Ion Cloud Model for a Linear Quadrupole Ion Trap

Abstract

If large numbers of ions are stored in a linear quadrupole ion trap, space charge causes the oscillation frequencies of ions to decrease. Ions then appear at higher apparent masses when resonantly ejected for mass analysis. In principle, to calculate mass shifts requires calculating the positions of all ions, interacting with each other, at all times, with a self-consistent space charge field. Here, we propose a simpler model for the ion cloud in the case where mass shifts and frequency shifts are relatively small ( ca 0.2% and 0.4%, respectively), the trapping field is much stronger ( ca × 102) than the space charge field and space charge only causes small perturbations to the ion motion. The self-consistent field problem need not be considered. As test ions move with times long compared to a cycle of the trapping field, the motion of individual ions can be ignored. Static positions of the ions in the cloud are used. To generate an ion cloud, trajectories of N ( ca 10,000) ions are calculated for random times between 10 and 100 cycles of the trapping radio frequency field. The ions are then distributed axially randomly in a trap of length four times the field radius, r0. The potential and electric field from the ion cloud are calculated from the ion positions. Near the trap center (distances r < 0.1 r0), the potential and electric fields from space charge are not cylindrically symmetric, but are quite symmetric for greater values of r. Trajectories of test ions, oscillation frequencies and mass shifts can then be calculated in the trapping field, including the space charge field. Mass shifts are in good agreement with experiments for reasonable values of the initial positions and speeds of the ions. Agreement with earlier analytical models for the ion cloud, based on a uniform occupation of phase space, or a thermal (Boltzmann) distribution of ions trapped in the effective potential [D. Douglas and N.V. Konenkov, Rapid Commun. Mass Spectrom. 26, 2105 (2012)]1 is quite good. All three models give similar electric fields to match experimental mass shifts.