Penning-trap measurement of the $Q$-value of the electron capture in $^{163}\mathrm{Ho}$ for the determination of the electron neutrino mass

The investigation of the absolute scale of the effective neutrino mass remains challenging due to the exclusively weak interaction of neutrinos with all known particles in the standard model of particle physics. Currently, the most precise and least model-dependent upper limit on the electron antineutrino mass is set by the KATRIN experiment from the analysis of the tritium \b{eta}-decay. Another promising approach is the electron capture in $^{163}\mathrm{Ho}$, which is under investigation using microcalorimetry within the ECHo and HOLMES collab orations. An independently measured Q-value of this process is vital for the assessment of systematic uncertainties in the neutrino mass determination. Here, we report a direct, independent determination of this $Q$-value by measuring the free-space cyclotron frequency ratio of highly charged ions of $^{163}\mathrm{Ho}$ and $^{163}\mathrm{Dy}$ in the Penning trap experiment \textsc{Pentatrap}. Combining this ratio with atomic physics calculations of the electronic binding energies yields a $Q$-value of $2863.2(0.6)\,\mathrm{eV}/c^{2}$ - a more than 50-fold improvement over the state-of-the-art. This will enable the determination of the electron neutrino mass on a sub-eV level from the analysis of the electron capture in $^{163}\mathrm{Ho}$.


The absolute scale of the neutrino mass
The observation of the neutrino flavor oscillations proves that neutrinos are massive particles, establishing that the weak neutrino flavor eigenstates are a superposition of three neutrino-mass eigenstates in contradiction to the Standard Model of particle physics [1,2].In oscillation experiments merely the differences of the squared neutrino mass eigenvalues can be investigated, leaving the absolute scale of the neutrino mass an open question.Thus, the absolute scale of the neutrino mass remains one of the most sought-after quantities in nuclear and particle physics, cosmology and beyond Standard Model theories that could potentially explain the origin of the neutrino rest mass [3][4][5][6].
Neutrinos are produced in weak nuclear decays; a model-independent measurement of their rest mass can be performed in a kinematic study of the decay products, where the neutrino itself is not directly detected.Relying on energy and momentum conservation, this is currently the most model-independent approach for neutrino mass determinations.Kinematic investigations constrain the effective rest mass of the electron neutrino or antineutrino m 2 νe = 3 i=1 |U ei | 2 m 2 i , where U f i are the elements of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, which describes the superposition of mass eigenstates m i (i ∈ {1, 2, 3}) in the flavor eigenstates ν f (f ∈ {e, µ, τ }).The individual mass eigenstates are not resolved in these experiments since the squared mass differences are well below current instrumental resolutions, with the largest one being ∆m 2  32 = (2.453± 0.033) • 10 −3 eV 2 /c 4 [7].The most stringent constraint on the neutrino mass scale comes from the analysis of the matter distribution in the universe which results in a limit on the sum of the neutrino masses of < 120 meV/c 2 [8] while the most stringent direct limit of 0.8 eV/c 2 (90 % C.L.) from a kinematic study of the tritium β-decay is set by the KATRIN collaboration [9,10].
Complementary to this approach, there are several experiments using calorimetric techniques to investigate the neutrino rest mass directly.Historically, the first calorimetric approaches were the MANU and MIBETA experiments investigating the 187 Re β-decay yielding upper limits of 19 and 15 eV 2 /c 2 (90% C.L.), respectively [11].Two current experiments, namely ECHo [12,13] and HOLMES [14,15], investigate the electron-capture in 163 Ho → 163 Dy + ν e + E cal , with E cal being the energy detected in a calorimeter.The current upper limit of the electron neutrino rest mass is on a level of < 150 eV/c 2 [13] and the ECHo and HOLMES collaborations aim to achieve sensitivities well below < 1 eV/c 2 [12].
Within the ECHo collaboration, metallic magnetic calorimeters are used for the measurement of the energy of all emitted radiation except for the energy carried away by the neutrino.This is obtained by implanting 163 Ho ions directly into the absorber material of the detector.The calorimetrically measured decay spectrum is subsequently analyzed by fitting it to a theoretical spectral shape from which the Qvalue as well as the effective electron neutrino mass m νe can be determined.In order to quantitatively investigate systematic effects in the interpretation of the calorimetrically measured spectra, that might arise due to the 163 Ho ions being implanted into a metallic material, this Q-value is best compared to one obtained from an independent direct measurement.The required accuracy of ∼ 1 eV/c 2 can currently only be reached using high-precision Penning-trap mass spectrometry (PTMS).In PTMS, the Q-value is addressed directly through a measurement of the mass difference of the mother and daughter nuclides, 163 Ho and 163 Dy, respectively [12,16], by measuring the free cyclotron frequency ratio of the two species in a strong homogeneous magnetic field B. Within a magnetic field, an ion with charge-to-mass ratio q/m is forced onto a circular orbit where it revolves with the free-space cyclotron frequency ν c = 1 2π q m B. In a Penning trap, a superimposed weak quadrupolar electrostatic potential confines the ion along the magnetic field lines and modifies the ion's radial motion: The free-space cyclotron motion splits into the magnetron motion with the frequency ν − and the modified cyclotron motion with frequency ν + .In addition, the quadrupolar electrostatic potential induces a harmonic oscillatory motion with frequency ν z along the magnetic field lines.From a measurement of all three motional eigenfrequencies, the free-space cyclotron frequency can be reconstructed using the invariance theorem . From subsequent measurements of the free-space cyclotron frequency the ratio R q+ = ν c ( 163 Dy q+ )/ν c ( 163 Ho q+ ) is determined, which finally allows the determination of the Q-value by including atomic physics calculations of the binding energy difference ∆E q+ B of the removed electrons: ∆E q+ B is given by the difference in the sum of the binding energies of the n missing electrons in the highly charged ions (HCIs) of both nuclides and m q+ Dy is the "reference" mass of the HCI of dysprosium.q = n • e is the charge of the ions, with e being the elementary charge and n the number of removed electrons ("charge state").In order to enhance the readability in formulas, sometimes q also denotes the number of missing electrons n.

Experimental setup
The measurement of the free-space cyclotron frequency ratio R of the two HCIs 163 Ho q+ and 163 Dy q+ was carried out with the high-precision Penning-trap mass spectrometer PENTATRAP located at the Max-Planck-Institute for Nuclear Physics in Heidelberg, Germany [18,19].An overview of the apparatus is given in Figure 1.
HCIs of the synthetic radioisotope 163 Ho, which was produced by neutron irradiation of stable 162 Er [21], and HCIs of the stable 163 Dy are produced in a compact room-temperature electron beam ion trap (EBIT) that is specifically designed and constructed for the production of HCIs from samples available only in limited quantities (TIP-EBIT) [20].For the measurements reported here only 2 • 10 15 atoms of 163 Ho were used, with a typical sample containing about 10 14 atoms of 163 Ho.HCIs of the two species are extracted with a kinetic energy of 4.4 keV/q from the EBIT and transported through an electrostatic beamline towards the Penning traps.Individual charge states n = {38, 39, 40} are selected using a Bradbury-Nielsen Gate and a fast switching electronic circuit [22,23] located about 1.5 m from the EBIT.Just before reaching the mass spectrometer, the HCIs are decelerated to a few eV/q by appropriately timed voltage pulses on two cylindrical drift tubes.
The mass spectrometer consists of a stack of five identical, cylindrical Penning traps located in the cold bore of a 7 T, actively shielded superconducting magnet [18,24].The voltages applied to the Penning-trap electrodes are supplied from an ultrastable voltage source [25].The Penning traps as well as the detection system are located inside a vacuum chamber immersed in liquid helium at a temperature of about 4 K. Two (trap 2 and trap 3, cf.Fig. 2 (a)) of the five Penning traps are equipped with a non-destructive image-current detection system [18,[26][27][28] and are used for the measurement of the ions' motional frequencies.Trap 1 and trap 4 serve as storage traps while trap 1 is also used as a capture trap when a new set of ions is loaded into the trap stack.
Environmental parameters affecting the magnetic field in the traps are stabilized e.g. the temperature in the laboratory to 0.1 K/day as well as the liquid helium level and pressure of helium gas inside the cold bore of the magnet.In these conditions the magnetic field exhibits a relative drift of a few 10 −10 per hour [29].Frequency

Data analysis
In order to extract frequency ratios R from the free-space cyclotron frequencies ν c , the magnetic field behavior has to be interpolated in-between the individual frequency measurement datapoints from one species to the time when the other species' frequencies were measured.
Fig. 2 (b) shows exemplary the free cyclotron frequencies from one measurement run performed on ions with the charge state q = 38 • e.The linear slope of the data points can be attributed to the slow decay of the magnetic field of the superconducting magnet due to the flux creep effect [30,31] and is on the order of a few 10 −10 per hour relative to the absolute magnetic field of ∼ 7 T.
In the data analysis, the frequency of 163 Dy is linearly interpolated between two datapoints to the time where 163 Ho was measured.From this interpolated datapoint the ratio R is determined as illustrated in Figure 3 (a).This procedure is followed for the full dataset.Residual non-linear behavior of the cyclotron frequency drift, originating from physical effects that alter the temperature and position of magnetic materials that surround the Penning traps and change the magnetic field within the For details on the analysis procedure see main text.Error bars correspond to the 1σ statistical uncertainty which is propagated by Gaussian uncertainty propagation.(a) Linear interpolation between two 163 Dy datapoints to the time at which 163 Ho was measured for the determination of the free-space cyclotron frequency ratio R. Please note that the vertical axis was broken for illustration purposes.(b) Exemplary estimation of the non-linearity by interpolation of the data onto itself.Here we linearly interpolate between the first and third datapoint and determine the difference between the measured datapoint in between and the interpolated one.The sum of these "residuals" divided by the number of residuals in the full dataset is taken into account as an additional uncertainty on the ratio.
traps is taken into account in the uncertainty of the interpolated R. For this, the frequency data points are interpolated back to themselves (see Figure 3 (b)), and the sum of the residuals divided by the number of residuals is included as an additional uncertainty in the ratio.The resulting ratios R i = ν c,i ( 163 Dy 38+ )/ν c,i ( 163 Ho 38+ ) for the two measurement runs are shown in Fig. 2 (c) for both traps.The ratios for the individual traps are consistent, therefore the final ratio is calculated as the weighted average and shown as a red line including the 1σ uncertainty band.For the calculation of the uncertainty of the final ratio, the inner error σ 2 int and the outer error σ 2 ext are calculated and the larger of the two is used as the final uncertainty [32,33]: Here, R i and σ i are individual cyclotron frequency ratios and their corresponding 1σ uncertainty, R is the weighted average and N is the total number of ratios.The total systematic uncertainty (e.g.field anharmonicities and inhomogeneity, image charge shift and relativistic shift) is strongly suppressed due to the fact that 163 Ho and 163 Dy in the same charge state form a unique mass doublet with a sufficiently small mass difference of about 2.8 keV.With a difference in mass-to-charge ratio of only about 10 −8 the same trapping potential is used for both 163 Ho and 163 Dy and the magnetron and axial frequencies are sufficiently equal.Thus, all systematic uncertainties in the free-space cyclotron frequency measurement cancel out to a large extent in the determination of the frequency ratio R and are smaller than 10 −12 .Extended Data Table 3 summarizes the considered systematic shifts.An additional systematic uncertainty can arise from the fact that HCIs might have long-lived low-energy atomic metastable states, as observed in previous measurements [34].This is undesirable since it will shift the determined Q-value by the energy of the metastable state.In Section 4 we compare the Q-values resulting from the measurements of three different charge states which allows us to exclude potential shifts of the Q-value due to long-lived electronic metastable states that would influence each charge state differently.The final ratios of free cyclotron frequencies of the ions in the different charge states are summarized in Table 2.

Configuration interaction method
In a first set of calculations, the binding energies are calculated in Quanty [35][36][37] using the configuration interaction (CI) method.The starting point is a fully relativistic density functional theory (DFT) calculation with the full-potential local-orbital minimum-basis code FPLO [38][39][40].The DFT calculation determines the groundstate density of the ion around which a CI expansion is made.The corresponding Kohn-Sham orbitals are used as single particle basis to construct the Slater determinants that span a configuration space.The Hamiltonian comprises Coulomb and static Breit interaction between the electrons as well as their relativistic kinetic energies and potential energies due to Coulomb attraction of the ion's nucleus.Diagonalization of this Hamiltonian on a given configuration space using the Lánczos algorithm determines the ground-state energy of an ion.
At first, only the space of the ground state configuration is considered.Subsequently the configuration space is iteratively expanded to include single, double and triple excitations of electrons into orbitals with higher principal quantum numbers.Details of these calculations are given in the Methods section.We arrive to the calculated binding energy differences given in Table 1.
In the MCDHF method, the atomic state function is modeled as a superposition of configuration state functions (CSFs) with fixed angular momentum, magnetic and parity quantum numbers.The CSFs are built as Slater determinants of Dirac orbitals in the jj coupling scheme.Using the parallel GRASP2018 codes [43], we expand the space of virtual orbitals used for the construction of CSFs by single and double electron exchanges in a systematic manner.The convergence of the energies with respect to the maximal principal quantum number of virtual orbitals is monitored, and the spread of values resulting from different correlation models is used as a measure of the leading contribution (90%) of the theoretical uncertainties.In case of the HCI, the set of CSFs is generated with exchanges including all occupied orbitals from 1s on, and with virtual orbitals up to typically 10h.Virtual orbitals are optimized in a layerby-layer fashion [43,44].Effects of the Breit interaction, recoil, and approximate quantum electrodynamic corrections are accounted for by the configuration interaction method using orbitals from the MCDHF procedure [43].More details are given in the Methods.We obtain the theoretical values of the binding energy differences listed in Table 1.
In a third set of calculations, we use the Multiconfiguration Dirac-Fock General-Matrix-Elements (MCDFGME) code [45], to check the previous results.The calculation is performed in the optimized level mode, where all correlation orbitals are fully relaxed instead of the layer by layer method.Convergence is much more difficult in this case and limits the number of extra orbitals that can be added in the evaluation of correlation.In this calculation, the magnetic and retardation interaction at the Breit level are included in the Dirac-Fock equations on the same footing as the Coulomb interaction, meaning that the Breit interaction is included to all orders in the correlation energy [46].The Uehling potential is also evaluated to all orders [47].Finally, self-energy screening is calculated using both the Welton method [48] and the model operator method [49].For the HCIs, energies obtained by exciting occupied orbitals from 3s or 3d to open shells (4f , 6p, 5d, 7s, 7p and 5g) were compared.For neutrals, values obtained by exciting the core from 3d and 4s were compared.Calculations included only single and double excitations, as triple excitations lead to unmanageably large numbers of magnetic and retardation integrals.All possible single excitations were included, even those obeying Brillouin's theorem [45].The results are given in Table 1 and are in good agreement with the GRASP2018 evaluation.Both sets of values are in agreement with the uncorrelated values [50], confirming the good compensation of correlation between the two ions.

Final values for the binding energy difference
The final binding energy ∆E q+ B for each charge state q is calculated as the weighted average of the values from the CI and MCDHF calculations (cf.Table 1).The uncertainty is determined by comparing the inner and outer errors and using the larger one as final uncertainty on ∆E q+ B .For the charge states q = {39, 40} • e, the larger of the two uncertainties is averaged with the uncertainty assuming correlations between the CI and MCDHF methods, i.e. with the uncertainty of 0.8 eV of the MCDHF method.The resulting ∆E q+ B are consistent with the MCDFGME calculations described above as well as with the calculations recently published in [51].Q-value determination of the 163   1 Summary of the electronic binding energy differences for the three charge states (first column) in electron volts (eV) from the three theory calculations: Configuration interaction (CI) method (second column), the Multiconfiguration Dirac-Hartree-Fock (MCDHF) method (third column) and the Multiconfiguration Dirac-Fock General-Matrix-Elements (MCDFGME) method (fourth column).The given uncertainties correspond to the 1σ uncertainty.In the last column the final ∆E B for the determination of the Q-value are given which were calculated as the weighted average of the CI and MCDHF methods.For details on the calculation of the uncertainties see main text.

Q-value determination
The Q-value of the electron capture in 163 Ho is determined from the measured ratio of the free cyclotron frequencies R (see Section 2) and the theoretically calculated binding energy differences ∆E q+ B (see Section 3 and Table 1) for each charge state q = {38, 39, 40} • e according to Equation 1.The (reference) mass m q+ Dy of 163 Dy q+ is calculated starting from the mass of atomic 163 Dy, m Dy [52], and subtracting the masses of the n missing electrons [53,54] and their binding energies [55].Table 2 lists the ratios R for the three measured charge states as well as the 1σ uncertainty δR which is computed using standard Gaussian uncertainty propagation.
Using Equation ( 1) and the binding energy differences, the final Q-values are calculated for the three charge states and are summarized in Table 2 Table 2 Summary of the main results for the three charge states (column one): Weighted averages of the ratios (second column) and their uncertainty (third column), the weighted averages of the binding energy differences (fourth column, c.f. Table 1) and the calculated Q-values (fifth column) for the three measured charge states.Uncertainties correspond to the 1σ statistical uncertainty.For details on the calculation of the uncertainties see main text.
The resulting Q-values for the different charge states agree within their 1σ uncertainties.Resulting from the very good agreement, systematic deviations from either the free cyclotron ratio measurement or from the calculation of the binding energy difference can be excluded to a large extent.Furthermore, also the influence of unknown metastable electronic states can be largely ruled out since it is very unlikely that an electronic metastable state has exactly the same excitation energy in all three of the measured charge states.
The final Q-value is calculated as the weighted average of the Q-values obtained for the three charge states resulting in: This value is in 1σ agreement with the previously measured value at SHIPTRAP of 2833 (34)eV/c 2 [56] but 50 times more precise.In Figure 4 the most recent measurements of the Q-value of 163 Ho from cryogenic microcalorimetry, PTMS and the Atomic Mass Evaluation (AME) 2020 are shown.The value from the AME 2020 is an average from three different microcalorimetric measurements.Our value is slightly higher than the current AME adjustment and agrees within 1.2σ.Fig. 4 Comparison of the most recent measurements of the 163 Ho EC Q-value from PTMS (SHIP-TRAP [56] and "this work"), microcalorimetry (ECHo 2017 [57] and ECHo 2019 [13]) and the most recent AME adjustment 2020 [52].
The 163 Ho Q EC -value was obtained by combining a high-precision measurement of the free-space cyclotron frequency of HCIs of the mother and daughter nuclide in a Penning trap and precise atomic physics calculations of the electronic binding energies of the missing electrons.Experiments investigating the electron neutrino mass by microcalorimetric measurements of the decay spectrum of 163 Ho such as those of the ECHo and HOLMES collaborations are now provided with an independently measured Q-value with an unprecedented precision of 0.6 eV, which allows the assessment of systematic uncertainties in the neutrino mass determination using cryogenic microcalorimetry on a level of < 1 eV.

Measurement preparation and sequence
The measurement preparation starts with loading a set of three ions, in the order 163 Dy, 163 Ho and 163 Dy into traps 2, 3, and 4, respectively (cf.Fig. 2 (a)).Each HCI is first loaded into trap 2 where its motional amplitudes are reduced by resistive cooling [17,58].Great care is taken to ensure that only a single HCI is captured in a trap and cooled.For this also the "magnetron cleaning" technique is applied [59].After being prepared in this way, the ion is moved to one of the following traps and stored until the set of ions for a measurement is complete.
In both measurement traps (traps 2 and 3), the motional frequencies of the HCIs are measured using the single-dip, double-dip and Pulse-and-Phase (PnP) techniques [58,60].The magnetron frequency is small compared to the other two motional frequencies and depends only very weakly on the ion's mass and is therefore measured only once a day using the double-dip technique prior to the main measurement sequence.Thus, the main measurement sequence reduces to a measurement of the modified cyclotron frequency (PnP technique) and the axial frequency (doubledip technique) which are performed simultaneously in traps 2 and 3. Compared to a single-trap measurement, this effectively doubles the statistics and furthermore allows different analysis methods to be employed as well as systematic checks by comparing the results obtained in both traps.
In the PnP cycle, the starting phase of the cyclotron motion is set by exciting it using a dipolar pulse with the frequency determined with the double-dip method during the preparation.The modified cyclotron motion then evolves freely during the phase evolution time T evol (about 40 s) while the axial frequency is determined using a dip measurement [19,29,61].Following the phase evolution time, the phase information that accumulated in the modified cyclotron motion is coupled to the axial motion using a π-pulse on the sideband frequency and the final phase is measured with the image current detection system [60].This is done in traps 2 and 3, starting with the ions in position 1.Subsequently, the ions are shuttled to position 2, which effectively swaps the ion species in traps 2 and 3 and the measurement is repeated (cf.Fig. 2 (a)).This sequence is repeated 24 times in one main measurement loop and can be continued in principle infinitely long.Typically, the measurement is stopped due to either external magnetic field perturbations or charge exchange of the HCIs.Lifetimes of the HCIs until a charge exchange process happens are up to 36 hours.Reloading ions is beneficial since it allows one to compare different sets of ions and therefore also systematic checks for contaminant ions that might be present in the Penning traps during the measurement or for possible metastable electronically excited states in the HCIs [34].

Fig. 1
Fig.1Rendered overview of the PENTATRAP experimental setup.The upper horizontal part of the beamline is located on the ground floor while the superconducting magnet is located in a dedicated laboratory in the basement.The TIP-EBIT is an electron beam ion trap specifically designed for very small samples sizes[20].Follwing the TIP-EBIT in the horizontal beamline, a Bradbury-Nielsen gate is used to separate a single charge state.HCIs produced in the TIP-EBIT are guided through the electrostatic beamline to the stack of five identical Penning traps in the superconducting magnet.For capturing the HCIs in the Penning traps deceleration electrodes with appropriately timed voltage pulses are used.A more detailed view of the Penning trap stack is shown on the right.

Position 1 Position 2 Trap 1 Fig. 2
Fig. 2 Overview of the measurement procedure and resulting data structure.Error bars correspond to the 1σ statistical uncertainty which is propagated by Gaussian uncertainty propagation.(a) Rendering of the stack of five identical, cylindrical Penning traps of the PENTATRAP experiment.Traps 2 and 3, with labels marked in red, are used as measurement traps and are equipped with a detection system.Shuttling the ions from Position 1 into Position 2 effectively swaps the ion species in traps 2 and 3 resulting in the alternating data structure as shown in panel (b).The traps 1 and 4 are used as storage traps while trap 5 is not used in this measurement.(b) Exemplary dataset of the measured free cyclotron frequencies νc of 163 Ho (orange) and 163 Dy (blue) in the traps 2 (upper panel) and 3 (lower panel) for one measurement run in charge state q = 38 • e.For trap 2 and 3 frequency offsets of 25081589 Hz and 25081620 Hz were subtracted.The linear drift of the free cyclotron frequency which can be attributed to the slow decay of the magnetic field of the superconducting magnet due to the flux creep effect [30, 31].Please note that the vertical axis is broken for illustrative purposes while there are no left and right sub-panels.(c) Ratios R i of the free cyclotron frequencies νc of 163 Dy and 163 Ho in traps 2 (filled circles) and 3 (empty circles) determined from the full dataset of two runs for the charge state n = 38.The data of each run is shown in a dedicated sub-panel where the ratios from (b) are shown in the left sub-panel.The horizontal black line indicates the weighted average of all measured ratios for this charge state with the light red band marking the 1σ uncertainty band.

Fig. 3
Fig.3Detailed plot of the first few datapoints of the cyclotron frequency νc from Figure2(b) in order to illustrate the data analysis procedure.From the frequency values an offset of 25081589 Hz is subtracted.For details on the analysis procedure see main text.Error bars correspond to the 1σ statistical uncertainty which is propagated by Gaussian uncertainty propagation.(a) Linear interpolation between two 163 Dy datapoints to the time at which 163 Ho was measured for the determination of the free-space cyclotron frequency ratio R. Please note that the vertical axis was broken for illustration purposes.(b) Exemplary estimation of the non-linearity by interpolation of the data onto itself.Here we linearly interpolate between the first and third datapoint and determine the difference between the measured datapoint in between and the interpolated one.The sum of these "residuals" divided by the number of residuals in the full dataset is taken into account as an additional uncertainty on the ratio.