Octupolar excitation of ion motion in a Penning trap: A theoretical study

• The interconversion of the radial motional modes of an ion in a Penning trap by a pulse of octupolar rf-radiation at twice the free cyclotron frequency is studied.
• A Hamiltonian formulation is given and Heisenberg's equations of motion for the creation and annihilation operators of the radial oscillator modes are derived.
• The model is reformulated in terms of the components of a “Bloch vector” and studied in the classical limit using quasi-classical coherent oscillator states.
• Equations of motion for the components of the normalized Bloch vector are derived and solved numerically for given initial values.
• Excitation functions and conversion line shapes are studied as functions of a time-like evolution parameter and the detuning of the octupolar rf-field.

High-precision Penning-trap mass spectrometry uses the resonant conversion of the magnetron motional mode into the cyclotron motional mode to determine the cyclotron frequency of the ions under investigation. Usually the conversion process is performed by interaction of the ions with external quadrupolar rf-fields. Recently it was found that conversion by means of octupolar rf-fields entails a tremendous increase in mass resolution and is thus of great interest. However, the conversion results depend in an intricate way on the amplitudes and phases of the octupolar rf-field and of the motional modes of the ions. Experimental progress was hampered by the lack of an underlying theory of the octupolar conversion process, in particular by the absence of theoretical predictions for the conversion line shape. This paper wants to fill this gap.Supported by quantum mechanical arguments the effective interaction responsible for the interconversion of the radial motional modes is identified and a Hamiltonian description of the conversion process is given. Unfortunately the equations of motion are highly non-linear and defy a simple solution. Here the concept of the Bloch vector operator, previously used with advantage to provide a theoretical basis for the two-pulse Ramsey technique for excitation by quadrupolar rf-fields, comes to help again. The expectation value of the 3-component of the Bloch vector is directly related to the “degree of conversion” which is the quantity that is measured in ion-cyclotron-resonance time-of-flight (ToF-ICR) experiments. A reformulation of the Hamiltonian description of the conversion process in terms of the normalized Bloch vector leads to a system of differential equations that are still non-linear, but are solvable in terms of Jacobi elliptic functions. The derivation of these equations reveals the complicated dependence of the conversion results on the amplitude and phase of the octupolar rf-field and on the starting parameters of the ion motion.In the second part of the paper numerical solutions of the differential equations are used to obtain excitation functions (conversion at the exact resonance frequency) as a function of a time-like evolution parameter and conversion line shapes (conversion at a fixed value of the evolution parameter) as functions of the detuning of the octupolar rf-field. These studies demonstrate the strong dependence of the conversion results on the initial phases of the octupolar field and the cyclotron and magnetron motional modes. For closer comparison with experimental data we can average over the unobserved phases. Moreover we observe an important influence of the initial cyclotron radius of the ion motion on the outcome of the conversion process.

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