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 closed-shell configurations, the best example of which is Z = 50 (tin), featuring the largest number of β-stable isotopes (10) of all elements. These nuclides lie between the closed neutron shells N = 50 and 82, conferring particular importance to the nuclides 100Sn and 132Sn. The neutron-rich 132Sn can be synthesized in comfortable quantities6. This is not so for 100Sn, 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 100Sn 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 100Sn offer important insight for understanding the single-neutron and proton states in this region and constitute an excellent proxy for the study of 100Sn itself. However, experiments have so far only been feasible with in-beam gamma-ray spectroscopy at fragmentation facilities4,5,7,8,9,10. 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 online (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 at the FRS Ion Catcher at GSI11 and the Cooler-Storage experimental Ring (CSRe) in Langzhou12 (both high-energy, heavy-ion fragmentation facilities) recently extended direct mass measurements to the 101In ground and isomeric states. However, the 100In mass value is still constrained 63% indirectly through its beta-decay link to 100Cd (ref. 13).

Thus, the first experimental challenge overcome in this work was the production and separation of the successfully studied 99,100,101g,101mIn 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 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 using ISOLDE’s high-resolution dipole mass separator and delivered to the ISOLTRAP online mass spectrometer15.

Fig. 1: High-precision mass measurements of neutron-deficient indium isotopes with ISOLTRAP.
figure 1

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, provided by the ISOLDE RILIS, which excited one electron above the indium ionization potential (IP). The extracted ion beam was mass separated and injected into a radiofrequency quadrupole (RFQ) ion trap sitting on a high-voltage (HV) platform, where it was bunched and cooled. The beam was then processed by an 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. A position-sensitive microchannel plate (MCP) detector was used to record the time of flight and/or the position of the ion after ejection from the precision Penning trap. In the case of 99In, 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).

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The ions were first accumulated in ISOLTRAP’s linear radiofrequency quadrupole cooler and buncher trap16. The extracted bunches were subsequently decelerated by a pulsed drift cavity to an energy of 3.2 keV before being purified by the multireflection time-of-flight mass spectrometer (MR-ToF MS)17, where multiple passages between two electrostatic mirrors rapidly separate the short-lived indium ions from much more abundant molecules of approximately the same mass. For all investigated isotopes, surviving molecular ions 80–82Sr19F+ were predominant in the ISOLDE beam. After a typical trapping time of about 25 ms, a resolving power in excess of m/Δm = 105 was achieved. This combination of speed and high resolving power enables the MR-ToF MS to perform precise mass measurements of very short-lived species (Methods). Because of its low production yield of <10 ions per second, the mass of 99In was measured with this latter method only (see typical MR-ToF MS spectrum in Fig. 2).

Fig. 2: Overview of the experimental results.
figure 2

Top: a typical A = 99 MR-ToF MS spectrum obtained after 1,000 revolutions. The solid black lines represent the Gaussian fit probability density function scaled to the histogram within the used fit range. The blue band indicates the restricted fit range used for the 80Sr19F peak analysis. Middle: unbinned time-of-flight data used to perform the mass evaluation. The red vertical bars represent the uncertainty of the mean of the time-of-flight distributions at the ±1σ confidence level. An overview of the experimental data can be found in Methods. Bottom: Δ2n(Z, N0) as a function of Z for N0 = 28 and N0 =50 (open grey symbols). The filled grey symbols show the corresponding value of the quantity Δ2n(Z, N0 + 2). At Z = 50, the filled red circle corresponds to the value of Δ2n(Z, N0 + 2) calculated using the masses from this work and the β-decay energy from ref. 4 and the open blue circle uses the value from ref. 5. The inset shows a 2.5-fold magnification of the Δ2n(Z, N0 + 2) curve from Z = 44 to 50. The error bars represent 1 s.d. The dashed vertical lines indicate the magic proton numbers 20, 28 and 50.

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The rate of 100In and 101In behind the MR-ToF MS was sufficient to perform Penning-trap mass measurements. For 100In the conventional time-of-flight ion-cyclotron-resonance (ToF-ICR) technique was used (Methods and Extended Data Fig. 1). Even-N neutron-deficient indium isotopes are known to exhibit long-lived isomeric states lying a few hundred kiloelectron-volts above the corresponding ground state, owing to the close energy proximity between the πg9/2 and πp1/2 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 phase-imaging ion-cyclotron-resonance (PI-ICR) technique18,19 had to be used to resolve them and ensure the accuracy of the ground-state 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 101In 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 101gIn is in excellent agreement with the value measured using PI-ICR. 100In is found to be 130 keV more bound, while the mass uncertainty is improved by almost a factor of 90.

Table 1 Summary of the mass values obtained in this work
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Since the 100Sn 2016 Atomic-Mass Evaluation (AME2016) mass excess value of −57,280(300) keV (ref. 20) is derived from that of 100In and the β-decay energy of ref. 4, our 100In result improves the 100Sn mass excess to −57,148(240) keV. However, combining our result with the more recently published β-decay Q-value (that is, the energy released in the decay) from ref. 5 yields a 100Sn mass excess of −56,488(160) keV. For both decay energies, the 100Sn mass is found to be more bound than previously inferred. In addition, the almost 2 s.d. between the Q-values from refs. 4,5 yields 100Sn 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 two-neutron empirical shell gap defined as Δ2n(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’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 100Sn, we adapt an approach proposed in ref. 22 using Δ2n(Z, N0 + 2), which is inversely correlated to Δ2n(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 Δ2n(Z, N0 + 2) with opposite sign. The case of N = 28 is shown in Fig. 2 for illustration. Our data allows extending Δ2n(Z, N0 + 2) to Z = 49 (indium) and indicates a slight downward trend towards Z = 50 (Fig. 2 inset), as expected for a doubly magic 100Sn. Eliminating the contribution of the 100In ground-state mass uncertainty in the calculation of the 100Sn mass directly allows to confront the nuclear-structure implications of the two Q-values from refs. 4,5, and a global picture now emerges for this region. As shown, the Q-value reported by Lubos et al.5 yields a 100Sn mass value that is at odds with the expected trend of Δ2n(Z, N0 + 2) to Z = 49 (open blue circle in the bottom panel of Fig. 2), whereas the value of Hinke et al.4 yields a 100Sn mass that agrees with the trend within experimental uncertainties and is in line with our observation for Z = 49. In other words, while the Q-value reported in ref. 4 follows the expectation of a doubly magic 100Sn, the more recent (and higher-statistics) Q-value reported in ref. 5 yields a 100Sn mass value that suggests quite the opposite. Such a conclusion is at odds with ab initio many-body calculations as discussed below.

In recent years, there has been great progress advancing ab initio calculations in medium-mass nuclei23,24 up to the tin isotopes2 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 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 state-of-the-art theoretical approaches. Among ab initio approaches, the valence-space formulation of the in-medium similarity renormalization group (VS-IMSRG)25 is able to access a broad range of closed- and open-shell nuclei in the nuclear chart26. In addition, we will explore the shell-model coupled-cluster (SMCC) method27 in this region. Both the VS-IMSRG and coupled-cluster calculations provide access to a broad range of observables, such as ab initio calculations of beta decays—up to 100Sn (ref. 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) isotopes28. Here we present VS-IMSRG and SMCC results that allow direct comparisons with the odd-Z nuclides adjacent to the iconic 100Sn nucleus.

We have performed cross-shell VS-IMSRG29 and SMCC calculations using the 1.8/2.0(EM) two-nucleon (NN) and three-nucleon (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 4He charge radius), and gives accurate results for ground-state energies of light and medium-mass nuclei26,31. To further explore the sensitivity to chiral effective field theory interactions, we also consider the NN + 3N(lnl) interaction32 that has proven to constitute a valuable addition to existing chiral Hamiltonians in medium-mass nuclei33 but has yet to be tested in heavier systems. Finally, we show results for the 100Sn region with the ΔNNLOGO(394) interaction34. Calculations with the ΔNNLOGO(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 Methods.

Figure 3a presents the experimental three-point empirical formula of the OES, Δ3n(Z, N) = 0.5 × (−1)N[ME(Z, N − 1) − 2ME(Z, N) + ME(Z, N + 1)] for the odd-Z indium isotopic chain. Figure 3a also shows the trends of Δ3n(Z, N) calculated with the ab initio methods described above. Both many-body methods using the 1.8/2.0 (EM) interaction yield Δ3n(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 ΔNNLOGO(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 500 keV, but with all methods yielding excellent trends.

Fig. 3: Comparison of experimental three-point estimators of the OES with theoretical results.
figure 3

a, Three-point empirical formula of the neutron OES Δ3n(Z, N) in the indium (Z = 49) isotopic chain as a function of the neutron number. b, Three-point empirical formula of the neutron OES Δ3n(Z, N) in the tin (Z = 50) isotopic chain as a function of the neutron number. The solid black line represents the same quantity computed considering the extrapolated 103Sn mass value given in AME2020 (refs. 37,38). c, Three-point empirical formula of the proton OES Δ3p(Z, N) along the N = 50 isotonic chain as a function of the proton number. The points resulting from the 100Sn mass deduced with the Q-value from ref. 4 are plotted as the filled red circles while the open blue circles show the value using the ref. 5 Q-value. The error bars represent 1 s.d. The dashed vertical lines indicate the magic proton/neutron number 50.

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Figure 3b shows the experimental trend of Δ3n(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 three-point empirical formula of the OES. This deviation is most likely explained by the AME2016 (ref. 20) 103Sn mass, which is known indirectly via its β-decay link to 103In (refs. 35,36). In fact, in the latest version of the Atomic-Mass 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 Δ3n(Z, N) trend for the tin chain obtained with the 103Sn 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 100Sn, only one nucleon away. At N = 51, the discrepancy observed between the Q-values 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 three-point 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 Q-value over that of Lubos et al.5.

Methods

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 mion1/2 = CToFΔref + 0.5Σref with Δref = mion,11/2 − mion,21/2, Σref = mion,11/2 + mion,21/2 and CToF = [2t − t1 − t2]/[2(t1 − t2)] (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 time-of-flight 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 (Fig. 2). The dependence of the time-of-flight 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 time-of-flight uncertainty and was found to be the dominant 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 per 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 99In, 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 99In was similar to that observed for 101In (that is, 25:1), because the two states in 99In 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 99In than 101In 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 100In 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 with the frequency νC,ref of a species of well known mass yields the frequency ratio rref,x = 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 mass41. 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 100In was measured using the well established ToF-ICR technique using both one-pulse excitation42 and two-pulse, Ramsey-type excitation43. 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 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 100In is shown in Extended Data Fig. 1a. For the same total excitation time, this method offers a threefold precision improvement when compared with 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 101In was likewise measured and agrees with a value determined by PI-ICR (see below) within one combined s.d.

PI-ICR mass measurements and analysis

To separate the A = 101 isomers, the recently introduced PI-ICR technique was used18. 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 multichannel plate detector. The PI-ICR technique offers several advantages over the regular ToF-ICR technique. First, it is a non-scanning 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 ToF-ICR. While the resolving power of the ToF-ICR method is entirely limited by the excitation time, the resolving power of PI-ICR depends on the observation time and the ion-distribution spot size projected on the 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 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 tacc and their position measured. Finally, the ions are prepared on a pure reduced cyclotron orbit, left to evolve freely during the same time tacc and their position again measured. The integer number of revolutions n− and n+ performed in steps 2 and 3 respectively, tacc and the angle Φ between the ions’ positions obtained in steps 2 and 3 can be related to νC following the relation νC = [2π(n− + n+) + Φ]/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 two-dimensional multivariate Gaussian distribution45. Extended Data Figure 1 shows a typical PI-ICR image obtained in step 3 after ~62 ms of phase accumulation. As in principle the angle Φgs,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 PI-ICR method was used to study the isomeric composition of the 100In beam. Hence, we can exclude the presence of a long-lived state with an excitation energy higher than 20 keV in the 100In beam delivered to ISOLTRAP’s measurement Penning trap.

VS-IMSRG calculations

The VS-IMSRG calculations25,46 were performed in a spherical harmonic-oscillator basis including up to 15 major shells in the single-particle basis with an oscillator frequency ħω = 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 IMSRG47 to construct an approximate unitary transformation to decouple a 78Ni 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 ref. 25, 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 code48. 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 reference-state dependence of the observables discussed above. For normal ordering with respect to either a filled neutron g7/2 or d5/2 orbit, we find approximately 1 MeV 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 100Sn reference state, computed in a harmonic-oscillator basis comprising up to 11 major oscillator shells and ħω = 16 MeV. The 3N interaction was restricted to E3max = 16ħω. The doubly closed-shell 100Sn core is decoupled by coupled-cluster calculations including single, double and the leading-order triple excitations (CCSDT-1 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 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 100Sn or 78Ni and their neighbours is ideally suited for the coupled-cluster method, because the reference state is closed shell2,46. Comparison of the SMCC results for 101Sn with those from ref. 2 exhibit differences in single-particle energies of about 0.2 MeV. We therefore estimate that our theoretical uncertainties on Δ3n(Z, N) are about ±0.2 MeV.