The effective potential for ion motion in a radio frequency quadrupole field revisited

Abstract

An effective potential to describe ion motion in radio frequency (rf) quadrupole fields is developed. It is based on the Mathieu equation instead of the standard averaging technique. An effective Hamiltonian series equation which is the extension of the classical effective potential approach allows determining the frequencies more accurately than in the case of the classical approach to the derivation of the effective potential; the latter uses only the first member of a Taylor series. Because the quadrupole potential is quadratic, time averaging of all the harmonics of the oscillation frequencies is possible and in this work all harmonics are considered using a strict mathematical background. The method can be applied to quadrupole operation at any value of the Mathieu parameters (a, q). With this method the ion motion in a quadrupole field is described by an effective quadratic potential and inside the stability zones this is a simple harmonic oscillator. It is shown that for a linear radio frequency (rf) quadrupole, with rf-only operation in the first stability region, the effective potential well depth, D, increases continuously with q and is given by the well-known relation D=qV0/4 where V0 is the amplitude of the rf voltage, pole to ground; however, here its validity is proved for q values up to the stability boundary at q=0.9080. The potential well size (i.e., the acceptable range of initial coordinates) is separated from the potential well depth (i.e., the acceptable range of initial kinetic energies). The theory also allows calculating the acceptable range of initial coordinates, and it is shown that these decrease with increasing q as ≈r01−q/q* (where q*=0.9080 is the high q boundary of the first stability zone, and r0 is the radius of the gap between electrodes) while the acceptable range of initial kinetic energies is still controlled by the potential well depth calculated as D=qV0/4. Preliminary numerical experiments and an alternate approach show the validity of the results.