Theory of ion cyclotron resonance mass spectrometry: resonant excitation and radial ejection in orthorhombic and cylindrical ion traps

Abstract

Most prior treatments of ion cyclotron resonance (ICR) radiofrequency excitation have been based on a uniform electric field (infinitely extended electrode) approximation. In this paper, we develop analytical representations (based on Taylor series or Fourier series expansions) for the radiofrequency excitation potential in finite orthorhombic traps of square cross-section. The rate of increase of ICR orbital radius in response to single-frequency on-resonance excitation was then integrated for these and other prior models to obtain the onset of radial ejection. Both present methods (and prior related methods by Rempel [Int. J. Mass Spectrom. Ion Processes, 70 (1986) 163] and van der Hart and van de Guchte [Int. J. Mass Spectrom. Ion Processes, 82 (1988) 17]) are nearly indistinguishable for ion cyclotron orbital radius out to about 40% of the maximum radius allowed by the trap, and diverge thereafter. The Fourier series method is highly accurate all the way out to the trap boundary for ions in the midplane of the trap; because the transverse component of the excitation field decreases away from that midplane, the Fourier series method represents an upper limit to the actual excited ICR orbital radius, and should thus accurately predict the onset of radial ejection. We find that the experimentally observed onset of ion ejection occurs at about 1.3 times the excitation voltage—duration product calculated from the uniform-field model, in excellent agreement with the Fourier series prediction. Moreover, since the experimentally observed onset of ejection is found to be essentially independent of trapping voltage and ion mass-to-charge ratio, m/z, the ejection onset can be interpreted as radial (rather than axial) ejection. The present theories are developed in full algebraic detail for a cubic ion trap. Extension of either method to orthorhombic traps of arbitrary relative dimensions is straightforward; explicit solutions are given for the Taylor series model for orthorhombic traps of square cross-section. In addition, the Fourier series method is extendable to right circular cylindrical traps of arbitrary aspect ratio; all relevant analytical expressions are listed in the text. Finally, for a given excitation voltage × duration product, the present results imply that the excitation period should exceed several ICR orbital periods in order to minimize magnetron radius; this constraint is especially important for ions of high m/z.