Ion Motion Stability Diagram or Distorted Square Waveform Trapping Voltage M. Sudakov and E. Nikolaev, Eur. J. Mass Spectrom. 8, 191–199 (2002) Ion motion stability diagram for distorted square waveform trapping voltage M. Sudakov Shimadzu Research Laboratory (Europe) Ltd., Manchester, UK E. Nikolaev The Institute for Energy Problems of Chemical Physics Russian Academy of Sciences, Leninskij pr. 38, k.2 , Moscow, Russia 117829. E-mail: nikolaev@chph.ras.ru Ion motion in a periodic radio frequency (RF) quadrupole electric field is analysed theoretically by the matrix method and direct tra- jectory calculation. General properties of the ion motion: stability condition, oscillation spectrum and secular frequency are derived analytically from the elements of the transformation matrix. Stability diagrams for ion motion in the Paul ion trap are presented for rectangular waveforms with different duty cycles (duration of pulse over period). Calculation of the secular frequencies of the ion motion in the ion trap is performed for the first time. The relation of radial and axial secular frequencies along the RF scan line was found to be practically identical in both the square waveform and harmonic voltage cases. Pulse shape distortions, due to resis- tive-inductive-capacitive filtering in real devices, are considered. Stability diagrams of ion motion in the ion trap with distorted volt- age waveforms are calculated. Distortion of the waveform is shown to introduce minor changes in the diagram shape with respect to the diagram for an ideal square wave. Within the first stable region, distortion of the waveform does not lead to any auxiliary para- metric resonances of the ion motion. Ion trapping with a pure random pulsed voltage is investigated by means of direct trajectory simulations. It is shown that, in this case, the ion motion can be conditionally stable for a considerable length of time. Keywords: ion trap, stability diagram, matrix method, trajectory calculations, square waveform Introduction The possibility of driving quadrupoles and ion traps with square waveform radio frequency (RF) voltages was demonstrated in the 1970s by Sheretov 1 and by Richards. 2 At a ratio of positive- to negative-pulse time duration equal to 1, there is no practical difference in the form of the stability diagrams for square waveform and sine waveform RF volt- ages. A square waveform voltage of relatively high ampli- tude can be obtained by alternating switching of positive and negative power supplies with a fast switch. 2,3 Theoretically, any voltage amplitude is achievable by this method but, cur- rently, a maximum amplitude around 1 kV at a frequency of 1 MHz has been demonstrated. 3 Rapid progress in fast-switch technology promises the possibility of achieving voltages comparable with those achieved in the conventional sine mode of operation. In the sine mode, resonant high Q circuits are used to obtain high amplitude RF voltages and precise tuning of the frequency to produce resonance is needed. With fast switching techniques, broadband excita- tion is possible without loosing RF amplitude. The Shimadzu Research Laboratory has recently demonstrated the applicability of rectangular waveform operation with the so-called digital ion trap. 3 By changing the ratio of positive- to negative-pulse time duration (see Figure 1) it is possible to © IM Publications 2002, ISSN 1356-1049 M. Sudakov and E. Nikolaev, Eur. J. Mass Spectrom. 8, 191–199 (2002) 191 Figure 1. Rectangular waveform function. Definition of the duty cycle d, definition of q 1 parameter and levels from which a 1 and a 2 parameters are calculated, are shown in the figure.