Theoretical Analysis of the Space Charge Effect in 3D Ion Trap

The strong Coulombrepulsion among a large number of ions stored in an ion trap leads to the so-calledspace-charge effect. This effect is of theoretical and practical importance inmass spectrometry since it significantly distorts the mass accuracy andsensitivity of an ion trap mass analyzer. However, it is very difficulty to getan analytical solutions for the ion motion in the presence of the space chargeeffect because of their many-body problem. In this article, a model for thespace charge effect is given and discussed. A spherically symmetric Gaussiandistribution of the ions in the center of a three-dimensional ion trap wasassumed, and the potential field due to the ion cloud was calculatedanalytically by solving the Poisson equation. Based on the small vibrationapproximation and the potential field calculation, a relationship between thefrequency shift of ion secular motion and the number of ions in the trap wasderived analytically.For comparison, the frequency shift was also calculatednumerically using Mathematica. The result was similar to that obtained with thesmall-vibration approximation. For an ion trap mass analysis using dipoleexcitation, the frequency of ion secular motion strongly affects the excitationvoltage, and in turn the accuracy and sensitivity of mass analyzer. Theperformance of an ion trap mass analyzer could be improved by reducing thenumber of stored ions, or using the corrections based on theoreticalcalculation.