Constructing a Paul Trap for Undergraduate Laboratories

Steven Ban, Matthew Bledsoe, Peter Chang, Garrett Wen May 4, 2016

Abstract For our project, we wanted to build an oscillating system to trap charged particles. We decided to build a Paul trap, which is a form of ion trap that uses alternating electric ﬁelds to conﬁne charged particles in the center of the device. We were able to accomplish this using spherical brass electrodes, thick copper wires, a wire ring, and a step up transformer to give us the high voltage we needed from the wall. Our Paul trap worked very well, trapping charged plaster dust, which we illuminated with a bright green laser so that we could easily see the particles in the trap. In this paper, we will primarily focus on how to recreate our experiment, and on our results.

1 Introduction

The quadrupole ion trap, or the Paul trap in honor of its inventor Wolfgang Paul who for which claimed the 1989 Nobel Prize, is a device that can confine charged particles using interactions between electric fields. Due to Earnshaw’s theorem, we know that charged particles cannot be trapped stably by static electric fields alone. There are several different methods for trapping charged particles, and the Paul trap accomplishes this using AC voltage to generate alternating electric fields on a pair of hyperbolic electrodes (in our case, two brass spheres), suspended on either side of a ring electrode (a simple wire ring in our setup). This setup, when the hyperbolic electrodes are connected such that the hyperbolic electrodes are always at the same potential, and with opposite polarity of the ring electrode, generates an electric field that is zero at the center of the ring and everywhere else provides a time- averaged restoring force back towards the center of the ring. This allows us to trap our charged particles within the ring. Ion traps are useful because they allow us to isolate individual charged particles over long periods of time, which we can then observe and use in experiments. This is much easier than attempting to gather data on a moving particle, and enables observation of particles that would otherwise be unobservable. Ion traps are also useful for controlling quantum states, and can be used in mass spectrometry and quantum computing.

1 Figure 1: Our ion trap with trapped particles illuminated by laser

2 Mathematical Background and Theory

Electrodes are strategically shaped and placed for desired electric field interactions. In our setup, the spherical brass balls are at the opposite potential of the ring electrode. The electric field then oscillates between two configurations at a frequency of 60 Hz, which is the frequency of the wall voltage. One of the configurations stretches the particles that are outside the center of the ring in the vertical direction and provides a restoring force towards the center in the horizontal direction, and the other configuration is the opposite, the particles are stretched horizontally, and there is a restoring force toward the center in the vertical direction. The rapid oscillation of the electric field between the two configurations provides an average confining force in all directions, because the trap field oscillates faster than the particles can escape the trap. Once particles settle into a place, they essentially oscillate in a fixed position very quickly. The oscillating and stretching in the horizontal and vertical directions causes the shape of the dust that we observe. The electric potential within the device has been long studied and can be described by the equation 2 2 (U0 + V0 cos Ωt)(r − 2z ) U = 2 d0

2 q 1 2 2 where Ω is the radio frequency and d = 2 r0 + z0. The restoring force that drives the charged particle back toward the center of the device increases as the ion deviates from the center of the device. The motion of the charged particle in an ideal Paul trap is most commonly described by the solutions to the Mathieu d2u equation: dt2 +(a−2q cos 2τ)u, where u is the coordinate axes, τ is a dimensionless parameter equaling to Ωt/2, and a and q are additional dimensionless parameters known as trapping parameters. Mathieu functions appear in physical problems involving periodic potentials or elliptical shapes, and they were ﬁrst introduced by French mathematician mile Lonard Mathieu in 1868 when analyzing the motion of elliptical membranes∗.

Figure 2: Electric ﬁeld conﬁgurations in a Paul trap

Figure 3: Alternating electric ﬁelds. Source: https://en.wikipedia.org/wiki/Quadrupole ion trap

3 Construction of the Trap

Our Paul trap was built using commercially available tools, making this an ideal project for student use. However, this project also utilizes high voltages, so proper construction should heavily emphasize the safety of the system.

∗The source of the mathematics of the Paul trap is from Trapping and Cooling from the Humboldt University of Berlin

3 3.1 Materials • A source of high voltage. We used a standard wall output and a 15X step-up transformer to achieve a working voltage of 2 kV AC at 60 Hz. High-voltage transformers might be diﬃcult to ﬁnd, and is perhaps the most challenging part of the set up.

• Current limiting resistors for the purpose of safety.

• An acrylic box. Our box’s dimensions are 600 × 600 × 800.

• A Variac.

• Small particles to be trapped. We used plaster dust.

• A laser pointer to illuminate trapped particles. We used a 30mW laser.

• Copper wires.

• 2 metal spheres (1-2 cm diameter) to be used as electrodes. Small spoons may also be used if desired.

• High speed camera.

• (Optional) Black spray paint and black construction paper.

• (Optional) Wooden stand to mount electrodes.

3.2 Set-up Acrylic Box The acrylic box serves two major purposes. The first is to isolate the system for the purpose of safety. We recommend constructing the acrylic box first so that it may be kept on top of the apparatus at all times except for when adjustments are made. The second purpose is to reduce air flow. We want to reduce the possibility that air currents could knock the particles out of the trap.

In our case, we laser cut the acrylic box with dimensions 600 × 600 × 800. These dimensions may vary depending on your particular setup. Each side should be cut independently and then either taped or glued together, except for a removable bottom. A medium size hole should be cut into one side to allow power cords to enter the system. A small hole should be drilled into the top of the box (above where the trap will be placed) so that the particles may be safely deposited into the trap.

Electric Source WARNING: For this portion, make sure that everything is turned oﬀ at all times and you should never touch any exposed metal or wiring.

4 First, the Variac is connected to wall voltage. The Variac allows us to gradually increase or decrase the voltage applied to the transformer. This gives us another layer of safety in the system, and it also allows us to test various aspects such as the minimum voltage needed to maintain the trapped particles. Next, the Variac is connected to the high-voltage transformer inside the acrylic box through the medium size hole. This will step up the voltage to the necessary minimum for trapping. Finally, the transformer will be connected to the nodes as described in the next section.

Figure 4: Order of setup

Electrodes WARNING: Again, for this portion, make sure that everything is turned oﬀ at all times and you never touch any exposed metal. Ensure that all wiring is covered by electrical tape.

In this setup, there are two electrodes that we will set up. For the first electrode, solder the two metal spheres to both ends of a thin, stiff copper rod. Bend the rod into a U shape, fix the electrode in place, and establish an electrical connect one wire from the transformer to the bent copper rod by either soldering it in place or wrapping the wire tightly around the copper rod. Cover the exposed wire with electrical tape.

For the second electrode, strip a thin metal wire of any insulation and bend it into a ring. Through the same procedure, electrically connect the metal wire to the wire coming from the transformer. Position the ring between the two metal spheres. Ideally, the diameter of the circle should be the same as the distance between the two metal spheres. Make sure that all wires are not in contact with one another.

In order to maintain the position of electrodes and wiring, a wooden stand or platform can be used (Figure 6). The hyperbolic electrodes are ﬁxed to the wooden stand using glue and tape, and the ring electrode is aﬃxed using glue.

5 Figure 5: Acrylic box setup Figure 6: Electrode setup

Visual Enhancements To improve the ability to see trapped particles, it is necessary to eliminate any reﬂec- tions that the laser might produce. Once the trap has been built, black spray paint may be applied to all surfaces of the apparatus, including the stand and the electrodes them- selves. Black spray paint or black construction paper can also be applied to the acrylic box to further reduce unwanted reﬂections from the illuminating laser light (See Figures 5 and 6).

Trapping Particles Once the setup has been complete, place a pile of plaster dust on top of the box next to the small hole that is drilled above the electrodes. Turn on the system using the Variac, and gently tap the box to release the particles into the device. Shine the laser pointer in between the ring. If done correctly, the laser should illuminate the particles that have been trapped inside. Turning oﬀ the lights may also help to visualize trapped particles.

3.3 Additional Information We would like to reiterate here that safety is a major concern when doing this project. The high voltages used in this trap make the system extremely dangerous. This is why we recommend building the acrylic box ﬁrst. This electrode apparatus (transformer, electrode, and wiring) should contained in the acrylic box as much as possible. Additionally, whenever adjustments are to be made on the apparatus, students should unplug the entire system

6 (unplug from the wall, turn Variac voltage to 0V , turn oﬀ Variac) to ensure that no accidents occur. It is also a good idea to connect the transformer wires to the electrodes with current limiting resistors. We used 1MΩ resistors so that the current ﬂowing through the wires was approximately 2 mA, which would not be deadly if we accidentally touched one of the electrodes or a wire.

4 Observations

The trapped particles assemble and oscillate along the electric ﬁeld lines of the system. In this setup, we have a restorative electric force that holds the particles in place. We noticed that the particles are more concentrated below the ring than above the ring. This is because gravity pulls the particles downward, which shifts the zero point of the system from the center of the ring to slightly below the center of the ring. This observation can also be contributed to the fact that our ring electrode is not perfectly circular, disturbing the symmetry of the system.

4.1 Minimum Voltage Requirement Below a certain voltage, gravity will overcome the electric forces holding the particles in place. The lowest voltage using which we were still able to trap the particles is at 40V from the Variac, which translates into around 600kV of post-transformer voltage. As we sweep the voltage down towards 40V , we can see particles start to drop out of the trap. The distribution of particles starts to get smaller and more concentrated toward the center, until all the particles drop out of the trap at around 600V .

4.2 Oscillations In this system, the particles will constantly be bouncing up and down due to the oscillations of the electromagnetic forces. If you would like to observe these oscillations, ﬁnd a camera with high frame rate and slow motion capabilities to record the system. A video that we took of the oscillations may be found here:

https://www.youtube.com/watch?v=9rGTlbpf-M0

7 Figure 7: Oscillating particles forming vertical strands to the naked eye

When you ﬁrst observe the trapped particles, it appears as though there are several strands of dust that are immobile. However, looking at the video, it becomes apparent that there are only individual particles and not strands of particles. This means that the oscillations are happening so fast that each trapped particle appears as a strand to the naked eye.

5 Conclusions

In this paper we have described how to construct a Paul trap using commercially available tool and materials. We have also included observations and rationales, along with a simple explanation of the theories behind the system. This setup can also be used to demonstrate the physics behind ion trapping, as well as serve as a potential oscillating system for students to study.

8 6 References

1. Werth, G. Basics of Ion Traps. Johannes Gutenberg University. https://www.icts.res.in/media/uploads/Old Talks Lectures/Document/ 1265634974GW- erth lecture notes.pdf

2. March, R.E. (1997) An Introduction to Quadrupole Ion Trap Mass Spectrometry. Jour- nal of Mass Spectrometry 32, 351-369 http://www.wiley.com/legacy/wileychi/ms/articles/351 a.pdf

3. Paul, W. (1990). Electromagnetic traps for charged and neutral particles. Rev. Mod. Phys. 62:3 http://journals.aps.org.ezp-prod1.hul.harvard.edu/rmp/pdf/ 10.1103/RevModPhys.62.531

4. Trapping and Cooling. Humboldt University of Berlin. https://www.physik.hu-berlin.de/de/nano/lehre/quantenoptik09/chapter13

5. https://en.wikipedia.org/wiki/Quadrupole ion trap

6. https://en.wikipedia.org/wiki/Ion trap

7 Acknowledgements

We would like to thank Prof. Mara Prentiss for working through the original idea of the project with us, helping us with the theoretical aspect of ion trapping, troubleshooting the set-up, and supervising the entire project. We would also like to thank Rob Hart for assistance ﬁnding materials, laser cutting the acrylic box, and ﬁnding a step-up transformer that functioned properly. Additionally, we would like to thank Lo¨ıc Anderegg and the Harvard Natural Sciences Lecture Demonstrations for their help, as well as Prof. Matthew Schwartz and Prof. Cora Dvorkin for making this experience possible.