First application of the phase-imaging ion-cyclotron resonance technique at TRIGA-Trap

The phase-imaging ion cyclotron resonance technique (PI-ICR) has been implemented at TRIGA-Trap together with a newly built five-pole cylindrical trap. In PI-ICR the total phase of trapped ions is measured by projecting the ion motion onto a position-sensitive delay-line micro-channel plate detector. The systematic uncertainties have been investigated and first mass measurements on stable Pb isotopes have been performed with PI-ICR. The new technique offers higher mass-resolving power, allows checking for the presence of contaminant ion species, and it proved useful in tuning the harmonicity of the trapping potential as well as in aligning the trap symmetry axis with respect to the magnetic field axis by visualizing the radial ion motion. This is a non-scanning technique where every detected ion contributes equally, therefore it is more sensitive than the previously used time-of-flight ion-cyclotron-resonance (ToF-ICR) technique, which is based on the scanning of the sideband-frequency of trapped ions and recording their time of flight after ejection. It will enable us to carry out high-precision mass measurements in the actinide region with uncertainties on the ppb level.


Introduction
Penning traps are used for the most accurate mass measurements to date contributing to a variety of topics in atomic and nuclear physics [1][2][3].
In the superimposed electric and magnetic fields of an ideal Penning trap, the motion of a charged particle con-S. Chenmarev and Sz. Nagy have contributed equally to this work. a e-mail: stanislav.chenmarev@mpi-hd.mpg.de b e-mail: szilard.nagy@mpi-hd.mpg.de (corresponding author) sists of a simple harmonic oscillation along the trap axis (z) with the axial frequency f z , the magnetron motion with the magnetron frequency f − , and the modified cyclotron motion with the reduced cyclotron frequency f + in the (x, y) plane [1]. The basis of a mass measurement is the cyclotron frequency determination of an ion with charge to mass ratio q/m, moving in a pure magnetic field B, f c = q B/(2π m). It can be measured in an ideal trap as sideband frequency f c = f − + f + , or it can be derived by measuring all three eigenfrequencies f 2 c = f 2 − + f 2 z + f 2 + [4,5]. The B-field is calibrated periodically by an additional cyclotron frequency measurement of the reference ion, which is a different ion species with sufficiently known mass.

Experimental setup
The introduction of the PI-ICR technique [6,7] to Penningtrap mass spectrometry (PTMS) on short-lived nuclei was a major breakthrough. The provided gain in sensitivity, resolving power, and accuracy was key to its success and fast spreading to nearly all online Penning-trap facilities worldwide [8][9][10][11]. Its capabilities have been demonstrated recently in a number of high-precision mass measurements ranging from resolving low-lying isomeric states [12], solving a Qvalue puzzle for neutrino studies [13] to sensitive measurements in the realm of superheavy elements [14][15][16][17].
This novel technique has recently also been implemented at TRIGA-Trap [18], where first test mass measurements have been performed and systematic uncertainties limiting the accuracy have been investigated. TRIGA-Trap, set up at the research reactor Training, Research, Isotopes, General Atomic Mainz (TRIGA Mainz) is one of the development platforms for the future Measurements of very short-lived Fig. 1 Sketch of the TRIGA-Trap double Penning-trap mass spectrometer in its current off-line configuration. 1 MiniRFQ laser-ablation ion source, 2 7 T superconducting magnet, 3 Drift section, 4 DLD40 delay-line MCP detector, 5 Purification trap, 6 Measurement trap nuclides using an Advanced Trapping System (MATS) experiment [19] at the Facility for Antiproton and Ion Research (FAIR) [20,21] in Darmstadt, Germany. MATS will be installed at the low-energy beam line behind the Superconducting FRagment Separator (Super-FRS) [22,23]. It will allow the investigation of nuclei far from stability and thereby expand the knowledge about their fundamental nuclear properties. Whilst FAIR is under construction, development platforms already exist in order to optimize the planned experiments, and to develop and to test new ideas. Being located at a research reactor [24], TRIGA-Trap can access fragments from neutron-induced fission [25], and can address offline measurements of long-lived radioisotopes e.g., in the region of the actinides [26].
TRIGA-Trap is a double Penning-trap mass spectrometer. In offline mass measurements it is served by a non-resonant laser-ablation ion source with singly-charged positive ions [27]. The layout of the experimental setup is shown in Fig. 1.
Ions are produced with the help of a frequency-doubled pulsed Nd:YAG laser (see Fig. 1). To this end, 5 ns long pulses with intensities ranging from 0.05 mJ to 0.5 mJ are shot on a target containing the elements of interest. These ions are stored and cooled in a miniature radio-frequency quadrupole structure (MiniRFQ) [27] in the presence of helium buffer gas for 5 milliseconds, and later extracted and transported by ion-optical elements biased at -1 kV.
The purification and measurement traps are operated in two homogenous regions of a 7 T superconducting magnet. Incoming ions are first captured and stored in the helium buffer-gas filled purification trap. Here mass selective buffergas cooling [28] is performed. The purification trap is a seven-pole cylindrical Penning trap. Connection is done through a 50 mm-long and 1.5 mm-diameter channel with the measurement trap to maintain differential pressure conditions. In the measurement trap the initial amplitudes of the eigenmotions can be further reduced before the measurement takes place. The ions are ejected from the measurement trap through the drift section towards the position-sensitive detector. The detector is a DLD40 delay-line Microchannel Plate (MCP) with a circular active area of at least 40 mm in diameter, combined with front-end electronics and a TDC8HP PCI card from Roentdek Handels GmbH.

New cylindrical trap and drift section
Initially, TRIGA-Trap used a hyperbolic measurement trap to favor Fourier-Transform Ion Cyclotron Resonance (FT-ICR) detection [18]. The hyperbolic trap had some restrictions though, which prevented the PI-ICR implementation. For example, the small ejection apertures limited the radius of ion motion that can be imaged on the position-sensitive MCP detector.
To overcome these limitations, we installed a new cylindrical measurement trap [29]. Additionally the drift section between the traps and the detector was rebuilt, featuring now 11 segments with constant 45 mm inner diameter. The fivepole Penning trap has 24 mm inner diameter and is fully open on the ejection side, as shown in Fig. 2. The main parameters of the trap are given in Table 1.
The trap consists of gold-plated oxygen-free high thermal conductivity copper electrodes and sapphire spacers. The center ring electrode is split in eight identical segments to  allow different radial excitations. The correction electrodes in the current configuration are split into two segments, making axial and radial motion coupling possible [30]. Five-pole Penning traps were first introduced by Gabrielse [31,32].

The PI-ICR technique
The first application of a position-sensitive detector in PTMS dates back to 2009 [33], and the first work on the PI-ICR technique, pioneered at SHIPTRAP by Eliseev et al., was published in 2013 [6]. This technique uses a position-sensitive detector to measure the total phase accumulated by a trapped ion in a measurement time t acc (see Fig. 3). The total phase is the sum of n full turns and the additional phase ϕ i ∈ [0, 2π ]. The radial frequency can be obtained as: In this way, both the magnetron and reduced cyclotron frequencies can be measured, making a mass determination possible. By applying various excitation patterns it is possible to determine either the radial frequencies individually or their sum directly [7]. Due to the magnetic field gradient between the trap and the position-sensitive MCP detector, a magnification of the motion takes place with no influence on the phase measurement. The number of turns n increases proportionally with t acc . The additional phase ϕ i at the final t acc can be calculated as the polar angle between the ion's position at t = t 0 and t = t acc (see Fig. 3). The center of the circular motion corresponds to the position of non-excited ions. Since n is an integer number, the statistical uncertainty is determined by the spread of the additional phase ϕ i . By increasing t acc the uncertainty of the measurement can be Illustration of the total phase measurement, after completing n full turns the additional phase ϕ i is shown reduced, not restricted by the Fourier limit. The Fourier limit implies that the linewidth is determined approximately by the inverse of the observation time.

Experimental procedure
In mass measurements we directly measure the sum of the reduced cyclotron and magnetron frequencies, f c = f − + f + , known as sideband frequency as described in [7].
The excitation scheme is executed with the help of arbitrary function generators (AFGs). All main timings are controlled by AFGs, which are synchronized by the time base signal with an SR-FS725 rubidium frequency standard, featuring a one second Allan Variance of < 2 · 10 −11 .
We are applying a phase accumulation time t acc ≈ 1 s in our mass measurements. In order to avoid systematic uncertainties due to projection (see Sect. 5.4) the phase accumulation time is selected as integer multiple of the period corresponding to the free cyclotron frequency, which means that the cyclotron and magnetron phase spots overlap on the position-sensitive detector.
Thanks to refined techniques of ion manipulation, we can prepare ions in the center of the trap with minimal initial amplitudes of the radial and axial motions. The initial magnetron motion radius, after capturing the ions in the measurement trap, is negligible, and no active reduction of the magnetron radius is needed. The small initial axial amplitude of captured ions is further reduced in the measurement trap by a dipolar excitation with properly chosen duration, amplitude, and phase [34]. The time of the ion ejection is varied to average possible position shifts in case of incomplete conversion of modified cyclotron motion to magnetron motion. The excitation signals are chosen rather short, 2.5 ms, in order to cover a broad frequency range.
The mass measurement begins by individually recording the detected positions corresponding to the center of the trap for the ion of interest and the reference ions. Afterwards, the magnetron and cyclotron spots are taken by recording about 300 ions per spot in about ten minutes, yielding circular normally-distributed spots (see Fig. 4). The center positions corresponding to the ion of interest and reference ions are registered periodically about every three hours.
Afterwards the x and y coordinates of the center spots are determined. The position of the center spot is linearly interpolated to correspond to the absolute time of the phase spot measurement taken as the middle time of such a recording. Next we switch to polar coordinates and calculate the average phase and radius. The uncertainties of the phases are calculated considering the widths of the angular and the radial distributions considering also the error contribution of the interpolated center spot. Knowing the phase accumulation time, the number of full turns during accumulation time, and the calculated phase, the cyclotron frequency ( f − + f + ) can finally be calculated (see Eq. 1). During the count-rate-class analysis described in Sect. 5.3, this evaluation is done for each count-rate class, and the cyclotron frequency is extrapolated to one trapped ion. Since the reference and ion of interest species are not measured simultaneously, the reference-ion frequencies are linearly interpolated to the midtime of the ion-of-interest measurement. In order to calculate the free cyclotron frequency ratios, needed for a mass determination, the frequencies of the reference ion and of the ion of interest require a small correction f (see Sect. 5.6). These frequency ratios are averaged and the final weighted-average ratio with inner and outer errors is calculated. The uncertainty of f is considered when calculating the inner and outer errors, but not during the averaging of the frequency ratios.
Taking the calculated ratio, the tabulated masses of the reference ion and the electron, the mass of the atom of interest can be calculated. The electron binding energy, E B , needs to be considered as well here (see Eq. 2).

Discussion of systematic effects
In the following we will discuss certain systematic effects that can influence the experimental results achieved with the upgraded TRIGA-Trap setup. A summary of different systematic error sources specific to the PI-ICR method can be found in Ref. [7]. Frequency-shifts caused by electric and magnetic imperfections have been calculated in Ref. [35].

Homogeneity and stability of the magnetic field
Besides the mass and charge, the magnetic field strength determines the ion's free cyclotron frequency, therefore it is of central importance. Ideally the magnetic field should be constant in time during the measurement and homogeneous in the volume probed by the trapped ions. The 7 T B-field is generated by a superconducting solenoid magnet, which consists of a main coil generating the main field and a set of smaller superconducting shim coils to generate cor- To calibrate the B-field we perform a frequency measurement with a reference ion species with a sufficiently wellknown mass value. According to the specifications of our superconducting magnet, the homogeneity of the magnetic field after shimming is better than 0.1 ppm inside a volume of 1 cm 3 [18], where the measurement trap is placed. This has been confirmed by our own independent field mapping using a Nuclear Magnetic Resonance (NMR) probe.
By cooling the ions in the purification trap, and further minimizing the axial and radial amplitudes in the measurement trap, we can confine the trapped ions in a relatively small volume <1 mm 3 , to further decrease the impact of these effects.
In Fig. 5 the radial dependency of the measured sideband cyclotron frequency for 208 Pb + is shown. In a mass measurement we typically use radii in the range of 0.7-1.0 mm. We obtain the radius of the ion motion in mm by performing a calibration of our position-sensitive detector. We can image apertures of known diameter at constant drift section voltage, or image the magnetron motion with radii known from a separate TOF-ICR evaluation.
The magnetic field can be decomposed into a Legendre series. The lowest order inhomogeneities of relevance are B 2 and B 4 . From a fit to the data points their values were obtained using the function [35] f c (ρ) = Residual inhomogeneities of the magnetic field cause amplitude-dependent shifts in the cyclotron frequency measurements. In Fig. 6 the relative deviation of the cyclotron frequency is shown as a function of the axial amplitude. At small amplitudes less than 0.1 mm this effect is negligible. The axial amplitude is measured as variation in flight time in μs, and it is converted to mm using a conversion factor derived by measuring frequency shifts with known C 4 and C 6 based on Ref. [35].
During a mass measurement we apply active axial damping in form of a dipolar radio-frequency excitation at the axial frequency with specific amplitude, duration, and phase, to minimize the initial axial-motion amplitude of the trapped ions. In this way the axial motion amplitude after active damping is reduced to values of less than 0.1 mm, where the corresponding relative deviation of the cyclotron frequency is in the 10 −10 range.
The long-term drift of our magnet is in the range 10 −10 /h, and does not play a significant role in a typical mass measurement [29,36].

Harmonicity of the electrostatic potential
Any deviation of the electric trapping potential from the quadratic form is unwanted, nevertheless it is not possible to construct a real trap with ideal quadrupole potential. Penning traps employ correction electrodes to tune out or compensate anharmonicities. High-precision frequency measurements of trapped ions are made possible by elaborated trap designs [37] and careful trap-tuning. Frequency shifts arising from residual anharmonicities are minimized by keeping the motional amplitudes small. The dependency of eigenfrequencies on the amplitude or radius of the motion can be probed by PI-ICR and is usually minimized with the help of correction electrodes.
By imaging ions with fixed magnetron phase and varying their radius, we obtain a picture called a magnetron line. This represents a powerful instrument to tune the harmonicity of the trap (see Fig. 7). A curved or distorted line is an indication for an anharmonic potential (see Ref. [7]).
The potential near the trap center can be expanded in Legendre polynomials, where the most relevant anharmonic terms are C 4 and C 6 . If C 4 = 0 the axial motion is anharmonic. C 6 is less important, a non-zero C 6 leads to frequency shifts proportional to the square of the energy of the motions. Assuming a grounded endcap, the ratio of the voltages of the correction electrode and ring electrode is called tunig ratio, T R. For C 4 = C 6 ≈ 0 the theoretical value for our trap is T R = 0.881. Fig. 8 shows the dependency of the magnetron frequency of 208 Pb + ions on the magnetron radius for different values of T R. The experimentally found optimal T R=0.8742 (1) is not far from the theoretical value.
Additionally the radial dependency of the axial motion can be checked. The axial frequency was measured after capturing ions in the measurement trap, employing the ToF&phase method using 1 s phase accumulation time (see Appendix 1). In Fig. 10 the radial dependency of the axial frequency is shown, at optimal T R.   In case different ion species are present in the trap, the different center-of-mass motions will interact, leading to frequency shifts [38]. The frequency shift depends on the total number of stored ions, the difference of the cyclotron frequencies, and the ratio between the number of wanted and unwanted ions [38]. This underlines the importance of a careful check for the presence of unwanted ion species. In general, the mass-selective buffer gas cooling technique applied in the purification trap deals with this problem. However, the PI-ICR technique offers tools to check for the presence of remaining unwanted ion species in the measurement trap [7]. For example, after capturing the ion ensemble, a long dipolar drive on the mass-dependent reduced-cyclotron frequency of the ion of interest is executed. After ejection the excited ions appear all on the same radius on the position-sensitive detector. Contaminant-ion species remain unexcited and will appear as a spot in the center (see Fig. 11).
In the left panel of Fig. 12 cyclotron frequency measurements for 208 Pb + are shown for different numbers of simultaneously trapped ions. The laser-power settings were altered for 10 minute-measurement periods. The black dots correspond to low count rate, i.e. between 0 and 3 detected ions. The green squares stand for moderate count rate, 0-6 detected ions per pulse, and the red diamonds represent high count rate, 0-20 detected ions per pulse. The real number of trapped ions can be significantly higher considering a detector efficiency of about 35 % for single ions, which worsens with increasing count rate [39]. Frequencies recorded with low and moderate ion count rates are similar, whereas frequencies measured with high ion count rate are systematically higher. In the right panel of Fig. 12 the distributions of frequencies around the mean values is shown. The mean values being 520052.2435(4) Hz, 520052.2463(4) Hz, 520052.2555(9) Hz for the low, medium and high count rates, respectively. While values corresponding to low and medium count rates are close, the value corresponding to high count rate indicates clearly the presence of the effect. In order to take into account the count-rate-dependent frequency shifts, the so called count-rate-class analysis [39,40] is performed, where the frequency is determined for different numbers of simultaneously trapped ions, and extrapolated to one trapped ion, considering the 35 % detection efficiency, see Fig. 13 for an example. We have applied this treatment to the data set with high count rate, and as a result the mean frequency was reduced by more than 10 mHz.
A successful ion detection generates five timing signals, one on the MCP and one on each end of the two delayline wires. The (x,y) position information is encoded in the four timing signals on the delay-lines, where missing signals would represent a problem for the position reconstruction [41]. The multi-hit performance of the detector and its spatial and temporal resolution is limited. The pulse-height distribution of the signals and limits on discriminator thresholds [41], furthermore the non-zero electronic, readout, and detector dead-times lead to loss of signals and non-reconstructed ion events as the count rate increases. Therefore, during the count-rate-class analysis of PI-ICR data we take the number of MCP signals as number of detected ions, instead of the number of reconstructed events having (x,y) position information. During mass measurements the count rates of the ion of interest and reference ion species are kept similarly low and the count-rate-class analysis is performed for TOF-ICR and PI-ICR data during the evaluation process.

Image distortions
Misalignment between the magnetic and electric field axes and a tilt of the position-sensitive-detector plane with respect to the symmetry axis of the trap electrodes may lead to distortion of the ion-motion projection as described in [7,39]. We investigate the distortion of the ion-motion projection onto the position-sensitive detector using the magnetron motion.
A set of 20 magnetron-motion-projection spots with constant phase shift was collected. The angular step size between points was constant, and the same magnetron drive was used to obtain a constant radius. In Fig. 14 we compare the measured and expected phase-spot positions on the detector and evaluate the phase shift. The same measurement has been repeated using a different magnetron drive leading to a smaller radius. The difference between measured and expected phases and radii was fitted by 3 harmonics.
We used the functions r = r 0 + n [A n · cos (n · ϕ + ϕ 0n )] and δϕ = m [A m · cos (m · ϕ + ϕ 0m )]. The first radial harmonic represents the shift of the center, when centers for circles with different radii are not the same. The second harmonic describes ellipticity, this can be due to a detector tilt. The third harmonic can be explained as the dependence of the image magnification factor on the radius.
In the right panel of Fig. 14 it is visible that the shift varies with the polar angle. To avoid frequency shifts during measurements, we collect ion spots on the same detector position, so that the magnetron and the cyclotron spots overlap. In other words, the phase accumulation time is chosen to be an integer multiple of the period of the motion. Moreover, we select for spot collection the region around the -90 • position on the detector, where the shift is close to zero.

Environmental effects
Changes in atmospheric pressure and room temperature can affect the mass measurement. The atmospheric pressure variations affect the boiling point of the liquid helium in the superconducting magnet cryostat. Thermal expansion or contraction of the materials used to build the experimental setup can lead to mechanical tensions altering the position or the alignment of the trap versus the superconducting magnet. The finite magnetic susceptibilities of certain construction materials in the trap region and their temperature dependence can also induce unwanted effects. The voltage sources connected to the trap electrodes have a non-zero temperature coefficient, and may thus drift or fluctuate with temperature.
To minimize the impact on our measurements, the construction materials were selected carefully, and active pressure, and temperature stabilization systems in the warm bore of the magnet are employed, keeping the temperature constant within ± 50 mK, and the cryostat pressure within ± 0.25 mbar. In our case the cyclotron frequency shifts about 0.1 Hz/K bore temperature change, and about 3.5 mHz/mbar magnet helium cryostat pressure change.
Effects due to slowly changing room temperature and cryostat pressure, on a few minutes time scale, are considered linear and taken into account by interpolating the reference frequency value to the midtime of the frequency measurement of the ion of interest. This is a classical procedure in PTMS [42]. We aligned our beamline first conventionally using optical levels, and the fine alignment of the beamline versus the magnet was done by minimizing the initial magnetron radius in the measurement trap using the position-sensitive detector.
The PI-ICR technique provides access to the radial motions' phases, and can measure the magnetron and reduced cyclotron frequencies. To determine the free cyclotron frequency, f c , for mass measurements we take their sum f c = f − + f + , and apply a correction f , since this relation holds only in an ideal case. In reality the eigenfrequencies depend upon a harmonic distortion of the trap potential, , and a tilt between the axes of the electric and the magnetic fields, θ and φ, so one measuresf [43,44].
If all three eigenfrequencies can be measured directly, the free cyclotron frequency is given by the Brown-Gabrielse invariance theorem [4,5]: which plays a crucial role, since f c determined in this way is also valid in the presence of misalignment and distortion of the trapping potential. Under real conditions the sideband frequencyf c [θ, φ, ] =f − [θ, φ, ] +f + [θ, φ, ] differs from the free cyclotron frequency value by f [θ, φ, ], so one can write [43,44] where f [θ, φ, ] ≈f − In the data evaluation the measured frequencies were linearly interpolated to the center time of the sideband frequency measurements. In each measurement cycle the frequency shift was determined according to Eq. 5.
The results of 71 measurement cycles are shown in Fig. 16, right panel. Averaging 71 values we obtain a mean value f [θ, φ, ] = (8.6 ± 0.7) mHz. Alternatively, the f correction can also be obtained by measuring cyclotronfrequency ratios between ions with well-known mass values.
In the case of a frequency ratio of ions with similar cyclotron frequencies, the systematic error due to f would cancel out to a large extent. When measuring mass ratios of ions with very different masses, f induces a significant systematic shift of the measured mass ratios, and it is frequently referred to as mass-dependent shift in the literature.  [45]. The grey shaded area represents the uncertainty of the literature ratios based on the uncertainties of the individual mass values. Triangles represent values measured by the PI-ICR technique, squares stand for data obtained with the classical ToF-ICR using a 1 s Ramsey-excitation scheme [46]. The smaller errors bars are statistical errors, the larger error bars represent the total uncertainties of our values only, including a systematic error of 2.5 · 10 −9 Table 2 Systematic errors considered during the evaluation of our PI-ICR and TOF-ICR data. Additionally, the count-rate-class analysis was applied to handle the ion count rate effect, and during the f correction the uncertainty of f was taken into account

First results on Pb isotopes with PI-ICR
To test the newly implemented PI-ICR technique at TRIGA-Trap and check possible systematic shifts from yet unknown sources, we measured the cyclotron frequencies of 206 Pb, 207 Pb and 208 Pb isotopes against each other, and derived their ion mass ratios in all combinations. Since their masses and cyclotron frequencies are similar, the f correction contributes only at the 10 −10 level of relative uncertainty. In Fig. 17 our results are compared to the literature values based on a recent work from Pentatrap by Kromer et al. [45]. Triangles represent our results using the PI-ICR technique, and squares represent our ToF-ICR results using 1 s Ramsey-excitation scheme [46]. The smaller and larger error bars represent the statistical and total uncertainties of our measurements, respectively. Table 2 includes the systematic errors taken into account during data evaluation. The y-axis is centered on the literature values, and the grey-shaded area represents their uncertainties. The PI-ICR and ToF-ICR mass measurements were performed with about 50 000 ions each. A good agreement with the literature values can be seen, and a good agreement between the two measurement techniques as well. The PI-ICR values have about 2 to 3 times lower statistical uncertainties than the ToF-ICR values, this becomes less pronounced after taking into account the systematic error. For a more precise comparison the values are listed in Table 3.

Conclusions
We have successfully implemented the PI-ICR technique at TRIGA-Trap. Based on the presented data, the PI-ICR technique reduces the statistical uncertainty by a factor of three compared to the ToF-ICR technique using the Ramseyexcitation scheme, our previously used standard technique. The main advantages of the PI-ICR technique are that it is not Fourier-limited, offers a superior mass resolving power, and it is more sensitive, since each detected ion contributes equally to the result.
The implementation of the position-sensitive detector opened up new ways to tune the trap potential. The overall performance of the setup allows us to perform mass measurements at the ppb level, as demonstrated with stable Pb isotopes.
Combined with the ToF-based axial-frequency measurement we derived all three eigenmotion frequencies and used the Brown-Gabrielse invariance theorem to obtain f (θ, φ, ) = (8.6 ± 0.7) mHz. This is important, since depending on the mass difference between reference ion and ion of interest, the non-zero f induces a systematic shift of the measured mass. The application of carbon cluster ions as mass reference offers the possibility to keep the mass difference between reference ion and ion of interest sufficiently small so that this effect becomes negligible [47,48].
Results of mass measurements carried out on stable Pb isotopes using PI-ICR show an excellent agreement with the high-precision values recently obtained with Pentatrap, encouraging us to attempt further mass measurements using the PI-ICR technique at TRIGA-Trap. 2018 codes. The developers of MM8 and Eva are acknowledged. We thank F. Köhler and C. Smorra for their valuable input to the new fivepole cylindrical trap design. The support by the mechanical workshop of the MPIK in Heidelberg for building the new trap is highly appreciated. Support by the technical staff of the TRIGA Mainz reactor is acknowledged. The crew of SHIPTRAP is acknowledged for fruitful discussions. Financial support by the Max Planck Society is gratefully acknowledged. This work has been financially supported in part by BMBF within the grant 05P21UMFN1.
Funding Open Access funding enabled and organized by Projekt DEAL.

Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: The experimental data used in this study are available from the corresponding author upon reasonable request.] Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/.

Appendix: Description of the phase&ToF method for axial frequency measurement
The measurement of the axial frequency is more challenging than the measurement of the radial frequencies using the PI-ICR technique. As the position-sensitive detector images the radial motions, one would need to couple the axial and one of the radial motions. A simpler axial frequency measurement can be performed as follows.
We start with a slightly detuned timing for the capture of ions in the measurement trap, so the captured ions gain about 0.3 mm axial motion amplitude upon capturing. On the one hand, the amplitude of the axial motion should be sufficiently small in order for the ion to remain in the well-tuned region of the trap. On the other hand, it should be large enough to be visible in the flight-time measurement. The axial ion motion in the trap means that ions are oscillating along the z-axis. By varying the ejection time delay in steps smaller than one period of the axial motion, the ions' phase and their axial velocity at ejection will change, resulting in different flight times from ejection until detection. Repeating this multiple times, a ToF spectrum is obtained where the average flight time behaves like a sinusoidal function (see Fig. 18).
The phase of that function varies as the axial motion phase, offering a tool to check this. In order to obtain the axial frequency with sufficient precision, we need to measure for about 1 s. In our case working with 208 Pb + ions, the period of the axial motion is about 30 μs. To extend the measurement time to 1 second and to record at least 6 phases per 30 μs period is not practicable. One scan would be longer than 33 000 steps. To overcome this problem, we record a short burst of about 2 periods to determine the start phase. After a certain phase accumulation time we record another short burst to determine the end phase. An example is shown on Fig. 18. This method has a well-known problem: the number of full periods between the two bursts is ambiguous, and results with frequency steps equal 1/T acc also fit with the data. To avoid this problem, it is necessary to increase the phase accumulation time during the measurement slowly, beginning with phase accumulation times of 3-4 periods, and increasing until the final phase accumulation time. Then it is possible to fit both bursts with an offset harmonic function, like T oF = T oF 0 + A · sin( f z · (t − t 0 )), or considering damping T oF = T oF 0 + A · e −t/τ sin( f z · (t − t 0 )) with time constant τ , to determine the axial frequency. Fig. 18 The burst to determine the start phase and the burst to determine the end phase separated by a phase accumulation time for an axial frequency measurement performed with 208 Pb + ions. Fitting an offset harmonic function to the data points yields f z = 33 828.31 ± 0.03 Hz