Magnetic moments of thallium isotopes in the vicinity of magic N = 126

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Magnetic moments of thallium isotopes in the vicinity of magic N = 126 © 2024 The Author(s). Published by Elsevier B.V. Funded by SCOAP³ Published version Yue, Z.; Andreyev, A.N.; Barzakh, A.E.; Borzov, I.N.; Cubiss, J.G.; Algora, A.; Au, M.; Balogh, M.; Bara, S.; Bark, R.A.; Bernerd, C.; Borge, M.J.G.; Brugnara, D.; Chrysalidis, K.; Cocolios, T.E.; De Witte, H.; Favier, Z.; Fraile, L.M.; Fynbo, H.O.U.; Gottardo, A.; Grzywacz, R.; Heinke, R.; Illana, A.; Jones, P.M.; Judson, D.S.; Korgul, A.; Köster, U.; Labiche, M.; Le, L.; Lica, R.; Madurga, M.; Marginean, N.; Marsh, B.; Mihai, C.; Nácher, E.; Neacsu, C.; Nita, C.; Olaizola, B.; Orce, J.N.; Page, C.A.A.; Page, R.D.; Pakarinen, J.; Papadakis, P.; Penyazkov, G.; Perea, A.; PiersaSiłkowska, M.; Podolyák, Zs.; Prosnyak, S.D.; Reis, E.; Rothe, S.; Sedlak, M.; Skripnikov, L.V.; Sotty, C.; Stegemann, S.; Tengblad, O.; Tolokonnikov, S.V.; Udías, J.M.; Van Duppen, P.; Warr, N.; Wojtaczka, W. Yue, Z., Andreyev, A.N., Barzakh, A.E., Borzov, I.N., Cubiss, J.G., Algora, A., Au, M., Balogh, M., Bara, S., Bark, R.A., Bernerd, C., Borge, M.J.G., Brugnara, D., Chrysalidis, K., Cocolios, T.E., De Witte, H., Favier, Z., Fraile, L.M., Fynbo, H.O.U., . . . Wojtaczka, W. (2024). Magnetic moments of thallium isotopes in the vicinity of magic N = 126. Physics Letters B, 849, Article 138452. https://doi.org/10.1016/j.physletb.2024.138452 2024 Phys. Lett. B 849 (2024) 138452 Available online 17 January 2024 0370-2693/© 2024 The Author(s). Published by Elsevier B.V. Funded by SCOAP³. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Contents lists available at ScienceDirect Physics Letters B journal homepage: www.elsevier.com/locate/physletb Letter Magnetic moments of thallium isotopes in the vicinity of magic 𝑁= 126 Z. Yuea, ,∗, A.N. Andreyeva,b,, A.E. Barzakhc,, I.N. Borzovc,, J.G. Cubissa,, A. Algorad, M. Aub,e,, M. Baloghf, S. Barag, R.A. Barkh, C. Bernerdb,g, M.J.G. Borgei,, D. Brugnaraf, K. Chrysalidisb, T.E. Cocoliosg, H. De Witteg, Z. Favierb, L.M. Frailej,, H.O.U. Fynbok, A. Gottardof, R. Grzywaczl, R. Heinkeb, A. Illanaj,m,n,, P.M. Jonesh,, D.S. Judsono, A. Korgulp, U. Kösterb,q, M. Labiche r, L. Le b, R. Licab,s, M. Madurga l, N. Marginean s, B. Marshb, C. Mihais, E. Nácherd, C. Neacsus, C. Nita s, B. Olaizola b,i,, J.N. Orce t, C.A.A. Pagea,, R.D. Pageo, J. Pakarinenb,m,n, P. Papadakis r, G. Penyazkov c,, A. Perea i, M. Piersa-Siłkowskab,p, Zs. Podolyákb,u,, S.D. Prosnyak c,, E. Reis b,v, S. Rothe b, M. Sedlakf,, L.V. Skripnikovc,, C. Sotty s, S. Stegemann b, O. Tengblad i, S.V. Tolokonnikovc,, J.M. Udías j, P. Van Duppen g,, N. Warr w,, W. Wojtaczkag aSchool of Physics, Engineering and Technology, University of York, York, YO10 5DD, United Kingdom bCERN, 1211 Geneva 23, Switzerland cAffiliated with an institute covered by a cooperation agreement with CERN dInstituto de Fisica Corpuscular, CSIC-Universidad de Valencia, E-46071 Valencia, Spain eJohannes Gutenberg-Universität, Saarstr. 21, 55099 Mainz, Germany fINFN, Laboratori Nazionali di Legnaro (LNL), Viale dell’Università 2, 35020 Legnaro (PD), Italy gInstituut voor Kernen Stralingsfysica, KU Leuven, B-3001, Leuven, Belgium hiThemba LABS, National Research Foundation, P.O. Box 722, Somerset West 7129, South Africa iInstituto de Estructura de la Materia, CSIC, 28006 Madrid, Spain jGrupo de Física Nuclear and IPARCOS, Universidad Complutense de Madrid, CEI Moncloa, E-28040 Madrid, Spain kDepartment of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark lDepartment of Physics and Astronomy, University of Tennessee, Knoxville, TN 37966, USA mDepartment of Physics, University of Jyväskylä, P.O. Box 35, FI-40014, Jyväskylä, Finland nHelsinki Institute of Physics, University of Helsinki, P.O. Box 64, FIN-00014, Helsinki, Finland oOliver Lodge Laboratory, University of Liverpool, Liverpool, L69 7ZE, United Kingdom pFaculty of Physics, University of Warsaw, Warsaw, PL 02-093, Poland qInstitut Laue-Langevin, F-38042, Grenoble, France rSTFC Daresbury Laboratory, Daresbury, WA4 4AD, Warrington, United Kingdom sHoria Hulubei National Institute of Physics and Nuclear Engineering (IFIN-HH), R-077125, Bucharest, Romania tDepartment of Physics, University of the Western Cape, P/B X17 Bellville 7535, South Africa uDepartment of Physics, University of Surrey, Guildford, GU2 7XH, United Kingdom vUniversität Duisburg–Essen, Duisburg, Germany wInstitut für Kernphysik, Universität zu Köln, Köln, D-50937, Germany A R T I C L E I N F O A B S T R A C T Editor: B. Blank Keywords: Laser spectroscopy Hyperfine structure Magnetic dipole moments Theory of finite Fermi systems The magnetic dipole moments (𝜇) of 209Tl𝑔(𝑁= 128) and 207Tl𝑚(𝑁= 126) have been measured for the first time using the in-source laser resonance-ionization spectroscopy technique with the Laser Ion Source and Trap (LIST) at ISOLDE (CERN). The application of the LIST suppresses the usually overwhelming background of the isobaric francium isotopes and allows access to heavy thallium isotopes with 𝐴 ⩾207. The self-consistent theory of finite Fermi systems based on the energy density functional by Fayans et al. well describes the 𝑁dependence of 𝜇for 1∕2+thallium ground states, as well as 𝜇for the 11∕2−isomeric states in europium, gold and thallium isotopes. * Corresponding author. E-mail address: [email protected] (Z. Yue). https://doi.org/10.1016/j.physletb.2024.138452 Received 20 July 2023; Received in revised form 15 November 2023; Accepted 8 January 2024 Physics Letters B 849 (2024) 138452 2 Z. Yue, A.N. Andreyev, A.E. Barzakh et al. The inclusion of particle-vibration coupling leads to a better agreement between the theory and experiment for 𝜇(Tl𝑔, 𝐼𝜋=1∕2 +). It is shown that beyond mean-field contributions to 𝜇cannot be neglected at least for thallium isotopes with 𝐼𝜋=1∕2 +. 1. Introduction Nuclear magnetic dipole moments (𝜇) provide direct information on the single-particle degrees of freedom in the nucleus and the underlying configurations of the valence nucleons [1]. Due to this, they are widely used to test the validity of nuclear theories (see, e.g. [2,3]). Of particular importance are data on the magnetic moments of states in doubly magic ±1 particle nuclei. The 𝜇value in these cases is expected to be determined by the last occupied single particle orbit (Schmidt moment, see Refs. [4–6]and references therein). However, in the majority of cases, the observed 𝜇values strongly deviate from the Schmidt predictions. This deviation can serve as a sensitive probe for the polarization induced by the unpaired valence nucleons, meson-exchange currents in nuclear medium and multiparticle excitations (see, for example, Ref. [5] and references therein). The doubly magic ±1 particle nuclei have simpler structure in comparison with other isotopes with odd number of protons or neutrons, therefore, the failure of the theory in their description can be more confidently connected with definite drawbacks in accounting for one or the other mechanism of the departure from the Schmidt value. There were many attempts to describe theoretically 𝜇for near-magic nuclei (see examples below in this Section). Nevertheless, some problems still remain, and new experimental data as well as theoretical analysis are necessary to obtain a self-consistent and comprehensive picture. In the present work we take a step toward this goal by studying nuclei in close vicinity to doubly-magic 208Pb. The magnetic moment of the 𝐼𝜋=1∕2 +ground state of 207Tl (𝑁= 126, 𝑍=81) was first measured nearly four decades ago (1.869(5) 𝜇𝑁 [7,8]) with a strong deviation from the Schmidt value (2.793 𝜇𝑁) and since then it was extensively used to test various approaches to 𝜇calculation (see e.g. the recent studies [9,10]). However, there is a long-lived (𝑇1∕2 =1.33 𝑠) 𝐼𝜋= 11∕2−isomer 207Tl𝑚with an unknown 𝜇value, which is expected to be well described by a single-hole configuration based on the 𝜋ℎ11∕2 orbital. In contrast with the 𝜋𝑠1∕2 ground state, which could have an appreciable contribution from other configurations, the 𝜋ℎ11∕2 orbit is a unique-parity state, thus is expected to have less mixing with other configurations. Therefore, different theoretical mechanisms can be tested by comparison with experiment for these two nuclear states. The measurement of 𝜇(207Tl𝑔)also revealed a marked discontinuity in the 𝜇isotopic trend for thallium ground states with 𝐼𝜋=1∕2 +. A noticeable drop was found when going from 207Tl𝑔to 205Tl𝑔(from 1.88 𝜇𝑁to 1.64 𝜇𝑁) [7,11], whereas for odd-𝐴179−205Tl𝑔isotopes, 𝜇remains nearly constant (between 1.60 𝜇𝑁and 1.64 𝜇𝑁; [12–14]). A similar reduction relative to 207Tl𝑔is expected for 209Tl𝑔(𝑁= 128) due to presumably the same underlying effect of the particle-vibration coupling [11]. This letter reports on the first measurement of 𝜇for 207Tl𝑚 and 209Tl𝑔with laser resonance-ionization spectroscopy at ISOLDE (CERN) [15]using the laser ion source and trap (LIST) [16–18]. The latter allowed production of isobarically-clean beams of thallium which was the key for the success of these measurements. Several self-consistent 𝜇calculations based on different energydensity functionals (EDF) were performed previously, such as the relativistic point-coupling model (see Refs. [6,19]and references therein), and Skyrme-Hartree-Fock Random Phase Approximation (RPA) calculations (see Ref. [20]and references therein). These and similar approaches restrict themselves as a rule to near-magic nuclei, and their applicability to the description of 𝜇in long isotopic chains is not obvious. Recent deformed EDF (DEDF) calculations with symmetry-restored wave functions and accounting for the time-odd terms in the functional, tackle the 𝜇-problem in an alternative way without going beyond the mean field level [9]. In Refs. [21–23], this method was successfully applied to high-spin states along several isotopic chains with 𝐼𝜋= 13∕2+, 11∕2−, and 9∕2+. In the present work, we use the self-consistent theory of finite Fermi systems (TFFS) [24]based on the Fayans DF3-a functional [25–27]for calculating 𝜇across long isotopic chains. The basics of our approach can be found in Refs. [10,28,29]. The same framework has also been applied for describing nuclear radii [25]and 𝛽decay [30]. Here, this approach has been substantially extended compared to its previous applications. It will be shown that the TFFS can qualitatively reproduce both the general trend, and the pronounced irregularities at 𝑁= 126 that our experiment revealed in the evolution of the 𝜇values in the thallium chain. 2. Experimental details Radioactive thallium isotopes were produced in spallation reactions by a 1.4-GeV proton beam (average intensity up to 2 𝜇A) from the CERN proton synchrotron booster bombarding a 50 g∕cm2UC𝑥target. The reaction products diffused through the target material, kept at a temperature of ∼ 2000 ◦C, and effused into the hot ion-source cavity as neutral atoms. At the masses of interest (𝐴 = 207−209), very strong isobaric contamination from surface-ionized francium is present, which prevented extensions of earlier experiments to heavier masses. The yields of francium isotopes at 𝐴 = 207−209 are >107ions/𝜇C [31,32], several orders of magnitude above the production rate of isobaric thallium nuclei (see Sec. 2.1). To overcome this problem, the LIST was used [16–18]. It separates the regions of laser and surface ionization by using a positively charged repeller electrode positioned immediately downstream of the hot cavity at the entrance of the LIST. This suppresses the flow of ions from the cavity, including those of surface-ionized francium. Only neutral atoms may diffuse into the ion guide of the LIST, where laser ionization of thallium isotopes takes place. Inside the LIST, thallium atoms were resonantly ionized when the laser beams were wavelength-tuned to the two-step thallium ionization scheme: 6𝑝2𝑃1∕2 277nm ←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←←→ LiopStar 6𝑑2𝐷3∕2 532nm ←←←←←←←←←←←←←←←←←←←←←←←←←→ Blaze continuum.(1) The excitation of the 6𝑝 2𝑃1∕2 →6𝑑2𝐷3∕2 transition was performed by a frequency-doubled tunable dye laser (LIOP-TEC LiopStar) beam at 277 nm with a linewidth of ≈3GHz. This transition was scanned to map out the hfs spectra. The subsequent ionization step was provided by a frequency doubled Nd:YVO4laser (Lumera Blaze). In our experiment, the francium ions were suppressed by a factor of ∼104(see Fig. 1 in Supplementary Materials). At the same time, the laser-ionized thallium isotopes were suppressed by a factor of ∼20, relative to normal RILIS operation without the LIST [17]. These losses are mostly due to the reduced spatial overlap of the thallium atoms and the laser beam inside the LIST cavity. The improved signal-to-background ratio was the key condition which allowed us to perform our measurements. After ionization, the thallium ions were extracted and accelerated by a 50-kV electrostatic potential, and mass-separated by the ISOLDE general purpose separator (GPS). The ions were then delivered to either a Faraday cup (FC) or the ISOLDE Decay Station (IDS) [33]for ion counting. The ion current of the stable isotopes 203,205Tl from a dedicated Physics Letters B 849 (2024) 138452 3 Z. Yue, A.N. Andreyev, A.E. Barzakh et al. Fig. 1. The hfs of studied thallium isotopes. The solid red lines are Voigt profile fits of the data. The nuclear spin and photoion detection method (FC or IDS), as well as 𝛾-ray energies [34]in the case of decay-based detection at IDS, are displayed for each isotope. The zero point on the frequency scale corresponds to a wavenumber of 36117.92 cm−1. The ion current of the stable isotopes 203,205Tl was produced by the heating of the dedicated oven. Table 1 Measured yields for thallium isotopes using the LIST. Uncertainties are estimated as 30%. Nucleus Ions/𝜇C 207Tl𝑔5.5×106 207Tl𝑚3.4×101 208Tl 1.5×104 209Tl 3.2×102 oven, as well as abundantly produced 207Tlg, were directly measured by the FC as reference scans to validate the stability of the laser. 205Tl was used to regularly check the stability of thallium production and LIST performance. At the IDS setup, the short-lived isotopes 207𝑚,209Tl were implanted onto a movable aluminized mylar tape which removed longer-lived contamination due to the decay products of the isotopes of interest (for example, 𝑇1∕2(209Pb) =3.2h). The production rate of thallium isotopes was measured via their 𝛽-delayed or internal transition (IT) 𝛾rays with six high purity Germanium clover detectors at IDS. Fig. 1shows examples of the hfs spectra, whereby the count rate in FC or IDS is plotted as a function of the frequency of the scanning laser. 2.1. Production yields for 207−209Tl Table 1summarizes the thallium yields measured in this work. We note that a target unit that had been employed in the previous experimental campaigns was used for the present study. Typically this re-use results in lower production yields than with the new targets. The sudden drop in yield between 208Tl and 209Tl was due to a difference in indirect, in-target production. A significant amount of 208Tl Table 2 Magnetic hfs constants for the atomic ground state 6𝑝 2𝑃1∕2 (𝑎1) and magnetic moments (𝜇) of thallium isotopes. Where available, literature values are shown in a second line for each isotope. 𝐴𝐼𝑎 1(MHz) 𝜇(𝜇𝑁) 203 1/2 21180(102) 1.622(8) 21105.4497638(5) [36] 1.616(2) [8] 205 1/2 21302(46) 1.631(4) 21310.835(5) [37] 1.632(2) [8] 207 g 1/2 24398(44) 1.868(6) 24690(300) [13] 1.869(5) [8] 207 m 11/2 8391(85) 7.045(84) 209 1/2 22650(330) 1.735(28) originated from the 𝛼decay of 212Bi (𝑇1∕2 ≈1 h, 𝑏𝛼≈ 36%), which was produced in the target both directly and via the in-target decay of the abundantly-produced precursors (216At, 220Fr). The analogous 213Bi→209Tl decay has a branching ratio of only 𝑏𝛼≈2%. 3. Data analysis and results 3.1. Fitting of the hyperfine-structure spectra The data analysis procedure was the same as in our previous studies of the neutron-deficient thallium isotopes (𝐴 = 179 − 207) [13,14]. The positions of the hyperfine components in the hfs spectra are determined by the standard relation [13]with five parameters: nuclear spin (𝐼), isotope shift relative to the stable 205Tl (𝛿𝜈𝐴,205 277nm), magnetic hfs constants (𝑎1and 𝑎2) for the first and the second level of the ionization scheme, and the electric quadrupole hfs constant 𝑏2for the second level. Note that 𝑏1≡0, since the first level in the ionization scheme has electronic angular momentum 𝐽=1∕2. Voigt profiles were fitted to the experimental hfs spectra [13,14], using a fixed 𝑎2∕𝑎1=−0.002013(19) ratio taken from the value for the stable isotope 205Tl [35], and 𝐼values from [34]. For 207Tl𝑚with 𝐼= 11∕2, the possible quadrupole splitting of the upper level 6𝑑2𝐷3∕2 of the scanned transition should be taken into account. There are no experimental data on the hfs quadrupole constant 𝑏2for this level in the thallium atom, therefore we made dedicated atomic calculations. The hyperfine constant 𝑏is related to the spectroscopic quadrupole moment 𝑄𝑠via the equation: 𝑏 =𝑒𝑄𝑠×𝑉, where 𝑉is the electric field gradient (EFG) produced by the electrons at the site of the nucleus. The value of the EFG for the 6𝑑2𝐷3∕2 level, 𝑉=41.2(23) MHz∕b, was calculated by employing the relativistic coupled cluster theory and the Dirac-Coulomb Hamiltonian (see details in Supplementary Materials). 207Tl𝑚is a nucleus with a single proton hole in a doubly magic 208Pb. Correspondingly, its quadrupole moment should be less than that of nuclei with the same configuration and moderate deformation (e.g. 𝐼𝜋= 11∕2−isomers in 197,195,193Au, all having |𝑄𝑠| <2b, see [38]). Thus, we assume |𝑄𝑠(207Tl𝑚)| <2b as an estimate for the calculations. Using the calculated EFG, one obtains 𝑏(6𝑑2𝐷3∕2; 207Tlm) <83 MHz. To test the sensitivity to the 𝑏2value, we fitted the 207Tl𝑚hfs with 𝑏2=0 and 𝑏2=83MHz assumptions. The resulting difference in the 𝑎constant is ∼3 MHz, thus, the influence of the 6𝑑2𝐷3∕2-state quadrupole splitting on the fitting result is negligible relative to our experimental precision. In Table 2, the 𝑎(6𝑝 2𝑃1∕2) constant for the studied thallium isotopes are shown, along with values from literature. Our results for 203,205,207Tl𝑔agree with literature data fairly well. 3.2. Magnetic dipole moments Based on the known values of 𝜇205 =1.632(2) 𝜇𝑁[8], 𝑎205 = 21310.835(5) MHz [37]and 𝐼205 =1∕2 of stable 205Tl, the 𝜇values for thallium isotopes were evaluated using the relation: Physics Letters B 849 (2024) 138452 4 Z. Yue, A.N. Andreyev, A.E. Barzakh et al. Fig. 2. The 𝜇values for 1∕2+ground states in thallium isotopes. The black triangles are experimental data from [41]and the present work (209Tl). The blue star is the theoretical value from [9]. The closed and open circles show our DF3-a+CQRPA calculations results with and without PC correction. The green dashed line marks the Schmidt value for the 𝜋𝑠1∕2 state. Fig. 3. The 𝜇values for the 11∕2−states in europium (𝑍=63), gold (𝑍=79), and thallium (𝑍=81) isotopes. The triangles are experimental data from [39, 41]and the present work (207Tl𝑚), the stars are theoretical values from [22], and the circles are theoretical values from the present work. The green dashed line marks the Schmidt value for the 𝜋ℎ11∕2 state. 𝜇𝐴=𝜇205 ⋅𝐼𝐴 𝐼205 ⋅ 𝑎𝐴(6𝑃1∕2) 𝑎205(6𝑃1∕2)⋅[1 +205 Δ𝐴(6𝑃1∕2)] (2) where 205Δ𝐴(6𝑃1∕2)is the relative hyperfine anomaly (RHFA) for the 6𝑝 2𝑃1∕2 atomic state. The RHFA for 207,209Tl𝑔were estimated in accordance with the procedure outlined in Ref. [39] taking into account that the hyperfine anomaly reveals itself in the change in the ratio of the magnetic hfs constants for different atomic states (see details in Supplementary Materials): |205Δ207(6𝑃1∕2)| <1.2 ⋅10−3, |205Δ209(6𝑃1∕2)| <1.2 ⋅10−3. The RHFA for the 𝜋ℎ11∕2 state in 207Tl was estimated using the RHFA for the 𝜋ℎ9∕2 state in thallium, deduced from experimental data [40], and the single-particle approximation for the ratio of the RHFA values for the 𝜋ℎ9∕2 and 𝜋ℎ11∕2 shell-model states (see details in Supplementary Materials): 205Δ207𝑚(6𝑃1∕2) =−0.0033(16). The magnetic moments of 209Tl and 207Tl𝑚were determined for the first time; they are presented in Table 2and in Figs. 2, 3along with the previously measured 𝜇values for the thallium ground states (Fig. 2), as well as gold and europium 11∕2−isomers (Fig. 3). The new result for 𝜇(209Tl; 𝑁= 128), shows that there is a maximum at 𝑁= 126, with a less pronounced decrease in 𝜇when going from 𝑁= 126 to 𝑁= 128, compared to that when going from 𝑁= 126 to 𝑁= 124. 3.2.1. Renormalization of the proton orbital 𝑔factor Our new data for 207Tl𝑚provide further insight into the proton orbital gyromagnetic factor 𝑔𝓁near 208Pb. Earlier studies showed that in order to describe 𝜇values associated with single-particle orbitals near 208Pb, an effective value 𝑔eff 𝓁∼1.1is needed instead of the free-proton value 𝑔𝓁=1 [24,42–44]. The magnetism connected with the orbital motion of nucleons is one of the few nuclear properties at low excitation energy directly connected with mesonic exchange currents (see, for example, Refs. [45–48]and references therein). Previous estimations of the “experimental” 𝑔eff 𝓁factor stemmed from considerations of single-particle high-spin states 𝑔factors (𝑔= 𝜇∕𝐼), but some were extracted from the measured 𝑔factors of twoparticles excited states using the additivity relation for magnetic moments ([42–44,49,50]and references therein) rather than from direct measurements. However, the usage of the additivity relation introduces a poorly defined uncertainty. Indeed, there are well-known cases where additivity does not work, see for example, the systematics of 𝑔factors for ℎ𝑛 9∕2 states in the 𝑁= 126 isotones [51–53], or the strong and so far unexplained violation of additivity in 196,198Au𝑚, 𝐼𝜋=12 −[54,55]. In contrast to this, our approach with directly measured 𝜇(207Tl𝑚; 𝜋ℎ11∕2), is free from this indeterminacy. We follow the suggestion made in Refs. [49,56], that the measurement of the 𝑔factors of the two spin-orbit partners (e.g. 𝜋ℎ11∕2 and 𝜋ℎ9∕2) in a doubly magic ±1 particle nuclei gives, with a good approximation, the value of the effective 𝑔𝓁factor: 𝑔(𝑗=𝓁+1∕2)+𝑔(𝑗=𝓁−1∕2)=2𝑔eff 𝓁.(3) Our measurement of 𝜇(207Tl𝑚; 𝜋ℎ11∕2)=7.045(84) 𝜇𝑁[𝑔(𝜋ℎ11∕2)= 1.281(15)] allows us to implement this procedure for the first time, taking into account that 𝜇(209Bi𝑔; 𝜋ℎ9∕2)=4.0900(15) 𝜇𝑁[𝑔(𝜋ℎ9∕2)= 0.9089(3)] was measured earlier [57]: 𝑔eff 𝓁=[𝑔(𝜋ℎ11∕2)+𝑔(𝜋ℎ9∕2)]∕2 = 1.095(11) (4) Thus, our result supports the previously proposed renormalized value 𝑔eff 𝑙=1.115(20) [49]. 4. Theoretical calculations The experimental data were compared with theoretical calculations performed within the framework of the self-consistent finite Fermi-system theory [24]. The nuclear ground state is constructed in the spherical EDF approach using the DF3-a functional by Fayans et al. [25–27]. The self-consistent feedback between the core deformation and static polarization induced by unpaired nucleon is given by the TFFS effective field. It is determined from the continuum quasiparticle random-phase-like (CQRPA) equations with the effective 𝑁𝑁interaction taken as the Landau-Migdal spin-dependent interaction augmented with the one-pion and rho-meson exchange modified by the nuclear medium [10]. Solving these equations directly in coordinate space allows one to exactly take into account the single-particle continuum in nuclei with pairing correlations. In our calculation, the CQRPA part of this framework has been substantially extended compared to its previous application in Refs. [10,28, 29]). A zero-range density-dependent surface pairing is used in diagonal HFB approximation on the extended base including the quasi-discrete states up to 𝐸Fermi =37MeV. On the top of CQRPA, the higher-order quasiparticle-phonon coupling (PC) correction was included. Notice, that there are two types of PC corrections: “regular”, which is supposed to be included into the Physics Letters B 849 (2024) 138452 5 Z. Yue, A.N. Andreyev, A.E. Barzakh et al. standard TFFS parameters, and “non-regular” (𝐴, 𝑍and state dependent) which should be calculated separately. In order to avoid double counting, regular corrections should be excluded in the PC-contribution calculations. We used a special procedure developed in [29]to account for non-regular PC contributions in the so-called 𝑔2 𝐿approximation of the perturbation theory (here 𝑔𝐿is the 𝐿-phonon creation amplitude, not to be confused with 𝑔𝓁gyromagnetic factor). To trace the 𝑁dependence of the 𝜇(11∕2−) values in the odd-𝑍 isotopic chains, we also calculated 𝜇for several gold and europium isomers. For the gold isotopic chain where deformation is non-negligible, the standard kinematic factors correction [10]is added within the particle-rotator model [58]. It should be stressed that in the present calculations we did not introduce or redefine any model parameters which were fixed in [10,28]. 5. Discussion In Fig. 2the results of our DF3-a+CQRPA calculations for the ground state 𝜇values of 187−211Tl (𝐼𝜋=1∕2 +) are compared to experimental data. The result for 209Tl𝑔(𝑁= 128) proves to be rather unexpected. Namely, it was suggested earlier [11]that in order to explain the jump in magnetic moments of the 1∕2+thallium state when going from 𝑁= 124 to 𝑁= 126, the particle-vibration coupling (PVC) should be taken into account. The justification of this suggestion is straightforward. Indeed, the energy of the 2+ 1phonon of the doubly-magic 208Pb core (4.085 MeV) is markedly higher than in lead isotopes with 𝑁<126 (0.803 MeV for 𝑁= 124; 0.899 MeV for 𝑁= 122 etc.). Keeping in mind that 𝐸(2+ 1, 210Pb128)= 0.800 MeV, i.e. is similar to 𝐸(2+ 1, 206Pb124), one can expect that the PVC effect will be the same for 209Tl128 and 205Tl124, that is, 𝜇(209Tl𝑔)should be close to 𝜇(205Tl𝑔). However, the decrease of 𝜇when going from 𝑁= 126 to 𝑁= 124 (0.24 𝜇𝑁) proves to be nearly two times larger than that when going from 𝑁= 126 to 𝑁= 128 (0.13 𝜇𝑁). This “asymmetry” casts doubt on the completeness of the conventional PVC explanation of the 𝜇irregularity in thallium isotopes near magic 𝑁= 126. Our calculations without the PC correction (open circles in Fig. 2) reproduce the general trend with mean deviation of ≈0.2𝜇𝑁(≈ 13%). However, in these variant the increase at 𝑁= 126 practically disappears. Accounting for the PC correction (full circles in Fig. 2) decreases the mean deviation from experiment to ≈0.13 𝜇𝑁. The reduction of 𝜇either side of the magic number is qualitatively reproduced, although the size of the decrease is still half of the experimental one. The smoother 𝑁-dependence of the DF3-a+CQRPA 𝜇values, as well as the better overall agreement with experiment compared to the earlier calculations in Ref. [29]has been achieved due to the inclusion of the quasi-discrete states with 𝐸⪅𝐸Fermi in the pairing calculation. Importantly, 𝜇expt (207Tl𝑔) =1.869(5) 𝜇𝑁is well described within the DF3-a+CQRPA approach (1.897 𝜇𝑁) with a deviation from the experiment of just 0.028(5) 𝜇𝑁. This agreement can be mainly ascribed to exact accounting for the particle-hole continuum and taking into account the meson exchange current effects in the external field operator, as well as the one-pion exchange in the effective 𝑁𝑁-interaction. Fig. 2also shows the calculated value of 𝜇(207Tl𝑔)from the recent DEDF study [9]. This value of 𝜇(207Tl𝑔) =2.61(2) 𝜇𝑁strongly deviates from the experimental data. This is highlighted in Fig. 2, which demonstrates the departure of this point from the experiment. As indicated in Ref. [9], 207Tl belongs to the group of “outliers” (57Ni, 133Sb, 207Tl, 209Bi) in their calculations, for which the calculated values strongly deviate from experiment (79%, 29%, 39%, 25%, respectively). At the same time, these nuclei do not present any peculiarity for DF3-a+CQRPA calculations and have the “standard” deviation from the experimental data (11%, 12%, 1%, 11%, respectively, see Ref. [10]). The key point of the DEDF calculations in Ref. [9]is the assumption that the magnetic moments of odd-𝐴nuclei can be analyzed in terms of the self-consistent polarization effects caused by the presence of the unpaired nucleon. In other words, the polarization responsible for the deviation of 𝜇from the Schmidt value is supposed to be exhaustively taken into account already at the deformed mean-field level. The disagreement with experiment for the “outliers” was attributed to the presence of configuration mixing that is not fully taken into account in the mean-field calculations [9]. In contrast with such an approach, in the TFFS calculations the first-order polarization produced by the odd nucleon is described by the CQRPA equations developed on the basis of the spherical self-consisted mean field. The quasiparticle-phonon coupling is responsible for the higher-order polarization effects. From the TFFS point of view the disagreement of the DEDF results with the experiment for the “outliers” could be mostly explained by the underestimation of the polarization in the pure mean-field approach. Indeed, in the earlier version of the TFFS [24]there was nearly the same strong discrepancy between the theoretical and experimental results for 207Tl𝑔(2.5 𝜇𝑁[24], 2.6 𝜇𝑁[9] versus the experimental value of 1.869(5) 𝜇𝑁). However, it was shown that a more accurate solving of the CQRPA equation with exact accounting for the single-particle continuum completely removes this discrepancy ([10]; see also Table 2). Our DF3-a+CQRPA calculation has shown that the CQRPA polarization is responsible for 66% of the deviation of 𝜇(207Tl𝑔) from the Schmidt value. For comparison, the TFFS calculation performed for Sly4 functional [59]results in a similar value of 61%. Note, that the absolute values of 𝜇(207Tl𝑔) in both calculations are in good agreement with the experiment (DF3-a+CQRPA: 1.897 𝜇𝑁; Sly4-TFFS: 1.884 𝜇𝑁). Another substantial mechanism of the 𝜇reduction within the TFFS is connected with the so-called local charges arising due to a nuclearmedium-induced change in the meson exchange currents and the multiparticle many-body diagrams [10,24]. In our DF3-a+CQRPA calculations, this “medium polarization” gives the remaining 34% of the deviation from the Schmidt value (the Sly4-TFFS calculation gives 33%). Both types of polarization seem to be missed in the pure mean-field DEDF approach. In Fig. 3the results of our DF3-a+CQRPA calculations for the europium, gold and thallium 11∕2−isomers are compared with experimental values. The calculations describe the 𝑁dependence of the 𝜇 values well, spanning the range of 𝑁=82 − 126 with the pronounced maxima at the magic numbers. In Fig. 3the results of the DEDF calculations [22]are also shown. The DEDF and TFFS approaches display a comparable accuracy in describing the 11∕2−isomers. Notice, that the relatively small CQRPA-polarization effect is induced by the 𝑀1-fields in this case. The absolute value of this polarization amounts to only 9% of the difference between the Schmidt value and the experiment for the Sly4-TFFS calculation of 207Tl𝑚[59]. This is in contrast to the much stronger CQRPA polarizaton for the 1∕2+state, which presents a greater challenge for theoretical models, as explained above. A “self-consistent” polarization correction due to time-odd fields was suggested earlier in [19]within the relativistic covariant density functional theory (CDFT). The contribution of this polarization to 𝜇was calculated as the difference between 𝜇of triaxially deformed CDFT and 𝜇of spherical CDFT. As in the DEDF, it turned to be insufficient to explain the deviation from the Schmidt value. An agreement with the experiment for 207Tl𝑔, as well as for other doubly magic ±1 particle nuclei, was achieved only after taking into account the beyond mean-field contributions (meson exchange currents, firstand second-order polarization corrections) [6,19]. 6. Conclusion The 𝜇values for 207Tl𝑚(𝐼𝜋= 11∕2−)and 209Tl𝑔(𝐼𝜋=1∕2 +)have been measured for the first time, using laser resonance-ionization spectroscopy in the LIST device at ISOLDE (CERN). Self-consistent DF3-a+CQRPA calculations taking into account the meson exchange in the external field operator and effective 𝑁𝑁- Physics Letters B 849 (2024) 138452 6 Z. Yue, A.N. Andreyev, A.E. Barzakh et al. interaction, as well as regular effects of 𝑛𝑝 −𝑛ℎ configurations and non-regular PC corrections, have shown an improved description of the 𝜇values in the long isotopic chain of 1∕2+ground states in thallium isotopes. The calculations agree fairly well with the general experimental trend, and qualitatively reproduce the “asymmetric” jump at 𝑁= 126 revealed by our measurement of 𝜇(209Tl). However, the calculations still underestimate the jump at 𝑁= 126. Note, that “regular” PC corrections were disregarded to avoid double counting, since these corrections are supposed to be included into the standard TFFS parameters. The discarding criteria should be refined in order to check whether the omitted corrections will enable better agreement with experiment. In this context, the development of the so-called “subtraction method” in Ref. [60]may be effective for ensuring a stable computational scheme for the 𝜇values. Fully consistent systematic calculations of the electromagnetic moments including all corrections beyond the 𝑔2 𝐿approximation remain to be developed. The calculations also describe the 𝑁dependence of the 𝜇values for the 11∕2−isomers fairly well, spanning the range of 𝑁=82 − 126. In order to check the predictive power of the theory it would be important to fill the gaps in the 𝜇(𝜋ℎ11∕2)systematics, namely, to measure magnetic moments for the known long-lived 11∕2−states in 167−173,193−197Ir90−96,116−120, 141Eu78, 205Au126. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Data availability Data will be made available on request. Acknowledgements We would like to acknowledge the support of the ISOLDE collaboration and technical teams. The work was supported by the STFC Grants Nos. ST/V001035/1, ST/V001108/1, ST/V001027/1 and ST/P004598/1, by the Romanian Nucleu project No. PN 23 21 01 02 and Institutul de Fizic˘ a Atomic˘ a (IFA) grant CERN/ISOLDE, by the Research Foundation Flanders (FWO, Belgium), by BOF KU Leuven (C14/22/104), by the Spanish funding agency MICIN/AEI (FEDER, EU) via projects Nos. RTI2018-098868-B-I00 and PID2021-126998OBI00, by German BMBF under contract 05P21PKCI1 and Verbundprojekt 05P2021, by the Polish Ministry of Education and Science under Contract No. 2021/WK/07, and by the Polish National Science Center under Grant No. 2020/39/B/ST2/02346. M. 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