Large Shape Staggering in Neutron-Deficient Bi Isotopes

The changes in the mean-square charge radius (relative to 209 Bi), magnetic dipole, and electric quadrupole moments of 187 ; 188 ; 189 ; 191 Bi were measured using the in-source resonance-ionization spectroscopy technique at ISOLDE (CERN). A large staggering in radii was found in 187 ; 188 ; 189 Bi g , manifested by a sharp radius increase for the ground state of 188 Bi relative to the neighboring 187 ; 189 Bi g . A large isomer shift was also observed for 188 Bi m . Both effects happen at the same neutron number, N ¼ 105 , where the shape staggering and a similar isomer shift were observed in the mercury isotopes. Experimental results are reproduced by mean-field calculations where the ground or isomeric states were identified by the blocked quasiparticle configuration compatible with the observed spin, parity, and magnetic moment.

neutron shell closures [2,[5][6][7][8]. One of the most spectacular examples of such irregularities was found via the isotope shift (IS) measurements of 177-186 Hg (Z ¼ 80) near the neutron midshell at N ¼ 104 [9][10][11][12]. It was shown that the radii of the I π ¼ 1=2 − ground states (gs) in 181;183;185 Hg are substantially larger relative to their even-A neighbors. This phenomenon was characterized as "one of the most remarkable discoveries in nuclear structure physics in the last 50 years" [13].
Fifty years after this discovery, we present in this Letter the second example of such an unusual behavior, now in the lightest bismuth (Z ¼ 83) isotopes. By performing laserspectroscopy studies of 187;188;189;191 Bi, we found a sharp radius increase for 188 Bi g , relative to the neighboring 187;189 Bi g . This dramatic change happens at the same neutron number, N ¼ 105, where the shape staggering started in the isotonic 185 Hg.
The experiments were performed at the ISOLDE facility (CERN) [14]. The bismuth nuclei were produced in spallation reactions induced by the 1.4-GeV proton beam from the CERN PS Booster impinging on a thick UC x target (50 gcm −2 of 238 U). The spallation products effused out of the high-temperature target (T ≈ 2500 K) as neutral atoms into the cavity of the Resonance Ionization Laser Ion Source, RILIS [15]. The bismuth atoms were resonantly ionized when the laser beams were wavelength-tuned to the three-step ionization scheme [16,17]. The produced photoions were extracted and accelerated by a 30-kV potential and mass separated by ISOLDE's General Purpose Separator.
The hfs and IS measurements were performed for the first-step transition 6p 34 S o 3=2 → 6p 2 ð 3 P 0 Þ7s 2 ½0 1=2 (λ 1 ¼ 306.9 nm) by scanning the frequency of the narrow band Titanium Sapphire laser [18], whilst the photoion current was monitored by the detection of characteristic α decays using the Windmill decay station [19]. Examples of the hfs spectra are shown in Fig. 1. It is the ultrasensitive insource resonance-ionization spectroscopy that enables us to study nuclei with the yield less than 1 ion per second [see Fig. 1 Fig. 1 one can see that the hfs centroid of 188 Bi g is strongly shifted in comparison with its neighbors and its own high-spin isomer. This behavior already demonstrates the essential structural changes in this nucleus (see below).
The positions of the hyperfine components in the hfs spectra are determined by the standard relation [20] with five parameters: nuclear spin (I), isotope shift relative to the stable 209 Bi (δν A;209 ), magnetic hfs constants (a 1 and a 2 ) for the first (6p 34 S o 3=2 ) and the second f6p 2 ð 3 P 0 Þ7s 2 ½0 1=2 g level of the ionization scheme, and electric quadrupole hfs constant b 1 for the first level. Note that b 2 ≡ 0 since the second level has total electronic angular momentum J ¼ 1=2.
Because of the limited resolution of RILIS, the nuclear spin cannot be determined by just considering number, positions, and relative intensities of individual hyperfine components. Instead, the "integration method" was used [17,21], which is based on a comparison of the ratio of the areas under each broad peak (see two peaks in each panel of Fig. 1) with theoretically predicted values for different spin assumptions. The theoretical ratio depends only on the nuclear spin: r theor ¼½ðI þ 1Þ=I [17,22], leading to r theor ¼ 2, 1.5, and 1.33 for the spin assumption of I ¼ 1, 2, and 3, respectively. The weighted mean value r expt ¼ 2.00ð12Þ for the six experimental hfs spectra of 188 Bi g gives a strong preference for an I ¼ 1 assignment.
As atomic spectroscopy cannot determine the parity of the nuclear states, we used nuclear-spectroscopic data. Based on the unhindered nature of the 6992-keV α decay of 188 Bi g followed by the 117.5-keV E1 decay feeding the adopted (2 − ) state of the daughter 184 Tl m2 [23], we inferred a positive parity for 188 Bi g . Thus, the proposed assignment [I π ¼ 1 ðþÞ ] should supersede the previous one [I π ¼ð3 þ Þ] made via α-decay studies by analogy with heavier even-A Bi nuclei [23].F o r 187;189 Bi g and 188 Bi m the adopted spin values from [24] were used when fitting the hfs spectra [I ¼ð9=2Þ and I ¼ð10Þ, respectively]. Experimental data were fitted by a Voigt profile (more details are in Ref. [17]). In the fit, ratios ρ ¼ a 2 =a 1 were fixed according to high-resolution hfs measurements for heavier bismuth isotopes with the same spins as for the aforementioned nuclei [25].
As there are no heavier isotopes with the spin I ¼ 1, ρð 188 Bi g Þ¼−11.01 was fixed (as in 209 Bi), with a possible variation of this value in the limits of 1% [25]. The deduced a 2 and b 1 hfs constants and δν A;209 are presented in Table I.
Magnetic moments were derived from the hfs constants a 2 by the standard relation [see Eq. (6) in [2] ]. The following reference values were used: a 2;209 ¼ 4922.3ð20Þ MHz and μ 209 ¼ 4.0900ð15Þ μ N [26]. The magnetic moments are presented in Table I.
The hyperfine constant b is related to the spectroscopic quadrupole moment Q s via equation b ¼ eQ s × V, where V is the electric field gradient (EFG) produced by the electrons at the site of the nucleus. Thus, when Q s is known for a reference isotope 209 Bi, its value for other isotopes can be determined by the scaling relation There are two independent Q s measurements for 209 Bi made in the 1970s: Q s ¼ −0.37ð3Þb (muonic x ray, Ref. [28]) and Q s ¼ −0.50ð8Þb (pionic x ray, Ref. [29]). The noticeable difference between these values, their large uncertainties, and a multitude of corrections needed in order to reliably estimate Q s by these methods (Refs. [28,30]) prevent us from using a muonic or pionic result as a reference value. The alternative way is to calculate the EFG and based on the measured bð 209 BiÞ, deduce Q s ð 209 BiÞ. However, the complicated electronic structure makes accurate calculations for atomic bismuth a challenging problem. The first ab initio calculations by the multiconfiguration Dirac-Fock method [31] gave the reference value Q s ð 209 BiÞ¼−0.516ð15Þb adopted in the recent compilation [32]. However, subsequent advanced molecular calculations resulted in −0.420ð8Þb [33] and −0.415b [34], questioning this value.
To overcome this longstanding discrepancy, we summarized the results of 33 atomic and molecular calculations for 209 Bi, either published in the last decade (2013-2021) [33][34][35][36][37], or made specifically for this study. In these calculations a variety of advanced theoretical methods [38][39][40][41][42][43][44] with various computational strategies [45][46][47][48][49] were used by several independent groups on five continents. Combining measured values of the hfs constant b of several electronic states in the neutral 209 Bi atom, the 209 Bi þ ion, as well as in several diatomic molecules (BiN, BiP, BiF, BiI, BiO, BiCl), with calculated EFG values for those particular systems, we obtained a sample of 33 values of Q s ð 209 BiÞ. The "world average" Q s ð 209 BiÞ¼−0.420ð17Þb was deduced by taking the average of all these results, and the uncertainty was evaluated as a standard deviation of this sample. The quadrupole moments for 187;188;189;191 Bi calculated using Eq. (1), "world average" Q s ð 209 BiÞ, and b 1 ð 209 BiÞ from Ref. [50] are presented in Table I. The details of methods and calculations will be published elsewhere.
In Fig. 2 the δhr 2 i A;209 values for bismuth nuclei are compared with the data for isotonic lead and mercury isotopes. For all three isotopic chains, one observes the similar smooth trend until N ¼ 106 along with the small and well-known odd-even staggering (OES). Surprisingly, a huge staggering in radii appears for 187-189 Bi isotopes manifested by a drastic increase for 188 Bi g in comparison with the neighboring 187;189 Bi g and with 188 Bi m . This dramatic change happens at the same neutron number, N ¼ 105, where the famous Hg shape staggering starts [10,11]. The latter was interpreted as a sharp change between nearly spherical shapes in the even-N cases and The statistical and systematical uncertainties are shown in round and curly brackets, respectively. The latter values originate from the uncertainty of the F and M factors for δhr 2 i A;209 and from the uncertainty of the hyperfine anomaly for μ that was estimated using data for heavier bismuth isotopes with the same spin and parity [25,26]. For μð 188 Bi g Þ a conservative 2% uncertainty was additionally included to account for unknown anomaly. b Reference [27].
PHYSICAL REVIEW LETTERS 127, 192501 (2021) strongly prolate deformed configurations in the odd-N isotopes [53]. In contrast to the mercury isotopes with spin 0 or 1=2 for which Q s ≡ 0, for 188 Bi g one can directly check this interpretation using the measured Q s value. Indeed, in the strong coupling scheme the quadrupole moment can be related to the quadrupole deformation parameter β [see, for example, Eq. (14) in Ref. [54] ]. Using this relation one obtains βð 188 Bi g Þ¼þ0.25ð7Þ, whereas βð 188 Bi m Þ¼−0.08ð2Þ, βð 189 BiÞ¼−0.10ð1Þ, and βð 187 BiÞ¼−0.08ð4Þ. Thus, 188 Bi g is strongly prolate deformed, whilst its 10 − isomer and the adjacent isotopes with lower radii have a small deformation. It was noted in Ref. [27] that the δhr 2 i values of the 9=2 − Bi gs's follow the same trend as the radii of the spherical Pb isotopes down to N ¼ 112, but start to deviate at N ¼ 110. This was interpreted as a possible onset of deformation in 193;191 Bi g . Our new data confirm this deviation from sphericity down to 187 Bi (N ¼ 104). This is consistent with suggestions from extensive nuclear-spectroscopy data on the possible structural changes in the N ≤ 110 bismuth isotopes (see Refs. [23,[59][60][61][62][63]).
The experimental results were analyzed on the basis of Hartree-Fock-Bogoliubov (HFB) calculations with the D1M Gogny interaction and the equal filling approximation for the odd-A and odd-odd nuclei. Potential energy surfaces (PESs) were calculated with the blocked quasiparticles (qp) compatible with the experimental spin and parity. The total number of PESs to compute is therefore particularly large for odd-odd systems, with, e.g., 100 PESs in 188 Bi with minima located at most 1 MeV above the HFB gs.
The further selection of the proper states was made by constraining the calculated magnetic moment to the experimental value. The magnetic moments were calculated as detailed in Ref. [64] on the basis of an effective magnetic moment operatorμ eff ¼ 0.82 × g sŝ þ 1.25 × g ll . The beyond mean-field and core-polarization effects were taken into account via the adopted effective coefficients (see Ref. [65]) in order to best reproduce all experimental moments [66]. For strongly deformed nuclei, a rotational contribution to the magnetic moment was also taken into account [64]. Following this approach, for a given combination (isotope, configuration) only the level with the lowest energy and a magnetic moment compatible with the experimental value within 50% was kept. The PESs obtained with this blocking and selection technique are shown in Fig. 3.
For the 9=2 − and 10 − states at A ≥ 193, only a single nearspherical minimum exists. In the lighter isotopes additional oblate and prolate minima appear, but they are still quite flat and shallow. In contrast, the 1 þ state in 188 Bi corresponds to the well-pronounced prolate minimum at β ¼ 0.28 [compare with β ¼ 0.25ð7Þ extracted from the Q s value]. The structure of this 1 þ state is determined by the blocked configuration π1=2½530 ⊗ ν1=2½521, which is the oddodd counterpart of the ν1=2½521 configuration in the isotonic 185 Hg g .
Note that the 1 þ minimum at β 2 ≃ 0.08, despite being lower than the β ¼ 0.28 minimum, has a magnetic moment of μ eff ¼ 0.0 μ N incompatible with experimental data and can therefore not be retained as a potential candidate for the gs configuration.
The calculated magnetic moments and δhr 2 i values obtained from the HFB minima constrained by the  magnetic moment (at β ≈−0.07 for 9=2 − and 10 − states and at β ≈ þ0.28 for 1 þ state) are compared with experimental data in Fig. 4. As shown in Fig. 4(b), the large δhr 2 i ( 188 Bi g ) value is well reproduced; however, the theory overestimates the slope of the δhr 2 i isotopic dependency for the 9=2 − and 10 − states at A ≤ 193. Such a discrepancy suggests possible configuration mixing (CM) between the various low-lying states connected with different minima in PES. Indeed, it is this kind of CM that was invoked to explain a small deviation from the spherical droplet model predictions in the core Pb nuclei around N ¼ 104 [57]. However, such a CM for odd-A or odd-odd nuclei cannot currently be modeled microscopically. Therefore, in order to mimic such a mixing between prolate, oblate, and near spherical shapes, a phenomenological approach, inspired from statistical physics [67,68], was followed. It consists of calculating the average value of the observable O by mixing its values over the various states q with different deformations, through the expression where E is the HFB energy along a given PES (Fig. 3) and T is a free parameter which enables the mixing of various low-lying configurations. A constant value of T ¼ 1 MeV was chosen in order to improve the description of the charge radii in neutron-deficient bismuth isotopes. As seen in Fig. 4(b) the mixing procedure preserves the large staggering in the neighborhood of 188 Bi but reduces its amplitude. The CM effect may consequently be overestimated for the 1 þ state. Only future beyond-mean-field calculations will be able to shed light on the exact impact of CM and in particular how it affects different blocked configurations. Note that CM does not significantly influence the μ values [see Fig. 4(a)]. At the same time, it results in a better agreement of the calculated δhr 2 i of the lightest bismuth isotopes with the experimental data. In particular, it reproduces the observed OES between the 9=2 − and 10 − states for light bismuth isotopes which is absent in the nonmixed HFB approach [see Fig. 4(b)]. Indeed, the nonmixed HFB minima predict the bismuth gs (apart from 188 Bi g ) to be quasispherical (Fig. 3), while the CM takes the contributions of the prolate and oblate minima into account, leading to a deviation from sphericity, hence an increase in δhr 2 i. The heaviest bismuth isotopes are not as affected by mixing since the PES becomes stiffer when approaching the N ¼ 126 closed shell (Fig. 3). The reproduction of the OES for the 9=2 − and 10 − states can be understood from contributions of the prolate wells that alter between these states for the light bismuth isotopes. The larger stiffness of the PES of the 10 − states compared to the 9=2 − states leads to less mixing for the former.
To summarize, ISs and hfs's were studied for neutrondeficient bismuth isotopes using the ultrasensitive (down to 0.1 ion/s) in-source resonance-ionization spectroscopy technique. A striking staggering in radii was observed for 188 Bi g relative to 187;189 Bi g at the same neutron number (N ¼ 105) as in the Hg case (see Refs. [10,11]). It is only the second example of such unusual behavior throughout the nuclide chart. The quadrupole moment of 188 Bi g confirms the strong prolate deformation in this nucleus with the newly established spin and parity of I π ¼ 1 ðþÞ .
This staggering was successfully explained by HFB calculations, where the ground state is identified by the blocked quasiparticle configuration compatible with the observed spin, parity, and magnetic moment. The departure from the trend for radii of Pb isotopes, found in light Bi's, was explained by invoking CM with states of different deformations.
We thank the ISOLDE Collaboration for providing excellent beams and the GSI Target Group for manufacturing the carbon foils for the Windmill. A. B. and M. L. R. would like to thank the Center for Information Technology of the University of Groningen for providing access to the Peregrine high performance computing cluster and for their technical support. S. G. acknowledges financial support from FNRS (Belgium). J. G. L. acknowledges financial