THE NONLINEAR ION-TRAP .5. NATURE OF NONLINEAR RESONANCES AND RESONANT ION EJECTION

The superposition of higher order multipole fields on the basic quadrupole field in ion traps generates a non-harmonic oscillator system for the ions. Fourier analyses of simulated secular oscillations in non-linear ion traps, therefore, not only reveal the sideband frequencies, well-known from the Mathieu theory, but additionally a commonwealth of multipole-specific overtones (or higher harmonics), and corresponding sidebands of overtones. Non-linear resonances occur when the overtone frequencies match sideband frequencies. It can be shown that in each of the resonance conditions, not just one overtone matches one sideband, instead, groups of overtones match groups of sidebands. The generation of overtones is studied by Fourier analysis of computed ion oscillations in the direction of the z axis. Even multipoles (octopole, dodecapole, etc.) generate only odd orders of higher harmonics (3, 5, etc.) of the secular frequency, explainable by the symmetry with regard to the plane z = 0. In contrast, odd multipoles (hexapole, decapole, etc.) generate all orders of higher harmonics. For all multipoles, the lowest higher harmonics are found to be strongest. With multipoles of higher orders, the strength of the overtones decreases weaker with the order of the harmonics. For z direction resonances in stationary trapping fields, the function governing the amplitude growth is investigated by computer simulations. The ejection in the z direction, as a function of time t, follows, at least in good approximation, the equation dz/dt = C(n-1)z(n-1) where n is the order of multipole, and C is a constant. This equation is strictly valid for the electrically applied dipole field (n = 1), matching the secular frequency or one of its sidebands, resulting in a linear increase of the amplitude. It is valid also for the basic quadrupole field (n = 2) outside the stability area, giving an exponential increase. It is at least approximately valid for the non-linear resonances by weak superpositions of all higher odd multipoles (n = 3, 5,...), showing hyperbolically increasing amplitudes, whereas the even multipoles strongly suppress their own z direction non-linear resonances. The hyperbolic increase of the amplitude, having a mathematical pole, explains the fast ejection processes possible with non-linear resonances.