Abstract
Penning traps offer unique possibilities for storing, manipulating and investigating charged particles with high sensitivity and accuracy. The widespread applications of Penning traps in physics and chemistry comprise e.g. mass spectrometry, laser spectroscopy, measurements of electronic and nuclear magnetic moments, chemical sample analysis and reaction studies. We have developed a method, based on the Green's function approach, which allows for the analytical calculation of the electrostatic properties of a Penning trap with arbitrary electrodes. The ansatz features an extension of Dirichlet's problem to nontrivial geometries and leads to an analytical solution of the Laplace equation. As an example we discuss the toroidal hybrid Penning trap designed for our planned measurements of the magnetic moment of the (anti)proton. As in the case of cylindrical Penning traps, it is possible to optimize the properties of the electric trapping fields, which is mandatory for high-precision experiments with single charged particles. Of particular interest are the anharmonicity compensation, orthogonality and optimum adjustment of frequency shifts by the continuous Stern–Gerlach effect in a quantum jump spectrometer. The mathematical formalism developed goes beyond the mere design of novel Penning traps and has potential applications in other fields of physics and engineering.
Highlights
Have been implemented in molecular electric dipole moment experiments [17], antihydrogen production [18, 19] or the most accurate test of the CPT symmetry in the lepton sector [20]
The curved shape of a toroidal ferromagnetic ring enhances the curvature of the magnetic bottle by more than one order of magnitude compared with a cylindrical ring of similar dimensions and the same material, making the resolution of the phase-sensitive Stern–Gerlach quantum jump spectrometer [28] big enough for the efficient determination of the spin state of a singleproton
Besides being our motivation for the measurement for theproton’s g-factor, the power of this calculation technique goes beyond the design of the hybrid Penning trap and could be used in many other problems involving the Laplace equation
Summary
The Green’s function formalism is a well-known and powerful technique for calculating electrostatic potentials. In the case of ion traps, the electrostatic potential (x) usually has to be calculated within a closed volume defined by some electrodes to which arbitrary voltages are applied This problem corresponds to solving the Laplace equation with Dirichlet boundary conditions: if the adequate Green’s function is available, the. Green’s functions for electrodes with various shapes (hyperboloids, oblate and prolate spheroids, toroids, flat-ring cyclide discs, etc) are well known in the literature [29, 30]. These kinds of electrodes can be put together in many different ways, so that a vast class of well-defined trapping volumes can theoretically be envisaged with them. Any such combination of differently shaped electrodes defining a closed trapping region is what we call a hybrid Penning trap
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