The motion of a single charged particle in a Paul trap in the presence of the damping force is investigated theoretically and the modified stability diagrams in the parameter space are calculated. The results show that the stable regions in the a-q parameter plane are not only enlarged but also shifted. Consequently, the damping force causes instability in some cases, contrary to intuition. As a by-product of the calculation, we derive new theoretical approximate expressions for the secular-oscillation frequency. In the limiting case of no damping, these formulas are in good agreement with early measurements done by Wuerker et al.
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