GENERAL-THEORY OF EXCITATION IN ION-CYCLOTRON RESONANCE MASS-SPECTROMETRY

The vector position of an ion as a function of time before, during, and after radio-frequency (rf) electric field excitation by an arbitrary time-domain waveform is derived, assuming that the magnetic field is uniform and keeping, in the differential equations of motion, terms through first order in the Taylor series expansion of the electric fields. The latter approximation yields a quadrupolar electrostatic potential and a spatially uniform rf excitation field whose magnitude is determined by the trap geometry and dimensions of the trap. Under these very general conditions, we find that the post-excitation cyclotron (magnetron) radius of an ion increases, in a vector sense, by an amount proportional to the magnitude-mode spectral peak height of the excitation waveform at the cyclotron (magnetron) frequency of the ion of interest; the proportionality constant depends on the trap geometry but is independent of the particular excitation waveform used. Thus, in order to compute (in advance) the ICR orbital radius resulting from excitation by a particular excitation waveform, one need simply evaluate the Fourier transform spectral magnitude of that excitation waveform at the ion's ICR orbital frequency (assuming negligible initial cyclotron radius). This linear theory does not address various nonlinear effects that are known to occur during the excitation event, most notably, effects due to the spatial inhomogeneity of the excitation electric field; however, the present theory provides a firm theoretical foundation for stored waveform inverse Fourier transform (SWIFT) excitation and two-dimensional FTICR mass spectrometry. The results obtained are for the special case of a tetragonal trap; however, they are generalizable to any trap geometry. With the aid of material presented in the Appendix, the generalization to cylindrical ion traps with the ring electrode divided into equal quadrants is simple.