Detailed study of the quadrupole mass analyzer operating within the first, second, and third (intermediate) stability regions. I. Analytical approach

The theoretical aspects of ion separation in imperfect fields of the quadrupole mass analyzer operating within the first, second, and third stability regions are discussed by analysis of the beam dynamics in phase-space. The analytical approach uses an approximate solution of the Hill equation with a small heterogeneous part. These calculations indicate that the trap mechanism of ion separation is conditioned by the properties of characteristic solutions. These solutions are reduced to an approximate solution in the form of a general solution of a homogeneous Mathieu equation with combined factors taking into account a small heterogeneous part that defines the region of beam capture (acceptance) in phase-space. The infringement of independence principle of ion oscillations about each of the positional axes caused by distortions increases the cross-sectional area of the beam. The beam is cut out by the mass analyzer aperture. This causes transmission losses that depend on phase. Therefore, the ion current at the mass analyzer exit is amplitude modulated by the frequency of the radiofrequency (rf) component of the field. The maximum current is at zero phase. The modulation depth is proportional to the relative value of the distortions