Abstract
The tin isotope ^{100}Sn is of singular interest for nuclear structure due to its closedshell proton and neutron configurations. It is also the heaviest nucleus comprising protons and neutrons in equal numbers—a feature that enhances the contribution of the shortrange proton–neutron pairing interaction and strongly influences its decay via the weak interaction. Decay studies in the region of ^{100}Sn have attempted to prove its doubly magic character^{1} but few have studied it from an ab initio theoretical perspective^{2,3}, and none of these has addressed the oddproton neighbours, which are inherently more difficult to describe but crucial for a complete test of nuclear forces. Here we present direct mass measurements of the exotic oddproton nuclide ^{100}In, the betadecay daughter of ^{100}Sn, and of ^{99}In, with one proton less than ^{100}Sn. We use advanced mass spectrometry techniques to measure ^{99}In, which is produced at a rate of only a few ions per second, and to resolve the ground and isomeric states in ^{101}In. The experimental results are compared with ab initio manybody calculations. The 100fold improvement in precision of the ^{100}In mass value highlights a discrepancy in the atomicmass values of ^{100}Sn deduced from recent betadecay results^{4,5}.
Main
The nuclear landscape is shaped by the underlying strong, weak and electromagnetic forces. The most salient features are the pillars of enhanced differential binding energy associated with closedshell configurations, the best example of which is Z = 50 (tin), featuring the largest number of βstable isotopes (10) of all elements. These nuclides lie between the closed neutron shells N = 50 and 82, conferring particular importance to the nuclides ^{100}Sn and ^{132}Sn. The neutronrich ^{132}Sn 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 selfconjugate nucleus on the nuclear chart.
Nuclei in the immediate vicinity of ^{100}Sn offer important insight for understanding the singleneutron and proton states in this region and constitute an excellent proxy for the study of ^{100}Sn itself. However, experiments have so far only been feasible with inbeam gammaray spectroscopy at fragmentation facilities^{4,5,7,8,9,10}. By direct determination of the nuclear binding energy, highprecision atomicmass measurements provide a crucial modelindependent probe of the structural evolution of exotic nuclei. Precision mass measurements are traditionally performed at isotope separation online (ISOL) facilities; however, the production of mediummass, neutrondeficient nuclides at such facilities is prohibitively difficult, explaining the lack of accurate mass values in the region. Measurements performed at the FRS Ion Catcher at GSI^{11} and the CoolerStorage experimental Ring (CSRe) in Langzhou^{12} (both highenergy, heavyion 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 betadecay link to ^{100}Cd (ref. ^{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 Isotope Separator On Line Device (ISOLDE) located at CERN. A 1.4 GeV proton beam impinged on a thick lanthanum carbide target, producing a swath of neutrondeficient radioactive species of various chemical elements. After diffusion from the heated target, the indium atoms of interest were selectively ionized using a twostep 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 using ISOLDE’s highresolution dipole mass separator and delivered to the ISOLTRAP online mass spectrometer^{15}.
The ions were first accumulated in ISOLTRAP’s linear radiofrequency 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 multireflection timeofflight mass spectrometer (MRToF 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/Δm = 10^{5} was achieved. This combination of speed and high resolving power enables the MRToF MS to perform precise mass measurements of very shortlived species (Methods). Because of its low production yield of <10 ions per second, the mass of ^{99}In was measured with this latter method only (see typical MRToF MS spectrum in Fig. 2).
The rate of ^{100}In and ^{101}In behind the MRToF MS was sufficient to perform Penningtrap mass measurements. For ^{100}In the conventional timeofflight ioncyclotronresonance (ToFICR) technique was used (Methods and Extended Data Fig. 1). EvenN neutrondeficient indium isotopes are known to exhibit longlived isomeric states lying a few hundred kiloelectronvolts above the corresponding ground state, owing to the close energy proximity between the πg_{9/2} and πp_{1/2} states and their large spin difference. As a result, the A = 101 indium beam delivered to ISOLTRAP was a mixture of two such states, so the phaseimaging ioncyclotronresonance (PIICR) technique^{18,19} had to be used to resolve them and ensure the accuracy of the groundstate mass value (Methods and Extended Data Fig. 1 for more details).
Table 1 summarizes our experimental results and compares them with 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 ToFICR measurement of ^{101g}In is in excellent agreement with the value measured using PIICR. ^{100}In is found to be 130 keV more bound, while the mass uncertainty is improved by almost a factor of 90.
Since the ^{100}Sn 2016 AtomicMass Evaluation (AME2016) mass excess value of −57,280(300) keV (ref. ^{20}) is derived from that of ^{100}In and the βdecay energy of ref. ^{4}, our ^{100}In result improves the ^{100}Sn mass excess to −57,148(240) keV. However, combining our result with the more recently published βdecay Qvalue (that is, the energy released in the decay) from ref. ^{5} yields a ^{100}Sn mass excess of −56,488(160) keV. For both decay energies, the ^{100}Sn mass is found to be more bound than previously inferred. In addition, the almost 2 s.d. between the Qvalues from refs. ^{4,5} yields ^{100}Sn mass values that 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 Δ_{2n}(Z, N_{0}) = M_{E}(Z, N_{0} − 2) − 2M_{E}(Z, N_{0}) + M_{E}(Z, N_{0} + 2), where M_{E}(Z, N_{0}) = M_{atomic}(Z, N_{0}) – (Z + N_{0}) × u (atomic mass unit) is the mass excess of a nucleus with Z protons and a magic neutron number N_{0}. It shows a local maximum at the crossing of a magic proton number, a phenomenon known as ‘mutually enhanced magicity’^{21}.
Since the lack of mass data for the N = 48 isotopes of In (Z = 49), Cd (Z = 48) and Ag (Z = 47) prevents derivation of this quantity out to ^{100}Sn, we adapt an approach proposed in ref. ^{22} using Δ_{2n}(Z, N_{0} + 2), which is inversely correlated to Δ_{2n}(Z, N_{0}) (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 Δ_{2n}(Z, N_{0} + 2) with opposite sign. The case of N = 28 is shown in Fig. 2 for illustration. Our data allows extending Δ_{2n}(Z, N_{0} + 2) to Z = 49 (indium) and indicates a slight downward trend towards Z = 50 (Fig. 2 inset), as expected for a doubly magic ^{100}Sn. Eliminating the contribution of the ^{100}In groundstate mass uncertainty in the calculation of the ^{100}Sn mass directly allows to confront the nuclearstructure implications of the two Qvalues from refs. ^{4,5}, and a global picture now emerges for this region. As shown, the Qvalue reported by Lubos et al.^{5} yields a ^{100}Sn mass value that is at odds with the expected trend of Δ_{2n}(Z, N_{0} + 2) to Z = 49 (open blue circle in the bottom panel of Fig. 2), whereas the value of Hinke et al.^{4} yields a ^{100}Sn mass that agrees with the trend within experimental uncertainties and is in line with our observation for Z = 49. In other words, while the Qvalue reported in ref. ^{4} follows the expectation of a doubly magic ^{100}Sn, the more recent (and higherstatistics) Qvalue reported in ref. ^{5} yields a ^{100}Sn mass value that suggests quite the opposite. Such a conclusion is at odds with ab initio manybody calculations as discussed below.
In recent years, there has been great progress advancing ab initio calculations in mediummass nuclei^{23,24} up to the tin isotopes^{2} based on modern nuclear forces derived from chiral effective field theory of the strong interaction. 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 these are the singleparticle 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 threepoint estimator. Odd systems thus provide a complementary and stringent testing ground for stateoftheart theoretical approaches. Among ab initio approaches, the valencespace formulation of the inmedium similarity renormalization group (VSIMSRG)^{25} is able to access a broad range of closed and openshell nuclei in the nuclear chart^{26}. In addition, we will explore the shellmodel coupledcluster (SMCC) method^{27} in this region. Both the VSIMSRG and coupledcluster calculations provide access to a broad range of observables, such as ab initio calculations of beta decays—up to ^{100}Sn (ref. ^{3}). The VSIMSRG was also recently shown to adequately describe both OES of nuclear masses and charge radii in neutronrich oddZ copper (Z = 29) isotopes^{28}. Here we present VSIMSRG and SMCC results that allow direct comparisons with the oddZ nuclides adjacent to the iconic ^{100}Sn nucleus.
We have performed crossshell VSIMSRG^{29} and SMCC calculations using the 1.8/2.0(EM) twonucleon (NN) and threenucleon (3N) interactions of ref. ^{30}. This interaction is fitted 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 groundstate energies of light and mediummass nuclei^{26,31}. To further explore the sensitivity to chiral effective field theory interactions, we also consider the NN + 3N(lnl) interaction^{32} that has proven to constitute a valuable addition to existing chiral Hamiltonians in mediummass nuclei^{33} but has yet to be tested in heavier systems. Finally, we show results for the ^{100}Sn region with the ΔNNLO_{GO}(394) interaction^{34}. Calculations with the ΔNNLO_{GO}(394) interaction and NN + 3N(lnl) were performed using the SMCC and VSIMSRG methods, respectively. Technical details regarding these computations can be found in Methods.
Figure 3a presents the experimental threepoint empirical formula of the OES, Δ_{3n}(Z, N) = 0.5 × (−1)^{N}[M_{E}(Z, N − 1) − 2M_{E}(Z, N) + M_{E}(Z, N + 1)] for the oddZ indium isotopic chain. Figure 3a also shows the trends of Δ_{3n}(Z, N) calculated with the ab initio methods described above. Both manybody methods using the 1.8/2.0 (EM) interaction yield Δ_{3n}(Z, N) trends that agree with our experimental results. The differences between the two methods are within estimated theoretical uncertainties (see Methods for details). Calculations performed with the ΔNNLO_{GO}(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 manybody method and nuclear Hamiltonian in a range of 500 keV, but with all methods yielding excellent trends.
Figure 3b shows the experimental trend of Δ_{3n}(Z, N) for the tin chain (solid grey line). The experimental N = 53 point in Fig. 3b deviates from the regular odd–even behaviour of the threepoint empirical formula of the OES. This deviation is most likely explained by the AME2016 (ref. ^{20}) ^{103}Sn mass, which is known indirectly via its βdecay link to ^{103}In (refs. ^{35,36}). In fact, in the latest version of the AtomicMass Evaluation (AME2020)^{37,38}, this experimental mass value was found to violate the smoothness of the mass surface in this region to such a degree that the evaluators recommended replacing its value by an extrapolated value. The Δ_{3n}(Z, N) trend for the tin chain obtained with the ^{103}Sn AME2020 extrapolated value (solid black line in Fig. 3b) appears more regular and is better reproduced by the various theoretical calculations. Hence, as for Z = 49, in Z = 50 the relative agreement of the theoretical predictions with experiment is good overall. The successful benchmarking of the ab initio calculations by our indium masses gives confidence in their predictions towards ^{100}Sn, only one nucleon away. At N = 51, the discrepancy observed between the Qvalues reported in refs. ^{4,5} is again highlighted, with that of ref. ^{4} more in line with our theoretical results. Since the uncertainties of the light tin masses are not as stringent as our indium results, we also compare our predictions with the threepoint proton OES as a function of proton number in Fig. 3c. Again, our calculations agree with the experimental trend all the way up to Z = 48, yielding a staggering of similar magnitude and differing only in absolute values. At Z = 49 the evolution of all theoretical trends clearly favours the Hinke et al.^{4} Qvalue over that of Lubos et al.^{5}.
Methods
MRToF MS mass measurement and analysis
The relation between the time of flight t of a singly charged ion of interest and its mass m_{ion} is given by t = a(m_{ion})^{1/2} + b where a and b are devicespecific calibration parameters. These can be determined from the measured flight times t_{1,2} of two reference ions with well known masses m_{ion,1} and m_{ion,2}. From the timeofflight information of all the singly charged species, the mass of an ion is then calculated from the relation m_{ion}^{1/2} = C_{ToF}Δ_{ref} + 0.5Σ_{ref} with Δ_{ref} = m_{ion,1}^{1/2} − m_{ion,2}^{1/2}, Σ_{ref} = m_{ion,1}^{1/2} + m_{ion,2}^{1/2} and C_{ToF} = [2t − t_{1} − t_{2}]/[2(t_{1} − t_{2})] (ref. ^{23}).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 timeofflight distribution corresponding to each ion species was estimated using the unbinned maximumlikelihood method, assuming a Gaussian probability density function (PDF). To cope with the pronounced asymmetries observed in the shape of the timeofflight distribution, a restricted fit range was used (Fig. 2). The dependence of the timeofflight fit to these tails was compared with an analysis using the asymmetric PDF from ref. ^{40}. The difference between the extracted mean time of flight was subsequently treated as a systematic timeofflight uncertainty and was found to be the dominant contribution in the final uncertainty. When too many ions are trapped in the MRToF MS, spacecharge effects can cause the timeofflight 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 per cycle, which has proven to be a safe limit from previous tests. Nonetheless, countrate 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 (that is, 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 halflives) 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 MRToF MS mass value for ^{100}In is in good agreement with our Penningtrap value (see Table 1).
Principle of Penningtrap mass spectrometry
Penningtrap mass spectrometry relies upon the determination of the free cyclotron frequency ν_{C} = qB/(2πm_{ion}) of an ion species stored in magnetic field B and charge q. Comparing ν_{C} with the frequency ν_{C,ref} of a species of well known mass yields the frequency ratio r_{ref,x} = ν_{C,ref}/ν_{C}, from which the atomicmass 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 m_{atom} = r(m_{atom,ref} − m_{e}) + m_{e}, where m_{e} is the electron mass^{41}. As contributions from electron binding energies are orders of magnitude smaller than the statistical uncertainty, they are neglected here.
ToFICR mass measurements and analysis
The mass of ^{100}In was measured using the well established ToFICR technique using both onepulse excitation^{42} and twopulse, Ramseytype 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 the trap and their time of flight to a downstream microchannel plate detector is measured. The response of the ions to the applied excitation is a resonant process whose resonance frequency is ν_{C} 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 Ramseytype ToFICR resonance for ^{100}In is shown in Extended Data Fig. 1a. For the same total excitation time, this method offers a threefold precision improvement when compared with the singlepulse ToFICR 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 PIICR (see below) within one combined s.d.
PIICR mass measurements and analysis
To separate the A = 101 isomers, the recently introduced PIICR 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 t_{acc} using the projection of their motion onto a positionsensitive multichannel plate detector. The PIICR technique offers several advantages over the regular ToFICR technique. First, it is a nonscanning technique, which greatly reduces the number of ions required to perform a measurement; that is, only five to ten ions are required, where a minimum of 50–100 are required for ToFICR. While the resolving power of the ToFICR method is entirely limited by the excitation time, the resolving power of PIICR depends on the observation time and the iondistribution spot size projected on the detector.
A threestep measurement scheme allows for the direct determination of ν_{C}. First, a position measurement is performed without preparing the ions on a specific motion radius, yielding the position of the centre of the ions’ motion. In a second step, the ions are prepared on a pure magnetron orbit, left to evolve freely during a time t_{acc} and their position measured. Finally, the ions are prepared on a pure reduced cyclotron orbit, left to evolve freely during the same time t_{acc} and their position again measured. The integer number of revolutions n_{−} and n_{+} performed in steps 2 and 3 respectively, t_{acc} and the angle Φ between the ions’ positions obtained in steps 2 and 3 can be related to ν_{C} following the relation ν_{C} = [2π(n_{−} + n_{+}) + Φ]/t_{acc}. 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 maximumlikelihood method, assuming a twodimensional multivariate Gaussian distribution^{45}. Extended Data Figure 1 shows a typical PIICR image obtained in step 3 after ~62 ms of phase accumulation. As in principle the angle Φ_{gs,m} between the ground and isomeric states directly reflects the energy difference between the two states, the mass of each state was measured separately to mitigate systematical effects. The PIICR method was used to study the isomeric composition of the ^{100}In beam. Hence, we can exclude the presence of a longlived state with an excitation energy higher than 20 keV in the ^{100}In beam delivered to ISOLTRAP’s measurement Penning trap.
VSIMSRG calculations
The VSIMSRG calculations^{25,46} were performed in a spherical harmonicoscillator basis including up to 15 major shells in the singleparticle basis with an oscillator frequency ħω = 16 MeV. The 3N interaction configurations were restricted up to e_{1} + e_{2} + e_{3} ≤ E_{3max} = 16 for the 1.8/2.0 (EM) interaction (to compare with SMCC calculations) and E_{3max} = 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 p_{1/2}, p_{3/2}, f_{5/2}, g_{9/2} and neutron s_{1/2}, d_{3/2}, d_{5/2}, g_{7/2}, h_{11/2} valence space. Using the ensemble normal ordering introduced in ref. ^{25}, we approximately include effects of 3N interactions between valence nucleons, such that a specific valencespace Hamiltonian is constructed for each nucleus to be studied. The final diagonalization is performed using the KSHELL shellmodel code^{48}. To estimate theoretical uncertainties in this framework, we note that in the limit of no IMSRG truncations the 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. For normal ordering with respect to either a filled neutron g_{7/2} or d_{5/2} orbit, we find approximately 1 MeV uncertainty for absolute or oneneutron 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 harmonicoscillator basis comprising up to 11 major oscillator shells and ħω = 16 MeV. The 3N interaction was restricted to E_{3max} = 16ħω. The doubly closedshell ^{100}Sn core is decoupled by coupledcluster calculations including single, double and the leadingorder triple excitations (CCSDT1 approximation). We note that triple excitations were performed in the full model space, without any truncations. This work was made possible by employing the Nuclear Tensor Contraction Library (NTCL)^{49} developed to run at scale on Summit, the US 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 pfg_{9/2} holes and neutron g_{7/2}sd singleparticle states. The SMCC decoupling only includes the one and twobody parts of the CCSDT1 similaritytransformed Hamiltonian. To estimate theoretical uncertainties, we note that the calculation of doubly magic nuclei such as ^{100}Sn or ^{78}Ni and their neighbours is ideally suited for the coupledcluster 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 in singleparticle energies of about 0.2 MeV. We therefore estimate that our theoretical uncertainties on Δ_{3n}(Z, N) are about ±0.2 MeV.
Data availability
Source data are provided with this paper.
Code availability
The analysis codes used for the ToFICR 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 https://github.com/jonaska/mrtofanalysis. The PIICR analysis code^{45} used in this study is available at https://github.com/jonaska/piicranalysis. The code used for the VSIMSRG calculations is available at https://github.com/ragnarstroberg/imsrg. The source code of KSHELL is available in ref. ^{48}.
Change history
21 February 2022
In the version of this Letter initially published, the following metadata was omitted and has now been included: Open access funding provided by Max Planck Institute of Nuclear Physics (MPIK) (2).
References
Faestermann, T., Górska, M. & Grawe, H. The structure of ^{100}Sn and neighbouring nuclei. Prog. Part. Nucl. Phys. 69, 85–130 (2013).
Morris, T. D. et al. Structure of the lightest tin isotopes. Phys. Rev. Lett. 120, 152503 (2018).
Gysbers, P. et al. Discrepancy between experimental and theoretical βdecay rates resolved from first principles. Nat. Phys. 15, 428–431 (2019).
Hinke, C. B. et al. Superallowed Gamow–Teller decay of the doubly magic nucleus ^{100}Sn. Nature 486, 341–345 (2012).
Lubos, D. et al. Improved value for the Gamow–Teller strength of the ^{100}Sn βdecay. Phys. Rev. Lett. 122, 222502 (2019).
Jones, K. L. et al. The magic nature of ^{132}Sn explored through the singleparticle states of ^{133}Sn. Nature 465, 454–457 (2010).
Lewitowicz, M. et al. Identification of the doublymagic nucleus ^{100}Sn in the reaction ^{112}Sn + ^{nat}Ni at 63 MeV/nucleon. Phys. Lett. B 332, 20–24 (1994).
Sümmerer, K. et al. Identification and decay spectroscopy of ^{100}Sn at the GSI projectile fragment separator FRS. Nucl. Phys. A 616, 341–345 (1997).
Park, J. et al. βdecays of the heaviest N = Z − 1 nuclei and proton instability of ^{97}In. Phys. Rev. C 97, 051301(R) (2018).
Park, J. et al. New and comprehensive β and βpdecay spectroscopy results in the vicinity of ^{100}Sn. Phys. Rev. C 99, 034313 (2019).
Hornung, C. et al. Isomer studies in the vicinity of the doublymagic nucleus ^{100}Sn: observation of a new lowlying isomeric state in ^{97}Ag. Phys. Lett. B 802, 135200 (2020).
Xu, X. et al. Masses of ground and isomeric states of ^{101}In and configurationdependent shell evolution in oddA indium isotopes. Phys. Rev. C 100, 051303(R) (2019).
Plettner, C. et al. βdecay of ^{100}In. Phys. Rev. C 66, 044319 (2002).
Fedosseev, V. et al. Ion beam production and study of radioactive isotopes with the laser ion source at ISOLDE. J. Phys. G 44, 084006 (2017).
Mukherjee, M. et al. ISOLTRAP: an online Penning trap for mass spectrometry on shortlived nuclides. Eur. Phys. J. A 35, 1–29 (2008).
Herfurth, F. et al. A linear radiofrequency ion trap for accumulation, bunching, and emittance improvement of radioactive ion beams. Nucl. Instrum. Methods Phys. Res. A 469, 254–275 (2001).
Wolf, R. N. et al. ISOLTRAP’s multireflection timeofflight mass separator/spectrometer. Int. J. Mass Spectrom. 349, 123–133 (2013).
Eliseev, S. et al. A phaseimaging technique for cyclotronfrequency measurements. Appl. Phys. B 114, 107–128 (2014).
Manea, V. et al. First glimpse of the N = 82 shell closure below Z = 50 from masses of neutronrich cadmium isotopes and isomers. Phys. Rev. Lett. 124, 92502 (2020).
Wang, M. et al. The AME2016 atomic mass evaluation (II). Tables, graphs and references. Chin. Phys. C 41, 030003 (2017).
Schmidt, K.H. & Vermeulen, D. in Atomic Masses and Fundamental Constants 6 (eds Nolen, J. A. & Benenson, W.) 119–128 (Springer, 1980); https://doi.org/10.1007/9781468437164_13
Welker, A. et al. Binding energy of ^{79}Cu: probing the structure of the doubly magic ^{78}Ni from only one proton away. Phys. Rev. Lett. 119, 192502 (2017).
Wienholtz, F. et al. Masses of exotic calcium isotopes pin down nuclear forces. Nature 498, 346–349 (2013).
Garcia Ruiz, R. F. et al. Unexpectedly large charge radii of neutronrich calcium isotopes. Nat. Phys. 12, 594–598 (2016).
Stroberg, S. R. et al. Nucleusdependent valencespace approach to nuclear structure. Phys. Rev. Lett. 118, 032502 (2017).
Stroberg, S. R., Holt, J. D., Schwenk, A. & Simonis, J. Ab initio limits of atomic nuclei. Phys. Rev. Lett. 126, 022501 (2021).
Sun, Z. H., Morris, T. D., Hagen, G., Jansen, G. R. & Papenbrock, T. Shellmodel coupledcluster method for openshell nuclei. Phys. Rev. C 98, 054320 (2018).
de Groote, R. P. et al. Measurement and microscopic description of odd–even staggering of charge radii of exotic copper isotopes. Nat. Phys. 16, 620–624 (2020).
Miyagi, T., Stroberg, S. R., Holt, J. D. & Shimizu, N. Ab initio multishell valencespace Hamiltonians and the island of inversion. Phys. Rev. C 102, 034320 (2020).
Hebeler, K., Bogner, S. K., Furnstahl, R. J., Nogga, A. & Schwenk, A. Improved nuclear matter calculations from chiral lowmomentum interactions. Phys. Rev. C 83, 31301 (2011).
Simonis, J., Stroberg, S. R., Hebeler, K., Holt, J. D. & Schwenk, A. Saturation with chiral interactions and consequences for finite nuclei. Phys. Rev. C 96, 14303 (2017).
Somà, V., Navrátil, P., Raimondi, F., Barbieri, C. & Duguet, T. Novel chiral Hamiltonian and observables in light and mediummass nuclei. Phys. Rev. C 101, 14318 (2020).
Mougeot, M. et al. Examining the N = 28 shell closure through highprecision mass measurements of ^{46–48}Ar. Phys. Rev. C 102, 14301 (2020).
Jiang, W. G. et al. Accurate bulk properties of nuclei from A = 2 to ∞ from potentials with Δ isobars. Phys. Rev. C 102, 54301 (2020).
Mukha, I. et al. Studies of βdelayed proton decays of N ≃ Z nuclei around ^{100}Sn at the GSIISOL facility. Nucl. Phys. A 746, 66–70 (2004).
Kavatsyuk, O. et al. Beta decay of ^{103}Sn. Eur. Phys. J. A 25, 211–222 (2005).
Huang, W. J., Wang, M., Kondev, F. G., Audi, G. & Naimi, S. The AME 2020 atomic mass evaluation (I). Evaluation of input data, and adjustment procedures. Chin. Phys. C 45, 30002 (2021).
Wang, M., Huang, W. J., Kondev, F. G., Audi, G. & Naimi, S. The AME 2020 atomic mass evaluation (II). Tables, graphs and references. Chin. Phys. C 45, 30003 (2021).
Audi, G., Kondev, F. G., Wang, M., Huang, W. J. & Naimi, S. The NUBASE2016 evaluation of nuclear properties. Chin. Phys. C 41, 030001 (2017).
Lan, K. & Jorgenson, J. W. A hybrid of exponential and Gaussian functions as a simple model of asymmetric chromatographic peaks. J. Chromatogr. A 915, 1–13 (2001).
Sturm, S. et al. Highprecision measurement of the atomic mass of the electron. Nature 506, 467–470 (2014).
König, M., Bollen, G., Kluge, H. J., Otto, T. & Szerypo, J. Quadrupole excitation of stored ion motion at the true cyclotron frequency. Int. J. Mass Spectrom. Ion Process. 142, 95–116 (1995).
George, S. et al. Ramsey method of separated oscillatory fields for highprecision Penning trap mass spectrometry. Phys. Rev. Lett. 98, 162501 (2007).
Kellerbauer, A. et al. From direct to absolute mass measurements: a study of the accuracy of ISOLTRAP. Eur. Phys. J. D 22, 53–64 (2003).
Karthein, J. jonaska/piicranalysis. Zenodo https://doi.org/10.5281/zenodo.3965767 (2020).
Stroberg, S. R., Hergert, H., Bogner, S. K. & Holt, J. D. Nonempirical interactions for the nuclear shell model: an update. Annu. Rev. Nucl. Part. Sci. 69, 307–362 (2019).
Morris, T. D., Parzuchowski, N. M. & Bogner, S. K. Magnus expansion and inmedium similarity renormalization group. Phys. Rev. C 92, 34331 (2015).
Shimizu, N., Mizusaki, T., Utsuno, Y. & Tsunoda, Y. Thickrestart block Lanczos method for largescale shellmodel calculations. Comput. Phys. Commun. 244, 372–384 (2019).
Jansen, G. NTCL—Nuclear Tensor Contraction Library. GitLab https://gitlab.com/ntcl/ntcl (2021).
Acknowledgements
We thank the ISOLDE technical group and the ISOLDE Collaboration for their support. We acknowledge the support of the Max Planck Society, the French Institut National de Physique Nucléaire et de Physique des Particules (IN2P3), the European Research Council (ERC) through the European Union’s Horizon 2020 research and innovation programme (grant agreement 682841 ‘ASTRUm’ and 654002 ‘ENSAR2’) and the Bundesministerium für Bildung und Forschung (BMBF; grants 05P15ODCIA, 05P15HGCIA, 05P18HGCIA and 05P18RDFN1). J.K. acknowledges the support of a Wolfgang Gentner PhD scholarship from the BMBF (05E12CHA). This work was supported by the US Department of Energy, Office of Science, Office of Nuclear Physics, under awards DEFG0296ER40963 and DEFG0297ER41014. This material is based upon work supported by the US Department of Energy, Office of Science, Office of Advanced Scientific Computing Research and Office of Nuclear Physics, Scientific Discovery through Advanced Computing (SciDAC) programme under award DESC0018223. TRIUMF receives funding via a contribution through the National Research Council of Canada, with additional support from NSERC. Computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) Program. This research used resources of the Oak Ridge Leadership Computing Facility located at Oak Ridge National Laboratory, which is supported by the Office of Science of the Department of Energy under contract DEAC0500OR22725. The VSIMSRG computations were performed with an allocation of computing resources on Cedar at WestGrid and Compute Canada, and on the Oak Cluster at TRIUMF managed by the University of British Columbia department of Advanced Research Computing (ARC). R.N.W. acknowledges support by the Australian Research Council under the Discovery Early Career Researcher Award scheme (DE190101137).
Funding
Open access funding provided by Max Planck Institute of Nuclear Physics (MPIK) (2).
Author information
Authors and Affiliations
Contributions
M.M., D.A., J.K., P.A., I.K., Y.A.L., V.M., T.S., A.W. and F.W. performed the experiment. M.M., D.A., J.K. and R.N.W. performed the data analysis. K.C. and S.G.W. set up the resonant laser ionization scheme. W.J.H. performed the update of the AtomicMass Evaluation with the latest experimental results. G.H., J.D.H., G.R.J., T.M., T.P., S.R.S. and Z.H.S. performed the theoretical calculations. K.B., V.M., D.L., A.S., L.S., K.Z. and M.M. prepared the manuscript. All authors discussed the results and contributed to the manuscript at all stages.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer review information Nature Physics thanks Nunzio Itaco and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended data
Extended Data Fig. 1 Overview of experimental results (continued).
(a), Ramsey ToFICR resonance of ^{100}In^{+} containing about 160 ions. A Ramsey pattern of T_{RF}^{on}T_{RF}^{off}T_{RF}^{on} = 50 ms – 500 ms – 50 ms was used for this measurement. The solid red line corresponds to the leastsquare adjustment of the theoretical line shape to the data. (b), PIICR ionprojection image of ^{101}In^{+}. (0,0) marks the center of the position sensitive detector. In a phaseaccumulation of about 62 ms a mass resolving power in excess of 5.10^{5} was reached allowing for the ground (blue) and isomeric (red) states to be separated by the angle Φ_{gs,m} which directly determines the nuclear excitation energy. The centre (black) of the projected ion motion is obtained in a separate measurement. The error bars represent one standard deviation.
Source data
Source Data Fig. 2
CSV file providing the data to reproduce Fig. 2c.
Source Data Fig. 3
CSV file providing the data to reproduce Fig. 3.
Source Data Extended Data Fig. 1
CSV file providing the data to reproduce Extended Data Fig. 1.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license 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 license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Mougeot, M., Atanasov, D., Karthein, J. et al. Mass measurements of ^{99–101}In challenge ab initio nuclear theory of the nuclide ^{100}Sn. Nat. Phys. 17, 1099–1103 (2021). https://doi.org/10.1038/s41567021013269
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s41567021013269
This article is cited by

Experimental studies on the doubly magic \(^{100}\)Sn and neighboring \(N \approx Z\) nuclei with EURICA
Journal of the Korean Physical Society (2023)

Close to the edge
Nature Physics (2021)