Comprehensive theory of the Fourier transform ion cyclotron resonance signal for all ion trap geometries

Abstract

An analytic solution for the charge induced on each of the detection electrodes of a Fourier transform ion cyclotron resonance (FT/ICR) ion trap has been derived from basic electrostatics for both tetragonal and cylindrical traps of arbitrary aspect ratio, by use of a Green’s function formalism. Dunbar has shown that the result of that calculation is in general equivalent to that obtained from prior ‘‘reciprocity’’-based methods (see text). A primary advantage of the present treatment is its variety of functional forms arising from the various forms of the Green’s function, some of which may converge much more rapidly in numerical evaluation. (Moreover, because the Green’s function is the potential field of a unit point charge, the Green’s function must be employed in any treatment of ion–ion repulsions.) The present results (a) exactly confirm prior analyses of the cubic and tetragonal traps; (b) provide the first complete analysis of the cylindrical trap; and (c) may be extended to any trap geometry for which the Green’s function is known. In the absence of an available Green’s function, the reciprocity-based treatment, either analytic or numerical, is the method of choice to solve for the induced charge for any ion trap geometry (e.g, unbroken or segmented hyperbolic). For circular orbits centered on the longitudinal axis of the trap, the presence of spectral components at odd multiples of the fundamental ICR orbital frequency is explained and a closed form solution for the relative magnitudes of these components is presented for tetragonal and cylindrical traps. The ratio of the spectral peak height at the third harmonic to that at the first (i.e., the ratio of the third to first Fourier coefficients) is a strong monotonic function of orbital radius; thus, measurement of that ratio provides a simple and direct means for determining the cyclotron orbital radius and hence its orbital translational energy. The presence and location of magnetron and trapping sidebands of the fundamental peak are also predicted. In addition, we show that a cylindrical trap whose ring electrode is divided into equal quadrants is only slightly more sensitive than a tetragonal trap of the same aspect ratio. Finally, we develop a general circuit model which relates the charge induced on one or more detection electrodes to the detected voltage (i.e., the unamplified signal). Since the effect of trapped-ion motion on each detection electrode is modeled as a charge (or current) source relative to ground, the net signal from any given electrode arrangement and interconnection scheme can be accommodated simply by adding, subtracting, or grounding the signal from each detection electrode, e.g., single-electrode detection, Comisarow’s differential detection between two electrodes, or any of various multiple-electrode configurations. From the measured ICR signal and ICR orbital radius, the number of coherently orbiting ions may be determined.