Analytical Solution for the Electric Field Inside Dynamically Harmonized FT-ICR Cell

Dynamically harmonized FT-ICR cell has a saddle-like hyperbolic field distribution inside when averaged over a cyclotron trajectory around the axis of the cell. Such a field distribution makes the motion along the magnetic field independent of the motion in the x,y-plane, as well as the cyclotron motion independent of the magnetron motion and prevents any disintegration of excited coherent ion clouds, which is ruining the resolution in the other types of FT-ICR cells providing by this ideal phasing of single-m/z ion clouds in the entire volume of the cell. FT-ICR instruments with such a cell show resolutions of more than ten million at m/z 1000 at relatively small magnetic fields like 7 Tesla in quadrupole detection mode, what is not reachable by any other type of modern mass spectrometers. We have found that for such ion traps, it is possible to find the analytical solution in the working volume of the trap without any averaging. The potential distribution for the almost whole volume of such a cell can be presented in the form ϕ(x, y, z) = αz ² + f _2D (x, y), where f _2D (x, y) is the solution of 2D Poisson equation, which could be found by the method of conformal transformation. This solution is applicable in the practical case and can serve as a base for an analytical theory of signal detection using such cells and as a standard for solutions obtained by numerical simulations of the cell field. [Figure not available: see fulltext.].