# Trying to understand Paul traps

Paul traps have been confusing me a lot recently. So sorry if there are too many questions

• I know that Paul traps confine ions with the help of rapidly switching quadruple electric fields, I've heard that Paul traps actually force the particle to move to the center where the electric field is zero. But I see no reason behind this. Wouldn't the particle just oscillate about a mean position? Why would they drift to the center?

• For Paul traps to work, the electric fields must switch faster than of which the particle reaches the wall. So, does any frequency above this value work fine? Just to make my point clear, would a Paul trap operating at 1 Thz work just as well as a one at 1 or 2 Mhz?

• If my assumption above is accurate, then can a Paul trap trap more than one type of species? Say electrons and protons? I understand that in order to confine electrons, we would need the switching of electric fields to be much higher than that needed to confine ions.

• Then can't we confine plasmas just as well? Debye shielding would hardly be a problem as the particles lack time for screening to occur?

1st question: Qualitatively, the reason the ions drift to the center is that the center of the trap is a point of dynamic stability for the ions, and ions experience a pondermotive force toward the center. I have explained this more in my answer to this question. The particle is always unstable in one direction but since that direction changes with the RF drive frequency, the particle is dynamically trapped.

The following is a typical trajectory of the ion in such a trap, which shows the closed path of the particle as well as small oscillations at the frequency of the RF field.

2nd and 3rd questions:

If the potential inside the Paul RF trap is exactly quadropolar (which is the case for hyperboloid electrodes), then the equations of motion will be in the form of Mathieu equations. These equations have two parameters $(a,q)$ which depend on the physical paramters of the problem: $$a_z = -\frac {8eU} {m r_0^2 \Omega^2} \qquad\qquad q_z = \frac {4eV} {m r_0^2 \Omega^2}$$ for the x direction equation (as stated in the Wikipedia article.). They depend on the RF voltage V, DC voltage U, RF frequency $\Omega$ and the particle's charge to mass ratio. Now, there are a couple of region of stabilities for the $(a,q)$ parameters in $(a,q)$ plane which are the allowed (a,q) parameters that lead to a stable motion of the ion in the trap. Therefore, the frequency $\Omega$ alone cannot lead to (in)stability of the motion. The lowest order (and largest) of these regions is this:

As you can see now, for a given frequency and voltage there are many possible charge to mass ratios that can be trapped. Note that practically you cannot increase $V$ or $\Omega$ too much as this can lead to other effects such as radiation or even electrical breakdown in the trap. The above plot is the 1st stability region. There are other stability regions too but they're rarely used because of the practical difficulties.

If you want to trap more than one particle in the trap simultaneously, you may need to consider Coulomb interactions among the particles which can cause their motion to become unstable. but other than that, it is possible to do that, and is actually done in practice because it's difficult to only insert one single particle into the trap.

• So If the voltage and the frequencies are just right, we can trap a plasma? And why is the stability dependent on frequency, As far as I understand, the trapping ability only depends on whether the frequency is higher than that of which the particles can hit the walls. So why does it matter at all? But then why are we not using paul traps to confine plasmas? Commented Apr 30, 2017 at 4:13
• About the frequency, you are right, as can be seen from the stability region. As you increase the frequency $\Omega$ you get smaller and smaller $(a,q)$ and approach the origin. In terms of the motion, this leads to smaller amplitudes for the secular motion of the ion (the small oscillatory motion on the main trajectory). Such $(a,q)$s are actually where they usually operate paul traps: near the origin with zero $U$ (and therefore zero $a$) and very small $q$s. Commented Apr 30, 2017 at 4:24
• About your question on trapping a plasma, I don't know. Commented Apr 30, 2017 at 4:25

A Paul trap can be imagined like an effective potential for the ion. If the right frequency is chosen this potential will have a minimum in the centre of the trap, unlike the normal quadrupole, which has a saddle point there. You can also see this from the equations of motion for the ion in the Paul trap.

There is a mechanical analogue of a Paul trap which is a saddle shaped surface that is rotating. If you place a small spherical mass in the middle and tune the rotation frequency correctly you can also achieve a stable particle.

To the OP's questions:

Wouldn't the particle just oscillate about a mean position? Why would they drift to the centre?

Initially the particle may oscillate, but remember that ions are charged particles and can radiate. This will cause them to fall to the ground state (or some low excited state) of the trap after some time. There may be other loss channels than radiation too, but I don't know the details of that.

So, does any frequency above this value work fine? Just to make my point clear, would a Paul trap operating at 1 Thz work just as well as a one at 1 or 2 Mhz?

No, in fact fine tuning a Paul trap is quite a delicate business. The mechanical analogue gives some intuitive insight here.

If my assumption above is accurate [...]

the assumption was not accurate, so there is no point in answering the other two questions.

How to explain Paul trap operation in simplest terms?

The electrode nearest to the particle first attracts it on corresponding AC halfcycle, then, as it gets closer it kicks it really hard on the opposing polarity next halfcycle. Because particle comes closer during attraction halfcycles the repelling interaction is stronger, and average force is repulsive. That true for all electrodes, so the particle is getting kicked around electrical center of the trap. However, if some viscosity or other friction is present, the particle will gradually settle in the electrical center of the trap. The AC driving period of the trap should be long enough to cause some noticable movement of the particle, on another hand too long period may cause the particle to hit the electrode or be thrown outside. If some disturbing force is present (e.g. gravity) it may demand some micromotion to maintain average force counteracting it.