Evolution of Octupole Deformation in Radium Nuclei from Coulomb Excitation of Radioactive 222 Ra and 228 Ra Beams
P.A. Butler, L.P. Gaffney, P. Spagnoletti, K. Abrahams, M. Bowry, J. Cederkäll, G. De Angelis, H. De Witte, P.E. Garrett, A. Goldkuhle, C. Henrich, A. Illana, K. Johnston, D.T. Joss, J.M. Keatings, N.A. Kelly, M. Komorowska, J. Konki, T. Kröll, M. Lozano, B.S. Nara Singh, D. O'Donnell, J. Ojala, R.D. Page, L.G. Pedersen, C. Raison, P. Reiter, J.A. Rodriguez, D. Rosiak, S. Rothe, M. Scheck, M. Seidlitz, T.M. Shneidman, B. Siebeck, J. Sinclair, J.F. Smith, M. Stryjczyk, P. Van Duppen, S. Vinals, V. Virtanen, N. Warr, K. Wrzosek-Lipska, M. Zielińska
EEvolution of Octupole Deformation in Radium Nuclei fromCoulomb Excitation of Radioactive
Ra and
Ra Beams
P.A. Butler , ∗ L.P. Gaffney , , P. Spagnoletti , K. Abrahams , M. Bowry , ,J .Cederk¨all , G. de Angelis , H. De Witte , P.E. Garrett , A. Goldkuhle ,C. Henrich , A. Illana , K. Johnston , D.T. Joss , J.M. Keatings , N.A. Kelly ,M. Komorowska , J. Konki , T. Kr¨oll , M. Lozano , B.S. Nara Singh ,D. O ' Donnell , J. Ojala , , R.D. Page , L.G. Pedersen , C. Raison , P. Reiter ,J.A. Rodriguez , D. Rosiak , S. Rothe , M. Scheck , M. Seidlitz , T.M. Shneidman ,B. Siebeck , J. Sinclair , J.F. Smith , M. Stryjczyk , P. Van Duppen ,S. Vinals , V. Virtanen , , N. Warr , K. Wrzosek-Lipska , and M. Zieli´nska University of Liverpool, Liverpool L69 7ZE, United Kingdom ISOLDE, CERN, 1211 Geneva 23, Switzerland University of the West of Scotland, Paisley PA1 2BE, United Kingdom University of the Western Cape, Private Bag X17, Bellville 7535, South Africa TRIUMF, Vancouver V6T 2A3 BC, Canada Lund University, Box 118, Lund SE-221 00, Sweden INFN Laboratori Nazionali di Legnaro, Legnaro 35020 PD, Italy KU Leuven, Leuven B-3001, Belgium University of Guelph, Guelph N1G 2W1 Ontario, Canada University of Cologne, Cologne 50937, Germany Technische Universit¨at Darmstadt, Darmstadt 64289, Germany Heavy Ion Laboratory, University of Warsaw, Warsaw PL-02-093, Poland University of Jyvaskyla, P.O. Box 35, Jyvaskyla FIN-40014, Finland Helsinki Institute of Physics, P.O. Box 64, Helsinki, FIN-00014, Finland University of Oslo, P.O. Box 1048, Oslo N-0316, Norway University of York, York YO10 5DD, United Kingdom Joint Institute for Nuclear Research, RU-141980 Dubna, Russian Federation Consejo Superior De Investigaciones Cient´ıficas, Madrid S28040, Spain IRFU CEA, Universit´e Paris-Saclay, Gif-sur-Yvette F-91191, France a r X i v : . [ nu c l - e x ] J a n Abstract
There is sparse direct experimental evidence that atomic nuclei can exhibit stable ‘pear’ shapesarising from strong octupole correlations. In order to investigate the nature of octupole collectivityin radium isotopes, electric octupole ( E
3) matrix elements have been determined for transitionsin , Ra nuclei using the method of sub-barrier, multi-step Coulomb excitation. Beams of theradioactive radium isotopes were provided by the HIE-ISOLDE facility at CERN. The observedpattern of E Ra aspear-shaped with stable octupole deformation, while
Ra behaves like an octupole vibrator.
There are many theoretical and experimental indications that atomic nuclei can exhibitreflection asymmetry in the intrinsic frame, and observation of low-lying quantum statesin many nuclei with even Z, N having total angular momentum and parity of I π = 3 − isindicative of the presence of octupole correlations (see [1] and references therein). Typically,the electric octupole ( E
3) moment for the transition to the ground state is tens of single-particle units, suggesting that the octupole instability arises from a collective effect andleads to a pear-shaped distortion of the nuclear shape. What is less clear, however, iswhether in some nuclei this distortion is stable, i.e. the nucleus assumes a permanent pearshape, or whether it is dynamic and the nucleus undergoes octupole vibrations. Evidencehas been presented that
Ra and
Ra have static octupole deformation on account of anenhancement in the E E Ra where stable beams have been used to obtain acomplete set of E Nd [6].In this Letter, results from a multistep, Coulomb-excitation experiment with radioactive , Ra beams are reported. By examining the pattern of E , Ra and
Nd, adistinction can be made between those isotopes having stable octupole deformation and thosebehaving like octupole vibrators. This observation is relevant for the search for permanentelectric dipole moments in radium atoms [7–9], that would indicate sizeable CP violationrequiring a substantial revision of the Standard Model.The radioactive isotopes
Ra (Z = 88, N = 134) and
Ra (Z = 88, N = 140) were C oun t s pe r k e V Ra X rays a K b K + fi + + fi + + fi + + fi + + fi + + fi - + fi - - fi + + fi - - fi - + fi - - fi - - fi + - fi + + fi - + fi - C oun t s pe r k e V Ra X rays a K b K + fi + + fi + + fi + + fi + + fi + + fi +
12 , + fi + + fi - + fi - + fi - + fi - + fi - + fi - + fi - - fi +2 + fi + g (4 + fi g + fi g + fi g FIG. 1. Spectra of γ rays emitted following the Coulomb excitation of Ra (upper) and
Ra(lower) using a
Sn target (blue), and Ni (red). The γ rays were corrected for Doppler shiftassuming that they are emitted from the scattered projectile. Random coincidences between Mini-ball and the silicon detector have been subtracted. The transitions that give rise to the observedfull-energy peaks are labelled by the spin and parity of the initial and final states. produced by spallation in a thick uranium carbide primary target bombarded by ≈ protons/s at 1.4 GeV from the CERN PS Booster. The ions, extracted from a tungstensurface ion source were stripped to charge states of 51 + and 53 + , respectively, for Ra and
Ra and accelerated in HIE-ISOLDE to an energy of 4.31 MeV/nucleon. The radioactivebeams, with intensities between 5 × and 2 × ions/s bombarded secondary targets of Ni and
Sn of thickness 2.1 mg/cm . Gamma rays emitted following the excitation ofthe target and projectile nuclei were detected in Miniball [10], an array of 24 high-puritygermanium detectors, each with sixfold segmentation and arranged in eight triple clusters.The scattered projectiles and target recoils were detected in a highly segmented silicondetector, distinguished by their differing dependence of energy with angle measured in thelaboratory frame of reference. Representative spectra from the Coulomb-excited , Raare shown in Fig. 1; in the spectra the γ -ray energies are corrected for Doppler shift assumingemission from the scattered projectile. The spectra were incremented when a target recoilwas detected in coincidence with γ rays within a 450-ns time window; these data werecorrected for random events. The fraction of the isobar Fr in the beam was estimatedto be about 20% by observing γ rays from the α -decay daughters at the beam dump. Bylowering the temperature of the transfer line from the ion source a nearly pure beam of Fr could be produced; apart from X-rays, no discernable structure was observed arisingfrom Coulomb excitation of the odd-odd nucleus in the particle-gated, Doppler-correctedspectrum. For the
Ra beam, the fraction of isobaric contamination was estimated to be ≈ Ra and
Ra the spectra reveal strong population of the ground-state band ofpositive-parity states, populated by multiple electric quadrupole ( E
2) Coulomb excitation,and substantial population of the octupole band of negative-parity states, populated by E γ -ray transitions detected in Miniball were measuredfor four ranges of the recoil angle of the target nucleus for each target, between 21 . ◦ and55 . ◦ for the Sn target and between 17 . ◦ and 55 . ◦ for the Ni target. The yield data werecombined with existing γ -ray branching ratios to provide input to the Coulomb-excitationanalysis code GOSIA [11–13]. The GOSIA code performs a least-squares fit to the Eλ ( λ = 1 , ,
3) matrix elements (m.e.s), which either can be treated as free parameters, can becoupled to other matrix elements, or can be fixed. Energy-level schemes that are includedin the analysis are given in [14]. A total of 114 data for
Ra were fitted to 42 variables,while for
Ra 121 data were fitted to 41 variables. The starting values of each of the freely-varied matrix elements were drawn randomly, within reasonable limits; the values obtainedfollowing the fitting procedure were found to be independent of the starting points. Examplesof fits to the experimental data can be found in the Supplemental Material, see below [14].For both nuclei the E E + → + 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 (cid:104) + || E || + (cid:105) and (cid:104) + || E || + (cid:105) m.e.s independently. The latter was therefore allowed to vary freely and the (cid:104) + || E || + (cid:105) matrix element was coupled to the (cid:104) + || E || + (cid:105) matrix element assuming the validity of therotational model; this assumption is based on the behaviour of nuclei where the lifetimesof the 2 + and 4 + states have been measured and for which the lowest transitions behavecollectively [14]. For the E (cid:104) I ± || E || I (cid:48)∓ (cid:105) , were coupled to m.e.s between lower-lying states, (cid:104) ( I − ± || E || ( I (cid:48) − ∓ (cid:105) , assuming the validity of the rotational model. E β λ [22]. E K π = 0 + and K π = 2 + bandswere also taken into account. The relative phase of Q and Q was investigated, as althoughthe overall phase of the E E E E E Q and Q having thesame sign for Ra and the opposite sign for
Ra, and these phases were adopted in thefinal fits. These values are consistent with macroscopic-microscopic calculations [23] andconstrained HFBCS calculations [24] that predict a decreasing value of Q with neutronnumber for radium isotopes, crossing zero for Ra as experimentally verified [25].Table I gives the values of E E Ra and
Ra obtainedin this work. The E E E I ( (cid:1) ) R a
R a | Q I , I -1)| (efm) FIG. 2. Absolute values of the intrinsic dipole moments, Q as a function of spin. The valuesare deduced from the measured matrix elements [14], and correspond to transitions between stateswith spin I and I − m.e.s except for those presented in Table I, which are independently determined. In theGOSIA fit the statistical errors for each fitted variable were calculated taking into accountcorrelations between all variables. Independent sets of fitted values were also obtained byvarying the constant hexadecapole moment used to calculate the E ± ±
1% ,the distance between the target and the particle detector by ± . E Ra the value of the intrinsic quadrupole moment, Q , derived from themeasured value of (cid:104) + || E || + (cid:105) , 770 ±
40 efm , agrees with the values determined fromthe 2 + lifetime, 775 ±
14 efm and the 4 + lifetime, 780 ± , as reported in Ref. [16].For Ra, the value is 590 ±
30 efm , significantly smaller than the value derived from themeasured lifetime of the 2 + state, 673 ±
13 efm [26]. It is noted that the value of Q for Ra I ( (cid:1) ) R a ( I , I - 2 ) R a ( I , I ) R a ( I , I - 2 ) R a ( I , I ) Q I , I' ) (efm2) FIG. 3. Values of the intrinsic quadrupole moments, Q plotted as a function of spin. Thevalues are deduced from the measured matrix elements given in Table I. The values correspond totransitions between states with spin I and I −
2; in some cases they are also derived from diagonalmatrix elements. The solid horizontal lines correspond to the values of Q obtained assuming thatthe matrix elements are related by the rotational model. extrapolated from the 2 + lifetime for Ra on the basis of B( E
2; 0 + → + ) systematics [27],is 593 ±
11 efm , in good agreement with the current measurement. Fitted values of Q and Q assuming that the Eλ matrix elements and Q λ are related by the rotational model arealso given in Table I. The values for λ = 3 indicate that the octupole collectivity in Ra issignificantly lower than for
Ra.The values of Q and Q for all the measured matrix elements are shown in Figs. 2 and3, respectively. The nearly constant values of Q as a function of spin for transitions inboth positive- and negative-parity bands is consistent with stable quadrupole deformation.Smaller values of Q , although with large uncertainty, were determined from the (cid:104) + || E || + (cid:105) matrix element for both nuclei. Such behaviour was also observed in Ra, interpreted asarising from deviations from axial symmetry [3]. The values of the intrinsic electric octupole
TABLE I. Values of E E Q λ , are derived from each matrix element (m.e.) using (cid:104) I i ||M ( Eλ ) || I f (cid:105) = (cid:112) (2 I i + 1) (cid:112) (2 λ + 1) / π ( I i λ | I f Q λ . The uncertainties include the 1 σ statistical error fromthe fit ( χ + 1 type) and the systematic contributions. The E (cid:104) + || E || + (cid:105) and (cid:104) + || E || + (cid:105) m.e.s are coupled. Values of Q λ fitted assuming that the m.e.s are related by the rotational model are also given. Ra Ra (cid:104) I || Eλ || I (cid:48) (cid:105) m.e. Q λ m.e. Q λ (eb λ/ ) (efm λ ) (eb λ/ ) (efm λ ) (cid:104) + || E || + (cid:105) − . ± . ± − . ± . ± (cid:104) + || E || + (cid:105) . ± .
15 590 ±
30 3 . ± .
19 770 ± (cid:104) + || E || + (cid:105) − . ± . ± (cid:104) + || E || + (cid:105) . ± .
18 559 ±
28 5 . ± .
26 800 ± (cid:104) + || E || + (cid:105) . ± .
23 560 ±
30 5 . ± .
29 790 ± (cid:104) + || E || + (cid:105) . ± . ±
60 7 . ± . ± (cid:104) + || E || + (cid:105) . +0 . − . +60 − (cid:104) − || E || − (cid:105) . ± .
22 560 ±
50 3 . ± . ± (cid:104) − || E || − (cid:105) . ± . ±
70 3 . +0 . − . +70 − (cid:104) − || E || − (cid:105) . ± . ±
60 4 . ± . ± (cid:104) − || E || − (cid:105) . ± . ±
120 5 . ± . ± Q (rotational model) 578 ±
18 798 ± (cid:104) + || E || − (cid:105) . ± .
09 3030 ±
240 0 . ± .
15 2300 ± (cid:104) + || E || − (cid:105) . ± .
24 2000 ±
600 1 . ± .
23* 3200 ± (cid:104) + || E || − (cid:105) − . ± . ± − . +0 . − . * 150 +360 − (cid:104) + || E || − (cid:105) . ± .
20 3100 ±
400 1 . ± .
23* 3000 ± (cid:104) + || E || − (cid:105) − . ± .
5* 4400 ± . +0 . − . * − +2300 − (cid:104) + || E || − (cid:105) . +0 . − . * 5500 +1300 − (cid:104) + || E || − (cid:105) − . ± .
0* 3200 ± (cid:104) + || E || − (cid:105) . +0 . − . * 4600 +500 − Q (rotational model) 3120 ±
190 2230 ± moment Q for transitions in Ra and
Ra are shown in Fig. 4. In the figure, the values I - , ( I + 1 ) + I + , ( I + 1 ) - I + , ( I + 3 ) - Q I ( (cid:1) ) I - , ( I + 3 ) + 2 2 2 R a
R a
R a
R a
N d
FIG. 4. Values of the intrinsic octupole moments, Q plotted as a function of spin. The valuesare deduced from the measured matrix elements given in Table I. Here the values of Q are shownseparately for transitions connecting I + → ( I + 1) − , I + → ( I + 3) − , I − → ( I + 1) + and I − → ( I + 3) + . The upper dashed line is the average value of Q (0 + , − ) for the radium isotopes. To aidcomparison the values of Q for Nd have been multiplied by 1.78. of Q are shown separately for transitions I + → ( I + 1) − , I + → ( I + 3) − , I − → ( I + 1) + and I − → ( I + 3) + , and are compared with values determined for the same transitions in , Ra [2, 3] and
Nd [6]. The values for
Nd are multiplied by a factor so that the valueof Q deduced from (cid:104) + || E || − (cid:105) is the same as the average value for the radium isotopes.It is observed that the values of Q for all transitions in , , Ra are approximatelyconstant, consistent with the picture of a rotating pear shape. In contrast, the values of Q corresponding to the 2 + → − and 1 − → + transitions in Ra are close to zero, asobserved for
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 E Ra (and
Nd) compared tothe lighter radium isotopes is also present in the behaviour of their energy levels, as shown0 (cid:1) w ( M e V ) D ix ( (cid:1) ) R a
R a
R a
R a
N d
FIG. 5. The difference in aligned angular momentum, ∆ i x = i − x − i + x , plotted as a function ofrotational frequency ω . The upper dashed line corresponds to the vibrational limit, ∆ i x = 3¯ h in Fig. 5. Here ∆ i x , the difference in aligned angular momentum between negative- andpositive-parity states at the same rotational frequency ω , is plotted as a function of ¯ hω for thefive nuclei. The behaviour of ∆ i x can reveal information regarding the nature of the octupolecorrelations [29, 30]. For Nd, the value of ∆ i x ∼ h for all values of rotational frequency,and for Ra it approaches 3¯ h when ¯ hω → .
15 MeV. This behaviour is expected foroctupole vibrators, where the octupole phonon aligns to the rotation axis. It is conjecturedhere that the observation of near-zero values of Q for some transitions in Ra (and
Nd)is consistent with the octupole-vibrator description. The interpretation of the behaviour ofenergy levels for , , Ra in terms of rotating pear shapes is less obvious as it is dominatedby pairing effects near the ground state; other interpretations of this behaviour, e.g. thecondensation of rotational-aligned octupole phonons [31], do not require the nucleus to havea permanent octupole distortion. On the other hand highly-collective E E Q provide compelling evidence that Ra together with , Ra havestable octupole deformation. This confirms theoretical predictions, e.g. [22, 32, 33], thatthe boundary of octupole deformation lies at Z ≈
88 and at N ≈ E , Ra are a consequence of the stability of the octupole shape for eachnucleus. Any model of quadrupole-octupole coupling that describes this behaviour shouldbe capable of calculating values of Q for different E − → + transition, as has been performed for Ra [35].We are grateful to Doug Cline and the late Tomek Czosnyka who led the developmentof the Coulomb excitation analysis technique used in this work, and to Niels Bidault,Eleftherios Fadakis, Erwin Siesling, and Fredrick Wenander who assisted with the prepa-ration of the radioactive beams. The support of the ISOLDE Collaboration and technicalteams is acknowledged. This work was supported by the following Research Councils andGrants: Science and Technology Facilities Council (UK) Grants No. ST/P004598/1, No.ST/L005808/1, No. ST/R004056/1; Federal Ministry of Education and Research (Germany)Grants No. 05P18RDCIA, No. 05P15PKCIA, and No. 05P18PKCIA and the “Verbund-projekt 05P2018”; National Science Centre (Poland) Grant No. 2015/18/M/ST2/00523;European Union’s Horizon 2020 Framework research and innovation programme 654002(ENSAR2); Marie Sk(cid:32)lodowska-Curie COFUND Grant (EU-CERN) 665779; Research Foun-dation Flanders and IAP Belgian Science Policy Office BriX network P7/12 (Belgium);GOA/2015/010 (BOF KU Leuven); RFBR (Russia) Grant No. 17-52-12015; the Academyof Finland (Finland) Grant No. 307685. ∗ [email protected][1] P.A. Butler and W. Nazarewicz, Rev. Mod. Phys. , 349 (1996).[2] L.P. Gaffney et al. , Nature (London) , 199 (2013).[3] H.J. Wollersheim et al. , Nucl. Phys. A , 261 (1993).[4] B. Bucher et al. , Phys. Rev. Lett. , 112503 (2016).[5] B. Bucher et al. , Phys. Rev. Lett. , 152504 (2017).[6] R.W. Ibbotson et al. , Nucl. Phys. A , 213 (1997).[7] N. Auerbach, V.V. Flambaum and V. Spevak, Phys. Rev. Lett. , 4316 (1996).[8] J. Dobaczewski, J. Engel, M. Kortelainen and P. Becker, Phys. Rev. Lett. , 232501 (2018).[9] M. Bishof et al. , Phys. Rev. C , 025501 (2016). [10] N. Warr et al. , Eur.Phys. J. A , 40 (2013).[11] T. Czosnyka, D. Cline, and C.Y. Wu, Bull. Am. Phys. Soc. , 745 (1983).[12] D. Cline, Nucl. Phys. A , 615c (1993).[13] M. Zieli´nska et al. , Eur. Phys. J. A. , 99 (2016).[14] See Supplemental Material (below) for details of the level-schemes, representative fitted yields, E E , 2851 (2011).[16] K. Abusaleem, Nuclear Data Sheets , 163 (2014).[17] T. Kib´edi, T.W. Burrows, M.B. Trzhaskovskaya, P.M. Davidson, and C.W. Nestor Jr., Nucl.Instrum. Methods Phys. Res. Sect. A , 202 (2008).[18] Evaluated Nuclear Structure Data File Search and Retrieval , 047302 (2004).[20] G. Thiamova, D.J. Rowe and J.L. Wood, Nucl. Phys. A 780 , 112 (2006).[21] B. Say˘gi et al. , Phys. Rev. C , 021301(R) (2017).[22] W. Nazarewicz et al. , Nucl. Phys. A429 , 269 (1984).[23] P.A. Butler and W. Nazarewicz, Nucl. Phys.
A533 , 249 (1991).[24] J.L. Egido and L.M. Robledo, Nucl. Phys.
A494 , 85 (1989).[25] R.J. Poynter et al. , Phys. Lett. B , 447 (1989).[26] R.E. Bell, S. Bjørnholm, and J.C. Severiens, Mat. Fys. Medd. Dan. Vid. Selsk. , no. 12(1960).[27] B. Pritychenko, M. Birch, and B. Singh, Nucl. Phys. A , 73 (2017).[28] K. Neerg˚ard and P. Vogel, Nucl. Phys. A , 33 (1970); A , 217 (1970).[29] J.F.C. Cocks et al. , Phys. Rev Lett. , 2920 (1997).[30] P.A. Butler, J. Phys. G , 073002 (2016).[31] S. Frauendorf, Phys. Rev. C , 021304 (2008).[32] L.M. Robledo and P.A. Butler, Phys. Rev. C , 051302 (R) (2013).[33] S.E. Agbemava, A.V. Afanasjev and P. Ring, Phys. Rev. C , 044304 (2016).[34] P.A. Butler et al. , Nat. Commun. , 2473 (2019).[35] S.Y. Xia, H. Tao, Y. Lu, Z.P. Li, T. Nik˘si´c, and D. Vretenar, Phys. Rev. C , 054303 (2017). SUPPLEMENTAL MATERIAL
Fig. 6 shows the partial level-schemes for , Ra used in the Coulomb-excitation anal-ysis. + ( ( ( ( ( ( Ra Ra + (4 + )(6 + )(8 + )1 + + + + + + + (2 + )(3 + )(4 + ) 0 + + + + + + + + + + + (4 + )(5 + )(6 + )(7 + )(8 + ) 4 + (6 + )2 + + (8 + )64141207263309348379846782835 381418451 474410474 297(762) ( + ( ( ( ( ( FIG. 1. This is the caption of a two-column width figure.
FIG. 6. Partial level-schemes for , Ra showing the excited states used in the Coulomb-excitation analysis. The dashed levels, also included in the fitting procedure, have either beententatively labelled with spin and parity or have been artificially constructed for this purpose.Arrows indicate γ -ray transitions that have been observed in the experiments described here; allenergies are in keV. In Ra no transitions to the higher lying collective bands were observed. Thelevel scheme data have been taken from [15, 16].
S n 4 + - 2 + + - 4 + - - 2 + - - 4 + f i t t e d v a l u e N i 4 + - 2 + + - 4 + - - 2 + - - 4 + f i t t e d v a l u e yield c . m . s c a t t e r i n g a n g l e ( d e g ) R a
S n 4 + - 2 + + - 4 + - - 0 + , 3 - - 2 + - - 4 + f i t t e d v a l u e N i 4 + - 2 + + - 4 + - - 0 + , 3 - - 2 + - - 4 + f i t t e d v a l u e yield c . m . s c a t t e r i n g a n g l e ( d e g ) R a
FIG. 7. Comparison of the experimental γ -ray yields and uncertainties with those calculated withGOSIA for selected transitions, based on the set of matrix elements resulting in the best overallagreement with all the available experimental data, including previously measured branching ratios.GOSIA takes into account internal conversion using the BrICC database [17]. Q to Q deduced from the transition strengths B( E
2; 4 + → + )and B( E
2; 2 + → + ), respectively, assuming the validity of the rotational model, for nucleiwith A >
130 where the lifetimes of both 2 + and 4 + states have been measured. Onlynuclei where B ( E
2; 2 + → + ) >
70 Wu and B ( E
2; 4 + → + ) >
70 Wu are included inthis compilation. The restriction to nuclei having collective low-energy transitions doesnot reveal departures from rotational behaviour, as observed in other studies, e.g. [19–21],and reenforces the assumption made in the fitting procedure that the (cid:104) + || E || + (cid:105) matrixelement can be coupled to the (cid:104) + || E || + (cid:105) matrix element assuming the validity of therotational model. Q Q A FIG. 8. Ratio of Q to Q deduced from the transition strengths B( E
2; 4 + → + ) andB( E
2; 2 + → + ), respectively, assuming the validity of the rotational model. This has been deter-mined for nuclei where the lifetimes of both 2 + and 4 + states have been measured. Only nucleiwhere B ( E
2; 2 + → + ) >
70 Wu and B ( E
2; 4 + → + ) >
70 Wu are included in this compilation.The lifetime data have been taken from [18]. E Ra and
Ra obtained in this work.
TABLE II. Values of E Q , are derived from each matrix element (m.e.) using (cid:104) I i ||M (E1) || I f (cid:105) = (cid:112) (2 I i + 1) (cid:112) / π ( I i | I f Q . The signs of the matrix elements are given by the rotationalmodel, on the assumption that Q has the same sign as Q for Ra and the opposite sign for
Ra, see the text of the main paper. The uncertainties include the 1 σ statistical error from thefit ( χ + 1 type) and the systematic contributions. Ra Ra (cid:104) I || E || I (cid:48) (cid:105) m.e. Q m.e. Q (eb / ) (efm) (eb / ) (efm) (cid:104) + || E || − (cid:105) . ± . . ± . − . +0 . − . − . +0 . − . (cid:104) + || E || − (cid:105) − . +0 . − . . +0 . − . . +0 . − . − . +0 . − . (cid:104) + || E || − (cid:105) . ± .
005 0 . ± . − . ± . − . ± . (cid:104) + || E || − (cid:105) − . ± .
026 0 . ± .
26 0 . +0 . − . − . +0 . − . (cid:104) + || E || − (cid:105) . +0 . − . . +0 . − . − . ± . − . ± . (cid:104) − || E || + (cid:105) . ± . . ± . − . +0 . − . − . +0 . − . (cid:104) + || E || − (cid:105) . ± . . ± . (cid:104) − || E || + (cid:105) . ± . . ± . − . +0 . − . − . +0 . − . (cid:104) + || E || − (cid:105) . ± .
004 0 . ± . − . +0 . − . − . +0 . − . (cid:104) − || E || + (cid:105) . ± .
005 0 . ± . (cid:104) + || E | − (cid:105) . ± .
006 0 . ± . (cid:104) − || E || + (cid:105) . ± .
004 0 . ± ..