Theoretical investigations of different excitation modes for Penning trap mass spectrometry

In Penning trap mass spectrometry the motion of trapped ions is manipulated by external radio-frequency fields. This paper describes a general theoretical framework to classify the various types of excitation of the ion's motional modes, to identify the resonance frequencies, and to find the effective interaction Hamiltonians which are valid in the vicinity of the resonances. Instead of Cartesian or cylindrical coordinates and momenta our theoretical approach uses the complex oscillator amplitudes of the cyclotron, magnetron, and axial oscillators as its basic dynamical variables. Equations of motion are set up, which can be simplified in the vicinity of resonances by the resonating wave approximation. Explicit analytical solutions of these equations of motion are obtained for dipolar and quadrupolar excitation. Results are stated as functions of a dimensionless evolution parameter and a dimensionless detuning parameter. The conversion of magnetron into cyclotron motion by a quadrupolar field using the two-pulse Ramsey technique is analyzed as an interference phenomenon between two complex amplitudes describing these motional modes. For octupolar excitation the equations of motion are transformed into a set of three ordinary differential equations which are amenable to numerical solution in a straightforward way. Some results on the line shapes for the conversion of magnetron into cyclotron motion by octupolar excitation are given. Simultaneous excitation of the ion motion at two different resonance frequencies is briefly discussed, using the example of dipole excitation of the magnetron motion and conversion of magnetron into cyclotron motion by quadrupolar excitation. Finally modifications due to frictional damping are reviewed.

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• Unified theoretical treatment of the dipolar, quadrupolar, and octupolar interactions between ions and external radio-frequency fields.
• Identification of resonance frequencies and effective interaction Hamiltonians valid in the vicinity of resonances.
• Ramsey's method of separated oscillatory fields applied to mass spectrometry is presented as an interference phenomenon.
• Equations for the interconversion of the cyclotron and magnetron modes due to octupolar rf-fields at twice the cyclotron frequency 2νc.
• Simultaneous dipolar and quadrupolar excitation applied for isobaric cleaning and mass-selective centering of the desired ion species.