An analytical approximation for the field in toroidal ion trap mass analysers having circular slits on endcap electrodes

Abstract

In this study, an analytical approximation that can be used to estimate the change in the electric field caused by the introduction of circular slits on endcap electrodes of toroidal ion trap mass analysers will be developed. A theory has been developed to estimate the contribution of circular slits on the electrodes of a toroidal ion trap. In this theory, the first potential due to an arbitrary aperture on an infinite ground plane will be derived. This will be specialised to obtain potential due to a circular slit on an infinite ground plane. The potential due to a circular slit on an infinite ground plane will be used to evaluate the field within the toroidal traps having circular slits. The approximation developed in this paper has been verified on two arbitrarily selected toroidal traps. One of these arbitrary traps has circular-shaped electrodes, and the other trap has asymmetric hyperbolic-shaped electrodes. Fields estimated by this approximation agree well with those obtained using the boundary element method. Additionally, the study reveals an interesting correlation between the secular frequency of ion motion and the width of the circular slit. The secular frequency reduces as the slit width is increased. The reduction in secular frequency is proportional to the square of the slit width.

Keywords

Toroidal ion trap secular frequency potential due to circular slit potential due to linear slit boundary element method (BEM)

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References

March

. Quadrupole ion traps. Mass Spectrom Rev 2009; 28: 961–989.

Cooks

Ouyang

. Geometry optimization for the cylindrical ion trap: field calculations, simulations and experiments. Int J Mass Spectrom 2005; 241: 119–132.

Lammert

Plass

Thompson

et al. Design, optimization and initial performance of a toroidal rf ion trap mass spectrometer. Int J Mass Spectrom 2001; 212: 25–40.

Koizumi

Whitten

Reilly

PTA

et al. The effect of endcap electrode holes on the resonant ejection from an ion trap. Int J Mass Spectrom 2009; 281: 108–114.

Plass

Cooks

. Theory, simulation and measurement of chemical mass shifts in rf quadrupole ion traps. Int J Mass Spectrom 2003; 228: 237–267.

Ivanov

Sysoev

Konenkov

. Mass-selective instability and resonance ejection modes for dit with rectangular asymmetric wave shape. Eur J Mass Spectr 2024; 30: 143–149.

Konenkov

. Dipole and quadrupole resonance excitation in linear quadrupoles. Eur J Mass Spectrom 2024; 30: 3–37.

Janulyte

Zerega

Carette

. Harmful influences of confinement field non-linearities in mass identifying for a fourier transform quadrupole ion trap mass spectrometer: Simulation studies. Int J Mass Spectrom 2007; 263: 243–259.

Cheng

Liu

et al. Concept and simulation of a novel dual-layer linear ion trap mass analyzer for micro-electromechanical systems mass spectrometry. Eur J Mass Spectrom 2024; 30: 150–160.

10.

Badman

Johnson

Plass

et al. A miniature cylindrical quadrupole ion trap: simulation and experiment. Anal Chem 1998; 70: 4896–4901.

11.

Chattopadhyay

Verma

Mohanty

. Composite field approximations for ion traps with apertures on electrodes. Int J Mass Spectrom 2009; 282: 112–122.

12.

Chattopadhyay

Mohanty

. Off-axis field approximations for ion traps with apertures. Int J Mass Spectrom 2009; 288: 58–67.

13.

Bier

Syka

JEP

. Ion trap mass spectrometer system and method. U.S. Patent No. 5420425 A, 1995.

14.

Ouyang

Song

et al. Rectilinear ion trap: concepts, calculations, and analytical performance of a new mass analyser. Int J Mass Spectrom 2004; 76: 4595–4605.

15.

Paul

Steinwedel

. Apparatus for separating charged particles of different specific charges. U.S. Patent No. 2939952, 1960.

16.

Jackson

. Classical electrodynamics. New Jersey, USA: John Wiley & Sons Inc., 1991.

17.

Smythe

. Static and dynamic electricityInternational series in pure and applied physicsOhio, USA: McGraw-Hill, 1950.

18.

Tallapragada

Mohanty

Chatterjee

et al. Geometry optimization of axially symmetric ion traps. Int J Mass Spectrom 2007; 264: 38–52.

19.

Kotana

Mohanty

. Computation of Mathieu stability plot for an arbitrary toroidal ion trap mass analyser. Int J Mass Spectrom 2017; 414: 13–22.

20.

Kotana

Mohanty

. Determination of multipole coefficients in toroidal ion trap mass analysers. Int J Mass Spectrom 2016; 408: 62–76.

21.

Weber

. Electromagnetic fields: Theory and applications. Vol. 1 – Mapping of fields. New York: John Wiley & Sons, 1950.

22.

Abramowitz

Stegun

. Handbook of mathematical functions. New York: Dover Publications Inc., 1970.

23.

Anderson

Bai

Bischof

et al. LAPACK users’ guide, 3rd ed. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1999, https://www.netlib.org/lapack/ .

24.

Atkinson

. An introduction to numerical analysis. 2nd ed. New York: John Wiley & Sons, 1989.

25.

Strang

. Linear agebra and its applications, 2nd ed. New York: Academic Press Inc., 1980.

26.

Folland

. Introduction to partial differential equations. Princeton, NJ: Princeton University Press, 2020.

27.

Muskhelishvili

. Singular integral equations. Dordrecht: Springer, 1958.

28.

Edwards

. A treatise on the integral calculus. London: Macmillan, 1930.

29.

Dwight

. Tables of integrals and other mathematical data. London: Macmillan, 1957.

30.

Kreyszig

. Advanced engineering mathematics, 8th ed. New York: John Wiley & Sons, 1999.

31.

http://ab-initio.mit.edu/wiki/index.php/Harminv (Updated in 2023, accessed 16 October 2025).

32.

Wall

Neuhauser

. Extraction, through filter-diagonalization, of general quantum eigenvalues or classical normal mode frequencies from a small number of residues or a short-time segment of a signal. I. Theory and application to a quantum-dynamics model. J Chem Phys 1995; 102: 8011–8022.

33.

Mandelshtam

Taylor

. Harmonic inversion of time signals and its applications. J Chem Phys 1997; 107: 6756–6769.