequency fields, are considered as the examples in investigating the relationship between discrete models with finite-difference equations and continuum models with point derivatives. In developing continuum models, integer indices are replaced by real parameters, and finite-difference relations are replaced by approximate differential relations involving derivatives. It is shown that this procedure is not reliable: the discrete model and the continuum model can diverge globally, even if for intuitive reasons it is expected that the models should be close. As a result, the transfer of the conclusions obtained for the approximate continuum model to the exact physical model can lead to conceptually significant errors and requires the investigator to be careful. In particular, the continuum model of the chemical kinetics of cluster formation and the continuum model of the pseudopotential of stroboscopic samples of coordinates and ion velocities in quadrupole radio frequency fields, considered as an example, are both insolvent.

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Notes

  1. Semi-invariants (cumulates) are the statistical characteristics of the distribution of a random variable that are obtained by repeatedly differentiating Eqs. (6) and (12) with respect to the parameter ω at ω = 0, that is, by expanding the logarithm of the characteristic function of a random variable in the Taylor or Maclaurin series at ω = 0 [53–56].

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ACKNOWLEDGMENTS

The work was supported by the Ministry of Education and Science of the Russian Federation, grant no. 3.9506.2017/VSN, and the Russian Foundation for Basic Research, project no. 07-17-00418.

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Authors and Affiliations

  1. Institute for Analytical Instrumentation, Russian Academy of Sciences, 190103, St. Petersburg, Russia

    A. S. Berdnikov

  2. MSC-CG (Mass Spectrometry Consulting), 85000, Bar, Montenegro

    A. N. Verentchikov

  3. Department of Physics and Mathematics, Ryazan State University, 390000, Ryazan, Russia

    N. V. Konenkov

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  1. A. S. Berdnikov
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  2. A. N. Verentchikov
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  3. N. V. Konenkov
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Correspondence to A. S. Berdnikov.

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Berdnikov, A.S., Verentchikov, A.N. & Konenkov, N.V. Methodological Problems in the Replacement of Discrete Mass Spectrometric Models by Continuum Models. J Anal Chem 73, 1229–1241 (2018). https://doi.org/10.1134/S1061934818130026

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