That is, we require an electric quadrupole field, say,
We can satisfy this in more than one way. The two of import are that associated with the linear Paul Trap, whose initial manifestations were not as a trap but as a focusing tunnel of sorts, but which can be turned into a `race track' ion trap,
Such potentials can be provided via hyperbolic electrodes. We can perform a successive over relaxation of a cross section of these electrodes and find that indeed a two-dimensional stable equilibrium is created at the center (though this is unstable in the third dimension, z) when we satisfy the above conditions (Figure 1.2, also see our report on the SOR method).
In both cases, we have a repulsive force in the z direction which must be avoided. This can be done via the clever mechanism of rotating the field so that the focusing and defocusing is applied alternatively in each direction. If done at the right set of frequencies, the ion will maintain a stable orbit near the center of the ion trap.
A way to visualize this is with W. Paul's mechanical analog [1,2]. Paul made an equivalent potential as that described above by carving an hyperbolic saddle surface out of plexiglass. Placing a ball on top of this surface would result in the ball falling off of it, of course. But if the surface is rotated at a proper rate, the ball will stay on the surface (Figure 1.3).
The applied potential is thus