Nuclear moments put a new spin on the structure of 131In

In spite of the high-density and strongly correlated nature of the atomic nucleus, experimental and theoretical evidence suggests that around particular ‘magic’ numbers of nucleons, nuclear properties are governed by a single unpaired nucleon 1,2 . A microscopic understanding of the extent of this behavior and its evolution in neutron-rich nuclei remains an open question in nuclear physics 3–5 . A textbook example is the electromagnetic moments of indium ( Z = 49) 6 , which are dominated by a hole with respect to the proton magic number Z = 50 nucleus. They exhibit a remarkably constant behavior over a large range of odd-mass isotopes, previously interpreted as pure “single-particle behaviour”. Here, we present precision laser spectroscopy measurements performed to investigate the validity of this simple single-particle picture. Observation of an abrupt change in the dipole moment at N = 82 reveals that while the single-particle picture indeed dominates at neutron magic number N = 82 2,7 , it does not for previously studied isotopes. We present state-of-the-art nuclear theory developed to investigate the details of the nuclear forces that describe the experimental results. The emergence and disappearance of single-particle behavior was reproduced from an ab initio theory, including challenging many-body correlations in these large, complex nuclei. The inclusion of time-symmetry-breaking mean ﬁelds is shown to be essential for a correct description of the nuclear electromagnetic properties within the Density Functional Theory framework. Until now, such time-odd channels have been poorly constrained, but are essential to provide accurate predictions of nuclear properties necessary for searches of new physics with neutrinos 8,9 and studies of fundamental symmetries 10,11 . These ﬁndings are key to under-stand the microscopic origin of nuclear electromagnetism and the emergence of single-particle phenomena from complex nuclei.


Introduction
The atomic nucleus is formed by strongly-interacting nucleons (protons, Z, and neutrons, N), packed tightly into a volume around a trillion times smaller than that of atoms. These highdensity systems are bound by the strong nuclear force which, in contrast to the well understood electromagnetic force that dominates the properties of atoms and molecules, is exceedingly more complex and not yet fully understood. Hence, describing the atomic nuclei and predicting their properties at extreme values of mass and charge are major long-standing challenges for nuclear science. Similar to electrons in an atom, the nucleons (protons and neutrons) in the atomic nucleus occupy quantum 'shells'. Thus, nuclei with a single valence particle or hole around a nuclear closed shell provide important foundations for our understanding of the atomic nucleus. Their simpler structure vastly reduces the complexity of the quantum many-body problem, providing critical guidance for the development of nuclear theory.
Recent advances in our understanding of the strong interaction and the development of many-body methods, combined with escalation in computer power, have enabled theoretical  12 . The evolution of single-particle properties of these isotopes are illustrated at the bottom of the figure. At neutron magic number N = 82, the electromagnetic properties approach the pure single-particle value (see text for more details).
descriptions of increasingly complex nuclei. Isotopes around the proton closed shell Z = 50, are now the frontier of ab initio calculations 13,14 . The properties of these nuclei can be described by complementary many-body methods such as configuration-interaction methods 4 and nuclear Density Functional Theory (DFT) 15 . This has lead to an increased focus on studying this region of the nuclear chart (around Z = 50, N = 50, 82) over the last decade 2,7,[16][17][18] .
Here we present measurements of two fundamental properties of indium isotopes using precision laser spectroscopy: the (spectroscopic) magnetic dipole moment, µ, and electric quadrupole moment, Q. Measurements were performed for the neutron-rich In (Z = 49) isotopes, reaching up to 131 In, which possesses a magic number of N = 82 neutrons (see Methods section for details). With a proton hole in the closedshell of Z = 50, the low-energy structure of the odd-mass In isotopes is expected to be governed by a single-hole configuration in the proton orbit π1g 9/2 .
As we show here, the nuclear magnetic dipole moments of odd-mass In isotopes are determined by the total spin distribution of the nucleus induced by the unpaired valence proton hole. This is schematically illustrated at the bottom right of Figure 1. The nuclear electric quadrupole moment provides a complementary measurement of the nuclear charge distribution, and is highly sensitive to the collective motion of all nucleons 19 . These observables together therefore probe distinct aspects of the nucleon distribution and measuring them across a large range of neutron numbers allows a unique insight into the evolution of the interplay between single-particle and collective nuclear phenomena.
Previously, the magnetic and quadrupole moments of the ground state I π = 9/2 − of indium isotopes were known to exhibit remarkably little variation over 22 isotopes, from A = 105 to 127 (see Figure 1b)) 12 . The constant value of the magnetic moment over such long range of isotopes has been presented as an archetypal example of the independentparticle behaviour of single-particle states near a proton shell closure 6,12 . 'How do these seemingly simple patterns emerge from complex interactions among protons and neutrons?' and 'Do they prevail at extreme number of neutrons?' are two major open questions that we address in this work.
In addition to the ground state, the indium isotopes can also exist in excited nuclear configurations with relatively long lifetimes -isomers -with spin I π = 1/2 − . These isomeric states 2/12 provide additional insight and are expected to be described by a single-hole configuration based on a different proton orbital (π2p 1/2 ). However, in contrast to the I π = 9/2 + states, the µ values of these isomeric states exhibit significant variations, posing a three-decades-long puzzle in our description of these nuclei 12 .
To unravel the microscopic origin of the electromagnetic properties of these isotopes, we compare our experimental results with two complementary state-of-the-art theoretical methods: i. ab initio valence-space in-medium similarity renormalisation group (VS-IMSRG) calculations 13,20 , which start from nucleon-nucleon interactions derived from chiral effective field theory, constrained by the properties of up to only four nucleon systems 21,22 ; and ii. symmetry-breaking nuclear Density Functional Theory (DFT) 15,23 . The latter assumes nucleons moving within their own self-consistently-generated spin-dependent broken-symmetry-confining potential. DFT provides a satisfactory description of bulk nuclear properties such as radii and binding energies across the whole nuclear chart [24][25][26] . Here, we have developed its symmetry-restored version 27 to be able to provide accurate calculations of spectroscopic µ and Q moments.

Experimental and Theoretical developments
Our measurements were performed using the collinear resonance ionisation spectroscopy (CRIS) technique at the ISOLDE facility of CERN 28 (see Methods section for details). From the hyperfine structure, we extracted the magnetic dipole and electric quadrupole parameters, A h f , B h f of the probed atomic states, corresponding to the two long-lived nuclear states, 9/2 + and 1/2 − , present in each isotope. Recent improvements in the sensitivity of the technique allowed us to achieve high-resolution spectroscopy measurements, despite production of the indium isotopes at rates below 1000 atoms/s in the presence of large isobaric contamination. Prior to the study of these exotic isotopes, sensitive ionisation schemes and atomic theory calculations had to be developed to accurately extract the nuclear properties from spectroscopy measurements 29,30 .
We performed ab initio VS-IMSRG calculations (see Methods for further details) using two different sets of initial two-(NN) and three-nucleon (3N) forces derived from chiral effective field theory 31, 32 ; the 1.8/2.0(EM) 21,22 and the more recent N 2 LO GO 33 . The 1.8/2.0(EM) set is constrained only by fitting to properties of two-, three-, and four-nucleon systems. N 2 LO GO was recently developed to include ∆-isobar degrees of freedom and is additionally fit to reproduce saturation properties of infinite nuclear matter 33 .
We performed DFT calculations using both Hartree-Fock (HF) or Hartree-fock-Bogilibov (HFB) approaches, corresponding to configurations of the nucleus constructed using single-nucleon (HF), or nucleon-hole pair excitations (HFB) as basis states, with HFB calculations introducing pairing correlations. The electromagnetic moments of semi-magic ± 1 nucleon systems, such as the indium isotopes, are well suited to study DFT time-symmetry-breaking (time-odd) contributions to the mean field 34,35 , which vary with the time-reversal operator. These fields are predominantly generated by the two-body spin-spin interaction terms, and up until now were poorly constrained within DFT theory. However, they are of particular interest as our understanding of time-reversalviolating mean fields is critical to tackle open problems of modern physics, e.g. the search for new physics [8][9][10][11] and dark matter searches 36 . Our experimental results presented an excellent opportunity to perform and test these developments. To investigate the relative importance of time-odd fields and pairing correlations, DFT calculations were performed by turning "on" and "off" of each effect. The results of both ab initio and DFT types of calculation are shown alongside the experimental Q and µ moments in Figure 1a) and 1b), respectively, for the 9/2 + states of 105−131 In. The µ moments of the 1/2 − states are shown in Figure 2. All experimental data are presented in Tables 1 and 2 of Methods, and compared with literature values that exist for 105−127 In.

Results and discussion
In the single-particle picture, a proton hole induces an intrinsic prolate deformation of the whole nucleus, as indicated schematically in Figure 1 (bottom left). A gradual decrease in the Q values of the 9/2 + states was previously observed up to N = 78. Our measurements reveal a notably larger decrease at N = 82, indicating a significant decrease in deformation ( Figure 1a). The VS-IMSRG calculations reproduce the experimental trends, i.e local variations in neutron number, a dip around N = 64, and a gradual decrease towards N = 82. However, the magnitude of the Q is underestimated. The reproduction of the magnitude of the quadrupole moments is a known challenge for ab initio nuclear theory, as the Q moments are a highly collective emergent property of the nucleus which can require the inclusion of extensive many-body correlations 37 . Traditionally, phenomenological calculations are fitted to match the experimental data by assigning empirical 'effective' charges to valence neutrons 38 . Neither effective charges nor effective g-factors were used in our ab initio or DFT calculations. This work therefore provides an important insight into how a correct description of these nuclei and single-particle behaviour emerges intrinsically from our calculations without the use of commonly used effective factors.
Conversely, our DFT calculations are able to closely reproduce the overall magnitude of the Q moments. As N = 82 is approached, the agreement with the calculations without pairing (HF), shows that describing individual neutron orbitals becomes important. However, due to effects induced by occupying individual neutron orbitals, an inaccurate staggering with neutron number is also produced, compared to HFB. As shown in Figure 1a), time-odd contributions have a negligible effect on Q moments.
In contrast to the Q moments, the µ moments of the 9/2 + states were known to exhibit little variation 12 , which is continued up to 129 In in our observation. However, we observe an abrupt increase at 131 In, see Figure 1b), and the atomic spectra in Figure 3. The extreme single-particle magnetic moment of a proton in the πg 9/2 orbit, the so-called 'Schmidt' limit 39 , is also indicated in Figure 1b). Deviations from the extreme single-particle value have usually been attributed to a mean effect of the other nucleons, and are typically taken into account using an empirical 'quenched' or 'effective' gfactor 12 . Our results now reveal a value much closer to the extreme single-particle limit at N = 82, indicating the µ moments for N < 82 in fact depart from the single-particle picture. This illustrates how the use of 'effective' operators can lead to contradictory conclusions and obscure critical details and changes in the underlying nuclear structure. In contrast to the quadrupole moments, we find that the inclusion of time-odd fields proves to be essential for an accurate description of magnetic moments. We observe that without the time-odd mean fields, the DFT value obtained for 131 In is close to the single-particle value. Thus, the quadrupole deformation of this nucleus cannot explain the experimental value. However, the addition of a spin-spin parameter corresponding to the isovector Landau parameter 40 of g ′ 0 = 0.82 generates time-odd mean fields that result in a perfect agreement between the DFT and the experimental 131 In magnetic moment. Similarly as for the Q moments, we find that the HF-variant of the DFT better matches experiment around N = 82, reproducing the magnitude of the jump in µ well in contrast to those including pairing correlations (HFB). In order to further investigate the abrupt change that was observed at N = 82 within the DFT framework, we traced back the properties of µ in indium directly to those of their deformed and polarised cores. To this end, we performed symmetry-restoration calculations by removing the deformed odd-proton orbitals from the self-consistent DFT states in indium or by filling in the corresponding odd-proton holes. In this way, we gained access to the corresponding 0 + , 1 + , 2 + , 3 + , 4 + , ... states of the cadmium or tin cores. At this point, it is essential to recall that without the time-odd mean fields the core is not spin-polarised and therefore its states conserve signature symmetry and odd-angular-momentum components vanish. The effect of the time-odd mean fields on magnetic moments thus proceeds through the coupling to the 1 + , 3 + , . . . , 9 + states of the polarised core. The DFT calculations indicate that the change in the 9/2 + moments can be attributed to the dominance of the effect of the spin-distribution at the N = 82 shell closure and the charge distribution for N < 82. Predictions for N > 82 using these DFT calculations are shown in Methods (Figure 4), supporting the suggestion of a possible new magic number at N = 90 41 .
Previously, the decreasing µ values of the 1/2 − isomeric states with increasing neutron number (see Figure 2) were not explained by nuclear theory. Our new experimental results for 129,131 In show a sudden increase, which crosses the extreme single-particle value at N = 82. The results are compared with literature values of 115−127 In in Table 1. We find that the DFT calculations without pairing correlations also provide a better description than those with pairing correlations (HFB). The introduction of time-odd fields improves the agreement with experiment. However, the pronounced increase at N = 82 is not reproduced within this framework, with predictions close to the single-particle limit. This suggests the 1/2 − isomers are highly collective in nature even at N = 82. Our VS-IMSRG calculations provide some insight here. Overall, these calculations provide a good description of the previously unexplained trend observed for the magnetic moments of the 1/2 − states. Analysis of the VS-IMSRG calculations indicates that the 1/2 − state contains a significant component resulting from the coupling between the 9/2 + single-particle configuration and the 5 − (anti-aligned neutrons in the h 11/2 and s 1/2 orbitals) state of the core. This therefore suggests that further refinement of the present DFT calculations would be needed to explicitly include other core configurations e.g. negative-parity states, to accurately describe the 1/2 − states.

Summary and Outlook
The indium isotopes have been considered a textbook example for the dominance of single-particle properties in heavy nuclei. Here, we show that their ground-state electromagnetic properties significantly differ at N = 82 compared to N < 82, despite the single unpaired proton (πg 9/2 proton orbit) dominating the behaviour of this complex many-body system.
Our new experimental results for 129,131 In show that upon reaching the neutron-rich magic number N = 82, abrupt changes of their electromagnetic properties are observed, and only at the neutron magic number the single-particle structure is almost fully recovered. This challenges our previous un-derstanding of these isotopes, which were assumed to have a single-particle description by invoking the use of empirical effective operators.
We presented two complementary nuclear models to investigate how these seemingly simple structures emerge from the complex interactions among nucleons. Both DFT and ab initio calculations provide a good description of the experimental trends. Within the DFT framework, the intricate isotopic dependence of the indium magnetic and electric ground-state moments turns out to require correct treatment of spin and shape polarization exerted by the proton-hole on the Z = 50 magic core. The inclusion of time-odd mean fields was shown to be essential to reproduce the experimental findings.
Indium nuclei can coexist in long-lived exited states with I π = 1/2 − . The structure of these states was suggested to be dominated by a single unpaired proton in the π p 1/2 orbit. However, in contrast to the simple trends observed for the ground states, the magnetic moments of the 1/2 − states exhibit a large variation with neutron number, previously considered a puzzle in the region 12 . These features and the new experimental observations are well described by our ab initio calculations. Our new measurements show that in addition to previous observations, the trend of their magnetic moments changes, with a sudden uptick at N = 82.
Although our ab initio calculations provide a good description of the experimental trends, they fall short in reproducing the magnitude of the electromagnetic moments. This may be attributed to the lack of inclusion of many-body currents, e.g. meson-exchange currents, which are known to be essential to describe the electromagnetic properties of light nuclei 42 . Future theoretical developments would be needed to include many-body currents and clarify the role that they play in heavy nuclei. On the other hand, a theoretical description of the nuclear quadrupole moments requires the extensive inclusion of neglected many-body correlations. Thus, our new experimental results at N = 82 provide critical data for future developments of both ab initio and DFT theory.
Our experimental results for 131 In allowed an investigation of the strength of the time-odd mean fields, which until now have been poorly constrained in DFT theories. Such time-odd channels are essential for a correct description of numerous nuclear properties, such as double beta-decay rates 8,9 , permanent electric dipole moment measurements 10,11 , and dark matter searches 36 . This provides strong motivation to extend experiments to other isotopes possessing single-hole (particle) with respect to suggested nuclear closed shells at extreme proton-to-neutron ratios, and to further pursue theoretical studies of these systems.

The collinear resonance ionisation setup
The indium isotopes were produced at CERN's on-line isotope separator facility, ISOLDE, by impinging 1.4-GeV protons onto the proton-to-neutron converter of a thick uranium-carbide target 8 . The converter suppressed nearby caesium mass contamination 9 . The indium isotopes diffused through the target material and their ionisation was significantly enhanced by the use of the resonance laser ion source, RILIS 10 . The produced indium ions were then accelerated to 40 keV and mass separated using the ISOLDE high-resolution mass separator before being cooled and bunched using a gas-filled linear Paul trap (ISCOOL) 11,12 . After a trapping time of up to 10 ms, ion bunches with a temporal width of 2 µs, were then re-accelerated to 40034(1) eV and deflected into the CRIS beamline 13,14 . The indium ions were then neutralised with a sodium-filled charge-exchange cell with an efficiency of up to 60% and predicted relative atomic populations of 57% and 37% respectively for the 5p 2 P 3/2 metastable state and 5p 2 P 1/2 ground state 15 . The remaining ion fraction was removed by electrostatic deflectors, and the neutralized atom bunch was collinearly overlapped with two pulsed lasers, one for excitation and another for non-resonant ionisation. The atoms were resonantly excited using two different UV transitions in separate measurements. The first using 246.8-nm laser light for the 5p 2 P 3/2 → 9s 2 S 1/2 atomic transition. The second using 246.0-nm laser light for the 5p 2 P 1/2 → 8s 2 S 1/2 atomic transition, which is more sensitive to nuclear magnetic moments, µ, but alone does not give the nuclear electric quadrupole moment, Q. The resonant laser light was produced by frequency tripling the light from an injection-locked Ti:Sapphire laser system 16 . This laser was seeded using a narrowband SolsTiS continuous-wave Ti:Sapphire laser, and pumped using a Lee Laser LDP-100MQ Nd:YAG laser, producing pulsed narrowband 740(738)-nm laser light at 1 kHz repetition rate. This light was then frequency tripled to 246.8(246.0)-nm light by the use of two non-linear crystals 17 . About 3 µJ of laser light was used to saturate both transitions. The excited atoms were then ionized by a final nonresonant 1064-nm step, provided by a Litron LPY 601 50-100 PIV Nd:YAG laser at 100 Hz repetition rate. The frequency of the resonant first step was scanned and the resulting ions were deflected onto a detector, producing the hyperfine structure containing spectra from which the hyperfine parameters were obtained. The wavelengths were measured using a HighFinesse WSU-2 wavemeter, which was In isotopes, measured using the 246.8-nm (5p 2 P 3/2 → 9s 2 S 1/2 ) transition. The 9/2 + ground and 1/2 − isomer states are indicated. b) Example spectra of the 1/2 − isomer structure measured with the 246.0-nm (5p 2 P 1/2 → 8s 2 S 1/2 ) transition.  (2)   drift stabilized by simultaneous measurement of a Toptica DLC DL PRO 780 diode laser locked to the 5s 2 S 1/2 → 5p 2 P 3/2 F = 2 → 3 transition of 87 Rb using a saturated absorption spectroscopy unit.

Evaluation of nuclear magnetic and quadrupole moments
The µ values were determined using a reference NMR value of µ re f = +5.5408 (2) µ N for 115 In 4 , and the relation where the differential hyperfine anomaly, ∆, is negligible for these atomic states of indium 18 .
Here A re f are our experimentally determined values for the 5p 2 P 3/2 , 9s 2 S 1/2 , 5p 2 P 1/2 and 8s 2 S 1/2 states of stable 115 In, and A are those of short-lived isotopes. The final µ values presented in Table 1 are a weighted average of the µ values from each atomic state, which were self-consistent within 2σ . The spectroscopic nuclear electric quadrupole moments, Q, were extracted using the relation where a value of B h f ( 2 P 3/2 )/Q = 576(4) MHz/b was used, obtained from relativistic coupled-cluster atomic calculations 7 . Here V ZZ is the electric field gradient produced by the electrons at the nucleus. The ratio of A h f factors is depends dominantly on atomic structure and therefore remains constant when with the correct nuclear spin assignment is used for fitting of spectra resulting from hyperfine structure 19 . Although reduced χ 2 fitting of the spectra (as shown in Figure 3) resulted in minima when we use I = 9/2; the spin assignment from ratio of upper and lower atomic state A h f factors (of the 5p 2 P 3/2 → 9s 2 S 1/2 transition) could not rule out nuclear spins 7/2, 11/2, outside of 1σ uncertainty. The spin of the I = 9/2 states are tentatively assigned by experiment and strongly supported by nuclear theory. The spin assignments of the I = 1/2 states were confirmed unambiguously from the number of peaks in their spectra.

VS-IMSRG calculations
The VS-IMSRG calculations start from the underlying NN and 3N interactions expressed in the harmonic-oscillator basis. We then construct an effective Hamiltonian designed for a particular valence space where the exact diagonalization is feasible 20 . The effective Hamiltonian was decoupled from the full A-body Hamiltonian through the application of an approximate unitary transformation derived from the Magnus expansion method 20 . Using the same transformation, the effective M1 and E2 valence-space operators were then decoupled consistently with the Hamiltonian. During the calculation, all of operators arising in nested commutator expansions were truncated at the two-body level, the IMSRG(2) approximation.
We begin in a spherical harmonic-oscillator basis with frequency of èω = 16 MeV and e = 2n + l ≤ e max = 14 for 1.8/2.0 (EM) (and 12 for N 2 LO GO ) with an additional cut of e 1 + e 2 + e 3 ≤ E 3max = 16 imposed due to storage limitations of 3N matrix elements. The valence space was taken as the proton {1p 1/2 , 1p 3/2 , 0 f 5/2 , 0g 9/2 } and neutron {2s 1/2 , 1d 3/2 , 1d 5/2 , 0g 7/2 , 0h 11/2 } single-particle orbits above 78 Ni core. The final valence-space diagonalization and evaluation of electromagnetic moments were performed with the KSHELL code 21 . The effect of fully accounting for translation invariance in the M1 operator was found to be less than 1%, largely due to the lack of radial dependence of the M1 operator.

DFT calculations
We performed the DFT calculations using code HFODD 22, 23 version (3.01m) and a standard Skyrme density functional UNEDF1 24 . The deformed DFT single-particle wave functions were expanded on the spherical harmonic-oscillator basis up to N 0 = 16 quanta. An A-dependent harmonic-oscillator frequency èω was fixed according to the prescription defined in Ref. 25 . For the state 9/2 + , the proton configuration (with no pairing correlations) was fixed by occupying 49 lowest axially deformed prolate orbitals, that is, the Nilsson orbital 26 [404] 9 2 with the angular-momentum projection on the symmetry axis Ω = −9/2 was left empty. Similarly, for the state 1/2 − , the hole was kept in the oblate Nilsson orbital [301] 1 2 with Ω = −1/2. It was essential to pick the proton-hole configurations with angular Figure 4. (color online) The a) nuclear magnetic dipole moments and b) electric quadrupole moments, respectively, of the 9/2 + states extracted by our experiment. Shown alongside our ab initio and DFT calculations, including predictions for N>82 momenta properly aligned along the axial-symmetry axis. Indeed, as discussed in Refs. 27,28 , the time-odd mean fields and core spin polarization depend on the relative orientation of the intrinsic angular momentum and shape.
For both proton configurations, the open-shell neutron configurations were constructed either with (HFB results) or without (HF results) pairing correlations included. For all deformed mean-field configurations, the angular-momentum symmetry was restored 27 , and in addition, for paired configurations the particle-number symmetry was restored as well 27 . Neither effective charges nor effective g-factors were used in the ab initio or DFT calculations. Figure 4 extends Figure 1, showing the DFT predictions of the magnetic dipole and electric quadrupole moments in indium, in an as-yet experimentally inaccessible region beyond the N = 82 shell closure. The confirmation of the moments at N = 90 would give strong evidence to support the existence of a new magic number of N = 90 in this neutron-rich tin region, for which theoretical and experimental evidence is beginning to hint towards [29][30][31] .