Inverse odd-even staggering in nuclear charge radii and possible octupole collectivity in 217 , 218 , 219 At revealed by in-source laser spectroscopy

Hyperﬁne-structure parameters and isotope shifts for the 795-nm atomic transitions in 217 , 218 , 219 At have been measured at CERN-ISOLDE, using the in-source resonance-ionization spectroscopy technique. Magnetic dipole and electric quadrupole moments, and changes in the nuclear mean-square charge radii, have been deduced. A large inverse odd-even staggering in radii, which may be associated with the presence of octupole collectivity, has been observed. Namely, the radius of the odd-odd isotope 218 At has been found to be larger than the average of its even- N neighbors, 217 , 219 At. The discrepancy between the additivity-rule prediction and experimental data for the magnetic moment of 218 At also supports the possible presence of octupole collectivity in the considered nuclei.


I. INTRODUCTION
The possible presence of octupole correlations has been conjectured for nuclei with the neutron and proton numbers N and Z in the 130 N 140, 86 Z 92 region, where the Fermi surface lies between shell-model states with j = l = 3, where j and l are the total angular and orbital moments. For the considered region, it is the proton f 7/2 , i 13/2 states and the neutron g 9/2 , j 15/2 orbitals that satisfy these conditions. In such cases the nucleus can assume octupole deformation, corresponding to reflection asymmetry in the intrinsic frame, either dynamically (octupole vibrations) or by having a static shape (permanent octupole deformation) [1,2].
Experimentally, the signs of octupole deformation were found in a number of ways (see Refs. [1,2]), including α decay to low-lying negative-parity states in the even-even nuclei [3], Coulomb excitation [4], or from the comparison of experimental nuclear masses with models where reflection asymmetry is taken into account [5].
Another sign of octupole effects in this region is the occurrence of a so-called inverse odd-even staggering (inverse OES) in the charge radii of an isotopic chain. Throughout the nuclide chart, an OES in charge radii is systematically observed, whereby an odd-N isotope has a smaller charge radius than the average of its two even-N neighbors. However, an inversion of OES has been found in some regions of the nuclide chart, in particular, for the N = 133−139 francium and radium isotopes [6].
The correlation between octupole deformation and inverse OES was qualitatively described by Otten [6] and corroborated by the calculations of Leander and Sheline [7], as due to an increase in the mean-square octupole deformation ( β 2 3 ) for odd-N nuclei, relative to their even-N neighbors. The schematic calculations by Talmi [8] also imply a normal OES for even-multipole deformations and inverse OES for odd-multipole deformations.
However, one cannot consider inverse OES as a definite "fingerprint" of octupole collectivity, despite the strong correlation between these two phenomena. In particular, the inverse OES in several francium and radium isotopes could also be qualitatively reproduced in the framework of the extended Thomas-Fermi approach without invoking odd-order contributions to the deformation [9] (see also the discussion on the octupole deformation and inverse OES in europium and barium isotopes near N = 88−90 [10][11][12][13][14]).
Recently, new information on the borders of the inverse-OES region at Z > 82 was obtained via laser-spectroscopy investigations of 84 Po [15], 87 Fr [16], and 88 Ra [17]. To better localize this region, in the present work we have undertaken isotope shift (IS) and hyperfine structure (hfs) investigations for 217,218,219 At. These nuclei lie in the vicinity of the presumed "octupole region" but so far have never been considered as reflection asymmetric.
The investigations presented in this paper are part of an experimental campaign at the ISOLDE facility (CERN) aimed at β-delayed fission, nuclear decay, and laser spectroscopy studies of the astatine isotopes. Partial results were reported in Refs. [18,19].

II. EXPERIMENTAL DETAILS
The present data came from the same experiment on the long chain of astatine isotopes at the ISOLDE facility, as described in detail in Ref. [19], which reported on charge radii and electromagnetic moments of 195−211 At. Therefore, here we provide only a short description and refer the reader to Ref. [19] for full details on the experiment and data analysis.
The in-source laser spectroscopy technique [20,21] was used for IS and hfs measurements of astatine atoms. Radioactive astatine isotopes were produced in spallation reactions, induced by 1.4-GeV protons from the CERN PS Booster in a 50 g cm −2 UC x target. The spallation products diffused through the target material as neutral atoms and effused into the hot cavity of the ion source. Laser beams were introduced into this cavity and performed selective ionization of the astatine isotopes of interest using a three-step ionization scheme [19]. The photoion current as a function of the laser frequency of the second excitation step (46234 cm −1 → 58805 cm −1 ; 795.2 nm) was measured by two methods. For the relatively short-lived isotopes, 217 At (T 1/2 = 32 ms) and 218 At (T 1/2 = 1.5 s), the α-decay rate was measured with the Windmill (WM) setup [22], whereas ion counting by the Multi-Reflection Time-of-Flight Mass Separator (MR-ToF MS) [23] was used for the longer lived 219 At (T 1/2 = 56 s).
A detailed account of hfs scanning with the Windmill setup and the MR-ToF MS device, and a description of the laser system can be found in Refs. [19,21,24]. Examples of experimental spectra are presented in Fig. 1.

III. RESULTS
The experimental spectra were fitted using the same method as described in detail in Ref. [19]. The fitting process requires the knowledge of a nuclear spin value, since the Doppler-limited resolution of the in-source laser-spectroscopy method does not allow an unambiguous determination of the spin in the astatine nuclei. Below we summarize available literature information on the possible spin and parity assignments based on the αand β-decay properties of 217,218,219 At.

A. Nuclear spins assumed from the literature data
According to the nuclear data evaluation [25], the ground state of 217 At was assigned a spin (I ) and parity (π ) of I π = (9/2 − ), based on the low hindrance factor [HF = 1.16(4)] of its α decay to the ground state of 213 Bi, which has a firmly  established I π ( 213 Bi g ) = 9/2 − . This spin and parity would correspond to a dominant (π h 9/2 ) 3 configuration for 217 At. Similarly, a value of I π = 9/2 − is suggested for the 219 At ground state, based on the low hindrance factor of the α decay of 219 At g to 215 Bi g (HF = 1.1) [26]. The ground state of 215 Bi is presumed to have I π = (9/2 − ), in view of both the strong population of the 11/2 + state in its β decay to 215 Po and the much lower population of the 5/2 + state [26].
Therefore, based on the available literature data we fixed the spin of 217,219 At g as I = (9/2).
As shown in the complementary paper [27], the most probable spin of 218 At is I = (3 − ). The latter is based on the unhindered nature of the 218 At α decay populating presumably the E = 63 keV, I = (3 − ) excited state in 214 Bi. However, the I π = (2 − ) assignment cannot be completely ruled out (see Ref. [27] for details). In the present study, due to the Dopplerlimited resolution, it was also impossible to choose between the fit results with I = 2 or 3 assignments [see Figs. 1(b) and 1(c)]. Thus, for 218 At g , both spin options, I ( 218 At g ) = (2, 3), must be considered.

B. Extraction of nuclear parameters
The data were analyzed with a fixed hfs a-constant ratio, ρ ≡ a(58805 cm −1 )/a(46234 cm −1 ) = −1.69(2) [19]. The hyperfine constants and isotope shifts for 217,218,219 At are presented in Table I. The magnetic dipole moments, μ A , were calculated using the scaling relation with 211 At as a reference: The following reference values were used: [19]. A possible hyperfine structure anomaly (HFA) was taken into account by increasing the uncertainty: by 1% for 218 At with I = 9/2 and by 0.1% for 217,219 At(I = 9/2) (see detailed discussion in Ref. [19] and compilation of the available HFA data in Ref. [28]).
To deduce the electric quadrupole moment Q S from the measured hfs b constant, the ratio b/Q S was calculated for the astatine atomic ground state by applying the multiconfiguration Dirac-Hartree-Fock (MCDHF) method [19]. The full description of numerical methods can be found in Ref. [29]. With the measured ratio of the b constants for the atomic ground state and the excited state at 46234 cm −1 [19], one The main   054317-3  TABLE II. Magnetic dipole and electric quadrupole moments, changes in mean-square charge radii, and staggering parameter γ A (see below) for 217,218,219 At. For 218 At, the results for the different possible spin assignments (see Sec. III A) are presented. Statistical and systematic uncertainties are given in round and curly brackets, respectively. The systematic uncertainties in δ r 2 A,205 , stem from the theoretical indeterminacy of the F and M factors; in μ, from the uncertainty in μ ref and the HFA indeterminacy; and in Q S , from the uncertainty in the theoretical b/Q S ratio. (45)  contribution to the uncertainty stems from the error in the b-constants ratio.
The changes in the mean-square charge radii, δ r 2 A,A , were deduced from the measured isotope shift δν A,A using the relations:  [19]. In this case a different approach to considering electron correlations in comparison with the MCDHF calculations of the b/Q S ratio was implemented (see details in Refs. [19,30]).
The magnetic dipole moments, electric quadrupole moments, and changes in the mean-square charge radii for 217,218,219 At are presented in Table II. The magnetic moment of 217 At was measured previously by the low temperature nuclear orientation method [31]: μ( 217 At) = 3.81 (18)μ N . Our result (see Table II) agrees with the literature value in the limits of uncertainties.

A. g factors for even-N isotopes
In Fig. 2 the g factors (g = μ/I ) for the odd-Z, even-N nuclei with N 126 are presented. Along with the results of the present work for 85 At isotopes, the data for 83 Bi ( [32] and references therein), 87 Fr ( [33,34]), and 89 Ac [35,36] isotopes are shown. Horizontal dotted lines mark the single-particle values of the g factors near doubly magic 208 Pb, for the relevant proton orbitals: π i 13/2 [37], π f 7/2 [38], and π h 9/2 (Schmidt estimation). All experimental g factors shown in Fig. 2 lie between the Schmidt value for the π h 9/2 orbital and the g factor for the 9/2 − ground state of semimagic 209 Bi. This suggests that the leading configuration for all of these nuclei is π h 9/2 , despite the change in spin from 9/2 to 5/2 in 221 Fr and 3/2 in 223,225 Fr and 227 Ac.
As is seen in Fig. 2, the g factors for different Z decrease at similar rate with increasing N, although there is a jump for Fr isotopes at N = 136 when the spin changes from 5/2 to 3/2. After this jump, the gradual decrease in the g factor with respect to N is restored, with the same rate of change as for the N = 126−132 isotopes of francium with I = 9/2. This jump may be explained by an admixture from a π f 7/2 configuration, which has a larger single-particle g factor (see Fig. 2). Indeed, the I π = 3/2 − ground state of 223 Fr is presumed to have a deformed 3/2 − [521] f 7/2 Nilsson configuration, which is strongly mixed with the 1/2 − [521]h 9/2 band [39].
Although the g factors for the odd astatine isotopes studied in the present work also decrease when going from N = 132 to N = 134 at the same rate as for the isotonic francium isotopes, its absolute value is markedly larger than that for 219,221 Fr 132,134 (see Fig. 2). This increase may be attributed to the admixture of other higherj proton configurations. It cannot be explained by the spherical f 7/2 − configuration admixture, due to the lower spin in this configuration. However, if one assumes the presence of octupole deformation, then mixing between opposite-parity orbitals stemming from π = −1 h 9/2 and π = +1 i 13/2 configurations becomes possible and the admixture of the i 13/2 configuration would result in an increase of the magnetic moment [g(i 13/2 ) > g(h 9/2 ), g(i 13/2 ) > g(9/2 − , 209 Bi)]. Thus, the deviation of g( 217,219 At) from the systematics may be connected with the appearance of octupole collectivity in these nuclei.

B. g factor for odd-N isotope
Comparisons of the experimental magnetic moment of odd-odd nuclei, with estimations from the additivity relation, μ add , [40] may aid in understanding nucleon orbital occupations. The most probable configuration for 218 At is (π 1h 9/2 ⊗ ν2g 9/2 ) 2−,3− . For the additivity-rule calculations, individual empirical g factors for the π 1h 9/2 and ν2g 9/2 orbitals were taken from the magnetic moments of the closest odd-A isotopes available: μ p = μ( 217 At) = 3.7(1)μ N (present work), μ n = μ( 217 Po) = −1.11 (14)μ N [15]. The results from the additivity-relation calculation are systematically lower than the experimental data for both possible spin assignments: μ add ( 218 At; 2 − ) = 0.58 (6)μ N , μ exp ( 218 At; 2 − ) = 1.20 (11)μ N and μ add ( 218 At; 3 − ) = 0.87(9)μ N , μ exp ( 218 At; 3 − ) = 1.25 (12)μ N . This may point to a possible admixture from other orbitals. Closely lying orbitals that could contribute to this admixture are π f 7/2 or π i 13/2 for the odd proton, and νh 11/2 for the odd neutron. However, for every two-particle combination of these orbitals other than that involving i 13/2 proton, the calculated μ add value is lower than that of a (π 1h 9/2 ⊗ 2g 9/2 ) 2−,3− configuration. In the case of the configurations with i 13/2 proton, μ add becomes significantly larger (μ add for 2 + or 3 + states with the i 13/2 proton is in the region of 4−7μ N , depending on the neutron state), thus, even a small admixture from these configurations would ensure an agreement between the additivity-rule estimations and the experimental results. Such an admixture is only possible in cases with mixing between opposite-parity states at nonzero octupole deformation. It is worth to note that the configuration mixing in 217,218,219 At with taking into account also the possible neutron excitations, may be probed by the large-scale shell-model calculations. Unfortunately, at present for astatine isotopes only the calculations with the limited number of the valence neutrons are available (A < 216; [41]).

C. Quadrupole moments
The quadrupole moments of the 9/2 − ground states in 217,219 At measured in the present work, indicate a small oblate deformation (β 2 ≈ −0.08), in the strong coupling scheme (see Ref. [19] and references therein on the applicability of this approach for weakly deformed nuclei).
It is instructive to compare the 217,219 At results with 219 Fr. The measured Q S values for 217,219 At 132,134 (see Table II) match within the limit of uncertainties with Q S ( 219 Fr 132 ) = −1.21(2) b [34]. However, there is a difference in the understanding of the nature of the astatine and francium ground states. The 9/2 − ground state in 219 Fr was proposed to be an anomalous member of the K = 1/2 − band (K is the projection of the intrinsic spin on the symmetry axis), based on the 1/2 − [521]h 9/2 orbital at a moderate prolate deformation [42]. The odd-proton spin is decoupled from the nuclear deformation axis in this nucleus, yielding a negative quadrupole moment at a positive deformation (see Ref. [34] and references therein). In contrast, the K = 1/2 − band was not observed in 217,219 At. The 9/2 − ground states in these nuclei are considered to be spherical h 9/2 states or 9/2 − [505]h 9/2 states with small oblate deformation (see Ref. [43] and references therein). Surprisingly, despite such an obvious difference in the interpretation of the 9/2 − ground states in the isotonic francium and astatine nuclei, the corresponding quadrupole moments have nearly the same values.
Similar to the use of the additivity rule for magnetic moments, spectroscopic quadrupole moments for oddodd nuclei can be estimated by applying a single-particle quadrupole additivity rule based on a general tensor coupling scheme (see Refs. [44,45] and references therein). Deviations from this quadrupole additivity rule may be attributed to the development of collectivity. We applied this additivity rule to 218 At with the following singleparticle quadrupole moments: Q S,p = Q S ( 217 At) (present work) and Q S,n = Q S ( 217 Po) [15]. To summarize, the analysis of the magnetic dipole and electric quadrupole moments of the 217,218,219 At isotopes shows that these nuclei are satisfactorily described within the framework of the spherical shell model. However, there are some indications for the possible presence of the octupole collectivity, which will be further emphasized by the discussion of the charge radii, presented in the next section.

A. Shell effect
In Fig. 3, changes in the mean-square charge radii δ r 2 for the astatine nuclei near N = 126 are shown (for N 126 the values from Ref. [19] are used), along with the droplet-model (DM) predictions [46]. As is seen in Table II, the radii obtained for 218 At with different spin assumptions, coincide within the experimental uncertainties and would be indistinguishable in Fig. 3, therefore only the δ r 2 218 (I=3), 211 value is presented.
A characteristic increase in the slope of the δ r 2 isotopic dependency when crossing the neutron magic number N = 126 is evident (although the IS for 212−216 At was not measured due to their short half lives). This shell effect in radii was found to be a universal feature of the δ r 2 behavior and was observed for different isotopic chains near N = 28, 50, 82, 126 [47]. To compare the shell effect in different isotopic chains, the dimensionless shell-effect parameter, ξ even , was used [32]: where the subscript indices are the neutron numbers. This parameter is independent of the uncertainties in the F factor (usually 5-10% in the lead region; note that in the case of astatine the uncertainty in ξ even due to the M-factor indeterminacy is less than 0.5%). The choice of the even-N isotopes with N = 124, 128, being the nearest to the neutron magic number N = 126, helps to avoid mixing of the shell effect with other effects which might contribute to the observed δ r 2 value. However, when there are no experimental data for N = 128 nuclei, it is instructive to assume linear interpolation and consider also the modified shell-effect parameter which takes into account heavier nuclei with known δ r 2 , where N 0 is the lowest even neutron number at N > 126 with measured IS: N 0 = 132 for 84 Po [48,49], 85 At (present work), 86 Rn [6,50], 87 Fr [16,33,51], 88 Ra [52], and N 0 = 138 for 89 Ac [53]. For 82 Pb and 83 Bi isotopes, data for nuclei with N = 124, 126, 128 were taken from Refs. [54] and [32], respectively.
In Fig. 4 the shell-effect parameters for lead-region nuclei are presented and compared with the results of relativistic mean-field (RMF) calculations with DD-PC1 energy-density functional [55]. It should be reminded that the standard nonrelativistic Hartree-Fock (NRHF) approach fails to explain this effect [ξ even (NRHF)] ∼ 1] [56,57]. A detailed discussion on the different theoretical descriptions of the shell-effect, as well as the analysis of the shell effect for odd-N nuclei can be found in Ref. [32]. Theoretical RMF calculations overestimate ξ even . A better description of the shell effect for lead nuclei was recently obtained with a new extended parameterization of the RMF model based on the effective field theory (see Fig. 2 in Ref. [58]), in the relativistic Hartree-Fock approach with nonlinear terms and density-dependent meson-nucleon coupling [59], and in the Hartree-Fock-Bogoliubov calculations using a density-dependent spin-orbit interaction [60].
The parameter ξ * even linearly increases with Z (by ∼20% when going from Z = 84 to Z = 89). It is unclear whether this trend is due to a Z dependence of the shell-effect, or it is connected with the different deformation contribution to δ r 2 N 0 , 126 at different Z (through δ β 2 2 and δ β 2 3 ), or it is a result of the linear interpolation used in the extraction of the modified shell parameter.

B. Odd-even staggering
The odd-even staggering in nuclear charge radii is quantified by the staggering parameter, introduced by Tomlinson and Stroke [61] (for odd N), When γ N = 1, there is no OES, whereas γ < 1 and γ > 1 correspond to normal and inverse OES, respectively. The γ parameter does not depend on the electronic factor and its uncertainty; therefore it is well suited to compare quantitatively the OES for the different isotopic chains.
The γ values for the isotopes of 85 At (present work, [19]), 82 Pb [54], 84 Po [24,48,49,62], 87 Fr [16,34,51], 88 Ra [52], and 86 Rn [6,50] are plotted as a function of neutron number in Fig. 5. The OES parameters for 83 Bi 123,125,127 [32] coincide within the limits of uncertainties with that of the isotonic 82 Pb isotopes and are excluded from the plot for clarity. The values of γ 123 and γ 125 for 88 Ra and 86 Rn isotopes also coincide with those for the isotonic nuclei and are also excluded. As is seen in Fig. 5, the OES at N < 131 and at N > 137 is normal for all isotopic chains (γ N < 1), whereas for 87 Fr, 88 Ra, and 86 Rn isotopes with 133 N 137, γ N > 1, which means the inverted OES effect. Our new data for 217−219 At testify to the retention of this effect for Z = 85. As described in Sec. I, there is a strong correlation between inverse OES and octupole deformation. Correspondingly, one can suppose the possible presence of octupole collectivity in astatine nuclei near N = 133.
This assumption is quite surprising, since the region of quadrupole-octupole deformation is supposedly confined between Z = 86 and Z = 92 (see Table I in Ref. [63]). Moreover, so far no evidence for parity doublet bands and their associated fast E1 transitions has been observed in 217,219 At [43], the appearance of which is usually regarded as a signature of octupole collectivity. In contrast with the isotonic francium isotopes where such signs of the octupole collectivity are firmly established [42], low-lying states in 217,219 At can be reliably described as parts of the seniority-three proton configurations: (h 9/2 ) 3 and (h 9/2 ) 2 f 7/2 , that is, as spherical shell-model states without invoking octupole or quadrupole deformation [42]. Unfortunately, the data on the excited states in 218 At are missing. Thus, it is unknown whether the parity doublet bands and the other signs of octupole collectivity are present in this nucleus.
In contrast with the majority of the presumed octupole deformed nuclei [63], spins of 217,218,219 At are reasonably well described by the spherical shell-model π h 9/2 and π h 9/2 ⊗ vg 9/2 configurations (see Sec. IV A). Thus, so far firm nuclear spectroscopic evidence for quadrupole-octupole deformation in 217,218,219 At has not been observed (see also Ref. [64]) and inverse OES remains the single definite experimental indication on the possible octupole deformation in these nuclei.
However, potential-energy surface (PES) calculations, in the framework of the macroscopic-microscopic approach, also testify to the presence of the quadrupole-octupole deformed minimum in the PES for 217−219 At [65]. This minimum is supposed to correspond to the ground state since it is deeper by 0.5-0.6 MeV than the minimum in the PES calculated with the assumption of zero octupole deformation [65].
To summarize, keeping in mind the OES systematics for nuclei with N 132, we believe that the observed large inverse OES in heavy isotopes of astatine is related to the presence of octupole collectivity, either for all three investigated isotopes ( 217,218,219 At), or only for odd-odd 218 At. This assumption is supported by the PES calculations. However, further nuclear spectroscopic information is required in order to substantiate this inference.

VI. CONCLUSIONS
Hyperfine structure parameters and isotope shifts have been measured for 217,218,219 At, using the 795-nm atomic transitions. Magnetic dipole and electric quadrupole moments, and changes in the nuclear mean-square charge radii have been deduced and discussed in the framework of the possible presence of octupole collectivity.
Analysis of the electromagnetic moments shows that 217,218,219 At are reasonably well described within the framework of the spherical shell model. However, the discrepancy between the additivity-rule prediction and experimental data for μ( 218 At) may qualitatively indicate the presence of the quadrupole-octupole collectivity.
The shell effect in the mean-square charge radii of the astatine isotopes when crossing N = 126 has been observed. The increase of the shell-effect parameter with Z may also be related to the increase in quadrupole-octupole collectivity at N = 132, when going from Z = 84 to Z = 88.
A large inverse odd-even staggering in radii has been found for 217,218,219 At. This result is surprising since for the isotonic 87 Fr isotopes the OES disappears, and the 85 At isotopes are expected to lie outside the region of the quadrupole-octupole collectivity, where inverse OES was previously established.
The assumption of the presence of the quadrupole-octupole collectivity in heavy astatine isotopes is supported by the potential-energy surface calculations, although so far there  is no nuclear-spectroscopic evidence from the excited states which could substantiate this inference.