Analytical study of electrostatic ion beam traps

The use of electrostatic ion beam traps requires one to set many potentials on the electrodes (ten in our case), making the tuning much more difficult than with quadrupole traps. In order to obtain the best trapping conditions, an analytical formula giving the electrostatic potential inside the trap is required. In this paper, we present a general method to calculate the analytical expression of the electrostatic potential in any axisymmetric set of electrodes. We use conformal mapping to simplify the geometry of the boundary. The calculation is then performed in a space of simple geometry. We show that this method, providing good accuracy, allows one to obtain the potential on the axis as an analytic function of the potentials applied to the electrodes, thus leading to fast, accurate, and efficient calculations. We conclude by presenting stability maps depending on the potentials that enabled us to find the good trapping conditions for O⁴⁺ at much higher energies than what has been achieved until now.