Can the effective potential of a linear quadrupole be extended to values of the Mathieu parameter q up to 0.90?

The motion of ions in a linear quadrupole is usually described by solutions to the Mathieu equation. A simplifying approximation to this theory that is widely used for low values of the Mathieu parameters a and q describes ion motion in an effective potential. In this work, we have calculated the effective potential for any q from displacements of calculated ion trajectories caused by a dipole DC electric field. It is assumed that the dipole DC electric field at the center of the displaced trajectory is countered by an "effective" electric field. For all q values, the effective electric field is found to increase linearly with the distance from the center of the quadrupole. The trapping forces probed in this way increase continuously with q up to the first stability region boundary at q=0.908. The well depth (D) at any q can be described by D = q[V(rf)/c], where c=3.955±0.005, very similar to the standard effective potential model with c=4.000.