The basic theoretical foundation of the Paul trap

To confine an ion, we should require a force such that $F=-kr$ . What would this entail for an electrical potential? Since the electric field is proportional to the force, and is equal to the divergence of the potential, we should require

$$\Phi \propto (\alpha x^2 + \beta y^2 + \gamma z^2)$$

That is, we require an electric quadrupole field, say,

$$\Phi = \frac{\Phi_0}{2r^2_0}(\alpha x^2 + \beta y^2 + \gamma z^2)$$ (1.1)

Equation 1.1 must obey that condition imposed on all potentials where there is no free charge distribution, namely that

$$\nabla^2\Phi=0 \rightarrow \alpha + \beta + \gamma =0$$

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,

$$\alpha=1=-\gamma, \beta=0 \rightarrow \Phi=\frac{\Phi_0}{2r_0^2}(x^2-z^2)$$ (1.2)

and that associated with the ``Ionenkäfig'', the chamber rf Paul ion trap

$$\alpha=1=\beta, \beta = 0 \rightarrow \Phi=\frac{\Phi_0}{r_0^2 + 2z^2_0}(r^2 -2z^2), 2z^2_0 =r^2_0$$ (1.3)

Figure 1.1: The linear rf Paul trap (a) and the chamber rf Paul trap (b). This Figure from [1]

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).

Figure 1.2: SOR calculation for hyperbolic electrodes. The outer box is held at ground, the horizontal electrodes held at V and the vertical ones held at -V. The grid was 1000 by 1000, our tolerance was 0.0001.

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).

Figure 1.3: The mechanical analog to the rf Paul trap

The applied potential is thus

$$\Phi_0=U + V \cos \omega t$$ (1.4)

If the particle has a charge e and mass m, then its equation of motion are

$$\ddot{x}+ \frac{e}{mr^2_0}(U + V\cos \omega t)x=0$$

$$\ddot{z} - \frac{e}{mr^2_0}(U + V\cos \omega t)z=0$$ (1.5)

Since cosine is an even function, we can generalize this to

$$\ddot{\eta}+(a-2q\cos(2\tau))\eta , a=\frac{4eU}{mr^2_0 \omega^2}, q=\frac{-2eV}{mr^2_0\omega^2}, \tau=\frac{\omega t}{2}$$ (1.6)

For the z equation, $a \rightarrow -a$ . The solution to this equation is simple enough though not trivial, and we give an informal derivation below.