Trajectory calculations of space-charge-induced mass shifts in a linear quadrupole ion trap

If too many ions are stored in a linear quadrupole ion trap, space charge causes the oscillation frequencies to decrease. Ions therefore appear at higher apparent mass-to-charge ratios in a mass spectrum. To further understand this process, we have used trajectory calculations of ions to determine mass shifts.

Two models of the ion cloud are used. The first assumes that the acceptance of the quadrupole is uniformly filled with ions. The second assumes that the ions have a thermal distribution trapped in the effective potential. Both give analytical descriptions of the field from space charge. Ion trajectories are calculated with and without space charge. Oscillation frequencies are determined with a Fourier transform. Shifts in oscillation frequency with space charge are then used to calculate mass shifts.

Both ion cloud models give similar results. More diffuse ion clouds or ion clouds that have higher temperatures produce lower electric fields near the center of the trap and hence lower mass shifts. Space charge produces a nonlinear field. As a result, the discrete resonant frequencies of ions in a pure quadrupole field become distributions of frequencies. Comparisons with experiments show agreement for reasonable values of the parameters of the two ion cloud models.

This relatively simple method for calculating the effects of space charge shows (i) that the spread of oscillation frequencies reduces mass resolution with axial ejection and (ii) that mass shifts are reduced with ion clouds with greater spatial extents or higher ion temperatures.