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

The superposition of higher order multipole fields on the basic quadru pole field in ion traps generates a non-harmonic oscillator system for the ions. Fourier analyses of simulated secular oscillations in non-l inear 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 sideban d, instead, groups of overtones match groups of sidebands. The generat ion of overtones is studied by Fourier analysis of computed ion oscill ations in the direction of the z axis. Even multipoles (octopole, dode capole, 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 lo west higher harmonics are found to be strongest. With multipoles of hi gher orders, the strength of the overtones decreases weaker with the o rder of the harmonics. For z direction resonances in stationary trappi ng fields, the function governing the amplitude growth is investigated by computer simulations. The ejection in the z direction, as a functi on 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 const ant. This equation is strictly valid for the electrically applied dipo le field (n = 1), matching the secular frequency or one of its sideban ds, resulting in a linear increase of the amplitude. It is valid also for the basic quadrupole field (n = 2) outside the stability area, giv ing an exponential increase. It is at least approximately valid for th e non-linear resonances by weak superpositions of all higher odd multi poles (n = 3, 5,...), showing hyperbolically increasing amplitudes, wh ereas 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.