Cylindrical Penning traps with orthogonalized anharmonicity compensation

Compensated Penning traps with cylindrical ring and compensation electrodes and flat endcaps are considered as alternatives to high precision traps with hyperbolic ring and endcap electrodes. Cylindrical and flat electrodes can be more easily and precisely constructed, especially for the very small traps which are desirable for ion trapping. They are easily studied since Laplace's equation can be solved by a series expansion which is discussed in most electricity and magnetism textbooks. A central new result is that a judicious choice of the height-to-diameter ratio for a cylindrical trap makes the axial oscillation frequency of a trapped particle independent of adjustments in the compensation potential, just as has been recently proposed for hyperbolic traps. Other properties of such orthogonalized cylindrical traps appear to be adequate for particle trapping so that properly designed cylindrical traps are promising alternatives for high precision work, ready for laboratory testing. Whether or not cylindrical traps will be able to replace hyperbolic traps entirely for the highest precision work, a major point of this paper is that orthogonalized traps can be built with any reasonable electrode geometry. Special access traps with orthogonalized anharmonicity compensation are thus completely feasible, though numerical calculations will be generally required for their design.