Abstract
There is strong circumstantial evidence that certain heavy, unstable atomic nuclei are ‘octupole deformed’, that is, distorted into a pear shape. This contrasts with the more prevalent rugby-ball shape of nuclei with reflection-symmetric, quadrupole deformations. The elusive octupole deformed nuclei are of importance for nuclear structure theory, and also in searches for physics beyond the standard model; any measurable electric-dipole moment (a signature of the latter) is expected to be amplified in such nuclei. Here we determine electric octupole transition strengths (a direct measure of octupole correlations) for short-lived isotopes of radon and radium. Coulomb excitation experiments were performed using accelerated beams of heavy, radioactive ions. Our data on 220Rn and 224Ra show clear evidence for stronger octupole deformation in the latter. The results enable discrimination between differing theoretical approaches to octupole correlations, and help to constrain suitable candidates for experimental studies of atomic electric-dipole moments that might reveal extensions to the standard model.
Main
The atomic nucleus is a many-body quantum system, and hence its shape is determined by the number of nucleons present in the nucleus and the interactions between them. For example, nuclei in their ground state in which the proton and neutron shells are completely filled (‘doubly magic’ nuclei) are spherical. If this configuration is excited, or if more nucleons are added, the long-range correlations between valence nucleons distort the shape from spherical symmetry and the nucleus becomes deformed. In most of these cases, it is well established that the shape assumed has quadrupole deformation with axial and reflection symmetry; that is, the nucleus is shaped like a rugby ball (prolate deformation) or as a discus (oblate deformation). For certain combinations of protons and neutrons, there is also the theoretical expectation that the shape of nuclei can assume octupole deformation, corresponding to reflection asymmetry or a ‘pear-shape’ in the intrinsic frame, either in a dynamic way (octupole vibrations) or having a static shape (permanent octupole deformation).
Octupole deformation and EDMs
Atoms with octupole-deformed nuclei are very important in the search for permanent atomic electric-dipole moments (EDMs). The observation of a non-zero EDM at the level of contemporary experimental sensitivity would indicate time-reversal (T) or equivalently charge–parity (CP) violation due to physics beyond the standard model. In fact, experimental limits on EDMs provide important constraints on many proposed extensions to the standard model1,2. For a neutral atom in its ground state, the Schiff moment (the electric-dipole distribution weighted by radius squared3) is the lowest-order observable nuclear moment. Octupole-deformed nuclei with odd nucleon number A ( = Z + N, see below) will have enhanced nuclear Schiff moments owing to the presence of the large octupole collectivity (spatial correlation between particle states) and the occurrence of nearly degenerate parity doublets that naturally arise if the deformation is static3,4,5. Because a CP-violating Schiff moment induces a contribution to the atomic EDM, the sensitivity of the EDM measurement to CP violation over non-octupole-enhanced systems such as 199Hg (ref. 2), currently providing the most stringent limit for atoms, can be improved by a factor of 100–1,000 (ref. 4). Essential in the interpretation of such limits in terms of new physics is a detailed understanding of the structure of these nuclei. Experimental programmes are in place to measure EDMs in atoms of odd-A Rn and Ra isotopes in the octupole region (see for example, ref. 6) but so far there is little direct information on octupole correlations in these nuclei.
Strong octupole correlations leading to pear shapes can arise when nucleons near the Fermi surface occupy states of opposite parity with orbital and total angular momentum differing by 3
. This condition is met for proton number Z ≈ 34, 56 and 88 and neutron number N ≈ 34, 56, 88 and 134. The largest array of evidence for reflection asymmetry is seen at the values of Z ≈ 88 and N ≈ 134, where phenomena such as interleaved positive- and negative-parity rotational bands in even–even nuclei7, parity doublets in odd-mass nuclei8, and enhanced electric-dipole (E1) transition moments9 have been observed. Many theoretical approaches have been developed to describe the observed experimental features: shell-corrected liquid-drop models10,11, mean-field approaches using various interactions12,13,14,15,16, models that assume α-particle clustering in the nucleus17,18, algebraic models19 and other semi-phenomenological approaches20. A broad overview of the experimental and theoretical evidence for octupole correlations is given in ref. 21.
In order to determine the shape of nuclei, the rotational model can be used to connect the intrinsic deformation, which is not directly observable, to the electric charge moments that arise from the non-spherical charge distribution. For quadrupole deformed nuclei, the typical experimental observables are the electric-quadrupole (E2) transition moments that are related to the matrix elements connecting differing members of rotational bands in these nuclei, and E2 static moments that are related to diagonal matrix elements for a single state. If the nucleus does not change its shape under rotation, both types of moments will vary with angular momentum but can be related to a constant ‘intrinsic’ moment that characterizes the shape of the nucleus. For pear-shaped nuclei, there will be additionally E1 and electric-octupole (E3) transition moments that connect rotational states having opposite parity. The E1 transitions can be enhanced because of the separation of the centre-of-mass and centre-of-charge. The absolute values of the E1 moments are, however, small (<10−2 single particle units) and are dominated by single-particle and cancellation effects9. In contrast, the E3 transition moment is collective in behaviour (>10 single particle units) and is insensitive to single-particle effects, as it is generated by coherent contributions arising from the quadrupole-octupole shape. The E3 moment is therefore an observable that should provide direct evidence for enhanced octupole correlations and, for deformed nuclei, can be related to the intrinsic octupole deformation parameters22. Until the present measurements, E3 transition moments have been determined for only one nucleus in the Z ≈ 88, N ≈ 134 region, 226Ra (ref. 23), so that theoretical calculations of E3 moments in reflection-asymmetric nuclei have not yet been subject to detailed scrutiny.
Experiments and discussion
Coulomb excitation is an important tool for exploring the collective behaviour of deformed nuclei that gives rise to strong enhancement of the probability of transitions between states. Traditionally, this technique has been employed by exciting targets of stable nuclei with accelerated ion beams of stable nuclei at energies below the Coulomb barrier, ensuring that the interaction is purely electromagnetic in character. Whereas E2, E1 and magnetic dipole (M1) transition probabilities dominate in the electromagnetic decay of nuclear states, and hence can be determined from measurements of the lifetimes of the states, E2 and E3 transition moments dominate the Coulomb excitation process allowing these moments to be determined from measurement of the cross-sections of the states, often inferred from the γ-rays that de-excite these levels. In exceptional cases, the Coulomb excitation technique has been applied to radioactive targets like 226Ra, which is sufficiently long-lived (half-life 1,600 yr) to produce a macroscopic sample. It is only comparatively recently that the technique has been extended to the use of accelerated beams of radioactive nuclei such as those from the Radioactive beam EXperimental facility at ISOLDE, CERN (REX-ISOLDE24). In the experiments described here, 220Rn (Z = 86, N = 134) and 224Ra (Z = 88, N = 136) ions were produced by spallation in a thick uranium carbide target bombarded by ∼1013 protons s−1 at 1.4 GeV from the CERN PS Booster. The ions were post-accelerated in REX-ISOLDE to an energy of 2.82 or 2.83 MeV per nucleon and bombarded secondary targets of 60Ni, 112Cd or 114Cd, and 120Sn of thickness approximately 2 mg cm−2 with an intensity of about 3 × 105 ions s−1 and 7 × 105 ions s−1 for Rn and Ra, respectively (Methods). The targets were chosen to give differing electromagnetic excitation (Ni, Z = 28 versus Sn, Z = 50; see Fig. 1) and, in the case of Cd, to provide a cross-check to the excitation of a target whose matrix elements are well known and can be measured in these experiments.

a, 60Ni (blue) and 120Sn (red) bombarded by 220Rn; b, targets as a but with bombardment by 224Ra. The differences in excitation cross-section for the targets with different Z are apparent for the higher spin states. The γ-rays are corrected for Doppler shift assuming that they are emitted from the scattered projectile. The asterisk in a marks an unassigned, 836(2) keV transition. A state at 937.8(8) keV is assigned Iπ = 2+ on the basis of its excitation and decay properties; it is assumed to be the bandhead of the γ-band in 220Rn.
The γ-rays emitted following the excitation of the target and projectile nuclei were detected in MINIBALL25, 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, distinguished by their differing dependence of energy with angle measured in the laboratory frame-of-reference, were detected in a highly segmented silicon detector26. Representative spectra for 220Rn and 224Ra, for which the γ-ray energy is corrected for Doppler shift assuming emission from the scattered projectile, are shown in Fig. 1. Here events were accumulated only if the target recoil was detected in coincidence with γ-rays within a 600-ns time window; these data were corrected for random events. The energy level schemes for 220Rn and 224Ra are shown in Fig. 2.
The spectra reveal strong population of the ground-state band of positive-parity states, excited by direct and multiple E2 Coulomb excitation, and substantial population of the octupole band of negative-parity states, excited by E3 excitation. The yields of the observed γ-ray transitions detected in MINIBALL were measured for two ranges of recoil angles of the target nucleus (28°–38° and 38°–52° for 220Rn; 24°–40° and 40°–54° for 224Ra) for each of the three targets, and combined with existing spectroscopic data (lifetimes of low-lying states27,28 and their γ-ray branching ratios7,29,30) to provide input to the Coulomb-excitation analysis code GOSIA31 (Methods). The separation of angular ranges increased sensitivity in the measurement by varying the relative excitation probabilities. For 220Rn, 34 independent data points determined 22 free parameters (16 matrix elements and 1 normalization constant for each combination of target and recoil angle range) whereas for 224Ra, 57 data points determined 23 free parameters (17 matrix elements). The analysis was also carried out for 220Rn independently of the previously measured lifetime of the Iπ = 2+ state (τ2+)27, and for 224Ra independently of τ4+ (ref. 28); here I is the total angular momentum of the state and π is its parity. (For 224Ra the previously measured value of τ2+ cannot be determined independently as the 2+ → 0+ transition is contaminated with the Ra X-rays.) In both cases the fitted matrix elements for the 2+ → 0+ E2 transition (220Rn) and for the 4+ → 2+ E2 transition (224Ra) were found to agree, within the experimental uncertainties, with the values obtained using the lifetime measurements.
The measured E1, E2 and E3 matrix elements for 220Rn and 224Ra are given in Table 1. The values of the intrinsic moments,
, are given in Fig. 3. These are determined from the experimental values of the reduced matrix element between two states having angular momenta I and I′ induced to undergo a transition by the electromagnetic operator Eλ, <I′||Eλ||I>, assuming the validity of the rotational model22. Here λ = 1,2,3 refers to E1, E2, E3 respectively. For the E2 and E3 matrix elements, the measured values are all consistent with the geo metric predictions expected from a rotating, deformed distribution of electric charge, although these data do not distinguish whether the negative-parity states arise from the projection of a quadrupole-octupole deformed shape or from an octupole oscillation of a quadrupole shape32. Table 2 compares the experimental values of
derived from the matrix elements connecting the lowest states for nuclei near Z = 88 and N = 134 measured by Coulomb excitation. It is striking that while the E2 moment increases by a factor of 6 between 208Pb and 234U, the E3 moment changes by only 50% in the entire mass region. Nevertheless, the larger Q3 values for 224Ra and 226Ra indicate an enhancement in octupole collectivity that is consistent with an onset of octupole deformation inthis mass region. On the other hand, 220Rn has similar octupole strength to 208Pb, 230,232Th and 234U, consistent with it being an octu pole vibrator. In the case of a vibrator, the coupling of an octupole phonon to the ground state rotational band will give zero values for matrix elements such as <1−||E3||4+>, because an aligned octupole phonon would couple the 4+ state to a 7− state. Although the present experiment does not have sensitivity to this quantity, this effect has been observed for 148Nd in the Z ≈ 56, N ≈ 88 octupole region33, while for 226Ra the intrinsic moment derived from the measured <1−||E3||4+> is similar to that derived from the value of <0+||E3||3−> (ref. 23). The deduced shapes of 220Rn and 224Ra are presented in Fig. 4. Here the values of quadrupole and octupole deformation β2 and β3 were extracted from the dependence of the measured Q2 and Q3 on the generalized nuclear shape34.

a, b, These values are derived from the matrix elements using the relation
for 220Rn (a) and 224Ra (b). Here
is the matrix element between states of angular momenta I and I′ defined in the text and
is a Clebsch-Gordon coefficient. The dashed lines have the value of the weighted mean for each Qλ, the error bars are ±1s.d.

a, 220Rn; b, 224Ra. Panel a depicts vibrational motion about symmetry between the surface shown and the red outline, whereas b depicts static deformation in the intrinsic frame. Theoretical values of β4 are taken from ref. 10. The colour scale, blue to red, represents the y-values of the surface. The nuclear shape does not change under rotation about the z axis.
The conclusions drawn from the present measurements are also consistent with suggestions from the systematic studies of energy levels7 (relative alignment of the negative-parity band to the positive-parity band) that the even–even isotopes 218–222Rn and 220Ra have vibrational behaviour while 222–228Ra have octupole-deformed character (see figures 12 and 13 in ref. 7). For odd-mass 219Ra there is no evidence35 for parity doubling, whereas for 221Ra a parity doublet of states with I = 5/2 separated by 103.6 keV has been observed36. In the Ba–Nd region with Z ≈ 56 and N ≈ 88, where the octupole states arise from vibrational coupling to the ground-state band, the evidence for parity doubling of the ground state arising from reflection asymmetry is inconclusive37,38. This suggests that the parity doubling condition that leads to enhancement of the Schiff moment15 is unlikely to be met in 219,221Rn. On the other hand 223,225Ra, having parity doublets separated by ∼50 keV (ref. 21), will have large enhancement of their Schiff moments.
The values of
, deduced from the measured transition matrix elements, are plotted in Fig. 5 as a function of N. The anomalously low value of Q1 for 224Ra, measured here for the first time, has been noted elsewhere9,13,39. The measured Q1 and Q2 values are in good agreement with recent theoretical calculations of the generator-coordinate extension of the Gogny Hartree–Fock–Bogoliubov (HFB) self-consistent mean field theory16, particularly using the D1M parameterization40. However, as remarked earlier, the trend of the experimental data is that the values of Q3 decrease from a peak near 226Ra with decreasing N (or A), which is in marked contrast to the predictions of the cluster model calculations17. It is also at variance with the Gogny HFB mean-field predictions of a maximum for 224Ra (ref. 16). It should be noted, however, that relativistic mean field calculations14 predict that the maximum value of Q3 occurs for radium isotopes between A = 226 and 230, depending on the parameterization, and Skyrme Hartree–Fock calculations15 predict that 226Ra has the largest octupole deformation. Both predictions are consistent with our data. We cannot completely eliminate the possibility that there are unobserved couplings from the ground state to higher-lying 3− states that should be added (without energy weighting) to the observed coupling to the lowest 3− state. However, in a detailed study33 of octupole strength in 148Nd, where these states lie closer in energy to the lowest state, such couplings were not observed.

Measured values for λ → 0 transitions are used for λ = 2, 3; for λ = 1 an average value over the measured spin range (≤5
) is used. Comparisons are made to the theoretical predictions of a cluster model17 (‘Ra cluster model’) and mean-field calculations16 (with two different parameterizations D1M and D1S40, ‘Rn D1M, D1S’ and ‘Ra D1M, D1S’). The points are connected by lines as a guide to the eye. The experimental data for Rn and Ra isotopes are denoted ‘Rn exp’ and ‘Ra exp’ respectively; the data for 220Rn (N = 134) and 224Ra (N = 136) are from the present work, other data are taken from refs 23, 49 and 50. The error bars are ±1s.d.
Outlook
We have demonstrated that radioactive beams of heavy nuclei with A ≈ 220 can be successfully accelerated with sufficient intensity to measure both even- and odd-order electric-multipole matrix elements with an accuracy of 10% or better. The extracted electric-quadrupole and electric-octupole moments are consistent with constant values over the range of measured angular momentum. Our data show that 220Rn has weaker octupole collectivity than 224Ra. We conclude that 219,221Rn are likely to have smaller octupole-enhanced EDMs than 223,225Ra, though more favourable Rn candidates may emerge from future studies of the low-lying structure of heavier isotopes. Comparing our data with predictions of the E3 strength from recent models, we find that the trend in octupole deformation extracted from the data presented here reveals detailed differences from some mean field predictions16 and opposes the trend predicted by the cluster model17. Our findings for the comparisons with models should be confirmed by extending studies to other radioactive isotopes in the Rn and Ra chain. It is interesting to note that the Gogny HFB calculations16 predict that Th and U isotopes with N = 134–136, already known to exhibit the characteristics of a rigid octupole shape7,41, should have significantly enhanced E3 transition strengths (70 Weisskopf units); however, tests of this prediction await major developments in radioactive beam technology.
Methods Summary
In our experiments, the 220Rn and 224Ra produced by spallation diffused to the primary target surface and were then singly ionized (q = 1+) in either an enhanced plasma ion-source42 with a cooled transfer line (Rn) or a tungsten surface ion-source (Ra), accelerated to 30 keV, separated according to A/q, and delivered to a Penning trap, REXTRAP43, at a rate of around 1.25 × 107 ions s−1 for 220Rn and 4.4 × 107 ions s−1 for 224Ra at the entrance. Inside the trap, the singly-charged ions were accumulated and cooled before allowing the ions to escape in bunches at 400 ms intervals into an electron-beam ion source, REXEBIS43. Here, the ions were confined for 400 ms in a high-density electron beam that stripped more electrons to produce a charge state of 52+, extracted as 400 μs pulses before being mass-selected again according to A/q, and injected at 2.5 Hz into the REX linear post-accelerator. The level of isobaric impurity (for example, Fr) in the Ra beam was estimated to be below 1% by observing radioactive decays at the end of the beam line. For Rn, observation of contaminant decays was more difficult because of the small α-decay branching ratios to excited states, and only an upper limit of 5% could be obtained.
The GOSIA code performs a least-squares fit to the matrix elements between all known states coupled by electromagnetic operators, which are treated as free parameters. Although the fit is sensitive to the relative E1/E2 decay rates, E1 (and M1) excitation is negligible at the beam energies used and can be ignored. The magnitudes of the values of the starting parameters were chosen to be random, within reasonable limits. The fit was found to be insensitive to many of the matrix elements; these were either fixed or coupled to other matrix elements assuming the validity of the rotational model.
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Acknowledgements
The support of the ISOLDE Collaboration and technical teams is acknowledged. This work was supported by the following Research Councils: STFC (UK), BMBF(Germany; 05P12RDCIA, 06DA9036I, 06KY9136I and 06KY205I), HIC for FAIR (Germany), FWO-Vlaanderen (Belgium), Belgian Science Policy Office (IAP-BriX network P7/12), Academy of Finland (contract no. 131665), DOE (US; DE-AC52-07NA27344 and DE-FG02-04ER41331), NSF (US), MICINN (Spain; FPA2009-08958 and FIS2009-07277), Consolider-Ingenio 2010 Programmes (Spain; CPAN CSD2007-00042 and MULTIDARK CSD2009-00064), Polish Ministry for Science and Higher Education (grant no. 589/N-G-POOL/2009/0), EC via I3-EURONS (FP6 contract no. RII3-CT-2004-506065), MC Fellowship scheme (FP7 contract PIEF-GA-2008-219175) and IA-ENSAR (FP7 contract 262010).
Author information
Affiliations
Oliver Lodge Laboratory, University of Liverpool, Liverpool L69 7ZE, UK
- L. P. Gaffney
- , P. A. Butler
- , M. Scheck
- , D. T. Joss
- & S. V. Rigby
Institut für Kernphysik, Technische Universität Darmstadt, Darmstadt D-64289, Germany
- M. Scheck
- , C. Bauer
- , S. Bönig
- , Th. Kröll
- & M. von Schmid
Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627-0171, USA
- A. B. Hayes
- & D. Cline
ISOLDE, CERN (Organisation Européenne pour la Recherche Nucléaire), Geneva CH-1211, Switzerland
- F. Wenander
- , T. E. Cocolios
- , J. Pakarinen
- , T. Stora
- & D. Voulot
Institut für Kernphysik, Universität zu Köln, Köln D-50937, Germany
- M. Albers
- , A. Blazhev
- , K. Moschner
- , M. Pfeiffer
- , D. Radeck
- , P. Reiter
- , M. Rudigier
- , M. Seidlitz
- , B. Siebeck
- , P. Thoele
- & N. Warr
GANIL (Grand Accélérateur National d’Ions Lourds), Caen, BP 55027, F-14076, France
- B. Bastin
Instituut voor Kern- en Stralingsfysica, KU Leuven, Leuven B-3001, Belgium
- N. Bree
- , H. De Witte
- , J. Diriken
- , M. Huyse
- , N. Kesteloot
- , K. Reynders
- , S. Sambi
- , P. Van Duppen
- & K. Wrzosek-Lipska
Department of Nuclear Physics, Lund University, LundPO Box 118, 221 00, Sweden
- J. Cederkäll
Department of Physics, University of Michigan, Ann Arbor, Michigan 48104, USA
- T. Chupp
School of Physics & Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, UK
- T. Davinson
SC•CEN (Studiecentrum voor Kernenergie - Centre d’Etude de l’énergie Nucléaire), Mol B-2400, Belgium
- J. Diriken
- & N. Kesteloot
Department of Physics, University of Jyväskylä, Jyväskylä FI-40014, Finland, and Helsinki Institute of Physics, PO Box 64, FI-00014 Helsinki, Finland
- T. Grahn
- , A. Herzan
- , J. Konki
- & J. Pakarinen
Department of Physics, University of York, York YO10 5DD, UK
- D. G. Jenkins
- & M. J. Vermeulen
Heavy Ion Laboratory, University of Warsaw, Warsaw 02-093, Poland
- M. Kowalczyk
- , P. Napiorkowski
- , K. Wrzosek-Lipska
- & M. Zielinska
Physics Division, Lawrence Livermore National Laboratory, Livermore, California 94551, USA
- E. Kwan
- & C. Y. Wu
Maier-Leibnitz-Laboratorium, Ludwig-Maximilians-Universität and Technische Universität München, Garching D-85748, Germany
- R. Lutter
Departamento de Fisica Teorica, Universidad Autónoma de Madrid, Madrid 28049, Spain
- L. M. Robledo
Physik Department E12, Technische Universität München, Garching D-85748, Germany
- K. Wimmer
DSM/IRFU/SPhN, CEA Saclay, Gif-sur-Yvette F-91191, France
- M. Zielinska
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Contributions
Instrument set-up: M.A., C.B., A.B., T.D., H.D.W., L.P.G., J.K., J.P., P.R., M. Seidlitz, B.S., M.J.V. and N.W. DAQ and on-line analysis: A.B., L.P.G., R.L. and N.W. Data analysis and interpretation: L.P.G., P.A.B., D.C., A.B.H., M. Scheck and M.Z. REX development and set-up: F.W., D.V. and J.C. Primary target: T.S. Preparation of manuscript: P.A.B., L.P.G., T.C., A.B., D.G.J., Th.K., J.P., P.R., M. Scheck, P.V.D. and N.W. Theoretical interpretation: L.M.R. All authors except L.M.R. took part in the experiments.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to P. A. Butler.
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Further reading
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1.
A new perspective on charge radii around Z = 82
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Recent advances in nuclear physics through on-line isotope separation
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. Note that the level at 938 keV in 220Rn is observed for the first time in this work.