Publisher Correction: The observation of vibrating pear-shapes in radon nuclei

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I t is well established by the observation of rotational bands that atomic nuclei can assume quadrupole deformation with axial and reflection symmetry, usually with the shape of a rugby ball. The distortion arises from long-range correlations between valence nucleons, which becomes favourable when the proton and/or neutron shells are partially filled. For certain values of proton and neutron number it is expected that additional correlations will cause the nucleus to also assume an octupole shape ('pear-shape') where it loses reflection symmetry in the intrinsic frame 1 . The fact that some nuclei can have pear-shapes has influenced the choice of atoms having nuclei with odd nucleon number A (=Z + N) employed to search for permanent electricdipole moments (EDMs). Any measurable moment will be amplified if the nucleus has octupole collectivity and further enhanced by static-octupole deformation. At present, experimental limits on EDMs, that would indicate charge-parity (CP) violation in fundamental processes where flavour is unchanged, have placed severe constraints on many extensions of the Standard Model. Recently, new candidate atomic species, such as radon and radium, have been proposed for EDM searches. For certain isotopes octupole effects are expected to enhance, by a factor 100-1000, the nuclear Schiff moment (the electric-dipole distribution weighted by radius squared) that induces the atomic EDM [2][3][4] , thus improving the sensitivity of the measurement. There are two factors that contribute to the greater electrical polarizability that causes the enhancement: (i) the odd-A nucleus assumes an octupole shape; (ii) an excited state lies close in energy to the ground state with the same angular momentum and intrinsic structure but opposite parity. Such parity doublets arise naturally if the deformation is static (permanent octupole deformation).
The observation of low-lying quantum states in many nuclei with even Z, N having total angular momentum ('spin') and parity of I π ¼ 3 À is indicative of their undergoing octupole vibrations about a reflection-symmetric shape. Further evidence is provided by the sizeable value of the electric octupole (E3) moment for the transition to the ground state, indicating collective behaviour of the nucleons. However, the number of observed cases where the correlations are strong enough to induce a static pear-shape is much smaller. Strong evidence for this type of deformation comes from the observation of a particular behaviour of the energy levels for the rotating quantum system and from an enhancement in the E3 moment 5 . So far there are only two cases, 224 Ra 6 and 226 Ra 7 for which both experimental signatures have been observed. The presence of a parity doublet of 55 keV at the ground state of 225 Ra makes this nucleus therefore a good choice for EDM searches 8 . In contrast to the radium isotopes, much less is known about the behaviour of radon (Rn) nuclei proposed as candidates for atomic EDM searches on account of possible enhancement of their Schiff moments [9][10][11][12][13][14][15][16][17] . For this reason, different isotopes of radon have been listed in the literature, for example 221,223,225 Rn 14 , each having comparable half-lives and ground state properties. The most commonly chosen isotope for theoretical calculations 9,10 and the planning of experiments [11][12][13] is 223 Rn.
In this work, we present data on the energy levels of heavy even-even Rn isotopes to determine whether parity doublets are likely to exist near the ground state of neighbouring odd-mass Rn nuclei. Direct observation of low-lying states in odd-A Rn nuclei (for example following Coulomb excitation or β-decay from the astatine parent) is presently not possible, as it will require significant advances in the technology used to produce radioactive ions. We observe that 224,226 Rn behave as octupole vibrators in which the octupole phonon is aligned to the rotational axis. We conclude that there are no isotopes of radon that have staticoctupole deformation, so that any parity doublets in the odd-mass neighbours will not be closely spaced in energy. This means that radon atoms will provide less favourable conditions for the enhancement of a measurable atomic electric-dipole moment.

Results
Measurement of the quantum structure of heavy radon isotopes. In the experiments described here, 224 Rn (Z = 86, N = 138) and 226 Rn (Z = 86, N = 140) ions were produced by spallation in a thick thorium carbide target bombarded by~10 13 protons s −1 at 1.4 GeV from the CERN PS Booster. The ions were accelerated in HIE-ISOLDE to an energy of 5.08 MeV per nucleon and bombarded secondary targets of 120 Sn. In order to verify the identification technique, another isotope of radon, 222 Rn, was accelerated to 4.23 MeV/u. The γ-rays emitted following the excitation of the target and projectile nuclei were detected in Miniball 18 , an array of 24 high-purity germanium detectors, each with six-fold segmentation and arranged in eight triple-clusters. The scattered projectiles and target recoils were detected in a highly segmented silicon detector 19 . See Methods.
Prior to the present work, nothing was known about the energies and spins of excited states in 224,226 Rn, while de-exciting γ-rays from states in 222 Rn had been observed 20 with certainty up to I π ¼ 13 À . The chosen bombarding energies for 224,226 Rn were about 3% below the nominal Coulomb barrier energy at which the beam and target nuclei come close enough in head-on collisions for nuclear forces to significantly influence the reaction mechanism. For such close collisions the population of high-spin states will be enhanced, allowing the rotational behaviour of the nucleus to be elucidated. This is the method described by Ward et al. 21 and has subsequently been coined unsafe Coulomb excitation 22 as the interactions between the high-Z reaction partners is predominantly electromagnetic. It is not possible to precisely determine electromagnetic matrix elements because of the small nuclear contribution. The most intense excited states expected to be observed belong to the positive-parity rotational band, built upon the ground state. These states are connected by fast electric quadrupole (E2) transitions. In nuclei that are unstable to pear-shaped distortion, the other favoured excitation paths are to members of the octupole band, negative-parity states connected to the ground-state band by strong E3 transitions.
The spectra of γ-rays time-correlated with scattered beam and target recoils are shown in Fig. 1. The E2 γ-ray transitions within the ground-state positive-parity band can be clearly identified, as these de-excite via a regular sequence of strongly-excited states having spin and parity 0 þ ; 2 þ ; 4 þ ; with energies h 2 2= I I þ 1 ð Þ. In this expression the moment-of-inertia = systematically increases with increasing I (reducing pairing) and with number of valence nucleons (increasing quadrupole deformation). As expected from multi-step Coulomb excitation the intensities of the transitions systematically decrease with increasing I, after correcting for internal conversion and the γ-ray detection efficiency of the Miniball array.
The other relatively intense γ-rays observed in these spectra with energies <600 keV are assumed to have electric-dipole (E1) multipolarity, and to depopulate the odd-spin negative-parity members of the octupole band. In order to determine which states are connected by these transitions, pairs of time-correlated ('coincident') γ-rays were examined. In this analysis, the energy spectrum of γ-rays coincident with one particular transition is generated by requiring that the energy of this 'gating' transition lies in a specific range. Typical spectra obtained this way are shown in Fig. 2. Each spectrum corresponds to a particular gating transition, background subtracted, so that the peaks observed in the spectrum arise from γ-ray transitions in coincidence with that transition.
The level schemes for 224,226 Rn constructed from the coincidence spectra, together with the known 20 scheme for 222 Rn, are shown in Fig. 3. For 226 Rn the energy of the stronglyconverted 2 þ ! 0 þ transition overlaps with those of the K β Xrays, but its value can be determined assuming that the relative intensity of K β , K α X-rays is the same as for 222,224 Rn. The E2 transitions connecting the states in the octupole band are not observed because they cannot compete with faster, higher-energy E1 decays. The only other plausible description for this band is that it has K π ¼ 0 þ or 2 + , implying that the K π ¼ 0 À octupole band is not observed. (Here K is the projection of I on the bodyfixed symmetry axis.) This is unlikely as the bandhead would have to lie significantly lower in energy than has been observed in 222,224,226 Ra, and inter-band transitions from states with I' > 4 to states with I and I-2 in the ground-state band and in-band transitions to I'-2 would all be visible in the spectra. The spin and parity assignments for the positive-parity band that is strongly populated by Coulomb excitation can be regarded as firm, whereas the negative-parity state assignments are made in accord with the systematic behaviour of nuclei in this mass region.
Characterisation of octupole instability. From the level schemes and from the systematics for all the radon isotopes (Fig. 4) at the same rotational frequency ω, as a function of ω. Here i x is approximately I for K=0 bands and hω is approximately ðE I À E IÀ2 Þ=2: For nuclei with permanent octupole deformation Δi x is expected to approach zero, as observed for several isotopes of Ra, Th, and U 5 . For octupolevibrational nuclei in which the negative-parity states arise from coupling an octupole phonon to the positive-parity states, it is expected that Δi x $ 3 h as the phonon prefers to align with the rotational axis. This is the case for the isotopes 218,220,222,224 Rn at values of hω (<0.2 MeV) where particle-hole excitations do not play a role, see Fig. 4. Thus we have clearly delineated the lower boundary at Z > 86 as to where permanent octupole deformation occurs in nature.

Discussion
The observation of octupole-vibrational bands in the even-even radon isotopes is consistent with several theoretical calculations [24][25][26] , which predict that only nuclei with Z > 86 have stable octupole deformation. Other calculations suggest that radon isotopes with A~222 will have non-zero values of the octupole deformation parameter β 3 27,28 . For such nuclei, which have a minimum in the nuclear potential energy at non-zero values of β 3 , the positive-and negative-parity states are projected from intrinsic configurations having K π ¼ 0 þ ; 0 À , which are degenerate in energy. In the odd-A neighbours parity doublets arise by     Fig. 4 Systematic behaviour of radon isotopes. a Systematics of the energies for different spins of low-lying positive-parity (black) and negative-parity states (red) in radon isotopes; b cartoon illustrating how the octupole phonon vector aligns with the rotation (R) vector (which is orthogonal to the rotating body's symmetry axis) so that I = R + 3 h and Δi x ¼ 3 h; c difference in aligned spin for negative-and positive-parity states in 218-224 Rn (re-analysed for [218][219][220][221][222] Rn that have been presented earlier 23 ). The dashed line at Δi x ¼ 0 is the expected value for static-octupole deformation coupling the odd particle to these configurations. This is not the case for reflection-symmetric nuclei that undergo octupole vibrations around β 3 ¼ 0: Bands of opposite parity with differing single-particle configurations can lie close to each other fortuitously 29,30 but in general those arising from coupling the odd nucleon to the ground state and octupole phonon will be well separated. The separation will be determined by the spacing of the bands in the even-even core,~500 keV in the case of 222-226 Rn (see Fig. 4), and will be in general much larger than that the value (~50 keV) observed for parity doublets in radium isotopes 1 . Quantitative estimates of Schiff moments for octupole-vibrational systems have yet to be made 31 . Nevertheless, it can be concluded that, if measurable CP-violating effects occur in nuclei, the enhancement of nuclear Schiff moments arising from octupole effects in odd-A radon nuclei is likely to be much smaller than for heavier octupole-deformed systems.

Methods
Production of radioactive radon beams. In our experiments, 222,224,226 Rn were produced by spallation in the primary target, diffused to the surface and then singly ionized (q = 1 + ) in an enhanced plasma ion-source 32 with a cooled transfer line. The ions were then accelerated to 30 keV, separated according to A/q, and delivered to a Penning trap, REXTRAP 33 , at a rate of around 8 × 10 6 ions s −1 for 222 Rn, 2 × 10 6 ions s −1 for 224 Rn and 10 5 ions s −1 for 226 Rn at the entrance. Inside the trap, the singly-charged ions were accumulated and cooled before being allowed to escape in bunches at 500 ms intervals into an electron-beam ion source, REXE-BIS 33 . Here, the ions were confined for 500-700 ms in a high-density electron beam that stripped more electrons to produce a charge state of 51 + ( 222 Rn) or 52 + ( 224,226 Rn) extracted as 1 ms pulses before being mass-selected again according to A/q, and injected at 2 Hz into the HIE-ISOLDE linear post-accelerator. The Rn beams then bombarded a 120 Sn target of thickness~2 mg cm −2 with an intensity of about 6 × 10 5 ions s −1 , 1.1 × 10 5 ions s −1 and 2 × 10 3 ions s −1 for 222 Rn, 224 Rn and 226 Rn, respectively. The total beam-times were respectively 8, 16 and 24 h. The level of Fr impurity in the Rn beams could be estimated for A = 222 as below 1% by observing radioactive decays at the end of the beam line.
Data selection and Doppler correction. Events corresponding to the simultaneous detection of γ-rays and heavy ions in Miniball and the silicon detector array respectively were selected if the measured energy and angle of either projectile or target satisfied the expected kinematic relationship for inelastic scattering reactions. This procedure eliminated any background from stable noble-gas contaminant beams produced in REX-TRAP having the same A/q as the radon beams. In the present setup the average angle of each of the 16 strips of one side of the silicon detector array ranged between 19.6°and 54.9°to the beam direction, corresponding to a scattering angular range of 140.9°-70.2°in the centre-of-mass. In order to reduce background from Compton scattering, events were rejected if any two germanium crystals in each triple-cluster registered simultaneous γ-ray hits, in contrast to the normal adding procedure which would substantially increase the probability of summing two γ-rays emitted in the same decay sequence ('true pileup'). Miniball was calibrated using 133 Ba and 152 Eu radioactive sources that emitted γ-rays of known energy and relative intensity. The relativistic Doppler correction was performed by determining the momentum vector of the projectile, using the energy and position information in the pixelated silicon detector, and the emission polar and azimuthal angle of the detected gamma-ray in the segment of Miniball where most energy was deposited. In the case of the latter the relative orientation of each segment to each other and to the beam axis was determined by employing d( 22 Ne,pγ) and d( 22 Ne,nγ) reactions. The Doppler corrected energies for transitions in 224 Rn and 226 Rn together with the deduced level energies are given in Table 1.

Data availability
The data that support the findings of this study are available from the corresponding author on reasonable request. The software used to sort the raw data is available at https://doi.