Evolution of Octupole Deformation in Radium Nuclei from Coulomb Excitation of Radioactive $^{222}$Ra and $^{228}$Ra Beams

There is sparse direct experimental evidence that atomic nuclei can exhibit stable pear shapes arising from strong octupole correlations. In order to investigate the nature of octupole collectivity in radium isotopes, electric octupole ($E3$) matrix elements have been determined for transitions in $^{222,228}$Ra nuclei using the method of sub-barrier, multi-step Coulomb excitation. Beams of the radioactive radium isotopes were provided by the HIE-ISOLDE facility at CERN. The observed pattern of $E$3 matrix elements for different nuclear transitions is explained by describing $^{222}$Ra as pear-shaped with stable octupole deformation, while $^{228}$Ra behaves like an octupole vibrator.

There are many theoretical and experimental indications that atomic nuclei can exhibit reflection asymmetry in the intrinsic frame, and observation of low-lying quantum states in many nuclei with even Z, N having total angular momentum and parity of I π = 3 − is indicative of the presence of octupole correlations (see [1] and references therein). Typically, the electric octupole (E3) moment for the transition to the ground state is tens of singleparticle units, suggesting that the octupole instability arises from a collective effect and leads to a pear-shaped distortion of the nuclear shape. What is less clear, however, is whether in some nuclei this distortion is stable, i.e. the nucleus assumes a permanent pear shape, or whether it is dynamic and the nucleus undergoes octupole vibrations. Evidence has been presented that 224 Ra and 226 Ra have static octupole deformation on account of an enhancement in the E3 moment in these nuclei [2,3]. Large E3 moments have also been recently measured for neutron-rich barium isotopes, suggesting that, within the experimental uncertainty, these nuclei could have octupole deformation [4,5]. The only example of an octupole unstable nucleus other than 226 Ra where stable beams have been used to obtain a complete set of E3 matrix elements is 148 Nd [6].
In this Letter, results from a multistep, Coulomb-excitation experiment with radioactive 222,228 Ra beams are reported. By examining the pattern of E3 matrix elements between different transitions in these nuclei and comparing them to those in 224,226 Ra and 148 Nd, a distinction can be made between those isotopes having stable octupole deformation and those behaving like octupole vibrators. This observation is relevant for the search for permanent electric dipole moments in radium atoms [7][8][9], that would indicate sizeable CP violation requiring a substantial revision of the Standard Model.
The radioactive isotopes 222 Ra (Z = 88, N = 134) and 228 Ra (Z = 88, N = 140) were  produced by spallation in a thick uranium carbide primary target bombarded by ≈ 10 13 protons/s at 1.4 GeV from the CERN PS Booster. The ions, extracted from a tungsten surface ion source were stripped to charge states of 51 + and 53 + , respectively, for 222 Ra and 228 Ra and accelerated in HIE-ISOLDE to an energy of 4.31 MeV/nucleon. The radioactive beams, with intensities between 5 × 10 4 and 2 × 10 5 ions/s bombarded secondary targets of 60 Ni and 120 Sn of thickness 2.1 mg/cm 2 . Gamma rays emitted following the excitation of the target and projectile nuclei were detected in Miniball [10], an array of 24 high-purity germanium detectors, each with sixfold segmentation and arranged in eight triple clusters.
The scattered projectiles and target recoils were detected in a highly segmented silicon detector, distinguished by their differing dependence of energy with angle measured in the laboratory frame of reference. Representative spectra from the Coulomb-excited 222,228 Ra are shown in Fig. 1; in the spectra the γ-ray energies are corrected for Doppler shift assuming emission from the scattered projectile. The spectra were incremented when a target recoil was detected in coincidence with γ rays within a 450-ns time window; these data were corrected for random events. The fraction of the isobar 222 Fr in the beam was estimated to be about 20% by observing γ rays from the α-decay daughters at the beam dump. By lowering the temperature of the transfer line from the ion source a nearly pure beam of 222 Fr could be produced; apart from X-rays, no discernable structure was observed arising from Coulomb excitation of the odd-odd nucleus in the particle-gated, Doppler-corrected spectrum. For the 228 Ra beam, the fraction of isobaric contamination was estimated to be ≈ 1%.
For both 222 Ra and 228 Ra the spectra reveal strong population of the ground-state band of positive-parity states, populated by multiple electric quadrupole (E2) Coulomb excitation, and substantial population of the octupole band of negative-parity states, populated by E3 excitation. The yields of the observed γ-ray transitions detected in Miniball were measured for four ranges of the recoil angle of the target nucleus for each target, between 21.5 • and 55.5 • for the 120 Sn target and between 17.8 • and 55.5 • for the 60 Ni target. The yield data were combined with existing γ-ray branching ratios to provide input to the Coulomb-excitation analysis code GOSIA [11][12][13]. The GOSIA code performs a least-squares fit to the Eλ (λ = 1, 2, 3) matrix elements (m.e.s), which either can be treated as free parameters, can be coupled to other matrix elements, or can be fixed. Energy-level schemes that are included in the analysis are given in [14]. A total of 114 data for 222 Ra were fitted to 42 variables, 5 while for 228 Ra 121 data were fitted to 41 variables. The starting values of each of the freelyvaried matrix elements were drawn randomly, within reasonable limits; the values obtained following the fitting procedure were found to be independent of the starting points. Examples of fits to the experimental data can be found in the Supplemental Material, see below [14].
For both nuclei the E1 couplings between the ground-state and negative-parity bands and the E2 couplings for transitions within the ground state and within the negative-parity bands, with the exception of the 2 + → 0 + transition, were treated as free parameters.
Under the experimental conditions described here, the probability of populating the 2 + state is > 90% and it was not possible to determine the 0 + ||E2||2 + and 2 + ||E2||2 + m.e.s independently. The latter was therefore allowed to vary freely and the 0 + ||E2||2 + matrix element was coupled to the 2 + ||E2||4 + matrix element assuming the validity of the rotational model; this assumption is based on the behaviour of nuclei where the lifetimes of the 2 + and 4 + states have been measured and for which the lowest transitions behave collectively [14]. For the E3 m.e.s the lowest couplings were treated as free parameters; m.e.s between higher-lying states, I ± ||E3||I ∓ , were coupled to m.e.s between lower-lying states, assuming the validity of the rotational model. E4 matrix elements were also included in the fitting procedure; these were calculated assuming the rotational model and a constant value of the hexadecapole moment, derived from the theoretical values of β λ [22]. E2 (and magnetic dipole) couplings to high-lying K π = 0 + and K π = 2 + bands were also taken into account. The relative phase of Q 1 and Q 3 was investigated, as although the overall phase of the E1 and E3 matrix elements is arbitrary, the fit is sensitive to the relative phase of E3 matrix elements as well as the phase difference between the E1 and E3 matrix elements. The difference in chi-square for the fit favoured Q 1 and Q 3 having the same sign for 222 Ra and the opposite sign for 228 Ra, and these phases were adopted in the final fits. These values are consistent with macroscopic-microscopic calculations [23] and constrained HFBCS calculations [24] that predict a decreasing value of Q 1 with neutron number for radium isotopes, crossing zero for 224 Ra as experimentally verified [25]. Table I gives the values of E2 and E3 matrix elements for 222 Ra and 228 Ra obtained in this work. The E1 matrix elements are given in [14]. Those E3 m.e.s marked with an asterisk are coupled to m.e.s between higher-lying states and as such are not completely independently determined; however the fit is mostly influenced by the value of the lowest matrix element. The diagonal E2 matrix elements are all coupled to the adjacent transition are deduced from the measured matrix elements [14], and correspond to transitions between states with spin I and I − 1.
m.e.s except for those presented in Table I, which are independently determined. In the GOSIA fit the statistical errors for each fitted variable were calculated taking into account correlations between all variables. Independent sets of fitted values were also obtained by varying the constant hexadecapole moment used to calculate the E4 m.e.s between zero and double the notional value, varying the target thickness by ±5%, the beam energy by ±1% , the distance between the target and the particle detector by ±7.5%, and the sign of the E2 couplings to the higher-lying collective bands.
The variations seen in the fitted values are included in the final uncertainties given in Table I. For 228 Ra the value of the intrinsic quadrupole moment, Q 2 , derived from the measured value of 2 + ||E2||4 + , 770 ± 40 efm 2 , agrees with the values determined from the 2 + lifetime, 775 ± 14 efm 2 and the 4 + lifetime, 780 ± 6 efm 2 , as reported in Ref. [16].
For 222 Ra, the value is 590 ± 30 efm 2 , significantly smaller than the value derived from the measured lifetime of the 2 + state, 673±13 efm 2 [26]. It is noted that the value of Q 2 for 222 Ra Smaller values of Q 2 , although with large uncertainty, were determined from the 2 + ||E2||2 + matrix element for both nuclei. Such behaviour was also observed in 226 Ra, interpreted as arising from deviations from axial symmetry [3]. The values of the intrinsic electric octupole  observed for 148 Nd. It is unlikely that this can be accounted for by K mixing [12] as the K π = 1 − band lies much higher in energy for these nuclei [28].
The contrast in the behaviour of the E3 moments of 228 Ra (and 148 Nd) compared to the lighter radium isotopes is also present in the behaviour of their energy levels, as shown    Arrows indicate γ-ray transitions that have been observed in the experiments described here; all energies are in keV. In 222 Ra no transitions to the higher lying collective bands were observed. The level scheme data have been taken from [15,16].