Concept of the effective potential in describing the motion of ions in a quadrupole mass filter

We propose a generalization of the effective potential theory for the motion of particles in a rapidly oscillating electric field for the stability parameters lying near the boundary of the diagram where the standard effective potential theory is inapplicable. We derive the dynamic equations describing the variation of the envelope of ion oscillations for the motion of ions near the stability vertex of the first zone of the quadrupole mass filter. We reduce them to the form of the Hamilton equations for oscillations of a material particle in the field of potential forces. We obtain expressions for the effective potential well. It is shown that in spite of the high kinetic energy of oscillations, the depth of the effective potential well for ions in the quadrupole is less than 1 eV in the case of filtration with a mass resolution exceeding 200 units. The acceptance of the mass filter is calculated as a function of the stability parameters and the resolving power.