Measurement of an unusually large magnetic octupole moment in 45 Sc challenges state-of-the-art nuclear-structure theory
R. P. de Groote, J. Moreno, J. Dobaczewski, I. Moore, M. Reponen, B. K. Sahoo, C. Yuan
MMeasurement of an unusually large magnetic octupole moment in Sc challengesstate-of-the-art nuclear-structure theory
R. P. de Groote, J. Moreno, J. Dobaczewski,
2, 3
I. Moore, M. Reponen, B. K. Sahoo, and C. Yuan Department of Physics, University of Jyv¨askyl¨a, PB 35(YFL) FIN-40351 Jyv¨askyl¨a, Finland ∗ Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom Institute of Theoretical Physics, Faculty of Physics,University of Warsaw, ul. Pasteura 5, PL-02-093 Warsaw, Poland Atomic, Molecular and Optical Physics Division,Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-Sen University, Zhuhai 519082, China (Dated: May 4, 2020)We measure the hyperfine C -constant of the 3 d s D / atomic state in Sc: C = − . d s D / state and second-order corrections are performed to infer the nuclear magnetic octupole moment Ω = 1 . µ N b .With a single valence proton outside of the doubly-magic calcium core, this element is ideally suitedfor an in-depth study of the many intriguing nuclear structure phenomena observed within theneighboring isotopes of calcium. We compare Ω to shell-model calculations, and find that theycannot reproduce the experimental value of Ω for Sc. We furthermore explore the use of DensityFunctional Theory for evaluating Ω, and obtain values in line with the shell-model calculations.This work provides a crucial step in guiding future measurements of this fundamental quantity onradioactive scandium isotopes and will hopefully motivate a renewed experimental and theoreticalinterest.
The application of laser spectroscopic techniques toelucidate the subtle perturbations of atomic energy levelsdue to the nuclear electromagnetic properties has givenrise to the study of fundamental nuclear structure, in par-ticular magnetic dipole moments ( µ ), electric quadrupolemoments ( Q ) and changes in the mean-squared nuclearcharge radii δ (cid:10) r (cid:11) . These methods, in combination withmodern radioactive ion beam (RIB) facilities, offer a pow-erful probe of changes in the structure of exotic nuclei.They provide information on nuclear shell evolution, nu-clear shapes and sizes, and single-particle correlationswhich emerge as, for example, local staggering variationsin charge radii [1–6]. The majority of the experimentaltechniques in current use at RIB facilities provide mea-surements of hyperfine frequency splittings with a pre-cision of 1 - 10 MHz [7]. This limitation restricts thesensitivity to higher order terms in the electromagneticmultipole expansion of the nuclear current densities, aswell as to higher order radial moments of the charge den-sity distribution. The progress in the development ofhigher precision methods along with ongoing develop-ment of theoretical tools has the potential to open upa new paradigm in our understanding of the atomic nu-cleus.Recently, high-precision isotope shift measurementscombined with improved atomic calculations were pro-posed for a determination of the fourth-order radial mo-ment of the charge density [8], which can in turn bedirectly linked to the surface thickness of nuclear den-sity [9]. The hyperfine anomaly, only measured for ahandful of radioactive isotopes (see e.g. [10–14]), wouldshed light on the distribution of magnetisation inside thenuclear volume [15, 16]. In addition to the M1 and E2 moments, µ and Q respectively, the M3 magnetic oc-tupole moment Ω is in principle accessible using existingtechniques for radioactive isotopes. To our knowledgehowever, this observable has only been measured for 18stable isotopes [17–30]. The general features of thesevalues can be understood in terms of the Schwartz lim-its [31]. However, there are notable exceptions: the re-cently measured Ω of Cs [24] and
Yb [27] are largerthan expected.The extraction of a higher-order electromagnetic mo-ment from the evaluation of atomic spectra in a nuclear-model independent manner has the potential to providenew insight into the distribution of protons and neutronswithin the nuclear volume. Ω is affected by correlations(core polarization and higher order configuration mix-ing) differently than the magnetic dipole moment, as washighlighted via calculations of the nuclear magnetizationdistribution of
Bi [32]. Thus, measurements of Ω mayserve to test our understanding of the nucleon-nucleonforces inside the nuclear medium in novel ways. Mea-surements of Ω may furthermore help to address openquestions related to e.g. effective nucleon g -factors andcharges. In this Letter, we report on the start of a wideexperimental and theoretical effort that will enable an in-terpretation of measurements of Ω and other higher-orderelectromagnetic observables on radioactive ions. Our ap-proach is threefold.Firstly, we demonstrate the feasibility of combiningthe efficiency of resonance laser ionization spectroscopy(RIS) [4, 33, 34] with the precision of radio-frequency(RF) spectroscopy. The efficiency provided by the RISmethod is vital for future applications on radioactive iso-topes due to limited production rates of radioactive ion a r X i v : . [ nu c l - e x ] M a y beams at on-line facilities. The combination with RFspectroscopy offers a dramatic improvement in the preci-sion, by at least three orders of magnitude. We demon-strate this with a high-precision measurement of threenuclear electromagnetic moments of Sc, including Ω.Secondly, we combine these measurements with state-of-the-art atomic-structure calculations to evaluate thesensitivity of the 3d4s D / state in neutral scandiumto the nuclear octupole moment Ω. We furthermore eval-uate the impact of off-diagonal hyperfine structure effectson the extraction of Ω. These calculations are essentialto extract Ω from the measurements. We note that themetastable D / state is expected to be well-populatedin a fast-beam charge exchange reaction [35]. There-fore, radioactive scandium isotopes could be studied us-ing collinear laser-double resonance methods [36, 37] inthe future.Thirdly, with a single proton outside a doubly-magiccalcium ( Z = 20) core, comparison of Ω for a chain ofscandium isotopes could help shed light on the manyintriguing nuclear structure phenomena observed in thecalcium isotopes [3, 38–40]. The proximity to proton- andneutron shell closures makes it possible to perform bothshell-model and Density Functional Theory (DFT) cal-culations. As we seek to eventually examine all existingvalues of Ω in one consistent framework, with measure-ments for nuclei scattered throughout the nuclear land-scape, developing a reliable global theory for magneticproperties would be highly advantageous. So far, verylittle is known regarding the overall performance of stan-dard nuclear DFT in describing µ , cf. Refs. [41–43], andnothing is known about the DFT values of Ω. Here, wethus start this investigation with Sc. The comparisonto nuclear shell-model calculations, which have a morewell-established track record in computing both µ and Ω(see e.g. [44]), serves to benchmark these developments.The value of Ω can be extracted from the first-ordershift ( E (1) F ) in the hyperfine structure interval given by: E (1) F = A I · J + B I . J ) + ( I · J ) − I ( I + 1) J ( J + 1)2 I (2 I − J (2 J − C (cid:20) I · J ) + 20( I · J ) I ( I − I − J ( J − J − I · J { I ( I + 1) + J ( J + 1) − N + 3 } − NI ( I − I − J ( J − J − (cid:21) , (1)where I · J = [ F ( F + 1) − I ( I + 1) − J ( J + 1)] and N = I ( I + 1) J ( J + 1). In these expressions, I , J and F are the nuclear, atomic, and total angular momentum,while A , B and C are the magnetic dipole (M1), electricquadrupole (E2) and magnetic octupole (M3) hyperfinestructure constants, respectively. These are all propor-tional to their corresponding nuclear moment, in a waywhich depends on the field distribution of the electronsat the site of the nucleus. Thus, accurate atomic struc-ture calculations of C/ Ω have to be performed to extract Ω from C .There are three stages in our experiment. First, bytuning a continuous wave (cw) laser into resonance witha transition from one of the hyperfine levels ( F ) ofthe atomic ground state into a corresponding hyperfinelevel of an excited J state, population may be opticallypumped. Through de-excitation from the excited stateinto either another level ( F (cid:48) ) of the ground-state hyper-fine manifold, or into other dark states, the state F isdepleted. If RIS is subsequently performed starting fromthe same F state, a reduced ion count rate is observed.If now, prior to the laser ionization stage, an RF field istuned into resonance with a ( F, m F ) → ( F (cid:48) , m F (cid:48) ) tran-sition, the observed ion count rate again increases. Byscanning the frequency of the RF and recording the ioncount rate, the hyperfine spacing between the levels F and F (cid:48) of the ground-state manifold can thus be mea-sured precisely.We produced an atomic beam of stable scandium byresistively heating a tantalum furnace. Just above theexit orifice of the furnace, up to 15 mW of cw laser lightcrossed the atom beam orthogonally, in order to opti-cally pump the atoms. This light was produced witha frequency-doubled Sirah Matisse Ti:Sapphire laser, fo-cused to a ∼ D / state at 168.3371 cm − , driving an opti-cal transition to the 25 014.190 cm − D )4 p D ◦ / state. The atomic beam then passed through a loopof wire through which an RF current was driven, ex-posing the atoms to an RF field for a few 10 µ s. Theatoms are then collimated and finally orthogonally over-lapped with the ionization lasers. A three-step resonantlaser ionization scheme was used to ionize the metastablescandium atoms, based on the scheme in [45]. The firststep is provided using a ∼ − → − transition, and to a broad auto-ionizingstate at ∼ − , respectively. Prior to performingany double-resonance measurements, an estimate of thehyperfine constants can be obtained by scanning the fre-quency of the first laser step, as shown in Fig. 1a.During the RF scans, the laser wavelength was keptfixed to pumping wavelengths suitable for the differentRF lines, and the RF field was introduced and scanned.RF resonances like the examples shown in Fig. 1c wereobtained. From the Zeeman splitting observed in wider-range scans (see Fig. 1b), the magnetic field was esti-mated to be 1.03 G for all but the (2 , → (1 ,
0) tran-sition, where the external field was partially shielded to80 mG using mu-metal foils. To avoid possible systemat-ics due to e.g. magnetic field inhomogeneities, only the m F (cid:48) = 0 → m F = 0 transitions, magnetically insensitivein first order, were used in the analysis. The final zero- a)c) (1,0) -> (2,0) RF detuning (kHz) (2,0) -> (3,0) (3,0) -> (4,0) (4,0) -> (5,0) (5,0) -> (6,0) b) FIG. 1. a) Hyperfine spectrum obtained without the opticalpumping and RF steps. b) A wide RF scan, which featuresseveral ( F = 4 , m F =4 ) → ( F = 3 , m F =3 ) transitions. For thisscan, the laser was set to the frequency indicated by the arrowin plot a. c) All five resonances are m F = 0 → m F = 0 lines. field values for each hyperfine transition are presented inTable I. This table also shows the zero-field corrections.Extracting accurate hyperfine constants requires atomicstructure calculations to estimate the second-order shift( E (2) F ) due to M1-M1, M1-E2 and E2-E2 interactions. Line Line center B-field Second-order shiftfrequency corr. M1-M1 M1-E2 E2-E2(1 , → (2 ,
0) 228 748 . − . − .
47 1 . − . , → (3 ,
0) 339 116 . . − . − .
70 0 . , → (4 ,
0) 444 676 . .
73 3 . − . − . , → (5 ,
0) 543 827 . .
96 5 .
79 0 . − . , → (6 ,
0) 634 966 . .
51 5 .
34 1 .
08 0 . F, m F ) → ( F (cid:48) , m F (cid:48) ) for the 3d4s D / state (shifted to zero B-fieldusing the correction given in the third column). Also givenare the zero-field corrections and the second-order shifts. Allvalues are in kHz. The relativistic coupled-cluster (RCC) theory, knownas the gold-standard of many-body theory [46], is usedto evaluate C/ Ω and the matrix elements involving thesecond-order hyperfine interaction Hamiltonians. In thiswork, we expand on earlier calculations [47] presenting
A/g I (with g I = µ/I ), B/Q and C/ Ω with a larger setof orbitals, using up to 19 s , 19 p , 19 d , 18 f , 17 g , 16 h and15 i orbitals in the singles- and doubles-excitation approx-imation in the RCC theory (RCCSD method). Due tolimitations in computational resources, we correlate elec-trons up to g -symmetry orbitals in the singles-, doubles-and triples-excitation approximation in the RCC the-ory (RCCSDT method). We quote the differences inthe results from the RCCSD and RCCSDT methods as‘+Triples’. Contributions from the Breit and lower-orderquantum electrodynamics (QED) interactions are deter-mined using the RCCSD method, and added to the fi-nal results as ‘+Breit’ and ‘+QED’, respectively. Con- tributions due to the Bohr-Weisskopf (BW) effect areestimated in the RCCSD method considering a Fermi-charge distribution within the nucleus and correctionsare quoted as ‘+BW’. We also extrapolated contributionsfrom an infinite set of basis functions and present theseas ‘Extrapolation’.The A and B hyperfine constants and the C/ Ω valuesof the 3d4s D / state are tabulated in Table II. To ob-tain A and B , literature values of the moments were used( µ = +4 . µ N [48] and Q = − . E (2) F , as defined in Ref. [49] as well as inSupplemental Material [50], are also quoted in Table II.We only consider the dominant contributing matrix ele-ments between the 3d4s D / state and the 3d4s D / state. Intermediate results from the zeroth-order calcu-lation using the Dirac-Fock method and the second-orderrelativistic many-body perturbation theory (RMBPT(2)method) are presented to demonstrate the propagation ofelectron correlation effects from lower to all-order RCCmethods. The final M1-M1, M1-E2 and E2-E2 shifts foreach of the lines are given in Tab. I. The final experimen-tal hyperfine constants are summarized in Table III. Anon-zero value for C is obtained once second-order cor-rections are included. Our results agree very well withliterature [17], and are an order of magnitude more pre-cise. We find Ω = 1 . µ N b . To our knowledge, thisis the first determination of Ω for a nucleus with a va-lence proton in the f / shell, thus providing a crucialnew constraint for nuclear theory.Since scandium has a single proton outside of the magicshell of Z = 20, the single-particle shell model estimatefor Ω [31] could be expected to be fairly good. We findΩ sm = 0 . µ N b , using (cid:10) r (cid:11) / = 4 .
139 fm (obtainedfrom DFT calculations discussed later). This value is3.5(17) times smaller than the experimental value. Forcomparison, for
Cs the discrepancy is even larger:Ω exp = 0 . µ N b [24], while Ω sm = 0 . µ N b using (cid:10) r (cid:11) / ≈ R = 4 . sd ) pf -shellmodel space for Sc [52–55], and in the g / sdh / shell [56–58] for Cs. The Ω for both Sc and
Cs isdominated by the proton contribution. In
Cs, the an-gular momentum and spin contribution to Ω have theopposite sign and thus largely cancel, while for Scthe angular momentum and spin contributions have thesame sign. For Sc, we obtain values in the range 0.41-0.49 µ N b with free g -factors and 0.28-0.35 µ N b with aspin-quenching factor of 0.6 for the different shell modelcalculations. The inclusion of cross-shell excitations fromthe sd -shell to the pf -shell enhances the correlation be-yond the single f / proton configuration, which results TABLE II. Theoretical hyperfine constants of the 3d4s D / state. The dominant off-diagonal reduced matrix elements (cid:104) J (cid:48) || T ( k ) e || J (cid:105) required for the estimation of the second-order corrections to the hyperfine intervals are provided in the last tworows. Dirac-Fock RMBPT(2) RCCSD + Triples + QED + Breit + BW Extrapolation Total A B -32.70 -37.55 -38.04 0.28 -0.01 -0.02 ∼ C/ Ω 0.78 2.33 - 17.09 0.58 ∼ ∼ − kHz/( µ N b) (cid:104) / || T (1) e || / (cid:105) µ N (cid:104) / || T (2) e || / (cid:105) g s = 1 g s = 0 . g s = 1 g s = 0 .
6A [MHz] 109.032(1) 109.0325(1) 109.033(1) 109.0329(1)B [MHz] -37.387(12) -37.390(1) -37.373(15) -37.371(1)C [kHz] 1.7(10) 0.06(12) 1.5(12) -0.25(12)Ω [ µ N b] -10.7(63) -0.4(8) -9.4(75) +1.6(8) 0.65 0.46 0.45(4) 0.32(4) 0.245(17) in small increases in Ω. In contrast to Sc, the calcu-lated Ω for
Cs (using a Hamiltonian to be publishedsoon) is very sensitive to the choice of effective g -factors:0.02 µ N b with free g -factors and 0.16 µ N b with a spin-quenching factor of 0.6. All shell-model results thus un-derestimate the value of Ω for Sc and
Cs. Since Ωfor , Cl are well reproduced by shell-model calcula-tions [44] even with free g -factors, strong quenching for Sc is not expected.To check if the large experimental value of Ω canbe reproduced by nuclear DFT calculations, we deter-mined values of µ , Q , and Ω for oblate states in Sc.We used constrained intrinsic mass quadrupole moments Q = (cid:104) z − x − y (cid:105) varying between 0 and − − I = 7 / − ground-state angular momentum. Config-urations were fixed at π and ν , where 3 n representsthe occupied n lowest oblate orbitals in the (cid:96) = 3 f / shell. No effective charges or effective g -factors were used.Details of calculations performed using the code hfodd (version 2.95j) [59, 60] are collected in the SupplementalMaterial [50]. For all functionals, the experimental valueof the electric quadrupole moment of Q = − . Q = − µ and Ω strongly depend onseveral input ingredients of the calculation. First, evenat Q = 0 these values lie far from the Schmidt andSchwartz [31] single-particle estimates. This can be at-tributed to a strong quadrupole coupling to the occu-pied neutron f / orbitals, which decreases both µ andΩ. Second, the spin polarization, which acts for the Lan-dau spin-spin terms included, also significantly decreases µ and Ω. The effect of spin polarization also dependson the Skyrme functionals. Third, with increasing in- trinsic oblate deformation, both µ and Ω increase. Thelatter effect can be removed by pinning down the intrin-sic deformation to the experimental value of Q , see starsin Fig. 2. The shaded area in Fig. 2 covers the rangeof results given by all starred points, and thus repre-sents a very rough estimate of the averages and rms de-viations of the DFT results: µ DFT = +4 . µ N andΩ DFT = +0 . µ N b. The value of Ω DFT is a factorof 6.5(33) smaller than the experimental value. Note thatΩ
DFT is about twice smaller than the effective Schwartzvalue, and in reasonable agreement with the shell modelvalue obtained with effective g -factors.In conclusion, we have measured a large magneticoctupole moment Ω in Sc, using high-precisionexperimental techniques and state-of-the-art atomiccalculations. The nuclear-structure interpretation ofthis new value poses a puzzle: while even the simplestsingle-particle calculations can successfully describe µ and Q , there appears to be no clear way to interpret thelarge values of Ω in Sc and
Cs. Further experimen-tal work is thus essential to independently verify ourresult, and to extend Ω-measurements to other elements.This experimental effort should be matched by accurateatomic structure and nuclear structure calculations. Asa next experimental step, we are currently designingand constructing a collinear RIS laser-RF apparatuswhich we will use to extend measurements to radioactiveisotopes. In addition to Sc, candidates for future studiesinclude In [21] and Bi [29]. Finally, we are currentlyin the process of experimentally verifying the largeΩ of
Yb [27], and are initiating systematic calcu-lations of nuclear magnetic dipole and octupole moments.RPDG received funding from the European UnionsHorizon 2020 research and innovation programme un-
UNEDF0UNEDF1 SkXcSIII SkMSLy4 SkO'SAMi D1SREG6d N )0.180.200.220.240.260.28 4.65 4.70 4.75 M a gn e t i c o c t upo l e Ω ( N b ) Landau spin-spinterms No spin-spinterms (a) (b)
FIG. 2. Values of µ and Ω of the I = 7 / − angular-momentum-projected ground states of Sc. Calculationswere performed for eight zero-range Skyrme-type functionals,UNEDF0 [61], UNEDF1 [62], SkXc [63], SIII [64], SkM* [65],SLy4 [66], SAMi [67] and SkO (cid:48) [68], and for two finite-rangefunctionals, D1S [69] and REG6d.190617 [70]. Panels (a) and(b) show results obtained with Skyrme functionals supple-mented by the Landau spin-spin terms or with no spin-spinterms, respectively. Arrows mark the experimental value of µ and point towards the experimental value of Ω, which isoutside the scale of the figure der the Marie Sk(cid:32)lodowska-Curie grant agreement No844829. BKS acknowledges use of Vikram-100 HPCcluster of Physical Research Laboratory, Ahmedabad foratomic calculations. 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