Mass measurements of 99-101In challenge ab initio nuclear theory of the nuclide 100Sn

100Sn is of singular interest for nuclear structure. Its closed-shell proton and neutron configuration exhibit exceptional binding and 100Sn is the heaviest nucleus comprising protons and neutrons in equal number, a feature that enhances the contribution of the short-range, proton-neutron pairing interaction and strongly influences its decay via the weak interaction. Decays studies in the region of 100Sn have attempted to prove its doubly magic character but few have studied it from the ab initio theoretical perspective and none have addressed the odd-proton nuclear forces. Here we present, the first direct measurement of the exotic odd-proton nuclide 100In - the beta-decay daughter of 100Sn - and 99In, only one proton below 100Sn. The most advanced mass spectrometry techniques were used to measure 99In, produced at a rate of only a few ions per second, and to resolve the ground and isomeric states in 101In. The experimental results are confronted with new ab initio many-body approaches. The 100-fold improvement in precision of the 100In mass value exarcebates a striking discrepancy in the atomic mass values of 100Sn deduced from recent beta-decay results.

can be synthesized in comfortable quantities 6 . This is not so for 100 Sn, forming the limit of proton stability due to its extreme neutron deficiency, only just staving off the Coulomb repulsion of the 50 protons. This rare combination of like closed shells causes 100 Sn to have one of the strongest beta transitions and makes it the heaviest self-conjugate nucleus on the nuclear chart.
Nuclei in the immediate vicinity of 100 Sn offer important insight for understanding the single neutron and proton states in this region and constitute an excellent proxy for the study of 100 Sn itself. However, experiments were so far only feasible with in-beam gamma-ray spectroscopy performed at fusionevaporation and fragmentation facilities 4,5,[7][8][9][10] . By direct determination of the nuclear binding energy, high-precision atomic mass measurements provide a crucial model-independent probe of the structural evolution of exotic nuclei. Precision mass measurements are traditionally performed at Isotope Separation On Line (ISOL) facilities, however the production of medium-mass, neutron-deficient nuclides at such facilities is prohibitively difficult, explaining the lack of accurate mass values in the region. Measurements performed recently at the FRS-Ion Catcher at GSI 11 and the CSRe storage-ring in Langzhou 12 (both high-energy, heavy-ion fragmentation facilities) recently extended direct mass measurements to the 101 In ground and isomeric states. However, the 100 In mass value is still constrained 63% indirectly through its beta-decay link to 100 Cd 13 .
Thus, the first experimental challenge overcome in this work was the production and separation of the successfully studied 99,100,101g,101m In states. A detailed schematic of the necessary stages, from radioactive ion beam production to beam purification, preparation and measurement, is shown in Fig. 1. The exotic indium isotopes were produced at the ISOLDE on-line isotope separator located at CERN. A 1.4-GeV proton beam impinged on a thick lanthanum carbide target, producing a swath of neutron-deficient radioactive species of various chemical elements. After diffusion from the heated target, the indium atoms of interest were selectively ionized using a two-step resonance laser ionization scheme provided by the ISOLDE Resonant Ionization Laser Ion Source (RILIS) 14 . The ion beam was extracted from the source and accelerated to an energy of 40 keV. The mass number (A = Z+N) of interest was selected by using ISOLDE's High-Resolution dipole mass Separator (HRS) and delivered to the ISOLTRAP on-line mass spectrometer 15 .

Figure 1 | High-precision mass measurements of neutron-deficient indium isotopes with ISOLTRAP at
ISOLDE/CERN. Radioactive atoms were produced by nuclear reactions of 1.4 GeV protons impinging on a thick lanthanum carbide target. Short-lived indium atoms diffusing from the target were selectively ionized using a two-step laser excitation scheme. The extracted ion beam was mass separated and injected into a radio-frequency ion trap, where it was bunched and cooled. The beam was then processed by a multi-reflection time-of-flight mass separator (MR-ToF MS) to separate the indium ions from the isobaric contaminants. When the precision Penning trap was used for the mass measurement, further cooling and purification of the beam was achieved using a helium buffer-gas-filled preparation Penning trap. In the case of 99 In, for which the production yield was too low, the MR-ToF MS was used to perform the mass measurement. Reference alkali ions were provided by the ISOLTRAP offline ion source (see text for details).
The ions were first accumulated in ISOLTRAP's linear radio-frequency quadrupole cooler and buncher trap 16 . The extracted bunches were subsequently decelerated by a pulsed drift cavity to an energy of 3.2 keV before being purified by the multi-reflection time-of-flight mass separator (MR-ToF MS) 17 , where multiple passages between two electrostatic mirrors rapidly separate the shortlived indium ions from much more abundant molecules of approximately the same mass. For all investigated isotopes, surviving molecular ions 80-82 Sr 19 F + were predominant in the ISOLDE beam. After a typical trapping time of about 25 ms, a resolving power in excess of m/Dm=10 5 was achieved. This combination of speed and high resolving power enables the MR-ToF to perform precise mass measurements of very short-lived species (see Methods). Because of its low production yield of < 10 ions s -1 , 99 In was measured with this latter method only (see typical MR-ToF MS spectrum in Fig. 2).
The rate of 100 In and 101 In after the MR-ToF MS was sufficient to perform Penning-trap mass measurements. For 100 In the conventional Time-of-Flight Ion-Cyclotron-Resonance (ToF-ICR) technique was used (see Methods). Even-N neutron-deficient indium isotopes are known to exhibit long lived isomeric states lying a few hundred keV above the corresponding ground state, owing to the close energy proximity between the pg9/2 and pp1/2 states and their large spin difference. As a result, the A = 101 indium beam delivered to ISOLTRAP was a mixture of such two states so that the novel phase-imaging ion-cyclotron-resonance technique (PI-ICR) 18,19 had to be used to resolve them and ensure the accuracy of the ground-state mass value (see Methods for more details). Table 1 summarizes our experimental results and compares them to the literature. The ISOLTRAP mass values for the ground and isomeric states of 101 In agree well with averages obtained from Refs. 11,12 . The excitation energy is determined to be 668(11) keV, reducing the uncertainty by a factor of four. The ToF-ICR measurement of 101g In is in excellent agreement with the value measured using PI-ICR. 100 In is found to be 130 keV more bound while the mass uncertainty is improved by almost a factor of 90. The mass of 99 In is measured for the first time.  a 100 Sn mass excess of -56488(160) keV. For both decay energies, the 100 Sn mass is found to be more bound than previously inferred. In addition, the almost two standard deviations between the Q-values from Refs. 4,5 yield 100 Sn mass values which differ by 650 keV. We examine the consequences below and resolve this inconsistency.
Because the binding energy is a large quantity, finite differences are commonly used for assessing changes in nuclear structure from the mass surface. Shown in Fig. 2 (open grey symbols) is the twoneutron empirical shell-gap defined as D2n(Z,N0)=ME(Z,N0-2)-2ME(Z,N0)+ME(Z,N0+2), where ME(Z,N0) = Matomic(Z,N0) -(Z+N0) × u(atomic mass unit) is the mass excess of a nucleus with Z protons and a magic neutron number N0. It shows a local maximum at the crossing of a magic proton number, a phenomenon, known as "mutually enhanced magicity" 22 . Hinke et al. 4 while the open blue circle is using the value from Lubos et al. 5 . The inset shows a 2.5 magnification of the D2n(Z,N0+2) curve from Z = 44 to 50.
Since the lack of mass data for the N=48 isotopes of In (Z = 49), Cd (Z = 48) and Ag (Z = 47) prevents deriving this quantity out to 100 Sn we adapt an approach proposed in Ref. 23 using D2n(Z,N0+2), which is inversely correlated to D2n(Z,N0) (filled grey symbols in Fig. 2c). With this difference, a local minimum is observed because the binding energy of the magic neutron number appears in D2n(Z,N0+2) with opposite sign. The case of N = 28 is shown in Fig. 2  In recent years, there has been great progress advancing ab initio calculations in medium-mass nuclei 24,25 up to the tin isotopes 2 based on modern nuclear forces derived from chiral effective field theory of the strong interaction, QCD. Most ab initio approaches are benchmarked on even-even nuclei, which are considerably simpler to compute, but this excludes from the benchmark effects that are only visible in odd nuclei. Among those are the single-particle states accessible to the unpaired nucleon and their interaction with the states of the even-even core, the blocking effect on pairing correlations and, in the case of odd-odd nuclei, the residual interaction between the unpaired proton and neutron. The latter two give rise to an odd-even-staggering (OES) of binding energies which can be quantified by a three-point estimator. Odd systems thus provide a complementary and stringent testing ground for novel theoretical approaches. Among ab initio approaches, the valence space formulation of the in-medium similarity renormalization (VS-IMSRG) group 26 is able to access a broad range of closed and open-shell nuclei in the nuclear chart 27 . In addition, we will explore the novel shell-model coupled-cluster method (SMCC) 28 for the first time in this region. Both the VS-IMSRG and coupled-cluster calculations provide access to a broad range of observables, such as first ab initio calculations of beta decays -up to 100 Sn 3 . The VS-IMSRG was also recently shown to adequately describe both OES of nuclear masses and charge radii in neutron-rich odd-Z copper (Z=29) isotopes 29 .
Here we present new VS-IMSRG and SMCC results that allow direct comparisons to the odd-Z nuclides adjacent to the iconic 100 Sn nucleus.
We have performed cross-shell VS-IMSRG 30 and SMCC calculations using the 1.8/2.0 (EM) twonucleon (NN) and three-nucleon (3N) interactions of Ref. 31 . This interaction is fit to the properties of nuclear systems with only A = 2, 3 and 4 nucleons (with 3N couplings adjusted to reproduce the triton binding energy and the 4 He charge radius) and gives accurate results for ground-state energies of light and medium-mass nuclei 27,32 . To further explore the sensitivity to chiral EFT interactions, we also consider the NN + 3N(lnl) interaction 33 that has proven to constitute a valuable addition to existing chiral Hamiltonians in medium mass nuclei 34 but has yet to be tested in heavier systems. Finally, we show the first results for the 100 Sn region with the DNNLOGO(394) interaction 35 . Calculations with the DNNLOGO(394) interaction and NN + 3N(lnl) were performed using the SMCC and VS-IMSRG methods, respectively. Technical details regarding these computations can be found in the Methods section.  using the 1.8/2.0 (EM) interaction yield D3n(Z,N) trends that agree with our experimental results. The differences between both methods are within estimated theoretical uncertainties (See Methods for details). Calculations performed with the DNNLOGO(394) and NN + 3N(lnl) interactions slightly underestimate the energy but closely follow the experimental trend, like the more explored 1.8/2.0 (EM) interaction. All in all, the predictions vary with the choice of many-body method and nuclear Hamiltonian in a range of 500keV but with all methods yielding excellent trends. In conclusion, we have measured the most exotic masses accessible with present-day facilities in the 100 Sn region. Our mass measurements of 99,100,101g,101m In were made possible only by employing the most advanced mass spectrometry techniques. The high-precision data is used to explore the trends of binding energies in the direct vicinity of 100 Sn. The 90-fold improvement in the 100 In mass amplifies a discrepancy existing in the b-decay Q-values used to derive the 100 Sn mass. The new precision mass data allows for the first time extending the experimental knowledge of binding energy to only one proton below 100 Sn. The 100 Sn region constitutes the frontier for the fast developing ab initio nuclear theoretical approaches. We performed systematic calculations for the first time in the odd-Z indium isotopic chain employing the VS-IMSRG and SMCC methods with three different chiral effective field theory Hamiltonians. Combined with calculations for the neighboring even-Z nuclides, we demonstrate good agreement between experiment and theoretical trends for the three-point neutron and proton odd-even staggering formulations. From these comparisons, we conclude that of the discrepant Q-values of Hinke et al. 4 and Lubos et al. 5

MR-ToF MS mass measurement and analysis
The relation between the time of flight t of a singly charged ion of interest and its mass mion is given by: t = a (mion) 1/2 + b where a and b are device-specific calibration parameters. These can be determined from the measured flight times t1,2 of two reference ions with well-known masses mion,1 and mion,2. From the time-of-flight information of all the singly charged species, the mass of an ion is then calculated from the relation mion 1/2 = CToF Dref + 0.5 Sref with Dref = mion,1 1/2 -mion,2 1/2 , Sref = mion,1 1/2 + mion,2 1/2 and CToF = [2t-t1-t2]/[2(t1-t2)] 24 . The ions' flight times were recorded with a 100 ps resolution.
The peaks corresponding to the indium ions of interest were unambiguously identified by their disappearance when blocking the RILIS lasers. The mean of the ToF distribution corresponding to each ion species was estimated using the unbinned maximum-likelihood method, assuming a Gaussian probability density function (PDF). To cope with the pronounced asymmetries observed in the shape of the time-of-flight distribution, a restricted fit range was used (see Fig. 2). The dependence of the time-of-flight fit to these tails was compared to an analysis using the asymmetric PDF from Ref. 40 . The difference between the extracted mean ToF was subsequently treated as a systematic time-offlight uncertainty and was found to be the dominating contribution in the final uncertainty. When too many ions are trapped in the MR-ToF MS, space-charge effects can cause the time-of-flight difference between two species to shift, affecting the accuracy of the mass determination. To mitigate this effect the count rate was always kept below 8 ions/cycle, which has proven to be a safe limit from previous tests. Nonetheless, count-rate effects were investigated and were found not to be statistically relevant. In the case of 99 In, an additional source of systematic uncertainty was considered. The sensitivity of the extracted time-of-flight to the presence of a possible isomeric state was studied employing a Monte Carlo approach. We assumed that the ratio of ground and isomeric states for 99 In was similar to that observed for 101 In (i.e., 25:1), because the two states in 99 In are expected to have the same spin and parity. Our procedure yields a conservative estimate since the target release efficiencies (expected to be lower for 99 In than 101 In due to shorter half-lives) are not taken into account. The result of this study was treated as an additional systematic uncertainty which was added in quadrature. Note that our MR-ToF MS mass value for 100 In is in good agreement with our Penning-trap value (see Table 1).

Principle of Penning-trap mass spectrometry
Penning-trap mass spectrometry relies upon the determination of the free cyclotron frequency νC=qB/(2πmion) of an ion species stored in magnetic field B and charge q. Comparing νC to the frequency νC,ref of a species of well-known mass yields the frequency ratio r = νC,ref/νC from which the atomic mass value of the ion of interest can be directly calculated. For singly charged ions, the atomic mass of the species of interest is thus expressed as matom = r (matom,ref -me) + me, where me is the electron mass 41 . As contributions from electron binding energies are orders of magnitude smaller than the statistical uncertainty, they are neglected here.

ToF-ICR mass measurements and analysis
The mass of 100 In was measured using the well-established ToF-ICR technique using both the onepulse excitation 42 and the two-pulse, Ramsey-type excitation 43 . In this method, the free-cyclotron frequency of an ion is directly determined. From one experimental cycle to the next, the frequency of an excitation pulse is varied. Following this excitation, the ions are ejected from trap and their timeof-flight to a downstream micro-channel plate detector is measured. The response of the ions to the applied excitation is a resonant process whose resonance frequency is νC and for which a minimum of the time-of-flight is observed. In the Ramsey scheme, two excitation pulses coherent in phase and separated by a waiting time are applied. The measured Ramsey-type ToF-ICR resonance for 100 In is shown in Fig. 4a. For the same total excitation time, this method offers a three-fold precision improvement when compared to the single-pulse ToF-ICR method. In both cases, the analysis was performed using the EVA analysis software and the various sources of systematic uncertainties were treated according to Ref. 44 . A mass value for 101 In was likewise measured and agrees with a value determined by PI-ICR (see below) within one combined standard deviation.

PI-ICR mass measurements and analysis
In order to separate the A = 101 isomers, the recently introduced Phase-Imaging Ion-Cyclotron-Resonance technique was used 18 . With this method, the radial frequency of ions prepared on a pure cyclotron or magnetron orbit is determined through the measurement of the phase they accumulate in a time tacc using the projection of their motion onto a position-sensitive multi-channel plate detector.  A three-step measurement scheme allows for the direct determination of νc. First, a position measurement is performed without preparing the ions on a specific motion radius, yielding the position of the center of the ions' motion. In a second step, the ions are prepared on a pure magnetron orbit, left to evolve freely during a time tacc and their position is measured. Finally, the ions are prepared on a pure reduced cyclotron orbit, left to evolve freely during the same time tacc and their position is again measured. The integer number of revolutions n-and n+, performed in steps 2 and 3 respectively, the phase accumulation time tacc and the angle F between the ions' positions obtained in steps 2 and 3 can be related to νc following the relation νc = [2p(n-+ n+) + F]/ tacc. In step 3, the phase accumulation is performed at the modified cyclotron frequency, so is mass dependent. The position of each ion spot was extracted using the unbinned maximum-likelihood method, assuming a 2D multivariate Gaussian distribution 45 . Figure 4 shows a typical PI-ICR image obtained in step 3 after ~65 ms of phase accumulation. If in principle the angle Fgs,m between the ground and isomeric state directly reflects the energy difference between the two states, the mass of both states was measured separately in order to mitigate systematical effects. The PI-ICR method was used to study the isomeric composition of the 100 In beam. Hence, we can exclude the presence of a long-lived state with an excitation energy larger than 20 keV in the 100 In beam delivered to ISOLTRAP's measurement Penning trap.

VS-IMSRG calculations
The VS-IMSRG calculations 26,46 were performed in a spherical harmonic-oscillator basis including up to 15 major shells in the single-particle basis with an oscillator frequency ℏw = 16 MeV. The 3N interaction configurations were restricted up to e1+e2+e3 £ E3max = 16 for the 1.8/2.0(EM) interaction (to compare with SMCC calculations) and E3max = 22 for the NN + 3N(lnl) interaction. We first transform to the Hartree-Fock basis, then use the Magnus Formulation of the IMSRG 47 to construct an approximate unitary transformation to decouple a 78 Ni core with a proton p1/2, p3/2, f5/2, g9/2 and neutron s1/2, d3/2, d5/2, g7/2, h11/2 valence space. Using the ensemble normal-ordering introduced in 26 , we approximately include effects of 3N interactions between valence nucleons, such that a specific valence-space Hamiltonian is constructed for each nucleus to be studied. The final diagonalization is performed using the KSHELL shell-model code 48 . To estimate theoretical uncertainties in this framework, we note that in the limit of no IMSRG truncations, results would be independent of the chosen reference state for the ensemble normal ordering procedure. Therefore, we examine the referencestate dependence of the observables discussed above. Normal ordering with respect to either a filled neutron g7/2 or d5/2 orbits, we find approximately 1MeV uncertainty for absolute or one-neutron separation energies. However, for all quantities shown in Fig. 3, this estimated uncertainty is approximately 0.1 MeV.

SMCC calculations
The SMCC approach generates effective interactions and operators through the decoupling of a core from a valence space. We start from a single Hartree-Fock 100 Sn reference state, computed in a harmonic oscillator basis comprising up to 11 major oscillator shells and an oscillator frequency ℏw = 16 MeV. The 3N interaction was restricted to E3max = 16ℏw. The doubly closed-shell 100 Sn core is decoupled by a coupled-cluster calculations including singles, doubles and the leading-order triples excitations (CCSDT-1 approximation). We note that triples excitations were performed in the full model-space, without any truncations. Thisx was made possible by employing the Nuclear Tensor Contraction Library (NTCL) 49 developed to run at scale on Summit, the U.S. Department of Energy's 200 petaflop supercomputer operated by the Oak Ridge Leadership Computing Facility (OLCF) at Oak Ridge National Laboratory. The SMCC calculations then proceed via a second similarity transformation that decouples a particle-hole valence-space defined by the proton pfg9/2 holes and neutron g7/2sd single-particle states. The SMCC decoupling only includes the one-and two-body parts of the CCSDT-1 similarity-transformed Hamiltonian. To estimate theoretical uncertainties, we note that the calculation of doubly-magic nuclei such as 100 Sn or 78 Ni and their neighbors are ideally suited for the coupled-cluster method, because the reference state is closed shell 2,46 . Comparison of the SMCC results for 101 Sn with those from Ref 2 exhibit differences on single-particle energies of about 0.2 MeV. We therefore estimate that our theoretical uncertainties on D3n(Z,N) are about ±0.2 MeV.
Code Availibity: The analysis codes used for the ToF-ICR and MRToF-MS data are available from the corresponding author upon reasonable request. A second MRToF-MS analysis code used in this study is available at this address: https://github.com/jonas-ka/mr-tof-analysis. The PI-ICR analysis code used in this study is available at this address: https://github.com/jonas-ka/pi-icr-analysis. The code used for the VS-IMSRG calculations is available at https://github.com/ragnarstroberg/imsrg. The source code of KSHELL is available in Ref. 48 . References: