Study of radial motion phase advance during motion excitations in a Penning trap and accuracy of JYFLTRAP mass spectrometer

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Study of radial motion phase advance during motion excitations in a Penning trap and accuracy of JYFLTRAP mass spectrometer © The Author(s) 2021 Published version Nesterenko, D. A.; Eronen, T.; Ge, Z.; Kankainen, A.; Vilen, M. Nesterenko, D. A., Eronen, T., Ge, Z., Kankainen, A., & Vilen, M. (2021). Study of radial motion phase advance during motion excitations in a Penning trap and accuracy of JYFLTRAP mass spectrometer. European Physical Journal A, 57(11), Article 302. https://doi.org/10.1140/epja/s10050-021-00608-3 2021 Eur. Phys. J. A (2021) 57:302 https://doi.org/10.1140/epja/s10050-021-00608-3 Special Article - New Tools and Techniques Study of radial motion phase advance during motion excitations in a Penning trap and accuracy of JYFLTRAP mass spectrometer D.A. Nesterenko1,a, T. Eronen1,Z.Ge 1, A. Kankainen1, M. Vilen1,2 1University of Jyväskylä, P.O. Box 35, 40014 Jyväskylä, Finland 2Present address: Experimental Physics Department, CERN, CH-1211 Geneva 23, Switzerland Received: 11 July 2021 / Accepted: 16 October 2021 © The Author(s) 2021 Communicated by Klaus Blaum Abstract Phase-imaging ion-cyclotron-resonance technique has been implemented at the Penning-trap mass spectrometer JYFLTRAP and is routinely employed for mass measurements of stable and short-lived nuclides produced at IGISOL facility. Systematic uncertainties that impose limitations on the accuracy of measurements are discussed. It was found out that the phase evolution of the radial motion of ions in a Penning trap during the application of radio-frequency fields leads to a systematic cyclotron frequency shift when more than one ion species is present in the trap during the cyclotron frequency measurement. An analytic expression wasderivedtocorrectly accountfor theshift. Cross-reference mass measurements with carbon-cluster ions have been performed providing the mass-dependent and residual uncertainties. 1 Introduction Penning-trap mass spectrometry is a widely used method for very accurate atomic mass measurements, applicable both for stable and radioactive isotopes down to short halflives (T1/2≈10 ms [1]). The phase-imaging ion-cyclotron- resonance (PI-ICR) technique [2] has become increasingly employed in the Penning-trap mass spectrometry [3–6]. It provides a higher accuracy, sensitivity and resolving power than the conventional time-of-flight ion-cyclotron-resonance (TOF-ICR) technique [7–11]. Higher accuracy is required e.g. for neutrino studies, where accurate mass differences are needed [12,13]. Due to the sensitivity of the PI-ICR method, more exotic nuclei with low production rates far from the stability can be explored. The superior resolving power of the PI-ICR technique enables the studies of low-lying isomeric ae-mails: [email protected];[email protected] (corresponding author) states with excitation energies of a few tens of keV [14,15], unavailable with other mass-spectrometry methods. Systematic effects specific to the PI-ICR technique were considered in detail in [3]. These include collisions of the stored ions with residual gas in a Penning trap, the temporal instability of the trapping voltage, the imperfections of the trapping potential and the uncertainty due to the conversion of the cyclotron motion to the magnetron motion limiting the maximum accuracy and resolving power. In addition to theseeffects,theeffectsduetotheion-ioninteractionsandthe mass-dependent motional phase advance prior to the phase accumulation time [16] have to be also taken into account when more than a single ion species is simultaneously stored in the trap. Ions with different masses gain a phase difference, which is significant only when more than one ion species is present. An analytic expression was derived to correct for this shift. Furthermore, we quantified systematic uncertainties in the mass measurements utilizing the PI-ICR technique at the JYFLTRAP double Penning trapmass spectrometer.The harmonizationprocedure ofthetrapping potentialat JYFLTRAP has already been described in [5]. For a broader range of masses and cyclotron frequencies, i.e., when the ion of interest and the reference ion have different mass numbers, cross reference mass measurements using carbon cluster ions were performed with the PI-ICR technique. The carbon clusters have been successfully employed to demonstrate the Penning trap performance at ISOLTRAP [17], SHIPTRAP [18], JYFLTRAP [19] and LEBIT [20] but these have been performed using the TOF-ICR technique. Here we report on the carbon-cluster measurements with the PI-ICR technique at JYFLTRAP. 0123456789().: V,-vol 123 302 Page 2 of 19 Eur. Phys. J. A (2021) 57:302 Fig. 1 Overview of the IGISOL facility, consisting of the target chamber, the 55◦ dipole magnet, the electrostatic beam switchyard, the gas-filled RFQ, the MR-TOF mass spectrometer and the Penning-trap setup JYFLTRAP. The offline ion source station upstairs and the offline ion source mounted at the quadrupole bender after MR-TOF MS can provide the stable ions for calibration and mass measurements with the Penning traps. Arrows show the direction of the ions 2 Experimental setup and method 2.1 JYFLTRAP double Penning trap mass spectrometer JYFLTRAP is a double Penning-trap mass spectrometer used for high-precision mass measurements of stable and radioactive ions and high-resolution ion beam purification [21]atthe Ion Guide Isotope Separator On-Line (IGISOL) facility [22] (Fig. 1). Radioactive ions are produced via fission, fusionevaporation or multi-nucleon transfer reactions at IGISOL, stopped in helium gas and extracted via a sextupole ion guide (SPIG) [23] to high vacuum, where they are electrostatically accelerated to 30qkV energy (qis the charge of ions). The ion beam is separated by the mass-to-charge ratio using a 55◦ dipole magnet with a mass resolving power (R=m/m) of about 500. The ions with the selected A/qare electrostatically decelerated to ∼100 eV and injected into a gas-filled radio-frequency quadrupole (RFQ) [24]. The ions are thermalized in helium gas at ambient temperature in the RFQ and collected into a potential well from where they are released as a temporally short (∼5μs) ion bunch. This allows an efficient injection of ions into the JYFLTRAP Penning trap. The RFQ is placed at the same + 30 kV high voltage platform as the Penning traps. The bunched ion beam can be additionally purified after the RFQ with the Multi-Reflection Time-of- Flight (MR-TOF) Mass Separator/Spectrometer [25] before injection into JYFLTRAP. Alternatively, ions of stable isotopes can be produced in the glow discharge, surface ionization or laser ablation ion sources placed in the IGISOL target chamber [26], at the offline ion source station [27], in front of the RFQ or in front of the Penning traps (discussed below). Typically, the singly charged ions constitute the main fraction of ions produced at IGISOL. The JYFLTRAP cylindrical double Penning trap system is placed inside a 7-T superconducting magnet (Magnex Scientific). The ions injected into the first (preparation) trap are thermalized, concentrated in the center of the trap both radially and axially and purified from unwanted ion species using the mass-selective buffer gas cooling technique [28]. This technique allows to separate individual isobars (R≤105) and prepare isobarically pure ion samples for high-precision mass measurements or post-trap decay spectroscopy. The ions are transferred through a 1.5 mm diameter diaphragm between the traps into the second (measurement) trap. There, the mass of the ion with mass mand charge qis determined based on its cyclotron frequency νc=1 2π q mB,(1) in the magnetic field B. For ion detection, a position-sensitive microchannel plate (MCP) detector with a delay-line anode (RoentDek GmbH, model DLD40) is located outside the strong magnetic field at ground potential. Thus, the ions extracted from JYFLTRAP hit the detector with 30qkV of beam energy. In 2020, the detector was moved further away from the magnet by about of 12 cm compared to the original configuration [5] to reduce the influence of the magnetic field and to gain a higher efficiency. The detector is now about 104 cm from the center of the measurement trap. The detection efficiency is discussed in Sect. 3.4.2. 2.2 Offline ion source for JYFLTRAP A new offline ion source consisting of a laser ablation and surface ionization ion source (Fig. 2) was mounted at the beam line between the MR-TOF mass separator and the Penning traps, see Fig. 1. A quadrupole ion beam bender, 123 Eur. Phys. J. A (2021) 57:302 Page 3 of 19 302 located in front of the offline ion source, can transport ions to JYFLTRAP either from the offline ion source (straight direction)or fromthe RFQ(deflection by90◦). Thepotential ofthe offline ion source is set 50 V lower than the potentials of the trap endcap electrodes to allow ions to be trapped by switching the voltage on the endcaps. The ions from the ion source have 800qV of energy. The position of the actuator allows to choose ions either from the laser ablation ion source (actuator at lens position) or from the surface ionization source (actuator at surface ionization ion source position). Both ion sources utilize a skimmer electrode and electrostatic lens to form an ion beam. The surface ionization source provides ions of stable isotopes 39,41K, 85,87Rb and 133Cs. They are embedded into the same heatable pellet. The current heating the filament is adjusted based on the required beam intensity and it is typically 1.2–1.5 A. The laser ablation ion source consists of a rotating target holder and Nd:YAG 535 nm laser which operates with up to 10 Hz repetition rate. The laser with the energy per pulse of >2 mJ, bombarding the target with a diameter of 16 mm, is focused to a spot size of about 1 mm in diameter. The target holder rotates changing direction within the angles that cover an individual target in order to prevent the laser continuously ablating the same spot. At its current configuration, three targets can be simultaneously installed in the laser ablation ion source. 2.3 Penning trap mass spectrometry 2.3.1 Basic principle of a Penning trap A Penning trap allows storing and manipulation of charged particles in the confining volume of the trap. At JYFLTRAP, predominantly singly charged ions are trapped for mass measurement or separation. An ion is confined in a Penning trap by the superposition of a strong homogeneous magnetic field  B=Bzezand a quadrupolar electrostatic potential, which has in cylindrical coordinates (z,ρ)the form V(z,ρ)=U0 2d2(z2−ρ2/2), (2) where zand ρ=x2+y2are the axial and radial distance from the trap center, respectively, U0is the trap potential (the potential difference between the ring and the endcap electrodes [21]) and dis the characteristic dimension of the trap defined by the trap geometry. The trajectory of an ion in the trap is a superposition of three independent, ideally harmonic, eigenmotions. One of the motions, called the axial motion, occurs along the magnetic field lines at axial frequency νz. The other two motions are radial motions perpendicular to the magnetic field with Fig. 2 Cut view of the offline ions source, showing the laser ablation source in use. The surface-ionization source is mounted on an actuator, which allows either having the source or a pass-through lens for the ions from the laser ablation source to be on the horizontal beam axis the frequencies ν−and ν+. The magnetron motion with the smaller ν−magnetron frequency is almost mass independent, while the cyclotron motion described by the modified cyclotron frequency ν+is mass dependent. The frequencies of eigenmotions are expressed as [29]: νz=1 2πqU0 md2,(3) ν±=1 2(νc±ν2 c−2ν2 z), (4) and the frequency hierarchy is νc≈ν+νzν−. The sum of the two radial motion frequencies is the ion cyclotron frequency νc: νc=ν++ν−.(5) This relation is valid in case of an ideal Penning trap, where the electric potential is fully harmonic, the magnetic field is perfectly homogeneous and there is no misalignment between the magnetic and electric field axis. A more robust relationship, called the invariance theorem [30,31], requiring measurement of all eigenfrequencies, significantly suppresses the contribution of certain imperfections and allows νcdetermination with high precision: ν2 c=ν2 ++ν2 −+ν2 z.(6) The ion is manipulated in a Penning trap by applying radiofrequency (rf) fields in different configurations. A dipolar rf excitation at an eigenfrequency of the ion (ν+,ν−,νz) canbe used to excitethecorresponding ion motion in thetrap. A quadrupolar rf field at the cyclotron frequency νcallows to convert one radial motion to the other [8]. The ring electrodes of the traps at JYFLTRAP are eight-fold segmented 123 302 Page 4 of 19 Eur. Phys. J. A (2021) 57:302 [21], allowing the application of the dipolar, quadrupolar and octupolar rf fields in the radial plane. For the dipolar excitation of the axial motion the voltages are applied to the endcap electrodes. 2.3.2 Cyclotron frequency determination with phase-imaging ion-cyclotron resonance technique The PI-ICR method [2,3] is based on observation of phase evolution of radial ion motions in a Penning trap. The radial position of ions in the trap is projected onto a positionsensitive MCP detector with a certain magnification factor. Ideally, the projection preserves the angular relations magnifying the relative positions of the ions on the detector. The PIICR method allows to determine the radial frequencies independently or measure directly their sum, i. e. the cyclotron frequency νc(Eq. (5)). For the direct cyclotron frequency determination two excitation patterns, differing by only one step, are applied alternately (Fig. 3). First, the ions that have been cooled, centered and purified in the preparation trap, are transferred to the center of the measurement trap (step 1). Then, the coherent components of the magnetron and axial motions are reduced by dipolar rf pulses at the corresponding frequencies (steps 2a and 2b). After these preparatory steps the cyclotron motion of the ions is excited via a dipolar rf pulse at the frequency ν+(step 3), imprinting a cyclotron phase. The step 4 is the one that differs in the two excitation patterns. This step is a qudrupolar rf pulse at the cyclotron frequency νc, converting the cyclotron motion to the magnetron motion.In pattern1, theconversionνc-pulseisapplied immediately after the ν+-pulse. After, the ions rotate freely and the magnetron motion accumulates a magnetron-motion phase φ−+2πn−=2πν−tacc during phase accumulation time tacc, where ϕ−∈[0,2π)is the phase of the last orbital period and n−the integer number of revolutions. Subsequently, the ions are ejected from the trap (step 5) and their position is projected onto the detector. Position of the ion image of the magnetron phase on the detector is described by the polar angle α−with respect to the trap center (projection of ions from the center of the trap). In pattern 2, the quadrupolar rf pulse at the cyclotron frequency νcis applied after the time tacc, allowing ion motion to accumulate a cyclotron-motion phase φ++2πn+=2πν+tacc, similarly to the magnetronmotion phase, where ϕ+∈[0,2π) and n+the integer number of revolutions. After conversion, the ions are projected onto the detector giving the image of the cyclotron phase at the polar angle α+with respect to the trap center. Note, that the conversion preserves the modulus of the angle of the accumulated cyclotron phase and flips the sign of the angle (see Sect. 2.3.3). Thus, the ion cyclotron motion and the movement of the image on the detector have opposite angular directions. Fig. 3 Measurement sequence in the measurement trap for the cyclotron frequency (νc) determination with the PI-ICR technique. The magnetron and cyclotron phases are accumulated in the patterns 1 and 2, respectively, differing by the position of the conversion pulse of the quadrupolar excitation We emphasize that it is the difference in time of conversion pulse at step 4 between pattern 1 and 2, i.e., the phase accumulation time tacc, that is critical fo νcmeasurement. It needs to be known with a sub-ns precision. With the exception of the timing of step 4, the two patterns are identical. The cyclotron frequency is determined as νc=ν−+ν+=αc+2πnc 2πtacc ,(7) where αc=α+−α−is the angle between the two phase images, ncis the full number of revolutions, which the studied ions would perform in a magnetic field Bin absence of electric field during a phase accumulation time tacc. For initial unambiguous ncdetermination, the cyclotron frequency is determined, e.g., by a quick TOF-ICR measurement with a moderate precision. The number nccan also be deduced with few preliminary PI-ICR measurements starting with short phase accumulation time to keep track of nc. It is enough to perform such a measurement only once every several days due to a rather small drift of the magnetic field (see Sect. 3.2). Precision of the cyclotron frequency determination depends on the duration of the phase accumulation time tacc, the angular sizes of the phase spots and the number 123 Eur. Phys. J. A (2021) 57:302 Page 5 of 19 302 of detected ions N. To understand the uncertainty of the frequency determination let us assume the magnetron and cyclotron phase spots are circular, have the same size, the same radial distance to the trap center (r−∼ =r+≡r) and the same number of detected ions. Therefore, the statistical uncertainty of the cyclotron frequency can be written as: δνc=δαc 2πtacc =r πrtacc√N,(8) where ris the standard deviation of ion distribution of the magnetron or cyclotron phase spot on the detector and Nis the total number of detected ions (N/2 for individual mangetron and cyclotron spots) and assuming no position uncertainty for the center. Initially, the ion distribution is defined by the cooling in the preparation trap. The instability of trapping potential and ion collisions with atoms of residual gas in the measurement trap are the main effects that change the radial motions and result in an increase in the spread of the phase spots on the detector [3]. Both of these effects are stronger with increasing phase accumulation time tacc, thus, limiting tacc typically to ≤1.2 s at JYFLTRAP. 2.3.3 Radiofrequency excitations The duration of the excitation pulses can be chosen as short as one period of the corresponding rf excitation frequency. In practice, the minimum duration is limited by the amplitude of the pulse that can be applied from the function generators and typically is several periods in order to gain the desired effect of the excitation. The dipolar excitation at the modified cyclotron frequency ν+is applied in the measurement scheme to excite the ion’s motion (Fig. 3, step 3). The dipolar excitation is formed by applying two rf-potentials with the phase shifted by πto two opposite segments of the ring electrode. The potential created by the dipolar excitation in a Penning trap is Vd(t)=aUdx ρ0 cos(2πνdt+φd), (9) where Ud,νdand φdare the amplitude, frequency and the initial phase of the dipolar excitation, ρ0is the inner radius of the ring electrode and ais a geometry factor for the trap electrodes [32]. The potential Vd(t)is added to the trapping potential V(z,ρ)(Eq. 2) of the trap. The equations for the radial motion in the trap can be solved by introducing the velocity vectors [29]:  V±=˙ ρ−2πν∓ρ׈z,(10) where ρ=(x,y,0). Following the procedure described in [32] the solution of the equations for  V±can be found using the ansatz  V±(t)= A±(t)e±i(2πν±t+φ0 ±).(11) Resonant cases when the frequency νdis close to the one of the radial eigenfrequencies, either ν+or ν−, are considered. The excitation at the frequency νd≈ν+does not affect the amplitude  A−of the magnetron motion and, similarly, the excitation at νd≈ν−does not affect the amplitude  A+of the cyclotron motion. It is assumed that the ion motion in the trap remains circular, i.e. A± x=∓iA ± yand the equations for the xand ycomponents are combined. Marking the solution for theamplitudenear theresonant modifiedcyclotronfrequency and magnetron frequency of excitation with index “+” and “-”, respectively, it can be written as 1 A± y(t)=A± y(0)+k 4πν± e±iφ±(1−e±i2πν±t), (12) where k=qaUd/(mρ0),ν±=νd−ν±is the frequency detuning parameter, φ±=φd−φ0 ±is the phase difference of the rf field and the existing ion motion, A± y(0)=2π(ν+− ν−)ρ±(0)and ρ±(0)are the initial radii of the cyclotron motion and magnetron motion. The component of the complex velocity vector V± y(t)= |V± y(t)|e±iφ±(t)describes the radial motion, where φ±(t)is the phase of the corresponding radial motion. The radii of the radial eigenmotions are given by: ρ±(t)=|V± y(t)| 2π(ν+−ν−)=1 2π(ν+−ν−)A±2 y(0) +k2 (2πν±)2sin2(πν±t)+A± y(0)k 2πν± ×(cos(φ±)−cos(2πν±t+φ±))1/2 −−−−→ ν±→0 1 2π(ν+−ν−)A±2 y(0)+k2t2 4 +A± y(0)kt sin(φ±)1/2 (13) Let us consider the situation when the ions injected into the center of the measurement trap have a zero radius of the cyclotron motion, i.e. A+ y(0)=ρ+(0)=0. Then, the dipolar excitation at the frequency νd≈ν+is applied at the time t=0. The amplitude |V+ y(t)|and the phase φ+(t)of the cyclotron motion over the excitation time are given by: 1It should be noted that the solution shown here is slightly different than the one given in [32]. 123 302 Page 6 of 19 Eur. Phys. J. A (2021) 57:302 |V+ y(t)|eiφ+(t)=| k 2πν+ sin(πν+t)|· ·ei(2φd+2π(νd+ν+)t+3π)/2−−−−→ ν+→0 kt 2ei(2φd+2π(νd+ν+)t+3π)/2.(14) Hence, the phase is φ+(t)=φd+π(νd+ν+)t+3π/2+2πp,(15) where pis an integer. The phase evolution in time depends on both νdand ν+frequencies, and when they are equal, it changes as φ+(t)∝2πν+t, equally to that of the free cyclotron rotation in absence of the dipolar rf field. The cyclotron phase accumulated during dipolar excitation is the same in both excitation patterns (step 3 in Fig. 3) and its effect is cancelled out. The effect of the finite duration of the dipolar excitation at ν+frequency becomes relevant, when more than one ion species is in the measurement trap. This is discussed in Sect. 3.3. Let us consider now the quadrupolar excitation of the conversion pulse that is achieved by the application of the rf voltage to four segments of the ring electrode at the frequency close to the cyclotron frequency νc, with the same phase to the opposite segments and with the phase shifted by πto the neighboring segments. The theoretical description of the conversion of ion’s radial motions in a Penning trap by an external rf quadrupolar field is given in [8,33]. The influence of the conversion pulse on the motion phase in PI-ICR measurement is discussed in [3]. In the ideal case the ions perform a pure cyclotron motion with the phase φ+(0)at time zero before the conversion. In this case, the conversion pulse initiated at time zero and lasting for a duration τconverts the cyclotron motion phase of the ions into the magnetron motion phase φ−(τ), which is given by φ−(τ) =2πν−+π(νq−νc)τ+φq−φ+(0)+3π 2+2πj, (16) where νqand φqare the frequency and initial phase of the quadrupolar excitation, and jis an integer. In case of the complete conversion, which can only occur when νq=νc and the excitation amplitude is chosen properly, the radius of the magnetron motion after the conversion is equal to the radius of the cyclotron motion before conversion. It can be seen from Eq. 16 that the conversion flips the sign of the initial phase of the cyclotron motion φ+(0).The difference of a certain final phase and the reference phase of the cyclotron motion φf +−φr +is converted to the phase difference of the final and reference phase of the magnetron motion φf −−φr −as follows φf −−φr −=−(φ f +−φr +). (17) Thus, the complete conversion preserves the modulus of the angle between the phases and flips the sign of the angle. A typicaldurationoftheconversionpulseis2msatJYFLTRAP. The excitation patterns for the magnetron and cyclotron phases are applied alternately, i.e. typically every 0.2−2s. Daily fluctuations of the magnetron ν−and cyclotron νcfrequencies at JYFLTRAP (see Sect. 3.2) are so small that the errors due to temporal instability of the ν−and νcfrequencies are negligible in the angle determination. The error of the angle determination occurs when the ions have a certain magnetron-motion amplitude before the conversion [3]. This error vanishes when 2πνctacc =πj,(18) where jis an integer, i.e. the angle error due to conversion is eliminated when the phase accumulation time tacc is a multiple of half the period of the cyclotron frequency [33]. The start times of the conversion pulses at JYFLTRAP are set by the delays td1and td2on the function generators for pattern 1 and pattern 2, respectively (Fig. 3). The delay times are chosen as the number of periods of applied rf excitation at the frequency νq≈νc, providing the condition of Eq. (18) for the phase accumulation time. It can be shown that at νq=νcthe magnetron and cyclotron phase spots have the same angular positions on the detector, i.e. α−=α+, which also reduces the error due to the distortion of ion motion projection (Sect. 3.1). Additionally, to eliminate the error due to the residual magnetron motion, the start time of the dipolar excitation pulse at the frequency ν+is scanned over the magnetron period to average the phase spot position on the detector over the magnetron phases. Similarly, to eliminate the shift due to the possible residual cyclotron motion in case the conversion is incomplete, the extraction time from the measurement trap is also scanned over the period of the modified cyclotron frequency. It allows to average the phase spot position on the detector over the phases of the residual cyclotron motion. This scan results in a smearing of the phase spot at the detector by the angle 2πν−/ν+1◦, which is practically less than the initial spatial distribution of the ions. Thus, a two-dimensional timing scan over the start time of the dipolar excitation (step 3 in Fig.3)andtheextractiontimefromthetrap(step 5inFig.3)is carried out at JYFLTRAP during the PI-ICR measurements. 2.3.4 Calibration of the magnetic field The mass of an ion of interest can be derived from the measured cyclotron frequency νcif the magnetic field is known 123 Eur. Phys. J. A (2021) 57:302 Page 7 of 19 302 (Eq. 1). For calibration of the magnetic field, reference ions with precisely-known mass values are used. The mass is determined based on the measured cyclotron frequency ratio rbetween the reference ion (νc,ref ) and the ion of interest (νc,ioi): r=νc,ref νc,ioi =qref qioi mioi mref ,(19) wheremioi andqioi arethemass andchargeof theionofinterest, respectively, and mref and qref are the mass and charge of the reference ion, respectively. When singly-charged ions are measured, the atomic mass Mioi can be determined as Mioi =(Mref −me)r+me+{rB e,ref −Be,ioi},(20) where Mref is the mass of the reference atom, meis an electron mass, Be,ref and Be,ioi are binding energies of valence electron in reference atom and atom of interest, respectively. Typically,the binding energyofa valence electron is lessthan 10 eV [34] and the term in curly bracket can be neglected. Stable nuclides 85Rb and 133Cs with mass uncertainties 5eV/c2and 8 eV/c2[35], respectively, are commonly used for the magnetic field calibration at the JYFLTRAP. Carbon clusters 12Cn, employed for the studies of systematic uncertainties in this work, have the following advantages. They cover a broad mass range of the chart of nuclides in the steps of 12 atomic mass units and their masses are very well-known, since the atomic mass unit uis defined as 1/12 of the 12C mass. The molecular binding energy per atom of the clusters are well known and range from 3.1 eV in C2to 7eVinC 60 [36]. The reference ion is selected with the mass as close as possible to the mass of the ion of interest to reduce the systematic uncertainties (Sect. 3.5). In order to determine the cyclotron frequency ratio rthe cyclotron frequencies of the reference ion νc,ref and ion of interest νc,ioi are measured alternately. The measurement time of the cyclotron frequency is defined as the midpoint of time between the start and end of the measurement. To temporallyoverlap theion-of-interestand referenceion measurements two calculation procedures have been recently used. In the first well-established (interpolation) method, the frequency νc,ref , measured before (t1) and after (t3) the measurement time t2of the frequency νc,ioi, is linearly interpolated to the time t2of νc,ioi measurement νinter c,ref (t2)=νc,ref (t1)+t2−t1 t3−t1 (νc,ref (t3)−νc,ref (t1)) (21) andasingle frequencyratiori=νinter c,ref (t2)/νc,ioi(t2)isdetermined. The final frequency ratioris a weighted mean ratio of individual ratios riwith the maximum of internal and external error [37]. In the second (polynomial) method (described, for example,in [38]) full setsof νc,ref and νc,int frequencies are simultaneously fitted with n-order polynomials Pn(t)and r·Pn(t), respectively, differing only by a coefficient of proportionality r. The cyclotron frequency ratio ris one of the fit parameters, the other parameters describe the magnetic field drift and, hence, the cyclotron frequency changes in time. The polynomial order (often <10) is chosen to reach the smallest reduced χ2. The interpolation and polynomial methods used in our analysis gave mutually agreeing results. 3 Investigation of the systematic uncertainties at JYFLTRAP 3.1 Distortion of the ion motion projection Distortion of the ion motion projection onto the detector can be due to misalignment between the magnetic and electric field axes and tilt of the detector plane with respect to the symmetry axis of the trap electrodes. Ions extracted from the measurement trap initially follow the diverging magnetic field lines in the region of constant electric field and, then, accelerated to 30qkV of energy, which creates an intermediate focus point in front of the ground electrode [5]. Thus, the assumption of the electric field-free drift region [3] is not applicable in the case of JYFLTRAP. After the focus point the beam freely develops and reach the detector. Distortion of the projection was studied using the magnetron motion, which has a period T−≈605 μs(ν−≈ 1653 Hz). The magnetron motion of 133Cs+ions was excited in the measurement trap by applying a dipolar rf pulse with a duration of 2 magnetron periods at the frequency ν−and amplitude A−before extracting them with different phases (altogether 63 phase points spaced apart from each other by delaying the extraction of ions from the trap in steps of 9.68 μs). The start time of the magnetron excitation was scanned over a magnetron period to average possible shifts of the phase spots due to the magnetron motion existing before the excitation. The center spot was collected after every two rounds of phase scans. This was done by applying no excitation and by scanning the extraction time from the trap over a magnetron period to average out any residual magnetron motion. Ideally, the projection would lead to a perfect circle withacertain constantradius,definedas thedistancebetween the phase spot and center spot on the detector. Figure 4shows the measured variation of radius at different angular positions of the phase spots. The radius variations are similar for different amplitudes of the magnetron excitation (0.7–1.3 V). Knowing the magnetron frequency (period) and the time difference between the phases in the measurement, the expected angles between the phase spots were calculated. The reference phase spot position was taken at the angular 123 302 Page 8 of 19 Eur. Phys. J. A (2021) 57:302 Fig. 4 a Measured projections of 133Cs+ions for 63 phases of the magnetron motion and trap center on the detector. The ion motion was excited in the trap by applying a 1-ms dipolar rf pulse at the magnetron frequency ν−with amplitude of 1.1 V. bMeasured radius as a function of the angular position of the phase spot on the detector for different excitation amplitudes. 0◦angle corresponds to the positive x-axis and angle increases counterclockwise position 0◦and deviation of the measured angle between the phase spot and reference phase spot from the calculated value was determined. This deviation as a function of the angular position of the phase spot on the detector is shown in Fig. 5 for the amplitude of magnetron excitation of 1.1 V. Note, the pulse generator (SpinCore, model PB24-100-4k-PCI), generating the timing triggers for the measurement cycle, has a resolution of 10 ns and the TTL-to-optical converter of the signal introduces the main uncertainty in the timings of 25 ns of jitter. However, this contributes to the uncertainty of the angle determination by an order of magnitude less than the statistical error. The deviation of the angles for each measurement with a certain amplitude of excitation was fitted with a periodicfunction f(α) =A0+9 k=1Aksin(kα−ak), where A0,Akand akare constants. Since the cyclotron frequency Fig. 5 Difference between the calculated and measured angles as a function of the angular position of the phase spot on the detector for the excitation amplitude of 1.1 V. The preferable range of angles used in the PI-ICR measurements is indicated by the vertical dashed lines is determined via the angle αc, which is the difference of the polar angles α+and α−(see Eq. 7), the uncertainty related to the angle shift depends on the value of αcand the angles α+ andα−ofthephase spotpositionsonthedetector. Totakeinto account the angle shift in the cyclotron frequency measurement the systematic uncertainty δsystαc=|f(α+)−f(α−)| can be quadratically added to the statistical uncertainty of the angle αc. Note, that the cyclotron frequency uncertainty caused by the angle shift decreases with increasing the phaseaccumulation time tacc (Eq. 8). In precision mass measurements the angle between the cyclotron and magnetron phase spots αcis tuned to be as close to zero as possible, i. e. the phase-accumulation time is as close to multiple of the period of the cyclotron frequency νc. In practice, the angle αccan remain within a few degrees for several hours with typical fluctuations of the magnetic field (Sect. 3.2). The position of the phase spots is also chosen to lie in the region where the angle shift is almost constant, and, thus, the systematic uncertainty for the angle αcis canceled out. For example, such region is in the ranges of polar angles of 231◦−271◦, where δsystαc≤0.001 rad. This translates to an upper limit for the relative uncertainty of the cyclotron frequency determination of, e.g., singly-charged ions of 133Cs to about of 2 ×10−10[s]/tacc. This uncertainty isseveraltimes smallerthan thetypicalstatistical uncertainty. At worst, the magnetron and cyclotron spot images can deviate by 0.05 rad. This translates the relative uncertainty of 1.2×10−8[s]/tacc and can introduce a significant systematic error. It was observed that the distortion of the projection depends on the preparation of ions in the preparation trap and can differ for different conditions. Thus, the mapping of 123 Eur. Phys. J. A (2021) 57:302 Page 15 of 19 302 Fig. 12 Single cyclotron frequency measurements performed for 12C+ 11 carbon cluster ions at JYFLTRAP. aTOF-ICR spectrum obtained with 400 ms of excitation time. The black points with error bars, represented the mean time-of-flight for each scanned frequency, are fitted with the theoretical curve [8] (red line). bProjection of the trap center and accumulated phase spots on the position-sensitive detector for a single cyclotron frequency measurement in the PI-ICR method with the phase accumulation time 400 ms by residual gas than the typically used monoatomic ions. For example, the ratio of the cyclotron-to-magnetron motion radii r+/r−in the PI-ICR measurement with 400 ms of the phase accumulation time was smaller by 22 % for the 12C+ 11 ions (A = 132) compared to the 133Cs+ions with the similar mass number at the same helium gas flow in the preparation trap. The cyclotron frequency ratio measurements have been performed for the singly-charged carbon cluster ions 12C+ n with 6 ≤n≤15 in three series, where the ions 12C+ 9,12C+ 11 and 12C+ 13 were used as reference ions (Table 1). A typical obtained statistical uncertainty for the frequency ratios was a few×10−9.Theuncertaintyrelated tothemagneticfieldfluctuations (Sect. 3.2) was added quadratically to the statistical uncertainty and made a very minor contribution. Count-rate class analysis for the measured cyclotron frequency ratios was performed [47]. No dependence of the frequency ratio on the number of detected ions was observed and data with detected 1–5 ions/bunch were taken into account in the analysis. The setup was optimized for a certain mass in the frequency ratio measurement. Especially the pressure of the purification trap is optimal for only a small mass range. Thus the masses significantly higher or lighter were not prepared in the preparation trap in an optimal way. Also, the damping effect was stronger for lighter carbon clusters at the same pressure in the trap, since the damping coefficient (Eq. 28) slightly increases with decreasing the cluster size. In addition to the measurements with the carbon clusters two previous PI-ICR measurements with 85,87Rb+and 170,172Yb+ions, which have very well-known mass values [35], were included in the analysis. The cyclotron frequency ratios r=νc(85Rb+)/νc(87Rb+)[5] and r= νc(172Yb+)/νc(170Yb+)[55] were measured with a relative uncertainty of 0.64 and 0.5 ppb, respectively. The molecular binding energy (ionization energy) of the carbon clusters gradually changes in range from about of 5.3 eV to 6.6 eV per atom for the clusters Cnwith 6 ≤n≤15 [36,56]. Since the molecular binding energy is almost a constant, its contribution to the calculated cyclotron frequency ratio of the cluster ions is negligible, less than 10−10 in our cases. The binding energy of valence electron is about 9−10 eV in the studied carbon clusters [57], about 4 eV in Rb+ions [34]and6eVinYb +ions [34] and, thus, its contribution to the frequency ratio is also negligible. The mass of the singly-charged carbon cluster ions was calculated as m(12C+ n)=n×m(12C)−me,whereanatomicmassof carbon m(12C)=12 u. The weighted mean ratios rof the measured individual cyclotronfrequencyratioswerecomparedwiththecalculated frequency ratios rcalc. If the measurements of the cyclotron frequencies results in values which deviate from the correct frequencies by a constant offset, the relative shift of the cyclotron frequency ratio ε(r) r=r−rcalc r∝(mref −mioi)(29) is proportional to the mass difference of the reference ion mref and ion of interest mioi. The cyclotron frequency offset, leading to a mass-dependent ratio shift, can be due to imperfections of the electric-quadrupolar field in a Penning trap or a misalignment of the electrostatic trapping field with respect to the magnetic field axis [58]. The relative deviation of the measured cyclotron frequency ratios from the calculated ratios ε(r)/ris plotted as a function of the mass difference m=(mref −mioi) between the reference ion and the ion of interest in Fig. 13. By fitting the data with a straight line, which is forced to pass 123 302 Page 16 of 19 Eur. Phys. J. A (2021) 57:302 Table 1 Carbon-cluster cross-reference measurements performed at JYFLTRAP with the PI-ICR method. The ions 12C+ 9,12C+ 11 and 12C+ 13 were chosen as the reference ions. The number of individual cyclotron frequency ratios ri=νc(12C+ n,ref )/νc(12C+ n,ioi)/ measured for the ions of interest is shown for each pair 12C+ n,ioi 6 7 8 9 10 11 12 13 14 15 12C+ n,ref A 72 84 96 108 120 132 144 156 168 180 9 108 36 27 26 27 49 11 132 26 45 22 23 29 28 13 156 32 29 44 23 Fig. 13 Relative deviation of the measured cyclotron frequency ratios from the calculated ones as a function of the mass difference between referenceion andionof interest.Thestraightlineis alinearleast-squares fit to the data. The measurements were performed with difference reference ions: 12C+ 9(blue rhombuses), 12C+ 11 (red triangles), 12C+ 13 (green squares), 85Rb+(violet circles) and 172Yb+(pink circles) Fig. 14 Relative deviation of the measured cyclotron frequency ratios after correction for the mass-dependent shift. The dashed lines indicate the residual uncertainty which was added quadratically to the uncertainties of the cyclotron frequency ratios to obtain a reduced chi-square χ2/N=1 through the origin, a mass-dependent shift δmr r=−2.35(81)×10−10/u×(mref −mioi)(30) was obtained. The cyclotron frequency ratios were corrected for the obtained mass-dependent effect and the reduced chi-square χ2/Nfor the (rcorr −rcalc)was greater than one, indicating the presence of an additional residual uncertainty. The residual uncertainty of δresr r=9×10−9(31) was quadratically added to the frequency ratios to satisfy the condition χ2/N≤1. The relative deviation of the corrected cyclotron frequency ratios with the included residual uncertainty is shown in Fig. 14. If only the carbon cluster measurements are taken into the analysis, the similar systematic uncertainties are obtained: δmr/r=−2.39(87)×10−10/u×mand δresr/r= 9.6×10−9. If all the data are restricted to |mref −mioi|≤ 12 u, the mass-dependent shift and residual uncertainty are −2.3(21)×10−10/u×mand 5.3×10−9, respectively. The previous cross-reference mass measurements with carbonclusterions performedatJYFLTRAPusingtheTOF-ICR method with Ramsey excitation patterns resulted in the systematic uncertainties δmr/r=−7.8(3)×10−10/u×mand δresr/r=1.2×10−8for the data with |mref −mioi|≤48 u and δmr/r=−7.5(4)×10−10/u×mand δresr/r= 7.9×10−9for the data with |mref −mioi|≤24 u [19]. The systematic uncertainties δmr/rand δresr/rimpose a limit on the accuracy of mass determination at JYFLTRAP. However, it is worth noting, that in mass measurements with the mass doublets (Aref =Aioi) these systematic uncertainties are cancelled out [47] and the accuracy level better than 10−9[59] can be reached. Such accuracy can also be obtained in mass measurements with ions differing by m=2u [5,55]. In general, it is very rare that reference mass is more than 12u away from the ion-of-interest. 123 Eur. Phys. J. A (2021) 57:302 Page 17 of 19 302 4 Conclusion Inthisworkthesystematicuncertaintiesofthe massmeasurements with the PI-ICR technique at the JYFLTRAP setup are discussed. The uncertainties related to the distortion of the ion motion projection onto the detector can be significantly reduced and maintained at a negligible level compared to the statistical uncertainties in the measurements with small angles αcbetween the magnetron and the cyclotron phase spots. The systematic uncertainty due to the fluctuations of the magnetic field is δB/(BδB)=2.01(25)×10−12 min−1, which is suitable for long-term measurements. A 5-h measurement between two reference measurements would introduce a relative uncertainty of 6.03(75)×10−10, which is much smaller than a typical statistical uncertainty obtained for weakly produced exotic nuclides. The effect of the collisions of the stored ions with residual gas in the measurement trap results in a significant damping of the cyclotron motion and increase in the size of the ion distribution of the cyclotron phase spot with increasing the phase accumulation time tacc. It limits the resolution and accuracy of the angle determination and, therefore, the cyclotron frequency determination. The phase accumulation times at JYFLTRAP are typically chosen up to 1.2 s. The damping effect is more pronounced for the carbon-cluster ions than for the monoatomic ions with the similar masses due to different ion mobilities. In the case when more than one ion species is simultaneously stored in the measurement trap the accumulated magnetron phase position has to be corrected due to finite durations of the excitation pulses. Derivation of an analytic expression of the phase evolution of radial ion motion during the application of rf fields made it possible to accurately correct for this effect. Also, a significant effect of the ion-ion interactions between different ion species in the measurement trap was observed in the count-rate class analysis. To take this effect into account more accurately the combined efficiency of the MCP detector and the data acquisition system was measured as a function of number of detected ions. Typically, the count-rate effect is not observed for a small number of detected ions (∼1−5 ions/bunch) of the same ion species at JYFLTRAP, but with different ion species present, the effect was found to be significant. The cross-reference mass measurements with the carboncluster ions allowed to determine the systematic uncertainties in the case when the ion of interest and the reference ion are not a mass doublet. The mass-dependent and residual uncertainties are δmr/r=−2.35(81)×10−10/u×m and δresr/r=9×10−9, respectively, for the data with |m|=|mref −mioi|≤36 u and δmr/r=−2.3(21)× 10−10/u×mand δresr/r=5.3×10−9, respectively, for the data with |m|≤12 u. For example, mass measurement of ions with m=20 u in the mass region of A∼100 the mass-dependent and residual uncertainties introduce a systematic uncertainty of about 1 keV/c2. This level of accuracy is enough for the mass values of radioactive nuclides far from stability needed for astrophysics or nuclear structure studies. For more precise mass measurement at the level of 10−9and better, the ion of interest and reference should be within |m|≤2u[55] or ideally A/qdoublets. The mass measurements with the carbon-cluster ions are in-line with the earlier measurements [19]. It is also worth noting that in most mass measurements at JYFLTRAP the mass difference between the reference ion and the ion of interest is |m|≤12 u, allowing 5.3×10−9precision to be reached. Acknowledgements This work has been supported by the European Union’s Horizon 2020 research and innovation program under grant agreement No.771036 (ERC CoG MAIDEN) and by the Academy of Finland under projects No.295207 and 327629. Funding Open Access funding provided by University of Jyväskylä (JYU). Data Availability Statement The manuscript has associated data in a data repository. [Authors’ comment: The experimental data used in this study are available from the authors upon reasonable request.] Open Access This article is licensed under a Creative Commons Attribution 4.0InternationalLicense, whichpermits 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/. References 1. M. Smith, M. Brodeur, T. Brunner, S. Ettenauer, A. Lapierre, R. Ringle, V.L. Ryjkov, F. Ames, P. Bricault, G.W.F. Drake, P. Delheij, D. Lunney, F. Sarazin, J. Dilling, First penning-trap mass measurement of the exotic halo nucleus 11 Li. Phys. Rev. Lett. 101, 202501 (2008). https://doi.org/10.1103/PhysRevLett.101.202501 2. S. Eliseev, K. Blaum, M. Block, C. Droese, M. Goncharov, E. MinayaRamirez, D.A. Nesterenko,Y.N. Novikov,L. Schweikhard, Phase-imaging ion-cyclotron-resonance measurements for shortlived nuclides. Phys. Rev. Lett. 110, 082501 (2013). https://doi. org/10.1103/PhysRevLett.110.082501 3. S. Eliseev et al., A phase-imaging technique for cyclotronfrequency measurements. Appl. Phys. B 114(1), 107–128 (2014). https://doi.org/10.1007/s00340-013-5621-0 4. R. Orford et al., Phase-imaging mass measurements with the canadian penning trap mass spectrometer. JPS Conf. Proc. 14, 011102 (2017). https://doi.org/10.7566/JPSCP.14.011102 5. D.A. Nesterenko et al., Phase-Imaging Ion-Cyclotron-Resonance technique at the JYFLTRAP double Penning trap mass spectrometer. Eur. Phys. J. A 54, 154 (2018). https://doi.org/10.1140/epja/ i2018-12589-y 123 302 Page 18 of 19 Eur. Phys. J. A (2021) 57:302 6. J. Karthein, Next-generation mass spectrometry of exotic isotopes and isomers, PhD thesis, CERN https://doi.org/10.17181/CERN. F9ZE.OM6X 7. G. Gräff, H. Kalinowsky, J. Traut, A direct determination of the proton electron mass ratio. Zeitschrift für Physik A Atoms and Nuclei 297, 35 (1980). https://doi.org/10.1007/BF01414243 8. M. König, G. Bollen, H.J. Kluge, T. Otto, J. Szerypo, Quadrupole excitation of stored ion motion at the true cyclotron frequency. Int. J. Mass Spectrom. Ion Processes 142(1–2), 95–116 (1995). https:// doi.org/10.1016/0168-1176(95)04146-C 9. S. George et al., Ramsey method of separated oscillatory fields for high-precisionPenning trapmass spectrometry.Phys. Rev.Lett.98, 162501 (2007). https://doi.org/10.1103/PhysRevLett.98.162501 10. S. George, K. Blaum, F. Herfurth, A. Herlert, M. Kretzschmar, S. Nagy, S. Schwarz, L. Schweikhard, C. Yazidjian, The Ramsey method in high-precision mass spectrometry with Penning traps: experimental results. Int. J. Mass Spectrom. 264(2–3), 110–121 (2007). https://doi.org/10.1016/j.ijms.2007.04.003 11. M. Kretzschmar, The Ramsey method in high-precision mass spectrometry with Penning traps: theoretical foundations. Int. J. Mass Spectrom. 264(2–3), 122–145 (2007). https://doi.org/10.1016/j. ijms.2007.04.002 12. D.A. Nesterenko et al., Direct determination of the atomic mass difference of 187 Re and 187 Os for neutrino physics and cosmochronology. Phys. Rev. C 90, 042501 (2014). https://doi.org/ 10.1103/PhysRevC.90.042501 13. S. Eliseev et al., Direct measurement of the mass difference of 163 Ho and 163 Dy solves the Q-value puzzle for the neutrino mass determination. Phys. Rev. Lett. 115, 062501 (2015). https://doi. org/10.1103/PhysRevLett.115.062501 14. M. Vilén et al., High-precision mass measurements and production of neutron-deficient isotopes using heavy-ion beams at igisol. Phys. Rev. C 100, 054333 (2019). https://doi.org/10.1103/PhysRevC. 100.054333 15. D. Nesterenko et al., Three beta-decaying states in 128 In and 130 In resolved for the first time using Penning-trap techniques. Phys. Lett. B 808, 135642 (2020). https://doi.org/10.1016/j.physletb. 2020.135642 16. R. Orford et al., Improving the measurement sensitivity of the Canadian Penning Trap mass spectrometer through PI-ICR. Nucl. Instrum. Meth. Phys. Res. B 463, 491–495 (2020). https://doi.org/ 10.1016/j.nimb.2019.04.016 17. A. Kellerbauer, K. Blaum, G. Bollen, F.H.H.-J. Kluge, M. Kuckein, E. Sauvan, C. Scheidenberger, L. Schweikhard, From direct to absolute mass measurements: a study of the accuracy of ISOLTRAP. Eur. Phys. J. D 22, 53–64 (2003). https://doi.org/10. 1140/epjd/e2002-00222-0 18. A. Chaudhuri et al., Carbon-cluster mass calibration at SHIPTRAP. Eur. Phys. J. D 45, 47 (2007). https://doi.org/10.1140/epjd/ e2007-00001-5 19. V.-V. Elomaa, T. Eronen, J. Hakala, A. Jokinen, A. Kankainen, I. Moore, S. Rahaman, J. Rissanen, C. Weber, J. Äystö, Systematic studies of the accuracy of the Penning trap mass spectrometer JYFLTRAP. Nucl. Instrum. Meth. Phys. Res. A 612(1), 97–102 (2009). https://doi.org/10.1016/j.nima.2009.09.002 20. S. E. Bustabad, From fundamental fullerenes to the cardinal calcium candidate : the development of a laser ablation ion source and its diverse application at the LEBIT facility, PhD thesis, Michigan State Universityhttps://doi.org/10.25335/M5P712 21. T. Eronen et al., JYFLTRAP: a Penning trap for precision mass spectroscopy and isobaric purification. Eur. Phys. J. A 48(4), 46 (2012). https://doi.org/10.1140/epja/i2012-12046-1 22. I. Moore et al., Towards commissioning the new IGISOL-4 facility. Nucl. Instrum. Methods Phys. Res. B 317, 208–213 (2013). https:// doi.org/10.1016/j.nimb.2013.06.036 23. P. Karvonen, I. Moore, T. Sonoda, T. Kessler, H. Penttilä, K. Peräjärvi, P. Ronkanen, J. Äystö, A sextupole ion beam guide to improve the efficiency and beam quality at IGISOL. Nucl. Instrum. Meth. Phys. Res. B 266(21), 4794–4807 (2008). https://doi.org/10. 1016/j.nimb.2008.07.022 24. A. Nieminen, J. Huikari, A. Jokinen, J. Äystö, P. Campbell, E. Cochrane, Beam cooler for low-energy radioactive ions. Nucl. Instrum. Meth. Phys. Res. A 469(2), 244–253 (2001). https://doi. org/10.1016/S0168-9002(00)00750-6 25. I. group, First isobaric mass separation with the JYFL multireflection time-of-flight mass separator at IGISOL, JYFL Accelerator Newsletter 28 (2) (2020) 1. https://www.jyu.fi/science/en/physics/ current/jyfl-accelerator-news/newsletter2_2020.pdf 26. S. Rahaman, V.-V. Elomaa, T. Eronen, J. Hakala, A. Jokinen, J. Julin, A. Kankainen, A. Saastamoinen, J. Suhonen, C. Weber, J. Äystö, Q values of the 76 Ge and 100 Mo double-beta decays. Phys. Lett.B 662(2), 111–116(2008). https://doi.org/10.1016/j.physletb. 2008.02.047 27. M. Vilén et al., A new off-line ion source facility at IGISOL. Nucl. Instrum. Meth. Phys. Res. Sect. B https://doi.org/10.1016/j.nimb. 2019.04.051 28. G.Savard,S. Becker, G. Bollen,H.J. Kluge, R.B.Moore,T. Otto,L. Schweikhard, H. Stolzenberg, U. Wiess, A new cooling technique for heavy ions in a Penning trap. Phys. Lett. A 158(5), 247–252 (1991). https://doi.org/10.1016/0375-9601(91)91008-2 29. L.S. Brown, G. Gabrielse, Geonium theory: physics of a single electron or ion in a Penning trap. Rev. Mod. Phys. 58, 233–311 (1986). https://doi.org/10.1103/RevModPhys.58.233 30. L.S. Brown, G. Gabrielse, Precision spectroscopy of a charged particle in an imperfect Penning trap. Phys. Rev. A 25, 2423–2425 (1982). https://doi.org/10.1103/PhysRevA.25.2423 31. G. Gabrielse, The true cyclotron frequency for particles and ions in a Penning trap. Int. J. Mass Spectrom. 279(2), 107–112 (2009). https://doi.org/10.1016/j.ijms.2008.10.015 32. K. Blaum, G. Bollen, F. Herfurth, A. Kellerbauer, H.-J. Kluge, M. Kuckein, S. Heinz, P. Schmidt, L. Schweikhard, Recent developments at ISOLTRAP: towards a relative mass accuracy of exotic nuclei below 10 8. J. Phys. B: At. Mol. Opt. Phys. 36(5), 921–930 (2003). https://doi.org/10.1088/0953-4075/36/5/311 33. M. Kretzschmar, On the phase dependence of the interconversion of the motional modes in a Penning trap by quadrupolar excitation. Int. J. Mass Spectrom. 309, 30–38 (2012). https://doi.org/10.1016/ j.ijms.2011.08.022 34. W. Lotz, Electron binding energies in free atoms. J. Opt. Soc. Am. 60(2), 206–210 (1970). https://doi.org/10.1364/JOSA.60.000206 35. M. Wang, W. Huang, F. Kondev, G. Audi, S. Naimi, The AME 2020 atomic mass evaluation (II). tables, graphs and references. Chin. Phys. C 45(3), 030003 (2021). https://doi.org/10.1088/1674-1137/ abddaf 36. D. Tománek, M.A. Schluter, Growth regimes of carbon clusters. Phys. Rev. Lett. 67, 2331–2334 (1991). https://doi.org/10.1103/ PhysRevLett.67.2331 37. R.T. Birge, The calculationof errors by the method of least squares. Phys. Rev. 40, 207–227 (1932). https://doi.org/10.1103/PhysRev. 40.207 38. D. Fink et al., Qvalue and half-lives for the double-β-decay nuclide110 Pd. Phys.Rev.Lett.108(2012).https://doi.org/10.1103/ PhysRevLett.108.062502 39. P.W. Anderson, Y.B. Kim, Hard superconductivity: theory of the motion of abrikosov flux lines. Rev. Mod. Phys. 36, 39–43 (1964). https://doi.org/10.1103/RevModPhys.36.39 40. L. Canete, High precision mass measurements for nuclear astrophysiscs, PhD thesis, University of Jyväskylä. http://urn.fi/URN: ISBN:978-951-39-7693-4 41. C. Droese, M. Block, M. Dworschak, S. Eliseev, E. Minaya Ramirez, D. Nesterenko, L. Schweikhard, Investigation of the mag- 123 Eur. Phys. J. A (2021) 57:302 Page 19 of 19 302 netic field fluctuation and implementation of a temperature and pressure stabilization at shiptrap, Nucl. Instrum. Meth. Phys. Res. A 632 (1) (2011) 157 – 163. https://doi.org/10.1016/j.nima.2010. 12.176 42. T. Eronen, V.-V. Elomaa, U. Hager, J. Hakala, A. Jokinen, A. Kankainen, S. Rahaman, J. Rissanen, C. Weber, J. Äystö, Preparing isomerically pure beams of short-lived nuclei at JYFLTRAP, Nucl. Instrum. Meth. Phys. Res. B 266 (19–20) (2008) 4527 – 4531, Proceedings of the XVth International Conference on Electromagnetic Isotope Separators and Techniques Related to their Applications. https://doi.org/10.1016/j.nimb.2008.05.076 43. M. Vilen et al., Exploring the mass surface near the rare-earth abundance peak via precision mass measurements at JYFLTRAP. Phys. Rev. C 101, 034312 (2020). https://doi.org/10.1103/PhysRevC. 101.034312 44. B.J. Mount, M. Redshaw, E.G. Myers, Qvalue of 115 In →115 Sn (3/2+): the lowest known energy βdecay. Phys. Rev. Lett. 103, 122502 (2009). https://doi.org/10.1103/PhysRevLett.103.122502 45. G. Bollen, R.B. Moore, G. Savard, H. Stolzenberg, The accuracy of heavy ion mass measurements using time of flight ion cyclotron resonance in a Penning trap. J. Appl. Phys. 68, 4355 (1990). https:// doi.org/10.1063/1.346185 46. G. Bollen, H.-J. Kluge, M. König, T. Otto, G. Savard, H. Stolzenberg, R.B. Moore, G. Rouleau, G. Audi, I. Collaboration, Resolution of nuclear ground and isomeric states by a Penning trap mass spectrometer. Phys. Rev. C 46, R2140–R2143 (1992). https://doi. org/10.1103/PhysRevC.46.R2140 47. C. Roux et al., Data analysis of Q-value measurements for doubleelectron capture with SHIPTRAP. Euro. Phys. J. D 67(7), 146 (2013). https://doi.org/10.1140/epjd/e2013-40110-x 48. M. Vilen, Mass measurements and production of ions at IGISOL for the astrophysical r- and rp-processes, PhD thesis, University of Jyväskylä. http://urn.fi/URN:ISBN:978-951-39-7838-9 49. A. Khanam, Yield measurements at IGISOL with a digital data acquisition system, Master’s thesis, University of Jyväskylä. http:// urn.fi/URN:NBN:fi:jyu-201709253810 50. M. Kretzschmar, Calculating damping effects for the ion motion in a Penning trap. Eur. Phys. J. D 48, 313–319 (2008). https://doi. org/10.1140/epjd/e2008-00125-0 51. S. George et al., Damping effects in Penning trap mass spectrometry. Int. J. Mass Spectrom. 299(2), 102–112 (2011). https://doi. org/10.1016/j.ijms.2010.09.030 52. L.A. Viehland, Zero-field mobilities in helium: highly accurate values for use in ion mobility spectrometry. Int. J. Ion Mobil. Spec. 15, 21–29 (2012). https://doi.org/10.1007/s12127-011-0079-4 53. G. von Helden, M.T. Hsu, N. Gotts, M.T. Bowers, Carbon cluster cations with up to 84 atoms: structures, formation mechanism, and reactivity. J. Phys. Chem. 97(31), 8182–8192 (1993). https://doi. org/10.1021/j100133a011 54. A.A. Shvartsburg, G.C. Schatz, M.F. Jarrold, Mobilities of carbon cluster ions: Critical importance of the molecular attractive potential. J. Chem. Phys. 108(6), 2416–2423 (1998). https://doi.org/10. 1063/1.475625 55. D. Nesterenko, R. de Groote, T. Eronen, Z. Ge, M. Hukkanen, A. Jokinen, A. Kankainen, High-precision mass measurement of 168 Yb for verification of nonlinear isotope shift. Int. J. Mass Spectrom. 458, 116435 (2020). https://doi.org/10.1016/j.ijms.2020.116435 56. F. Manby, R. Johnston, C. Roberts, Predatory genetic algorithms, Commun. Math. Comp. Chem. 38 (2001) 111. https://match.pmf. kg.ac.rs/electronic_versions/Match38/match38_111-122.pdf 57. L. Belau, S. E. Wheeler, B. W. Ticknor, M. Ahmed, S. R. Leone, W. D. Allen, H. F. S. III, M. A. Duncan, Ionization thresholds of small carbon clusters: tunable vuv experiments and theory, J. Am. Chem. Soc. 129 (33) (2007) 10229–10243. https://doi.org/ 10.1021/ja072526q 58. G. Bollen et al., ISOLTRAP: a tandem Penning trap system for accurate on-line mass determination of short-lived isotopes. Nucl. Instrum. Meth. Phys. Res. A 368(3), 675–697 (1996). https://doi. org/10.1016/0168-9002(95)00561-7 59. D. Nesterenko, L. Canete, T. Eronen, A. Jokinen, A. Kankainen, Y. Novikov, S. Rinta-Antila, A. de Roubin, M. Vilen, High-precision measurement of the mass difference between 102Pd and 102Ru. Int. J. Mass Spectrom. 435, 204 (2019). https://doi.org/10.1016/j.ijms. 2018.10.038 123