Doubly-magic character of 132 Sn studied via electromagnetic moments of 133 Sn

We report the ﬁrst measurement of the magnetic dipole and electric quadrupole moment of the exotic nucleus 133 Sn by high-resolution laser spectroscopy at ISOLDE / CERN. These, in combination with state-of-the-art shell-model calculations, demonstrate the single-particle character of the ground state of this short-lived isotope and, hence, the doubly-magic character of its immediate neighbor 132 Sn. The trend of the electromagnetic moments along the N = 83 isotonic chain, now enriched with the values of tin, are discussed on the basis of realistic shell-model calculations.

In nuclear physics, certain numbers of protons (Z ) or neutrons (N ), such as 2, 8, 20, 28, 50, 82, and 126, are known as "magic."These numbers endow the nucleus with a special Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Open access publication funded by the Max Planck Society.
stability analogous to the chemical stability associated with noble gases.Its existence led to the hypothesis that the nucleus contains shells of nucleons that are similar to the shells of electrons in an atom.About 250 species, of approximately 3000 discovered to date, have magic numbers of protons or neutrons, and only ten of them have magic numbers of both.Among this exclusive group, five nuclides are due to their radioactive nature notoriously difficult to access experimentally.However, thanks to state-of-the-art techniques, detailed spectroscopic information can nowadays be obtained for 132 Sn (50 protons and 82 neutrons), the heaviest radioactive doublymagic nucleus.
During the past decade, many experimental studies have aimed to investigate whether 132 Sn, eight neutrons away from the heaviest stable tin isotope, retains its doubly-magic character [1][2][3][4][5][6][7].It is, in fact, well recognized, that the singleparticle ordering which underlies nuclear shell structure may change in those nuclei with a large N/Z ratio, leading to the disappearance of classic shell gaps and the appearance of new magic numbers.Clear evidence of this phenomenon has been found for light-and medium-mass nuclei.For instance, it has been shown that in 42 Si N = 28 is no longer a magic number [8], whereas N = 16 does appear to be magic in neutronrich oxygen isotopes [9][10][11] and the same is suggested for N = 32 and 34 in calcium isotopes [12][13][14] although the doubly-magic nature of 52 Ca is challenged by recent laserspectroscopy work [15].As for 132 Sn, two transfer-reaction experiments have provided leading information through measurements of spectroscopic factors [16] and lifetimes [17] of ground and excited states in 133 Sn.Both experiments have shown that, regardless of its large neutron-to-proton ratio, this nucleus can be considered a very robust doubly-magic core.Such a finding is of importance in current nuclear structure research as the persistence of (double-) magicity despite an unbalanced N/Z ratio may shed light into the detailed mechanism causing the unexpected shell evolution in other areas of the nuclear landscape.Furthermore, it validates the choice of 132 Sn as a closed core in shell-model calculations, making them a reliable tool to describe this mass region, which is important for the rapid neutron-capture process creating elements in merging neutron stars [18,19].
In this Rapid Communication, we report new evidence of the doubly-magic character of 132 Sn through a measurement of the electromagnetic moments of 133 Sn using high-resolution collinear laser spectroscopy.The experimental data, in combination with state-of-the-art shell-model calculations, clearly show that 132 Sn plays a prominent role as a closed core and can, therefore, be used to describe more complex systems in this region.This is confirmed on highermass isotones (N = 83) for which experimental moments are found to be well described by theory.
The beam of 133 Sn was produced at ISOLDE/CERN.High-energy protons impinging on a tungsten rod generated spallation neutrons, which, in turn, induced fission in a uranium carbide target [20].Following laser ionization [21], electrostatic acceleration to 40 or 50 keV and mass selection, the ions were injected into a linear Paul trap [22], which provided bunched beams with a temporal width of about 5 μs.Fast ion bunches were released to the collinear laser spectroscopy beam line, postaccelerated and neutralized by charge exchange with sodium vapor [23,24].A continuous-wave laser beam was collinearly superimposed with the bunched atomic beam.The laser frequency was kept fixed whereas the Doppler-shifted frequency was scanned by varying the potential applied to the charge-exchange cell.The fluorescence emitted from the laser excited atomic beam was imaged by telescopes of aspheric lenses onto four photomultiplier tubes.To suppress background events a time gate corresponding to the laser and atom-bunch interaction time was applied to the photon signal.Details concerning the experimental setup can be found in the review by Neugart et al. [25].A sketch of the collinear laser spectroscopy beamline is given in Ref. [26].
Hyperfine structures were measured in two complementary transitions of the neutral tin atom, shown in Fig. 1.The transition 5p 2 1 S 0 → 5p6s 1 P 1 at 453 nm offers a large quadrupole splitting whereas the transition 5p 2 3 P 0 → 5p6s 3 P1 at 286 nm provides high sensitivity to magnetic moments.The laser light was produced by frequency doubling the fundamental light of Example hyperfine spectra of 133 Sn are presented in Fig. 1.Simultaneous analysis of the two transitions was conducted within the ROOT framework [27].A combined χ 2 was built and minimized using the WrappedMultiTF1 class and the MINUIT2 minimization package.The hyperfine A and B coefficients of the triplet state ( 3 P 1 ) and singlet state ( 1 P 1 ), respectively, were free parameters of the fit since these exhibit the larger response to the nuclear moments.The resonances were defined by where c 1 and c 2 are constants that depend on the nuclear, electronic, and total angular momentum quantum numbers [28].
The ratios of hyperfine coupling constants were defined with the aid of additional spectra, obtained in the same experimental run as explained below.R A = A( 1 P 1 )/A( 3 P 1 ) = 0.0517(2) was determined with high accuracy from simultaneously fitting the 1/2 + states, which do not undergo quadrupole splitting in 115,117,119 Sn [26].It was then used as a constraint in the fitting of the spectra of 133 Sn and 109 Sn also performed simultaneously.The addition of 109 Sn in the analysis aided the precision of the extracted R B = B( 3 P 1 )/B( 1 P 1 ) = −0.25(2)which was then adopted as a constraint in the analysis of the odd-mass isotopes 117-131 Sn [26].
The line profiles for the fitting were described by a symmetric Voigt function [29].The linewidth and background level were kept free and independent for each spectrum.The relative peak intensities were fixed to the Racah coefficients [30] for the 1 P1 state and were free parameters for the fully resolved 3 P1 state.
The effect of the hyperfine anomaly in 133 Sn due to the extended distribution of the magnetization over the nuclear volume [31] and the extended nuclear charge distribution [32], was estimated using a developer version of the General Relativistic Atomic Structure Package GRASP2K [33].The two-parameter Fermi model was used as the charge distribution and the magnetic distribution was approximated with the square of the harmonic-oscillator wave function of the last uncoupled neutron with h/(mω) = A 1/3 .The resulting hyperfine anomaly, is smaller than the uncertainty of the magnetic moment.It was, therefore, neglected during the fit and further treated as a contribution to the experimental error.By linking two independent measurements of the hyperfine structure in two J: 0 → 1 transitions we were able to confirm the spin I = 7/2 for the ground state.The electromagnetic moments, presented in Table I, were evaluated from the measured hyperfine parameters A( 3 P 1 ) = −965.2(5)and B( 1 P 1 ) = −102(3) MHz, through the following expressions: The constants above are taken from Ref. [26] and represent the average magnetic field per unit angular momentum and the electric field gradient generated by the electron cloud at the position of the nucleus, respectively.This measurement completes the sequence of ground-state moments of N = 83 isotones from tin to dysprosium as shown in Fig. 2. In the extreme single-particle shell model, the 7/2 − ground state of all these isotones are expected to be dominated by configurations with one valence neutron in the 1 f 7/2 orbital which can be considered to be almost entirely responsible for the magnetic and quadrupole moment of the nucleus.Consistent with the expectation for a doubly-magic-plus-one-neutron nucleus, both the magnetic and the quadrupole moment of 133 Sn are indeed very close to the single-particle estimates for a single neutron in the 1 f 7/2 orbital [34], indicated by the straight dotted lines in Fig. 2. On the other hand, for the higher-mass N = 83 isotones with an open proton shell, the single-particle shell model is a too crude approximation and the moments deviate from the dotted lines.These observations point to the single-particle character of 133 Sn and, hence, confirm the robustness of the 132 Sn core.In the following paragraphs, these qualitative findings will be supported by realistic shell model calculations.Note that the case of 135 Te does not follow the general trend of the other isotones and will be discussed at the end of this section.
The experimental magnetic dipole and electric quadrupole moments shown in Fig. 2 are also summarized in Table I and compared with theoretical results obtained by performing a realistic shell-model calculation.An effective Hamiltonian was derived from the high-precision CD-Bonn NN potential FIG. 2. Magnetic and quadrupole moments (dots) of the I = 7/2 − ground state of N = 83 isotones with even Z up to Z = 66 compared with shell-model calculations using microscopic (squares) and empirical (diamonds) effective operators.Gray dotted lines represent the effective single-particle estimates for a single neutron in the 1 f 7/2 orbital.These single-particle values were obtained by using standard prescriptions for the effective neutron charge and the effective neutron spin gyromagnetic factors, namely, e(ν ) = 0.7e and g s (ν ) = 0.7g free s (ν ), a choice supported by the microscopic calculations presented in this work.[44] renormalized by means of the V low-k approach [45] with the addition of the Coulomb term for the proton-proton interaction.This Hamiltonian has already been adopted in several previous studies of neutron-rich nuclei beyond 132 Sn [46].
The doubly-magic 132 Sn was considered as a core and all the neutron orbits of the 82-126 shell (0h 9/2 , 1 f 7/2 , 1 f 5/2 , 2p 3/2 , 2p 1/2 , 0i 13/2 ) and all the proton orbits of the 50-82 shell (0g 7/2 , 1d 5/2 , 1d 3/2 , 2s 1/2 , 0h 11/2 ) were included in the model space.The two-body matrix elements of the effective shell-model Hamiltonian for the chosen model space were derived using the Q box folded-diagram approach [47,48], including in the perturbative diagrammatic expansion of the Q box one and two-body diagrams up to second order in the interaction.The single-proton and single-neutron energies appearing in the one-body term of the Hamiltonian were taken where possible from experiment, namely, from the level schemes of 133 Sb and 133 Sn [49].The proton 2s 1/2 and neutron 0i 13/2 energies, which were not available, have been determined by reproducing the experimental energy of the 2150-keV 1/2 + state in 137 Cs and of the 2423-keV 10 + state in 134 Sb, respectively.
The electromagnetic properties are calculated by employing microscopic (Calc-M) as well as empirical (Calc-E) effective operators.A standard prescription is adopted for the empirical M1 operator, namely, the spin gyromagnetic factors for both protons and neutrons are quenched to 70% of their bare values, whereas the orbital ones are not modified [50].The empirical E 2 operators are obtained by choosing an effective proton charge of e p = 1.7e and an effective neutron charge of e n = 0.7e, which reproduce the experimental B(E 2; 0 + 1 → 2 + 1 ) values in 134 Te and 134 Sn [49].On the other hand, the single-particle matrix elements of the effective microscopic M1 and E 2 operators are calculated within the same framework of the shell-model Hamiltonian by employing the Suzuki-Okamoto formalism [51], that is an extension of the Q box-plus-folded diagram method for transition operators.Details on this procedure can be found in Ref. [52].Shell-model calculations have been carried out using the shell-model code KSHELL [53].Within the adopted model space, with 132 Sn as a core, 133 Sn is a one-valence system.Therefore, the results for 133 Sn from Calc-E coincide with the single-particle estimates shown in Fig. 2.
The values predicted by both calculations are very close to each other.In fact, the renormalization of the bare one-body matrix elements of the M1 and E 2 operators derived within the perturbative approach are consistent with the corrections introduced by using empirical effective charges and gyromagnetic factors.In particular, from Table I we see that the empirical and microscopic M1/E 2 operator produces about the same diagonal single-particle matrix element for the 1 f 7/2 neutron orbit, that is very close to the experimental value of 133 Sn.From a quantitative point of view the predictions of both calculations are in good agreement with the experimental data also for the higher-mass isotones, except for 135 Te which will be discussed at the end of the paper.In fact, discrepancies for magnetic moments are less than 0.1μ N in most of the cases, reaching the maximum value of 0.13μ N in 137 Xe, whereas for the electric moments the largest difference between theory and experiment is 0.14b in 139 Ba.
It is worth noting that, starting from 137 Xe with four valence protons, the observed overall trends of the magnetic dipole and electric quadrupole moments are well reproduced by the theory as shown in Fig. 2. The behavior of the two curves essentially reflects the effects of valence protons, which give a positive contribution to the magnetic moments and a negative contribution to the quadrupole moments, determining their respective decrease and increase in magnitude as compared to the values of 133 Sn.The ground state of a N = 83 nucleus can be written in terms of a neutron coupled to the N = 82 neighbor.Since protons coupled to a spin 0 do not contribute to the magnetic or quadrupole moment, these proton contributions arise mainly due to π 2 + 1 ⊗ ν f 7/2 configurations as shown by our calculations.Actually, we find that, whereas the π 0 + gs ⊗ ν f 7/2 component accounts for ≈85% of the calculated wave functions of the N = 83 ground states, a non-negligible percentage -ranging from 5 to 6%-, comes also from the π 2 + 1 ⊗ ν f 7/2 component.This 5 to 6% contribution in the wave function, indeed, results in an increase in magnetic moment and the amount depends on the magnetic moments of the yrast 2 + state in the N = 82 isotones.By using the experimental 2 + magnetic moments, which are known from 136 Xe up to 144 Sm and range in value between +1.5 and +1.9μ N [49], the magnetic moments of the N = 83 ground states are reproduced quite well in a simple two-level mixing calculation, which confirms the reliability of our predictions for their wave functions.Similar considerations are not possible, unfortunately, for the electric quadrupole moments.In fact, the required quadrupole moments of the yrast 2 + state in the N = 82 isotones have been measured only for 138 Ba and the sign of the π 2 + 1 |E 2|π 0 + gs matrix element, which also comes into play, is unknown.
In concluding, it is worth underlining that, although both microscopic and empirical calculations give a quite reasonable account of the experimental data, they are not able to reproduce the observed staggering for the magnetic moments, which may be related to changes in the structure of the proton wave functions not accounted by the adopted theoretical approach.Furthermore, the location of both theoretical curves for the quadrupole moment, which is slightly above the experimental one by 0.1b, suggests the need for a further small renormalization of the proton charge.
Regarding 135 Te, we observe that both experimental magnetic and quadrupole moments show a strong deviation from the experimental systematics and the calculated values.As suggested by the above discussion, the disagreement between theory and experiment implies that our calculations for the ground-state wave function of 135 Te underestimate the percentage of components including excited states of 134 Te.This conclusion, however, is not in line with the results of higher-mass N = 83 isotones.On this basis, a remeasurement of the electromagnetic moments of 135 Te is required in order to clarify the true structure of its ground state.
We have presented the first measurement of the magnetic dipole and electric quadrupole moment of 133 Sn by highresolution laser spectroscopy.The obtained electromagnetic moments approach the single-particle estimates for a single neutron in the 1 f 7/2 orbital suggesting a single-particle behavior on top of a closed 132 Sn core.Both magnetic and quadrupole moments are very well reproduced by theory, which gives also a good description of the moments of the higher-mass N = 83 isotones.We have also shown that the trend along the isotonic chain can be explained in simple terms by decomposing the ground-state wave functions of the N = 83 isotones as an 1 f 7/2 neutron coupled to the yrast 0 + and 2 + states of the N = 82 neighbors.The perturbative approach used to derive the microscopic effective M1 and E 2 operators, which does not need the introduction of adjustable parameters, induces the correct renormalizations.

FIG. 1 .
FIG.1.Hyperfine spectra of133 Sn: (a) in the 5p 2 1 S 0 → 5p6s 1 P 1 and (b) in the 5p 2 3 P 0 → 5p6s 3 P1 transitions.The horizontal axis is relative to the fine-structure transition.Solid lines represent a simultaneous fit of the two transitions.

TABLE I .
Electromagnetic moments of the I = 7/2 − ground state of N = 83 isotones from this work and from literature.Shell-model calculations using microscopic (Calc-M) as well as empirical (Calc-E) effective operators are included in the table.
a Statistical uncertainty is shown in a first set of parentheses and systematic uncertainty due to the electric-field gradient is shown in a second set of parentheses.