EQUILIBRIUM SPACE-CHARGE DISTRIBUTION IN A QUADRUPOLE ION-TRAP

A simple model provides a basis for evaluating the ion spatial distrib ution in a quadrupole (Paul) ion trap and its effect on the total pote ntial (trap potential plus space charge) acting on ions in the trap. B y combining the pseudopotential approximation introduced by Dehmelt 25 years ago with the assumption of thermal equilibrium (leading to a Bo ltzmann ion energy distribution), the resulting ion spatial distributi on (for ions of a single mass-to-charge ratio) depends only on total n umber 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 Poi sson's equation subject to a Boltzmann ion energy distribution. Howeve r, because the ion distribution depends in tum on the total potential, solving for the potential and the ion distribution is no longer a sim ple boundary condition differential equation problem; an iterative pro cedure is required to obtain a self-consistent result. For the particu larly convenient operating condition, a(z) = q(z)2/4 [in which a(z) = -8qU/mrho0(2) OMEGA2, and q(z) = -4qV/mrho0(2) OMEGA2, where U and V a re direct current and radiofrequency (frequency, OMEGA) voltages appli ed to the trap, m/q is ion mass-to-charge ratio, and rho0 is the radiu s of the ring electrode at the z = 0 midplane], both the pseudopotenti al and the ion distribution become spherically symmetric. The resultin g one-dimensional problem may be solved by a simple optimization proce dure. The present model accounts qualitatively for the dependence of t otal potential and ion distribution on number of ions (higher ion dens ity or lower temperature flattens the total potential and widens the s patial distribution of ions) and pseudopotential (higher pseudopotenti al increases ion density near the center of the trap without widening the ion spatial distribution).