Equilibrium space charge distribution in a quadrupole ion trap

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Abstract
A simple model provides a basis for evaluating the ion spatial distribution in a uadrupole (Paul) ion trap and its effect on the total potential (trap potential plus space charge 3 acting on ions in the trap. By combining the pseudopotential approximation introduced by Dehmelt 25 years ago with the assumption of thermal equilibrium (leading to a Boltzmann ion energy distribution), the resulting ion spatial distribution (for ions of a single mass-to-charge ratio) depends only on total number of ions, trap pseudopotential, and temperature. (The equilibrium assumption is justified by the high helium bath gas pressure at which analytical quadrupole ion traps are typically operated.) The electric potential generated by the ion space charge may be generated from Poisson’s equation subject to a Boltzmann ion energy distribution. However, because the ion distribution depends in turn on the total potential, solving for the potential and the ion distribution is no longer a simple boundary condition differential equation problem; an iterative procedure is required to obtain a self-consistent result. For the particularly convenient operating condition, (a z = -8qU/mϱ 0 2 Ω2, and q z =-4qV mϱ 0 2 Ω2, where U and V are direct current and radiofrequency (frequency, ω) voltages applied to the trap, m/q is ion mass-to-charge ratio, and ϱ0 is the radius of the ring electrode at the z=0 midplane], both the pseudopotential and the ion distribution become spherically symmetric. The resulting one-dimensional problem may be solved by a simple optimization procedure. The present model accounts qualitatively for the dependence of total potential and ion distribution on number of ions (higher ion density or lower temperature flattens the total potential and widens the spatial distribution of ions) and pseudopotential (higher pseudopotential increases ion density near the center of the trap without widening the ion spatial distribution).