# User:Anu Kankainen/Proposed/The ion traps in nuclear physics

An Ion trap is a device that can confine electrically charged particles into a small volume by electric and/or magnetic fields. These devices allow not only storage of charged particles but also manipulation of their motion inside a trap. Ion traps are used in nuclear physics in many ways, commonly as ion beam improvement devices, high-resolution mass separators and perhaps most importantly, in high resolution mass spectrometry.

The most common types of traps used in nuclear physics are

• Paul traps
• Penning traps
• Electron beam ion traps (EBIT)

In short, Paul traps utilize both time-varying (radio-frequency) and static electric fields while Penning traps use additionally also magnetic fields. Electron beam ion trap (EBIT) is a Penning traps with an additional intense beam of electrons traversing through the trapping volume used for charge breeding.

## Contents

In a Paul trap the trapping effect is achieved solely with electric fields. They consist of a ring electrode and two endcap electrodes that in ideal case are hyperboles of revolution. Confinement of ions is achieved by using both DC and AC electric fields. Motion of ions is described with Mathieu equations which in short describes the suitable combinations of frequency and amplitude of the electric field for storing ions with certain mass-over-charge ratio. Often, Paul trap is filled with low-pressure inert gas to allow for ion motion cooling.

In nuclear physics, Paul traps are used mainly for storing and cooling ions. Some trap structures are prepared so that the center of the trap is exposed for example for lasers and particle detectors.

A special type of Paul trap is a so-called linear Paul trap, also known as RadioFrequency Quadrupole or RFQ. In such a device the ring electrode is significantly extended compared to a Paul trap. The extended space offers significantly more space for cooling than ordinary Paul traps and for this reason, gas-filled RFQs are used as high-efficiency beam coolers and bunchers in nuclear physics.

Figure 1: A quadrupole electrode configuration.

A radiofrequency quadrupole consists of four (usually cylindrical) rods to which AC potential is applied. See figure. The frequency and amplitude is appropriately chosen so that ions remain radially confined within the RF field. Quite often the rods are segmented to allow application of different DC potential along the rod segments and thus it is possible to form potential gradients and well(s) to the axis of the quadrupole structure. An RFQ filled with dilute buffer gas the ions can be cooled in a series of collisions with buffer gas atoms. This way the ions cool both radially to the center of the trap and also axially to the potential well.

Figure 2: Electric potential along the axis of an RFQ. Once an ion enters the RFQ, its longitudinal and transverse momenta is reduced due to collisions with buffer gas. Finally the ion thermalizes to the same temperature than the buffer gas and thus remains in the well.

Gas-filled radiofrequency quadrupoles were introduced to nuclear physics in the 1990s. These devices are commonly used to improve quality (emittance and energy spread) of low-energy (1-100 keV) ion beams and also to convert a (semi-)continuous beam to bunched. Typically ions are fully cooled within 1-20 milliseconds of time allowing cooling of even very short-living ions. Taking advantage of bunching capability, the RFQs are excellent devices for delivering bunched beams for example to Penning trap or to collinear laser spectroscopy experiments.

#### Benefits to collinear laser spectroscopy

Collinear laser spectroscopy requires very well defined beam energy to avoid Doppler broadening. Using an RFQ, the energy spread of the ion beam can be reduced to below 1-eV level. Some ion sources already provide very well defined beam energy but in case of bad beam quality an RFQ can be used to improve the beam quality. Ion beam bunching is also essential. Having a bunched beam, the background that is most commonly caused by scattered photons can be suppressed on the order of $$10^4$$ since observation time can be significantly reduced to the duration of bunches.

#### RFQ as an injector for Penning trap

Penning traps require high-quality and also bunched ion beams for efficient capture of the ions. Low-quality beam can have energy spread well exceeding the depth of the trap; also bunching is essential since capturing a continuous beam is rather inefficient.

## Penning traps

An ideal Penning trap consists of a strong homogenous magnetic field and a weak quadrupolar electrostatic potential. In contrast to a Paul trap, full confinement is achieved with static trapping fields. Like a Paul trap, a Penning trap also consists of ring and endcap electrodes. Quite often so-called guard or correction electrodes are placed between the endcaps and the ring to compensate for the truncation of the hyperbolic electrodes. Two types of geometry configurations are commonly used: hyperbolic and cylindrical. Both constructs have their own benefits although in precision experiments usually hyperbolic are favored due to better production of quadrupolar electric field. On the other hand, cylindrical electrodes are easier to manufacture and sometimes more open geometry offer other benefits such as better conductance of gas.

Figure 3: A schematic illustration of a hyperbolic Penning trap. The endcap electrodes are draw in blue color and the ring electrode in the middle with orange color. The two characteristic dimensions, $\rho_0$ and $z_0$, denote the shortest distance from the trap center to the ring and endcap electrodes, respectively.

Figure 3 shows a schematic illustration of a hyperbolic Penning trap. Applying a static voltage $$U_0$$ between the ring and endcap electrodes will create a quadrupole potential inside the trap.

### Ion motion in a Penning trap

A charged particle in a Penning trap exhibits three different eigenmotions. One is in the direction of the magnetic field lines (axial motion)) and two of them (magnetron and cyclotron) are perpendicular to the magnetic field. Ideally the axial oscillation is independent of magnetic field and has a frequency $\nu_z = \frac{1}{2\pi} \sqrt{\frac{qU_0}{md^2}},$ where $q$ and $m$ are the charge and mass of the particle, $U_0$ the voltage across the ring and endcap electrodes and $d$ the characteristic trap parameter defined as $$d = \sqrt{\frac{1}{2} \left (z_0^2 + \frac{\rho_0^2}{2} \right) }$$. The frequencies of the radial modes -- denoted $$\nu_-$$ for magnetron and $$\nu_+$$ for trap-modified cyclotron frequency are $\nu_\pm = \frac{1}{4\pi} \left ( \nu_c \pm \sqrt{\nu_c^2 -2\nu_z^2} \right ),$ where $$\nu_c = \frac{1}{2\pi}\frac{q}{m}B$$ is the so-called free-space cyclotron frequency which gives the frequency of ion in absence of any electric fields.

In first order, the magnetron frequency is independent of the ions mass-over-charge ratio $$m/q$$ and only depends on the trap geometry $$d$$ and the applied voltage $$U_0$$.

### Ion motion excitation

Any of the three eigenmotions can be modified by using time-varying electric fields. Usually the ring electrode (or one of the correction electrodes between is radially split in order to create electric fields that have azimuthal and/or axial component. Using a time-varying dipolar electric field with a frequency of one of the eigenmodes will increase or decrease the radius of the corresponding motion.

Motion excitation with quadrupole fields is another commonly used technique. With a quadrupole field one can couple two motions like cyclotron and magnetron motion with a quadrupole field having frequency $$\nu_+ + \nu_-$$. With ion motion excitations, mass-over-charge ratio of trapped particles can be linked to frequency, which can be easily determined with high accuracy.

### Penning traps in nuclear physics

Penning traps in nuclear physics have been found to be extremely useful. Common uses are

• Atomic mass measurements
• High-resolution mass separation for post-trap decay spectroscopy
• In-trap decay spectroscopy

Presently many radioactive ion beam facilities have Penning trap setups:

Also Penning traps dedicated for particular nuclear physics studies include

• FSU-trap at Florida State University, Florida, USA
• THe-trap at Max-Planck-Institut für Kernphysik, Heidelberg, Germany

### Mass separation with a Penning trap

Penning traps have been used to separate ions having different mass-over-charge ratio for years. In short, unwanted ions are driven to large orbits and rejected, typically by letting the contaminants hit the electrode surface.

#### Sideband cooling technique

One well established separation is technique is the so-called sideband cooling technique [Sav1991], which allows separation of ions down to about ten parts-per-million level. This technique was pioneered at ISOLTRAP and later adopted to other trap setups such as SHIPTRAP and JYFLTRAP. In order to perform mass separation, the trap needs to be filled with dilute gas to allow fast ion motions (axial and cyclotron) to be cooled. Another requirement is a small extraction aperture for the ions which is usually accommodated by having a small hole in one of the endcap electrodes.

The separation procedure is rather simple. The first step after capturing the bunch of ions is to let the ions cool down. Depending mostly on the gas pressure, the cooling typically takes 10-100 ms. Next, a dipolar RF field at magnetron frequency is switched on in order to radially drive all ions to such a large magnetron orbit that ions would, if extracted towards the extraction side, hit the electrode having only small aperture. After establishing a large magnetron orbit for the ions, a quadrupole electric field with frequency $$\nu_+ + \nu_-$$ for the desired ion species is switched on to convert the established magnetron motion to cyclotron motion. Since the sideband frequency $$\nu_+ + \nu_-\ \approx \nu_c = \frac{1}{2\pi}\frac{q}{m}B$$ is highly mass selective, only ions near the resonance frequency get their magnetron motion converted to cyclotron motion. A frequency scan varying the quadrupolar field frequency is shown in Figure 4.

Figure 4: A sideband cooling resonance. The trap was loaded with ions produced in proton-induced fission having atomic mass number 104. When the excitation frequency matches the sideband frequency of the ion, the ions are centered and transmitted through the narrow aperture.

In this particular case, the trap was loaded with ions produced in proton-induced fission reaction. When the quadrupole frequency matches the ions' sideband frequency $$\approx \frac{1}{2\pi}\frac{q}{m}B$$, that particular species of ions get centered in the trap as the magnetron motion is converted to cyclotron motion which eventually cools away. Once the cooled bunch is ejected through the narrow aperture in the extraction side of the trap, the contaminants are lost to the electrode and the desired ion species is transmitted.

#### High-resolution cleaning

The sideband cooling technique has its limits and typically can separate ions with about 10 part per million resolution, mostly limited by the applied gas and the number of simultaneously stored ions. A common and well established way to increase the resolution is to apply a dipolar RF field in a gas-free Penning trap at the contaminant ions' $\nu_0$ frequency. This will drive the ion to large orbit and eventually the ions will be lost. This way, one can reach 1 part per million or better resolution. Often, resolution of this magnitude is enough to separate nuclear isomeric states.

### Atomic mass measurements

Measurement of ion's mass in a Penning trap is based on determining ion's so-called free-space cyclotron frequency $\nu_c$ (see Eq. XX), which depends on magnetic field $B$ and ion's charge-over-mass ratio $q/m$. The charge state of the ion is usually known before it enters into a trap. Thus, in order to obtain the mass of the ion of interest, it is enough to determine the resonance frequency of the ion and the magnitude of the magnetic field at the moment of the measurement. For short-living ions, the cyclotron resonance frequency in a Penning trap is typically measured with time-of-flight ion cyclotron resonance (TOF-ICR) technique described below. The magnetic field strength is determined by measuring the resonance frequencies for a reference ion with a well-known atomic mass shortly before and after the ion of interest (or interleavedly) and interpolating to the time when the ion of interest was measured. As a summary, the main objective in high-precision Penning-trap measurements is to measure the frequency ratio between the reference ion and the ion of interest, which is inversely proportional to their mass ratio.

In the full TOF-ICR measurement cycle, ions undergo the following steps:

1. Motion excitation by a short (about 10 ms) dipolar RF electric field at magnetron frequency to increase the ions' magnetron radius
2. A coupling excitation with a quadrupolar RF electric field near the coupling frequency $\nu_+ + \nu_-$ of the ion. Duration of this excitation (with constant amplitude) can vary between milliseconds up to several seconds, usually limited by the ion's lifetime. The excitation duration and amplitude is selected to match one full conversion from magnetron to reduced cyclotron motion for the ions excited at their sideband coupling frequency $\nu_+ + \nu_-$. For other frequencies, the conversion is only partial. Thus, ions in the resonance have the highest radial energy after the excitation.
3. When the ions are extracted from the trap, their time-of-flight to a detector outside the magnetic field of the trap is measured. The strong magnetic field gradient exerts an axial force on the ions which is proportional to the ions’ radial energy. Therefore, the ions at the resonance have a shorter time-of-flight to the detector than the ions off the resonance. An example of a TOF-ICR spectrum is shown in Figure 5.
Figure 5: A TOF-ICR curve obtained with 54Co ions. The quadupolar excitation frequency is on the horizontal axis while time of flight of the ions is on the vertical axis. The black datapoints with errorbars are the average time-of-flight of the ions excited with that frequency and red solid curve is a fit to the data. The blue pixels with variable denseness show the detected number of ions in the pixel. Darker pixels denote more detected ions. Here, an excitation time of 200 ms was used.

The excitation can be replaced by an excitation via time-separated oscillatory fields. Instead of a continuous fixed-amplitude excitation, the excitation is divided e.g. into two short excitation pulses separated by a longer waiting time. For example, instead of a continuous 200 ms excitation, two 25 ms excitations separated by a 150 ms waiting time can be applied. In this Ramsey method, steeper sideband minima and a narrower resonance width is obtained for the TOF-ICR spectra compared to the conventional method. This is illustrated in Figure 6.

Figure 6: Same as in Figure 5 with exception that the excitation field is split to two parts of 25 ms separated with a waiting time of 150 ms. Compared to Figure 5, the sidebands are more pronounced and also the width of the central minimum is reduced.

Another excitation method which shows great potential is excitation with octupolar electric fields. In past few years, octupolar excitation method has obtained both theoretical and experimental interest. Instead of exciting ions at $\nu_c = \nu_+ + \nu_-$, in octupolar excitation ions are excited at $2\nu_c$. Although frequency is doubled, the obtained precision is more than ten-fold more precise or even more.

Several corrections have to be taken into account when determining the final mass value from the measured resonance frequencies. These include: - frequency shifts due to contaminant ions - fluctuations in the magnetic field strength as a function of time - mass-dependent frequency shifts - characteristic relative residual uncertainty of the trap The frequency shifts due to contaminant ions can be studied via count-rate class analysis. The resonance frequency is plotted as a function of the number of ions present in the trap and it is extrapolated into one ion in the trap. Time-dependent fluctuations of the magnetic field have been studied via long-term measurements of well-known reference ions, such as 133Cs. Mass-dependent uncertainties have been studied via carbon-cluster measurements. An additional relative residual uncertainty may be needed for an ion trap to be in agreement with carbon cluster measurements.

• Link to Scholarpedia: G. Bollen: Mass measurements?

## Electron beam ion traps (EBIT)

Electron beam ion traps (EBIT) are relatively new addition to nuclear physicist toolbox and so far only one such device at TITAN Penning trap setup at Vancouver, Canada is in active use. An EBIT produces and traps highly charged ions (HCI) by means of a high-current electron beam compressed to a high current density by a strong few-Tesla magnetic field. Similarly as in Penning traps, the ions are confined in EBIT by means of electrostatic quadrupole potential and strong axial magnetic field. In addition, ions are confined radially by the electron-beam space-charge induced potential.

Penning trap mass spectrometry greatly benefits from usage of highly charged ions since obtainable precision is directly proportional to the charge state. Even with modest charge breeding of singly charged ions to doubly-charged will increase the precision by a factor of two. With increasing charge states other effects start to play more important role like uncertainties in the electron binding energies or ion loss due to charge-exchange with background atoms.

Another feasible application of EBIT usage is to use them as a storage device for spectroscopy. As EBITs are usually used not with solenoids but more open-access Helmholz-coils, one can mount radiation detectors around the trap center.

## Applications to Nuclear Physics

### Atomic mass spectrometry

#### Nuclear structure studies

Figure 7: Two-neutron separation energies as a function of neutron number for isotopic chains from Kr (Z=36) to Pd (Z=46).

Precise atomic mass values serve as an important input to study nucleon binding energies and their behavior as a function of neutron and proton numbers. Two-proton and two-neutron separation energies describe how much energy is required to remove two protons or neutrons from a nucleus. In general, the two-nucleon separation energies follow a smooth trend as a function of proton (Z) or neutron (N) number. Shell closures at Z or N = 8, 20, 28, 50, 82 or 126 show up as a steep decrease in the separation energies as the binding energy at the magic nucleon number is increased. This is illustrated in Figure 7 which shows the two-neutron separation energies as a function of neutron number.

Sudden changes in the smooth behavior of two-nucleon binding energies can be due to a shape transition. The shape of a nucleus - whether it is a spherical, prolate (elongated spheroid like a rugby ball), oblate (flattened spheroid) or triaxial - has an effect on its binding energy, and thus, shape changes are seen as a discontinuity in the two-nucleon binding energies. Figure 7 shows two-neutron separation energies as a function of the neutron number for different isotopic chains from Kr (Z=36) to Pd (Z=46). The shape transition to a prolate shape is seen as a kink e.g. in Y, Zr, and Nb isotopic chains at around N=60. A steep increase in the mean-square charge radius has also been observed in this same region in laser spectroscopy experiments in agreement with the onset of large prolate deformation.

High-precision mass measurements have also enabled detailed studies on pairing effects in nuclei, reflected for example in odd-even staggering of nuclear masses and in nucleon binding energies. The charge independence of the nuclear force - meaning that the force between two nucleons should be the same independent of the nucleon pair in question (proton-proton, neutron-neutron or proton-neutron) - can also be studied via high-accuracy mass measurements. This has been investigated by testing the validity of the isobaric multiplet mass equation for different multiplets at several facilities.

### Beta-neutrino correlation measurements

Dr. Tommi Eronen, Department of Physics University of Jyväskylä, was invited on 26 November 2009.