Nuclear Moments of Germanium Isotopes around N = 40
A. Kanellakopoulos, X. F. Yang, M. L. Bissell, M. L. Reitsma, S. W. Bai, J. Billowes, K. Blaum, A. Borschevsky, B. Cheal, C. S. Devlin, R. F. Garcia Ruiz, H. Heylen, S. Kaufmann, K. König, Á. Koszorús, S. Lechner, S. Malbrunot-Ettenauer, R. Neugart, G. Neyens, W. Nörtershäuser, T. Ratajczyk, L. V. Rodríguez, S. Sels, S. J. Wang, L. Xie, Z. Y. Xu, D. T. Yordanov
NNuclear Moments of Germanium Isotopes around N = 40 A. Kanellakopoulos, X. F. Yang,
2, 1, ∗ M. L. Bissell, M. L. Reitsma, S. W. Bai, J. Billowes, K. Blaum, A. Borschevsky, B. Cheal, C. S. Devlin, R. F. Garcia Ruiz, † H. Heylen, S. Kaufmann,
8, 9
K. K¨onig, ‡ ´A. Koszor´us, § S. Lechner,
7, 10
S. Malbrunot-Ettenauer, R. Neugart,
5, 9
G. Neyens,
1, 7
W. N¨ortersh¨auser, T. Ratajczyk, L. V. Rodr´ıguez,
5, 11, ¶ S. Sels, ∗∗ S. J. Wang, L. Xie, Z. Y. Xu, †† and D. T. Yordanov Instituut voor Kern- en Stalingsfysica, KU Leuven, B-3001, Leuven, Belgium School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China Department of Physics and Astronomy, The University of Manchester, Manchester, M13 9PL, United Kingdom Faculty of Science and Engineering, Van Swinderen Institute for Particle Physics and Gravity,University of Groningen, 9747 AG Groningen, The Netherlands Max-Planck-Institut f¨ur Kernphysik, D-69117 Heidelberg, Germany Oliver Lodge Laboratory, Oxford Street, University of Liverpool, Liverpool, L69 7ZE, United Kingdom Experimental Physics Department, CERN, CH-1211 Geneva 23, Switzerland Institut f¨ur Kernphysik, TU Darmstadt, D-64289 Darmstadt, Germany Institut f¨ur Kernchemie, Universit¨at Mainz, D-55128 Mainz, Germany Technische Universit¨at Wien, Karlsplatz 13, AT-1040 Wien, Austria Institut de Physique Nucl´eaire, CNRS-IN2P3, Universit´e Paris-Sud, Universit´e Paris-Saclay, 91406 Orsay, France (Dated: November 4, 2020)Collinear laser spectroscopy measurements were performed on , , Ge isotopes ( Z = 32)at ISOLDE-CERN. The hyperfine structure of the 4 s p P → s p s P o transition of thegermanium atom was probed with laser light of 269 nm, produced by combining the frequency-mixingand frequency-doubling techniques. The hyperfine fields for both atomic levels were calculatedusing state-of-the-art atomic relativistic Fock-space coupled-cluster calculations. A new Gequadrupole moment was determined from these calculations and previously measured precisionhyperfine parameters, yielding Q s = − Ge have been revised: µ = +0.920(5) µ N and Q s =+0.114(8) b, and those of Ge have been confirmed. The experimental moments around N = 40 areinterpreted with large-scale shell-model calculations using the JUN45 interaction, revealing rathermixed wave function configurations, although their g -factors are lying close to the effective single-particle values. Through a comparison with neighboring isotones, the structural change from thesingle-particle nature of nickel to deformation in germanium is further investigated around N = 40. I. INTRODUCTION
Over the years, structural changes have beenintensively investigated in the region between the semi-magic Ni and the doubly-magic Ni [1–6]. Theinvestigation involves multiple experimental methodsas well as theoretical models, which provide variousnuclear properties (masses, spins, life-times, transitionprobabilities, excitation energies, moments and radii),aiming to get a global view of the nuclear structure inthis region. As studies deepen, more interesting nuclear ∗ [email protected] † Present address: Massachusetts Institute of Technology,Cambridge, MA, USA ‡ Present adress: National Superconducting CyclotronLaboratory, Michigan State University, East Lansing, Michigan48824, USA § Present address: Oliver Lodge Laboratory, Oxford Street,University of Liverpool, Liverpool, L69 7ZE, United Kingdom ¶ Present address: Experimental Physics Department, CERN, CH-1211 Geneva 23, Switzerland ∗∗ Present address: Instituut voor Kern- en Stalingsfysica, KULeuven, B-3001, Leuven, Belgium †† Present address: Department of Physics and Astronomy,University of Tennessee, 37996 Knoxville, TN, USA phenomena appear. Some examples are: collective effectsoccur around and above N = 40 [7–10]; the proton p / and f / orbitals invert as neutrons are fillingthe neutron g / orbital due to the tensor part of themonopole interaction [5, 8, 11]; a spherical shape coexistswith a deformed state around the neutron closed shells N = 40 ,
50 [12–14]; the indication of a weak sub-shelleffect observed at N = 40 disappears quickly with moreprotons added above Z = 28 [3, 7].The nuclear properties of ground and isomericstates, such as spins, moments and charge radii, havemade significant contributions in elucidating the above-mentioned nuclear phenomena. This was achieved bylaser spectroscopy measurements on the isotopic chainsof nickel ( Z = 28), copper ( Z = 29), zinc ( Z = 30)and gallium ( Z = 31) [3, 5–8, 11, 13–18]. Nuclear spinsand magnetic dipole moments are sensitive probes ofthe single-particle nature or configuration mixing of thewave functions [5, 6, 8, 11], while the electric quadrupolemoment and charge radii tell us more about the nuclearshape and collectivity [3, 13, 14].Deformation, triaxiality and shape coexistence havebeen observed for zinc isotopes, which have also beenreported for germanium isotopes around N = 40 basedon γ -spectroscopy and reaction experiments [7, 19–21].Germanium ( Z = 32) has four protons outside the a r X i v : . [ nu c l - e x ] N ov + + + 532 nmFiber laserNarrowband cw (Ti:Sa)1550 nm824 nmFrequency mixing538 nmFrequency doubling269 nm I S C O O L ++ + + + + Photomultiplier tubes(PMTs)Charge exchange cell(CEC)Post-accelerationDoppler tuning electrodesRFQ cooler-buncher50 keV Ge beamfrom HRS Gated PMT signal → 1 000 times PMT background reductionElectrostaticdeflector FIG. 1. A schematic view of the COLLAPS experimental setup and the laser frequency-mixing and frequency-doubling system.For details see text. Z = 28 major shell closure, which may induce additionalcorrelations leading to a complex wavefunction andcollective effects for the low-lying states of germaniumisotopes around N = 40 [19, 22, 23]. This collectivefeature can be further investigated with the nuclearmoment measurements of germanium isotopes around N = 40 in combination with wave function calculationsusing large-scale shell-model interactions.Until now, collinear laser spectroscopy has not beenapplied to study germanium isotopes, due to the fact that(1) germanium species cannot be easily produced at ISOLfacilities, and (2) the suitable fine structure transitionsare not easily accessible with standard laser equipment.In this article, we report the first hyperfine structure (hfs)measurement in the 4 s p P → s p s P o transitionof , , Ge atoms by combining the frequency-mixingand frequency-doubling laser techniques. State-of-the-art relativistic Fock-Space Coupled-Cluster (FSCC)calculations of the atomic hyperfine fields and electricfield gradients have been performed, showing a goodagreement with the experimental values, and thusbenchmarking these atomic theories. The calculationsalso reveal that incorrect nuclear moments have beenreported for Ge from earlier atomic beam magneticresonance studies [24]. The extracted nuclear momentsare compared with large-scale shell-model calculationsusing the JUN45 interaction in the f / pg / modelspace in order to understand the collective effects. Thesystematic comparison of the nuclear moments of the9/2 + states in zinc ( Z = 30), germanium ( Z = 32)and selenium ( Z = 34) allow the investigation of thestructural evolution from single-particle to collectiveeffects around N = 40 as more protons are added abovethe Z = 28 shell closure. II. EXPERIMENTAL PROCEDURE
The experiment was performed at the collinearlaser spectroscopy setup, COLLAPS [25], located atISOLDE-CERN. The radioactive germanium isotopeswere produced by a 1.4-GeV proton beam impinging on a ZrO target with a sulphur leak. The sulphurvapour was introduced into the target material wherethe sulphur atoms bind to form volatile GeS molecules.These molecules easily diffuse out of the target and thendisassociate under electron impact to form germaniumions inside a plasma source. The germanium ions wereaccelerated to 50 keV, mass separated using the high-resolution isotope separator (HRS) and subsequentlycooled and bunched in a gas-filled linear Paul trap(ISCOOL) [26]. The accumulated ions were released inshort bunches with a typical temporal width of 5 µ s every5 ms. Due to the large isobaric contamination in all ofthe Ge beams [27], the accumulation time was optimizedat 5 ms to avoid the overfilling of the ISCOOL.As shown in Fig. 1, the bunched ion beam wasthen deflected into the COLLAPS beamline, where itwas neutralized in-flight by passing through a chargeexchange (CE) cell filled with sodium vapor. Thestate 4 s p P of the germanium atom was populatedin the neutralization process [30]. A frequency fixedcontinuous-wave (cw) laser beam, overlapped with thegermanium beam in a collinear geometry, was usedto probe the 4 s p P − s p s P o (269.13411 nm)transition of the germanium atom. By applying a varyingvoltage to the germanium ions before entering the CEC,neutralized germanium atoms were resonantly excitedto the 4 s p s P o state through Doppler tuning. Theemitted fluorescence photons from the resonantly excitedatoms were detected as a function of the tuning voltage,by the use of 4 photomultiplier tubes (PMTs) [31]. Asoftware time gate was applied in order to select photonsonly when each bunch of germanium atoms passes andde-excites in front of the PMTs. This resulted in areduction of the background from laser stray light, non-resonantly scattered photons and PMT dark counts by afactor of about 10 .To take full advantage of ISCOOL and to retain asmuch information on the beam structure in the recordeddata as possible, a new data acquisition system (DAQ)was introduced at COLLAPS. It was developed at theTRIGA-SPEC setup in Mainz [32] and here we reportits online-operation. The main advantage of this DAQ is TABLE I. Magnetic and electric hfs constants ( A and B ) for both atomic states (4 s p P and 4 s p s P o ) obtained fromthis work. The numbers from Refs. [24, 28, 29] are also summarized as a comparison. A N I π T / A litl (MHz) A l (MHz) A u (MHz) B litl (MHz) B l (MHz) B u (MHz)69 37 5/2 − ∓ − ± a − a
71 39 1/2 − − + stable +15.5480(18) [29] − − a Correlation C( B l , B u ) = -0.445 C o un t s hfs spectrum with time gate 2(d)50100 hfs spectrum with time gate 1(c)-2000 -1500 -1000 -500 0 500Relative frequency [MHz]4050607080 T O F [ s ] (a) 1500 2000 2500CountsTOF spectrumgate 1gate 2(b) 0246810121416 FIG. 2. Color-coded 2D plot with time vs frequency(voltage step) vs counts acquired for Ge with the newDAQ system (TILDA) of COLLAPS. The two componentspresented in the TOF spectrum are corresponding to thegermanium atom bunch (gate 1 in red) and the possiblemolecular contamination bunch (gate 2 in green). Gatingon the germanium atom bunch allows the hfs spectrum to bereconstructed with a significantly reduced background. the time-resolved photon detection. In order to reducethe photon background, the detection of photons hasto be restricted to the time window when the bunchedbeam is traversing the detection region. Previously,this was realized by hardware gating, which had to beadapted in advance for each isotope according to thetime-of-flight (TOF) recorded. With the new DAQ,photons are recorded with a time stamp relative to theion extraction trigger of ISCOOL. This is shown for Ge in Fig. 2(a): the x -axis represents the scanningvoltage, converted into frequency, while the y -axis isthe TOF since the ISCOOL extraction pulse. Thenumber of photons detected within a 500-ns intervalduring n -time extractions from ISCOOL are color-coded.These numbers are integrated along a specified timein Fig. 2(b), which provides the time structure of thebunch. Integration along a fixed frequency reveals thehfs resonance spectrum of the isotope, as illustrated inFig. 2(c). The spectra in Fig. 2(c)-(d) are obtained byselecting events within the specific time window indicatedwith dashed horizontal lines in Fig. 2(a) and with their (a) Ge C o un t s (b) Ge Relative frequency [MHz] (c) Ge FIG. 3. Hyperfine structure spectra of , , Ge in the4 s p P → s p s P o transition. The red lines show thebest fit with a Voigt line profile. Note that the productionof the known short-lived isomers of , Ge (T / ( m Ge) =20.41(18) ms and T / ( m Ge) = 499(11) ms) [33, 34] is atleast 100 times lower than that of the ground state [35]. Thus,these isomers could not be observed in this experiment. corresponding colored dashed lines in Fig. 2(b).In Fig. 2(a) three regions can be clearly distinguished.Outside of the dashed lines, the background is dominatedby stray laser light, which is weak compared to themore intense signal within the time gate of the bunch.One can clearly see that the bunch is separated intotwo parts, the first one arriving after about 52 µ s, whilethe second one is delayed by another 5 µ s. The timeinterval between two bunches released from ISCOOL isabout 1000 times longer. Thus, the two consecutive‘bunches’ seen in Fig. 2(a) are from photons emittedfrom two different species. Those with a heavier massarrive later in front of the PMT’s. In the first timewindow (gate 1) clear resonance spots are seen, as wellas an additional beam-related but frequency-independentbackground (see also Fig. 2(c)). This background isattributed to the de-excitation of all contaminant neutralparticles that were excited in the CE process. The secondpart of the bunch (gate 2) does not exhibit any resonance-like structures, only beam-related photon background,as seen in Fig. 2(d). It is assumed that this beam-related photon background is due to laser-light scatteringfrom the molecular contamination in the beam and de-excitations of these molecules excited via the CE process.With the latest DAQ, these individual parts of thebunch can be clearly separated in online and offlineanalysis. Therefore, by gating on the first component(gate 1) instead of the plotted range (40 µ s) of the TOFspectrum, the hfs spectrum of Ge was obtained with amuch improved signal-to-background ratio, as shown inFig. 2(c) and Fig. 2(e), respectively.One challenge for the laser spectroscopy measurementof germanium atoms is the production of the 269 nmcw laser light. It requires frequency-doubling of thewavelength 538 nm. This wavelength is lying in the‘green gap’ region, which is not covered by the commonlyused continuous wave (cw) Ti:Sa and dye laser systems.To bridge this wavelength gap, a frequency mixingmethod was employed. As shown in Fig. 1, a 824 nm laserbeam from a Ti:Sa laser cavity (Sirah Matisse-2) anda 1550 nm laser beam generated by a single-frequencyfiber laser (Koheras Boostik E 15) are superimposedand single-pass through a periodically poled crystal ina frequency mixing unit (Sirah MixTrain) to generatethe sum frequency of 538 nm ( λ = λ + λ ). Thislight is then coupled into a frequency doubling unit(Sirah WaveTrain) to produce 269 nm light with anoutput power around 90 mW. A small fraction of theoutput light from the frequency mixing unit was lockedto a wavelength meter (HighFinesse WSU10), which wasregularly calibrated with a stabilized diode laser (TopticaDLPRO780) locked to the F = 2 → F = 3 hyperfinetransition of the D1 line in Rb.
III. RESULTS
The obtained hfs spectra of , , Ge isotopes, asshown in Fig. 3, were fitted with a χ -minimizationPython routine using the SATLAS package [37]. A Voigtprofile with one side peak was used to compensate forthe slightly asymmetric resonance peaks, which resultedfrom the energy loss due to the population of higheratomic states or the collision-excitation in the chargeexchange process [38, 39]. All the recorded hfs spectraof each individual isotope were fitted simultaneously,generating one optimal reduced χ , with magnetic dipoleand electric quadrupole hyperfine constants ( A and B )as common fit parameters for each isotope.The A and B parameters for the lower atomic state(4 s p P ), named A l and B l in the following, arereported in literature with high precision for each of
69 71 73Mass number A -21.0-20.0-19.0-18.0-17.0-16.0 A r a t i o R A (a) Free mean: -17.1(3)Theory: -17.8(16) A l fixed to literatureAll free37 39 41Neutron number N -3.0-2.5-2.0-1.5-1.0-0.5 B r a t i o R B (b) Free mean: -0.7(1)Theory: -0.75(5) B l fixed to literatureAll free FIG. 4. Ratio of the A (a) and B hyperfine constants(b) for the 4 p P (lower) and 4 p s P o (upper) atomicstates, deduced with two fitting procedures. One with the A l and B l values for each isotope fixed (green) to the high-precision value reported in literature [24, 28, 36], and one withall hyperfine constants as free fit parameters (black). Thehyperfine constants ratios calculated with atomic relativisticFSCC are shown by red dashed lines. the three odd- A germanium isotopes from atomic beammagnetic resonance experiments [24, 28, 29, 36], assummarized in Tab. I. We therefore fitted our data intwo ways: (1) with the A l and B l values fixed to theliterature value and upper hyperfine constants as freefit parameters for each isotope, and (2) with all lowerand upper hyperfine constants as free fit parameters.The ratio of the hyperfine parameters of the upper andlower atomic states ( R A = A u / A l , R B = B u / B l ) reflectthe ratio of the atomic hyperfine fields of both levels.These ratios should be the same for all isotopes, apartfrom a possible small hyperfine anomaly that can causesome scatter of the order of at most a few percent in TABLE II. Deduced R A = A u /A l and R B = B u /B l for , , Ge isotopes with the A l and B l fixed to literature valuesor with all hyperfine constants as free fit parameters. R A R B All free A l fixed to lit. All free B l fixed to lit. Ge − − − − Ge − − Ge − − − − R A . In Table II, as well as in Fig. 4, the ratio of R A and R B resulting from the two fit procedures are shown.When using the first fitting procedure, both the A and B factor ratios for Ge deviate strongly from thoseobserved for the other isotopes, suggesting a problemwith the reported hyperfine parameters of Ge [24],from which the literature nuclear moments have beendeduced. When all four hyperfine parameters are freefit parameters, the deduced ratios of R A and R B areconsistent for all three isotopes, as shown in Tab. II andFig. 4. To corroborate these conclusions, state-of-the-artatomic calculations were performed, as discussed in thefollowing section. A. Atomic hyperfine field calculations
To further investigate the aforementioned discrepancybetween the hyperfine constants from literature andour experimental results for Ge, we performed atomiccalculations to obtain the hyperfine fields of three atomicfine structure levels (4 s p P , 4 s p P and 4 s p s P o ), for which experimental information is available forthe stable Ge isotope [29].High-quality treatment of relativistic and electroncorrelation effects is required to obtain accurateand reliable computational results. Therefore, themultireference relativistic FSCC method [42, 43] isemployed for the investigation of the electric fieldgradients (EFGs) ( q th ) and the hyperfine magnetic fieldconstants ( A th0 ). This method was shown to be one of themost powerful approaches for treatment of spectra andproperties of heavy many-electron atoms [44].The FSCC method requires a closed shell referencestate from which the ground state and excited states canbe reached by adding or removing electrons. Neutralgermanium has an open shell ground state electronconfiguration [Ar]3 d s p and thus the closed shellGe system ([Ar]3 d s ) is used as the reference state.The ground state and excited states of interest arereached by adding two electrons to the correspondingvirtual orbitals, which comprise the model space. Theintermediate Hamiltonian (IH) approach is applied toavoid the intruder-state problem [45]. The finite fieldmethod [46] is used to determine the q th and A th0 properties, as described in Refs. [47, 48]. All calculationsare carried out in the framework of the Dirac-CoulombHamiltonian and the nuclear charge distribution ismodeled by a Gaussian function as described in Ref.[49]. The q th calculations were carried out using theDIRAC15 program package [50], while DIRAC17 wasused for the A th0 calculations [51]. The final values areobtained using the full 4-component DC Hamiltonianand the relativistic core-valence 4-zeta basis set of Dyall[52] (cv4z), augmented by three diffuse functions in eachsymmetry in an even-tempered fashion. A large modelspace was used, consisting of the 4 p s (4 d p s f d p s f g p d d p g s f ) orbitals, where the orbitals in parentheses are in the intermediate space P i .All the electrons were correlated and virtual orbitals withenergies up to 500 a.u. were included in the calculation.The finite field perturbation strength λ was 1 × − forthe q th calculations and 1 × − for A th0 .To estimate the uncertainty on the calculated values,we have investigated the effect of various computationalparameters: the error from the limited size of thebasis set, model space and virtual space, while alsoestimating the contribution from higher order excitationsand the relativistic Gaunt term. These sources of errorare combined by assuming them to be independent togive a total conservative uncertainty estimate for eachcalculated property. The uncertainty of q th is 3.0 % forthe 4 p P and 4 p states and 6.4 % for 4 p s P o ,while for A th0 the uncertainties are 7.8 %, 3.5 % and3.4 % respectively for these three states. The finalrecommended values and uncertainties for q th and A th0 are shown in Tab. III. Further details on the procedureused for the calculation of these properties and theuncertainties can be found in Ref. [53].The calculated atomic observables, A and q , areproportional to the magnetic hyperfine field ( B ) createdby the electrons at the point of the nucleus, A = B /J ,and to the electric field gradient ( V zz ) created bythe electronic cloud, q = eV zz , respectively. Here, J is the atomic spin. These atomic observablesare related to the measured A and B parametersas follows: A = AI/µ and q = B/ Q s , with µ and Q s the nuclear magnetic dipole moment and electricquadrupole moment, respectively, and I the nuclearspin. From the experimental hyperfine constants of thethree atomic states taken from Refs. [29] and measuredin this work, as well as the experimental magneticand quadrupole moments from Refs. [40, 41], we cancalculate the experimental A and q parameters for Ge.These can be directly compared with atomic theorycalculations, as presented in Tab. III. A remarkableagreement is achieved for both A and q , for all threestates within less than 5%. We further compared theratios ( R A , R B ) of hyperfine constants of two atomicstates (4 p P and 4 p s P o ) measured in this workwith the atomic calculation, as shown in Fig. 4. Theratios of the hyperfine constants from theory show a goodagreement with experimental values for all three isotopesof , , Ge obtained with fitting procedure (2). Thisfurther supports our experimental result for the A l and B l hyperfine constants of Ge.
B. Hfs constants and nuclear moments
As mentioned above, both our experimental resultsand the atomic calculations are inconsistent with theliterature hyperfine constants for the 4 p P atomic stateof Ge. A revision of the lower state hfs parameters of Ge is therefore suggested. From our measurements,the upper-state hyperfine parameters, A u and B u , can TABLE III. Atomic A and q parameters extracted from Ge experimental results and calculated with atomic relativisticFSCC method.At. state A exp (MHz) A exp0 (MHz) A th0 (MHz) B exp (MHz) q exp (MHz/b) q th (MHz/b)4 p P +15.5480(18) [29] − − − p P − − − p s P o − a +1343(6) +1314(45) +40(6) a − − a Hyperfine parameters measured in this work.
TABLE IV. Magnetic dipole and electric quadrupole moments of , , Ge isotopes compared with JUN45. The numbers fromRefs.[24, 29, 40, 41] are also summarized as a comparison.
A N I π µ lit ( µ N ) µ exp ( µ N ) µ JUN45 ( µ N ) Q lits (b) Q exps (b) Q JUN45s (b)69 37 5/2 − − +0.54606(7) [29] +0.547(5) +0.43873 41 9/2 + − a − − a − a Weighted mean of the moments extracted from the measured hyperfine parameters and calculated hyperfine fields of the 3 transitionsin Tab. III. also be determined for the first time for all isotopes,yielding an independent measurement of the nuclearmoments. For , Ge, the final A u and B u are obtainedwith A l and B l fixed to the literature values [28, 29],since they are known with high-precision. In Tab. I, therevised hyperfine parameters of the lower atomic statefor Ge, and the newly measured hyperfine constants ofthe upper atomic state for , , Ge are shown togetherwith literature values.A precise nuclear magnetic moment of Ge hasrecently been redetermined using gas-phase NMRmeasurements on GeH [40] and a precise electricquadrupole moment was extracted from molecularmicrowave data of GeO and GeS [41]. Thus, themagnetic dipole µ and electric quadrupole Q s moments of , Ge can be calculated from the experimental A and B parameters in a model independent way relative to thoseof Ge, by using: µ = IAI ref A ref µ ref (1) Q s = BB ref Q s , ref (2)The large A u hyperfine parameter is used to extractthe magnetic moments of , Ge via Eq. (1). Sincethe hyperfine B parameters for both atomic states arecomparable in magnitude, two sets of electric quadrupolemoments are extracted from B l and B u using Eq. (2).We use the weighted average of the two quadrupolemoments as the final value, after taking into account thecorrelation between the two hyperfine B parameters [54].The results are summarized in Tab. IV together with theliterature values.For the reference isotope Ge, we also determine anindependent set of nuclear moments, using the measured hyperfine parameters and calculated hyperfine fields foreach of the three atomic states shown in Tab. III. Thesemoments are fully consistent with the literature values(Tab. IV), in particular for the quadrupole momentreaching almost similar precision. This illustrates thesignificant progress that has been made in atomiccalculations in the last few years.
IV. DISCUSSION
The relevant proton and neutron shells in the nickelmass region are displayed in Fig. 5. For the even- Z elements (like nickel, zinc and germanium), with neutronnumbers between N = 28 and N = 50, the neutronsare expected to dominate the ground state structure ofthe odd- A isotopes in a naive single particle shell-modelpicture. By filling the ν p / , ν f / , ν p / , ν g / orbitals, this would lead to ground-state (g.s.) spins !/ $/ %/ &/ '/ '/ FIG. 5. Proton and neutron orbitals around the
Z, N = 28and the
Z, N = 50 major shell closures according the nuclearshell-model. The model space of the effective interactionJUN45 is also marked.
FIG. 6. Ground and low-lying excited (isomeric) states of − Zn (top) and − Ge (bottom) compared to results fromJUN45 calculations. States with half-lives of at least 10 ns and a firm spin assignment are plotted. Thick lines represent stateswith half-lives longer than 1 ms. of 3/2 − , 5/2 − , 1/2 − and 9/2 + for the odd- A isotopes,respectively. Thus, for Ge and its isotones, Zn and Ni, with N = 37, the unpaired neutron is expected tooccupy the νf / orbital, resulting in a g.s. spin of 5/2 − ,which is consistent with the experimental assignment foreach of these isotones [55–57]. For isotones with N = 39,a spin 1/2 − is then expected and also experimentallyconfirmed for Ni, Zn and Ge [33, 56, 57]. So,below N = 40, the single-particle (SP) shell-model fillingseems to be respected in germanium, despite havingfour valence protons, which could induce significantcorrelations and deformation. Moving to N = 41 andbeyond, the filling of the νg / orbital is then beginningand the odd- N isotopes are all expected to have g.s. spinof 9/2 + . This is indeed the case for Ge and Ni but notfor Zn [33, 34, 57]. This might suggest some stabilizingeffect from a completely filled πp / orbital in Ge, andonset of correlations between protons in the open πp / orbital in Zn.Recent studies have suggested that the structure of thezinc isotopes between N = 40 and N = 50 [5, 13] presentssome complexity with long-lived SP-like or deformedisomers occurring in each of the odd- N isotopes. Forthe germanium isotopes, deformation was proposed forthe even- A isotopes around N = 40 [22, 23] but asystematic discussion of the nuclear moments of the odd- N isotopes has not yet been done. We first investigatethe nuclear structure around N = 40 by looking at thelow-lying energy levels of − Zn and − Ge isotopes, as summarized in Fig. 6. Note that only the energy levelswith a firm spin assignment and with measured half-liveslonger than 10 ns are plotted, as nuclear moments havebeen measured for most of them [58]. These energy levelsare also compared to large-scale shell-model calculationsusing the JUN45 effective interaction [59] in the f / pg / model space starting from a Ni core, as illustrated inFig. 5.While the shell-model reproduces well the levelordering in germanium and zinc isotopes below N = 40,it fails to reproduce the complex level structure in bothisotopic chains beyond N = 40 [13, 19–21]. This isapparent from the spins of the low lying states in , Geand Zn, which may not be easily understood fromthe normal filling of the shell-model orbitals. In the Zn isotope, the isomeric character of the 5/2 + stateallowed a measurement of its nuclear moments, and anunambiguous assignment of its spins and parity to bemade, which was debated before in literature [5]. Fromthe magnetic moment, it was clear that the unpairedneutrons mostly occupy the g / orbital, but the largequadrupole moment could be only explained by includingproton and neutron excitations across Z = 28 and N = 50, leading to a triaxial shape [13]. Investigating theexperimental electromagnetic properties of the low-lyingstates in the odd- A germanium isotopes will thus help toget a better understanding of their structure, which willbe discussed in the following sections. FIG. 7. (a) g -factor of ground and low-lying isomeric statesof − Ge, compared with the shell-model calculations usingJUN45 interaction. (b) Experimental g -factors of ground andlow-lying isomeric 9/2 + states of the isotones of even- Z , zinc,germanium and selenium isotopes. A. Magnetic Moments
The magnetic dipole moment, and more specifically therelated g -factor ( g = µ/I ), is an excellent probe of theorbital that is occupied by the unpaired particles (in thiscase neutrons). The available experimental g -factors ofthe ground and isomeric states of − Ge are comparedwith the effective SP g -factors of relevant orbitals, aspresented in Fig. 7(a). The effective SP g -factors forthe orbitals of νf / , νp / , νg / are calculated usingeffective g -factors of g eff s = 0 . g free s and g eff l = g free l , whichare the typical values used in the region [14, 60]. The newly measured magnetic moment (and g -factor)of the 5/2 − g.s. of Ge ( N = 37) is in excellentagreement with the νf / effective SP value, intuitivelypointing to its simple shell-model character of anunpaired valence neutron in the νf / orbital. Theavailable experimental g -factors for the states with spin9/2 + in , , Ge lie close to the effective SP g -factorof the g / orbital, as shown in Fig. 7(a). Againthis suggests a configuration dominated by an unpairedneutron in the νg / orbital. More interestingly, the g -factor of the 9/2 + state of Ge deviates more from theeffective SP g -factor (enhanced in Fig. 7(b)), pointingto a possible onset of collective effects from N = 41onwards. As for the g.s. of , Ge, both having aspin/parity of 1/2 − , the structure is likely dominatedby an odd neutron in the p / orbital as the g -factorsare close to the νp / SP value, yet deviate somewhat.Similar deviations have been observed for other 1/2 − states, e.g. in the neighbouring zinc isotopes [5], as wellas for heavier isotopes with spin 1/2 − with a valenceparticle filling the p / orbital (e.g. Ag and In) [61].These experimental g -factors are also compared withthe shell-model calculations using the JUN45 interaction,as presented in Fig. 7(a). The calculations nicelyreproduce the new experimental value of the 5/2 − g.s. of Ge ( N = 37) as well as the g -factors of the 9/2 + statesin − Ge. However, the calculated wave functionsfor these states are found to be very fragmented. Thecalculated dominant configuration of the wave functionfor the 5/2 − g.s. of Ge is about 26%. For the 9/2 + states of − Ge, the main component of the wavefunction is, in all cases, less than 50%. For all theisotopes, the leading configuration is the same as thatconcluded from the effective SP g -factors. Surprisingly,the calculated g -factors of the 1/2 − g.s. of , Ge areclose to the SP values, so they also deviate from theexperimental values. Further theoretical investigation isneeded to understand this deviation for the 1/2 − states.Earlier studies have shown that the N = 40 subshelleffect observed in the nickel and copper isotopes, quicklydisappeared with more protons added above Z = 28, asdescribed by various experimental observables: magneticand quadrupole moments [5, 6, 8, 11], charge radii [3, 7,16], nuclear masses [62, 63], E (2 + ) excitation energies,and B ( E
2) transition rates [9, 10, 64, 65]. This indicatesthat an increase in collectivity around and above N = 40is expected when going away from Z = 28, as has beenobserved in the zinc and gallium isotopes [5, 7–10, 13, 62–65].In order to have a systematic investigation of thestructural evolution from single-particle nature to thecollective effect, we compare the experimental magneticmoments of the 9/2 + states for even- Z nuclei (zinc,germanium and selenium) around N = 40, as shown inFig. 7(b). The g -factors of all the 9/2 + states in theregion around N = 40 are close to the effective SP g -factor, clarifying the leading configuration of an unpairedneutron in the g / orbital in their wave functions. FIG. 8. (a) Quadrupole moments of ground and low-lyingisomeric states of − Ge, compared with the shell-modelcalculations using JUN45 interaction using effective charges e eff p = 1 . e p and e eff n = 1 . e n . (b) Experimental quadrupolemoments of ground and low-lying isomeric 9/2 + states ofeven- Z zinc, germanium and selenium. However, a continuously increased deviation from the SPvalue is obvious from zinc, to germanium and selenium,giving a clear sign of the increased collectivity as moreprotons are added above Z = 28 closed shell. This isfurther confirmed by comparing the contribution of themain configuration of wave functions (single neutron in g / orbital), which is about ∼
35% for Zn and ∼ Ge.
B. Quadrupole moments
The known experimental quadrupole moments ofground and isomeric states in − Ge are presented inFig. 8(a). The new value of the quadrupole moment ofthe 5/2 − g.s. in Ge is well reproduced by the shell-model calculation. The same agreement can be found forthe 9/2 + states of − Ge. It is known from the above-discussed magnetic moments that the main configurationfor these 9/2 + states comes from the unpaired valenceneutron in the νg / orbital.In Fig. 8(a), we see a linear trend in the quadrupolemoments of these 9/2 + states, when more neutrons fillthe νg / orbital. A similar trend has been observed inthe neighboring zinc isotopes [5, 13], which are comparedto those of germanium in Fig. 8(b). In the case ofthe zinc isotopic chain, the quadrupole moments of the9/2 + states in Zn ( N = 39) and Zn ( N = 49)reflect a relatively pure configuration of single neutronparticle and single neutron hole in the νg / orbital,respectively [5, 13], from which we estimated the SPquadrupole moment of νg / orbital (see Ref. [5] formore details). A linear trend for quadrupole momentswith increased neutron number for a seniority-1 ( νg / ) n configuration is then calculated, as shown with the blackdash line in Fig. 8(b).For the Ge isotopes, with only three measuredquadrupole moments, it is difficult to make a firmconclusion. It appears that the slope of the curve issomewhat steeper than for the Zn isotopes, which wouldpoint to some enhanced deformation (correlations) forthese isotopes with four protons outside the Z = 28 shell.However, more data on neutron-rich isotopes are neededto make a firm conclusion. V. SUMMARY
The hfs of germanium atoms was measured for the firsttime in the 4 s p P → s p s P o atomic transition,taking advantage of the development of the laserfrequency-mixing technique. A clear discrepancy withliterature magnetic and electric hyperfine parameters isobserved for Ge. The systematic analysis of the hfsof , , Ge isotopes obtained in this work requires arevision of the hyperfine constants for Ge, resultingin new magnetic and quadrupole moments. State-of-the-art atomic relativistic Fock-Space Coupled-Clustercalculations were performed for the hyperfine fields inthree atomic fine structure levels, allowing to determinethe quadrupole moment of Ge to a precision of 2%,in agreement with the precision value obtained frommolecular theory calculations and experiments.The available experimental nuclear moments of groundand isomeric states of − Ge around N = 40 areinterpreted through a comparison to large-scale shell-model calculations in the f / pg / model space. Acomparison of the g -factors with the effective SP g -factors0reveals the nature of the orbital occupied by the unpairedneutrons. Yet, the calculated wave functions reveala very mixed configuration. Through a systematiccomparison of the nuclear moments of germaniumisotopes with its isotones (zinc and selenium) around N = 40, an increase in collectivity is observed, whenprotons are added to Z = 28, reflecting a sign ofstructural change from single particle to deformationwhen moving away from Z = 28.To shed more light on the structural evolution inthe region, further study of germanium isotopes upto N = 50 and even beyond is necessary, with thewell-established laser systems and atomic transitionstudied in this work. This will allow us to have a betterunderstanding of the structure changes in the nickelregion, particularly, the gradual emergence of collectivity. ACKNOWLEDGMENTS
We acknowledge the support of the ISOLDEcollaboration and technical teams. We would like to acknowledge the Center for Information Technologyof the University of Groningen for their supportand for providing access to the Peregrine high-performance computing cluster. This work wassupported by the National Key R&D Program ofChina (No. 2018YFA0404403), the National NaturalScience Foundation of China (No:11875073, U1967201);the BriX Research Program No. P7/12, FWO-Vlaanderen (Belgium), GOA 15/010 from KU Leuven;the UK Science and Technology Facilities Council grantsST/L005794/1 and ST/P004598/1; the NSF grantPHY-1068217, the BMBF Contracts No 05P18RDCIA,the Max-Planck Society, the Helmholtz InternationalCenter for FAIR (HIC for FAIR); the EU Horizon2020research and innovation programme through ENSAR2(No. 654002); [1] O. Sorlin et al. , Phys. Rev. Lett. , 092501 (2002).[2] Z. Y. Xu et al. , Phys. Rev. Lett. , 032505 (2014).[3] M. L. Bissell et al. , Phys. Rev. C , 064318 (2016).[4] R. Taniuchi et al. , Nature , 53 (2019).[5] C. Wraith et al. , Phys. Lett. B , 385 (2017).[6] R. P. de Groote et al. , Phys. Rev. C , 041302(R)(2017).[7] L. Xie et al. , Phys. Lett. B , 134805 (2019).[8] B. Cheal et al. , Phys. Rev. Lett. , 252502 (2010).[9] N. Aoi et al. , Phys. Lett. B , 302 (2010).[10] C. J. Chiara et al. , Phys. Rev. C , 037304 (2011).[11] K. T. Flanagan et al. , Phys. Rev. Lett. , 142501(2009).[12] B. Crider et al. , Phys. Lett. B , 108 (2016).[13] X. F. Yang et al. , Phys. Rev. C , 044324 (2018).[14] X. F. Yang et al. , Phys. Rev. Lett. , 182502 (2016).[15] P. Vingerhoets et al. , Phys. Rev. C , 064311 (2010).[16] T. J. Procter et al. , Phys. Rev. C , 034329 (2012).[17] S. Kaufmann et al. , Phys. Rev. Lett. , 132502 (2020).[18] R. P. de Groote et al. , Nat. Phys. , 620 (2020).[19] A. D. Ayangeakaa et al. , Phys. Lett. B , 254 (2016).[20] J. J. Sun et al. , Phys. Lett. B , 308 (2014).[21] Y. Toh et al. , Phys. Rev. C , 041304(R) (2013).[22] K. Heyde and J. L. Wood, Rev. Mod. Phys , 1467(2011).[23] K. Heyde and J. L. Wood, Phys. Scr. , 083008 (2016).[24] A. F. Oluwole, S. G. Schmelling, and H. A. Shugart,Phys. Rev. C , 228 (1970).[25] R. Neugart et al. , J. Phys. G: Nucl. Part. Phys. ,064002 (2017).[26] E. Man´e et al. , Eur. Phys. J. A , 503 (2009).[27] U. K¨oster et al. , Nuclear Instruments and Methods inPhysics Research Section B: Beam Interactions withMaterials and Atoms , 303 (2003). [28] W. J. Childs and L. S. Goodman, Phys. Rev. , 245(1963).[29] W. J. Childs and L. S. Goodman, Phys. Rev. , 15(1966).[30] A. Vernon et al. , Spectrochimica Acta Part B: AtomicSpectroscopy , 61 (2019).[31] K. Kreim et al. , Phys. Lett. B , 97 (2014).[32] S. Kaufmann et al. , J. Phys. Conf. Ser , 012033(2015).[33] K. Abusaleem and B. Singh, Nucl. Data Sheets , 133(2011).[34] B. Singh and J. Chen, Nucl. Data Sheets , 1 (2019).[35] M. L. Bissell and X. Yang, Ground and isomeric statespins, moments and radii of Ge isotopes across the N =40 subshell closure via laser spectroscopy at COLLAPS ,Tech. Rep. CERN-INTC-2016-035. INTC-P-472 (CERN,Geneva, 2016).[36] W. J. Childs and L. S. Goodman, Phys. Rev. C , 750(1970).[37] W. Gins, R. de Groote, M. Bissell, C. Granados Buitrago,R. Ferrer, K. Lynch, G. Neyens, and S. Sels, Comput.Phys. Commun. , 286 (2018).[38] R. Neugart, S. L. Kaufman, W. Klempt, G. Moruzzi, E.-W. Otten, and B. Schinzler, in Laser Spectroscopy III ,edited by J. L. Hall and J. L. Carlsten (Springer BerlinHeidelberg, Berlin, Heidelberg, 1977) pp. 446–447.[39] N. Bendali, H. T. Duong, P. Juncar, J. M. S. Jalm, andJ. L. Vialle, J. Phys. B: At. Mol. Phys. , 233 (1986).[40] W. Makulski, K. Jackowski, A. Antuˇsek, andM. Jaszu´nski, J. Phys. Chem. A , 11462 (2006).[41] V. Kell¨o and A. Sadlej, Mol. Phys , 275 (1999).[42] U. Kaldor and E. Eliav, Adv. Quantum Chem. , 313(1998).[43] L. Visscher, E. Eliav, and U. Kaldor, J. Chem. Phys , 9720 (2001).[44] A. Borschevsky, E. Eliav, and U. Kaldor, “High-accuracy relativistic coupled cluster calculations for theheaviest elements,” in Handbook of Relativistic QuantumChemistry , edited by W. Liu (Springer Berlin Heidelberg,2016) pp. 825–855.[45] A. Landau, E. Eliav, and U. Kaldor, in
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