Electromagnetic transition probabilities have been measured for the intraband and interband transitions in the two sequences in the nucleus Nd135 that were previously identified as a composite chiral pair of rotational bands. The chiral character of the bands is affirmed and it is shown that their behavior is associated with a transition from a vibrational into a static chiral regime

Almehed, D.

Carpenter, M.P.

Frauendorf, S.

Garg, U.

Ghugre, S.S.

Gros, S.

Hecht, A.

Janssens, R.V.F.

Kondev, F.G.

Lauritsen, T.

Li, T.

Mukhopadhyay, S.

Rao, P.V.M.

Seweryniak, D.

Wang, X.

Zhu, S.

Carolina Digital Repository

From Chiral Vibration to Static Chirality in 135NdS. Mukhopadhyay,1,2 D. Almehed,1 U. Garg,1 S. Frauendorf,1 T. Li,1 P. V. Madhusudhana Rao,1 X. Wang,1,3 S. S. Ghugre,2M. P. Carpenter,3 S. Gros,3 A. Hecht,3,4 R. V. F. Janssens,3 F. G. Kondev,5 T. Lauritsen,3 D. Seweryniak,3 and S. Zhu31Physics Department, University of Notre Dame, Notre Dame, Indiana 46556, USA2UGC-DAE Consortium for Scientific Research, Kolkata Centre, Kolkata 700098, India3Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA4Department of Chemistry and Biochemistry, University of Maryland, College Park, Maryland 20742, USA5Nuclear Engineering Division, Argonne National Laboratory, Argonne, Illinois 60439, USA(Received 26 June 2007; published 23 October 2007)Electromagnetic transition probabilities have been measured for the intraband and interband transitionsin the two sequences in the nucleus 135Nd that were previously identified as a composite chiral pair ofrotational bands. The chiral character of the bands is affirmed and it is shown that their behavior isassociated with a transition from a vibrational into a static chiral regime.DOI: 10.1103/PhysRevLett.99.172501 PACS numbers: 21.10.Tg, 11.30.Rd, 21.60.Ev, 27.60.+jChirality is a well-known phenomenon in chemistry andbiology as a geometric property of many molecules, inparticular of complex biomolecules. Particle physics isanother domain of chirality, where it describes a kinemati-cal feature of massless particles. Both in chemistry andparticle physics, space inversion changes left-handed intoright-handed systems. Nuclei have been considered asachiral, because their shapes are, generally, too simple.However, it was pointed out some time ago that a triaxialnucleus becomes chiral if it rotates about an axis that liesoutside the three planes spanned by the principal axes of itstriaxial ellipsoidal shape [1,2]. The short, intermediate, andlong axes form a screw with respect to the angular mo-mentum vector. In contrast with molecules and masslessparticles, space inversion leaves nuclear chirality un-changed. The left-handed configuration is converted intothe right-handed one by the time reversal operation, whichchanges the sign of all linear and angular momenta. Sinceboth chiral structures have the same energy and are relatedby time reversal, one expects to observe two identicalbands of the same parity. A number of pairs of bandshave been identified in nuclei in the A 130 and A100 regions of the nuclear chart and have been suggestedas candidates for chiral partners [3–11]. A small observedenergy difference between the states of the same spin, I, inthe chiral partners indicates rapid conversion between theleft-handed and right-handed configurations (chiral vibra-tion). With decreasing energy splitting between the partnerbands, the left-right mode changes from soft chiral vibra-tion to tunneling between well-established chiral configu-rations (static chirality).In addition to close level energies, the two chiral part-ners should deexcite in a very similar way via electromag-netic radiation. Thus, in this context, a measurement of E2transition probabilities is an essential probe of nuclearchirality. Only two lifetime measurements have been pub-lished so far. For 128Cs [12], it was established that theelectromagnetic transition probabilities are similar, but theexcitation energies of the two partner bands never ap-proach each other. In 134Pr, the two bands come very closearound spin 14; however, the intraband BE2 values werefound to differ by about a factor of 2 [13,14]. The latterobservation was interpreted as suggesting that the changein orientation of the angular momentum vector must beaccompanied by a change in shape [13], or even that theresults rendered the chiral interpretation itself doubtful[14].The study of magnetic rotation [1] suggests that chiralityis better developed if the nonplanar geometry is generatedby three excited quasiparticles (instead of two, as in thecase of 134Pr and 128Cs). Such a case was realized in thenucleus 135Nd, where two partner bands consistent withchirality have been observed [8]. This expectation is nowborne out by the measurements and calculations presentedbelow as the electromagnetic transition rates in the chiralbands of 135Nd were found to be nearly identical. For thefirst time, the splitting between the chiral partners is calcu-lated in a microscopic way by extending the tilted-axis-cranking model (TAC) by the random phase approximation(RPA). The observed transition probabilities and energiesare in very good agreement with the calculations, whichindicate, furthermore, that a transition from chiral vibra-tion to static chirality occurs with increasing spin.The experiment was carried out at the ATLAS accelera-tor facility at the Argonne National Laboratory, and em-ployed a 175-MeV 40Ar beam to populate high-spin statesof 135Nd with the 100Mo40Ar; 5n reaction. The target wasa 1:14 mg=cm2-thick, isotopically enriched, 100Mo foilbacked by a 17:9 mg=cm2-thick layer of Pb in order toslow down and stop the recoiling nuclei. A total of about2:5 109 fivefold and higher-fold coincidence events wereaccumulated using the Gammasphere array [15]. The levelscheme of 135Nd is already well-established [16,17] and thechiral bands were presented in Ref. [8]; in the remainder ofthis Letter we refer to the partial level scheme presentedtherein.PRL 99, 172501 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending26 OCTOBER 20070031-9007=07=99(17)=172501(4) 172501-1 © 2007 The American Physical SocietyFor the analysis of these thick-target data with theDoppler-shift attenuation method (DSAM), the power ofthe BLUE database approach [18] has been employed toefficiently sort the data angle-by-angle. After building thedatabase for each fold, and performing appropriate back-ground subtraction [19], double-gated coincidence spectrawere obtained for each ring (angle) of Gammasphere forfurther analysis of the relevant Doppler shifts. In the caseswhere the gating transition was not fully stopped, the gatewas made sufficiently wide to ensure that the full peakshape was included in the gate; this eliminated a possiblebias in the analysis that could be introduced by omittingpart of a peak’s line shape, thereby favoring a specific timecomponent of the gating transition.Lifetimes of states in both chiral bands were extractedusing the LINESHAPE analysis codes of Wells and Johnson[20]. In each case, a total of 5000 Monte Carlo simulationsof the velocity history of the recoiling nuclei traversing thetarget and the backing material were generated in timesteps of 0.002 ps. Electronic stopping powers were calcu-lated with the code SRIM [21]. More details about the fittingprocedures can be found in Refs. [22,23].The lifetimes of the levels with spins, I, from 29=2 to43=2 in Band A, and from 31=2 to 39=2 in Band B,have been deduced in this work. Representative examplesof DSAM fits are displayed in Fig. 1, and the extractedlifetimes are listed in Table I. Uncertainties in the lifetimeswere determined from the behavior of the 2-fit in thevicinity of the minimum [22–24].Reduced transition probabilities, BM1 and BE2,were derived from the resulting lifetimes and are presentedin Table I and in Figs. 2 and 3. Within the experimentaluncertainties, the BE2 and BM1 values for the intra-band transitions of the two bands are essentially the same,pointing to their identical nature. In addition, the BM1values of the intraband transitions exhibit a characteristicstaggering with increasing spin. A similar staggering, butopposite in phase, is observed in the BM1 values of theinterband transitions. All these results are fully compatiblewith a pure chiral interpretation of these bands [2,25].We have performed new calculations for the twin bandsin 135Nd in the framework of TAC complemented by RPA.As proposed in Ref. [8], these bands are built on thefh211=2; h11=2g configuration. TAC is a microscopicmean field method that has been shown to describe theenergy and the inband transition rates of the lower of thetwo chiral partner bands very well (see, e.g.,Refs. [5,11,26]). However, to describe the energy splittingbetween the two partner bands, one needs to go beyond themean field approach. This has previously been done only ineither the two-particle-plus-rotor model [2,27] or the core-quasiparticle-coupling model [28]. But, these models can-not be applied to the three-quasiparticle configuration of135Nd. Moreover, both these approaches assume that themoments of inertia are of the irrotational-flow type, whichis not consistent with microscopic cranking calculations[13]. RPA, as a microscopic approach, does not have theserestrictions since it describes the excitations of the equi-librium mean field as harmonic vibrations.We use the self-consistent TAC with the QQ-force and aconstant pair gap (named PQTAC in Ref. [29]) in twomajor N shells (4 and 5): H0  h0 X;NN2QN QN   P  P  ~!  ~J(1)where h0 is the spherical Woods-Saxon energy [26] and478 483 488 493 498 503 508Energy (keV)0200400600Counts0200400600800Counts430 435 440 445 450Energy (keV)01002003004000100200300400 494-keV transition (43/2- ---> 41/2-,                   Band A)θ=500        494-keV transitionθ=1300 440-keV transition  (39/2- ---> 37/2-,                   Band B)θ=69.80   440-keV transitionθ=110.20FIG. 1 (color online). Line shape fits for representative transitions in the chiral bands of 135Nd. Left: the 494-keVtransition in Band A. Right: the 440-keV transition in Band B.TABLE I. Derived lifetimes and electromagnetic transitionprobabilities in the chiral bands of 135Nd.Spin I 	@ Lifetime [ps] BM1 [2n] BE2 [e2b2]Band A292 1.00(5) 3.2(2) 0.32(2)312 0.75(8) 2.5(3) 0.32(3)332 0.44(2) 2.2(2) 0.32(3)352 0.28(2) 2.4(3) 0.32(3)372 0.23(2) 1.7(3) 0.32(4)392 0.22(2) 2.1(3) 0.13(3)412 0.18(2) 2.1(3) 0.19(3)432 0.16(2) 2.0(3) 0.21(4)Band B312 1.46(2) 2.7(3) 0.28(3)332 0.87(6) 2.1(2) 0.28(3)352 0.64(5) 2.2(2) 0.28(4)372 0.48(3) 1.7(2) 0.29(4)392 0.24(4) 1.9(3) 0.11(3)PRL 99, 172501 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending26 OCTOBER 2007172501-2QN are the five quadrupole operators for eachN shell withan N-dependent force strength [30]: N  0NL  BN  B2ZNA1=3: (2)Here, NL  4 and the parameters 0 and B 0:5 arechosen to reproduce the Strutinsky potential energy sur-face. The rotational frequency in the cranking term ~!  ~J isdefined as !1  ! sin# cos’, !2  ! sin# sin’, and!3  ! cos#, where # and ’ are the two tilt anglesdefined in Ref. [26]. We use a proton (neutron) pairinggap n;p that is 50% (80%) of the odd-even mass differ-ence. Since the proton configuration corresponds to aparticle-hole excitation of the even-even system, we haveused a smaller pairing gap than that of an odd-neutronsystem.The RPA calculates the harmonic excitations around themean field minimum. This chiral vibrational regime ap-plies well below the transition point to chiral rotation,where the RPA solution goes to zero energy and the tiltedmean field gets a nonzero ’ value. We solved the RPAequations: 	HRPA; O   ERPAO (3)where HRPA is the Hamiltonian in RPA order [31] derivedfrom the corresponding self-consistent mean field wavefunction, and O are the RPA eigenmode operators.HRPA describes the orientation, shape, and two quasipar-ticle degrees of freedom in the harmonic approximation.The RPA phonon energy gives the energy splitting betweenthe zero-phonon lower band and the one-phonon excitedband, and the RPA wave functions give the interbandtransition rates. Since RPA describes harmonic vibrationsaround the equilibrium, it does not contribute to the ex-pectation values of the operators; i.e., the intraband tran-sition rates remain the same as those given by the TACmodel. Details of the RPA calculations will be provided ina future publication [32].The TAC results reproduce the rotational energies of thebands rather well, as can be seen in Fig. 4, and the one-phonon energy matches the energy splitting between thetwo bands.The angular momentum where the RPA energy goes tozero and static chirality emerges in the TAC calculations isone unit higher than the point at which the two bands comeclosest in the data. This coming together of the two bandshas been interpreted as the onset of chiral rotation [8]. Byanalyzing the RPA wave functions, we can determine thatthe lowest phonon is dominated by fluctuations in theorientation of the shape relative to the angular momentum13 14 15 16 17 18 19 20 21 22 23Spin I [h]01234B(M1) [µ n2 ]Band ABand BBand B ---> Band ATAC (Band A and Band B)RPA (Band B ---> Band A)FIG. 2 (color online). Evolution of the BM1 transition rateswith spin for the two chiral partner bands in 135Nd. A compari-son with the calculations described in the text is shown as well.13 14 15 16 17 18 19 20 21 22 23Spin I [h]-2-10E(I)-E0-0.01*I(I+1) [MeV]TAC (Band A)TAC+RPA (Band B)Band ABand BFIG. 4 (color online). Comparison, as a function of spin, of themeasured and calculated energies, relative to the band head, E0,of the chiral bands, with a rotor reference subtracted.13 14 15 16 17 18 19 20 21 22 23Spin I [h]00.10.20.30.40.5B(E2) [e2b2]Band ABand BBand B ---> Band ATAC (Band A and Band B)RPA (Band B ---> Band A)FIG. 3 (color online). Variation of the BE2 transition rateswith spin for the two chiral partner bands in 135Nd, together withthe results of the calculations described in the text.PRL 99, 172501 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending26 OCTOBER 2007172501-3vector, with a small admixture of  vibration. Details ofthis analysis also will be published in Ref. [32].The calculated intraband and interband transition prob-abilities are also displayed in Figs. 2 and 3, respectively.There is a good agreement with the data. Below the criticalfrequency, the TAC calculations give ’  0 and a slowincrease of #, resulting in a slowly decreasing intrabandBE2. After the critical frequency, the rapid increase of ’causes the steep rise in the BE2. The calculated interbandBE2’s are somewhat smaller than the observed values butexhibit the correct functional form. In accordance with theexperiment, the calculated interband BM1 rates are muchsmaller than the intraband BM1’s; so the observed differ-ent behavior with spin is not of much consequence. Thus,the calculations reproduce the data rather well, even closeto the point of transition to static chirality where theTAC RPA approach breaks down.The nucleus 135Nd (3 quasiparticles) differs from 134Pr(2 quasiparticles) by the additional h11=2 quasiproton. TheBE2 values within the two bands are about the same, aslong as the chiral vibration remains near harmonic. As willbe shown in a forthcoming paper [32], they may differsubstantially near the instability point, where strong anhar-monicities make the average tilt-angle, #, different in thetwo bands. The resulting difference in the BE2 values ismost pronounced near #  45, where the BE2’s go tozero. As compared to 134Pr, the longer proton angularmomentum in 135Nd delays the instability; i.e., the chiralvibration is near harmonic for a longer spin range. Theadditional quasiproton also leads to an increase in # from50 to 70. These two effects explain why the BE2values for the twin bands are different in 134Pr, but almostthe same in 135Nd.In summary, we have measured electromagnetic transi-tion probabilities for intraband and interband transitions inthe two bands in 135Nd that were previously identified as acomposite chiral pair. The intraband transition probabil-ities in the two bands are nearly identical, establishingthem as chiral partners. A microscopic calculation basedon the combination of TAC with RPA reproduces allexperimental observables quite well, substantiating thetheoretical interpretation: At the bottom of the bands, theangular momentum vector oscillates perpendicular to theplane spanned by the long and short axes of the triaxialnuclear shape (chiral vibration). These oscillations slowdown with increasing angular momentum, resulting in adecreasing energy splitting between the bands and an in-crease in the interband BE2 values. The vibration, then,becomes strongly anharmonic, changing into tunnelingbetween well-established left- and right-handed configura-tions (chiral rotation), the best realization of chirality innuclei so far.One of the authors (S. M.) expresses his gratitude toDr. C. J. Chiara for several helpful discussions on theLINESHAPE codes. This work has been supported in partby the U.S. National Science Foundation (GrantsNo. PHY04-57120 and No. INT-0111536); by theDepartment of Science and Technology, Government ofIndia (Grant No. DST-NSF/RPO-017/98); by the U.S.Department of Energy, Office of Nuclear Physics, underContracts No. DE-AC02-06CH11357 and No. DE-FG02-95ER40934; and by the University of Notre Dame and theANL-UND Nuclear Theory Initiative.[1] S. Frauendorf, Rev. Mod. Phys. 73, 463 (2001).[2] S. Frauendorf and J. Meng, Nucl. Phys. A 617, 131(1997).[3] C. M. Petrache et al., Nucl. Phys. A 597, 106 (1996).[4] K. Starosta et al., Phys. Rev. Lett. 86, 971 (2001).[5] A. A. Hecht et al., Phys. Rev. C 63, 051302 (2001).[6] T. Koike et al., Phys. Rev. C 63, 061304(R) (2001).[7] D. J. Hartley et al., Phys. Rev. C 64, 031304(R) (2001).[8] S. Zhu et al., Phys. Rev. Lett. 91, 132501 (2003).[9] C. Vaman et al., Phys. Rev. Lett. 92, 032501 (2004).[10] P. Joshi et al., Phys. Lett. B 595, 135 (2004).[11] J. Timar et al., Phys. 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Dönau, Phys. Rev.Lett. 84, 5732 (2000).[27] R. A. Bark et al., Nucl. Phys. A 691, 577 (2001).[28] K. Starosta et al., Phys. Rev. C 65, 044328 (2002).[29] S. Frauendorf, Nucl. Phys. A 677, 115 (2000).[30] M. Baranger and K. Kumar, Nucl. Phys. A 110, 490(1968).[31] P. Ring and P. Schuck, The Nuclear Many Body Problem(Springer-Verlag, Berlin, 2004).[32] D. Almehed and S. Frauendorf (to be published).PRL 99, 172501 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending26 OCTOBER 2007172501-4

From chiral vibration to static chirality in Nd135

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From chiral vibration to static chirality in Nd135

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