A superdeformed rotational band has been identified in 36Ar, linked to known low-spin states, and observed to its high-spin termination at Iπ = 16+. Cranked Nilsson-Strutinsky and spherical shell model calculations assign the band to a configuration in which four pf-shell orbitals are occupied, leading to a low-spin deformation β2 ≈ 0.45. Two major shells are active for both protons and neutrons, yet the valence space remains small enough to be confronted with the shell model. This band thus provides an ideal case to study the microscopic structure of collective rotational motion

Aberg, S.

Appelbe, D.E.

Austin, R.A.E.

Baktash, C.

Ball, G.C.

Carpenter, M.P.

Caurier, E.

Clark, R.M.

Cromaz, M.

Deleplanque, M.A.

Diamond, R.M.

Fallon, P.

Furlotti, M.

Galindo-Uribarri, A.

Janssens, R.V.F.

Juodagalvis, A.

Lane, G.J.

Lee, I.Y.

Lipoglavsek, M.

Macchiavelli, A.O.

Nowacki, F.

Paul, S.D.

Poves, A.

Radford, D.C.

Ragnarsson, I.

Sarantites, D.G.

Seweryniak, D.

Stephens, F.S.

Svensson, C.E.

Tomov, V.

Vetter, K.

Ward, D.

Yu, C.H.

HAL-IN2P3

Superdeformation in the N = Z Nucleus $^{36}$Ar: experimental, Deformed Mean Field, and Spherical Shell Model Descriptions

Åberg, S.

Yu, C.-H.

Carolina Digital Repository

VOLUME 85, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 25 SEPTEMBER 2000Superdeformation in the N 5Z Nucleus 36Ar: Experimental, Deformed Mean Field,and Spherical Shell Model DescriptionsC. E. Svensson,1 A. O. Macchiavelli,1 A. Juodagalvis,2 A. Poves,3 I. Ragnarsson,2 S. Åberg,2 D. E. Appelbe,4R. A. E. Austin,4 C. Baktash,5 G. C. Ball,6 M. P. Carpenter,7 E. Caurier,8 R. M. Clark,1 M. Cromaz,1M. A. Deleplanque,1 R. M. Diamond,1 P. Fallon,1 M. Furlotti,9 A. Galindo-Uribarri,5 R. V. F. Janssens,7 G. J. Lane,1I. Y. Lee,1 M. Lipoglavsek,5 F. Nowacki,10 S. D. Paul,5 D. C. Radford,5 D. G. Sarantites,9 D. Seweryniak,7 F. S. Stephens,1V. Tomov,9 K. Vetter,1 D. Ward,1 and C. H. Yu51Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 947202Department of Mathematical Physics, Lund Institute of Technology, S-22100 Lund, Sweden3Departamento de Física Teórica, Universidad Autónoma de Madrid, E-28049 Madrid, Spain4Department of Physics and Astronomy, McMaster University, Hamilton, Canada L8S 4M15Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-63716TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, Canada V6T 2A37Argonne National Laboratory, Argonne, Illinois 604398Institut de Recherches Subatomiques, IN2P3-CNRS-Université Louis Pasteur, F-67037 Strasbourg Cedex 2, France9Chemistry Department, Washington University, St. Louis, Missouri 6313010Laboratoire de Physique Theorique, Université Louis Pasteur, F-67084 Strasbourg Cedex, France(Received 8 May 2000)A superdeformed rotational band has been identified in 36Ar, linked to known low-spin states, andobserved to its high-spin termination at Ip  161. Cranked Nilsson-Strutinsky and spherical shell modelcalculations assign the band to a configuration in which four pf-shell orbitals are occupied, leading toa low-spin deformation b2  0.45. Two major shells are active for both protons and neutrons, yet thevalence space remains small enough to be confronted with the shell model. This band thus provides anideal case to study the microscopic structure of collective rotational motion.PACS numbers: 21.10.Re, 21.60.Cs, 23.20.Lv, 27.30.+ tThe microscopic description of collective motion is afundamental challenge in quantum many-body physics. Aclassic example in the field of nuclear structure is the desireto understand, within a spherical shell model framework,the origins of nuclear deformation and collective rotation.For deformed nuclei in the first half of the sd shell, this isaccomplished in Elliott’s SU(3) model [1]. For heavier nu-clei, SU(3) symmetry is destroyed by the strong spin-orbitinteraction and the relevant coupling scheme is less clear.The microscopic degrees of freedom contributing to therotational motion can be determined by comparing experi-mental data, the intuitive results of deformed mean fieldcalculations, and rigorous shell model (SM) diagonaliza-tions. However, the contradictory requirements of havinga sufficiently large number of active particles for collec-tive rotation to develop, and a sufficiently small number topermit SM calculations, severely limit the opportunities forsuch direct comparisons. Considerable effort has recentlybeen focused on 48Cr [2–9], in large part because it is oneof a few nuclei where these criteria are satisfied. Rotorslike 48Cr are, however, the exception rather than the rule,in that particles from only a single major shell are active,whereas rotational motion in heavier nuclei quite generallyinvolves two major shells for both protons and neutrons.In this Letter, we report the observation of a superde-formed (SD) rotational band in the N  Z nucleus 36Ar.Cranked Nilsson-Strutinsky and spherical shell model cal-culations lead to a configuration assignment in which four0031-90070085(13)2693(4)$15.00particles are promoted to the pf shell. This band thusprovides an ideal opportunity to extend the microscopicdescription of collective rotation from the single-shell ro-tors like 20Ne, 24Mg, and 48Cr to a case involving thecross-shell correlations characteristic of rotational motionin heavier nuclei. The detailed spectroscopic data for the36Ar band should also provide constraints on the effec-tive sd-pf cross-shell interactions important for the onsetof deformation in the neutron-rich “island of inversion”around 32Mg [10,11].High-spin states in 36Ar were populated via the24Mg20Ne, 2a36Ar reaction, with an 80-MeV 20Nebeam provided by the ATLAS facility at Argonne Na-tional Laboratory directed onto a 440 mgcm2 24Mg foil.Gamma rays were detected with 101 HPGe detectors ofthe Gammasphere array [12], in coincidence with chargedparticles detected with the Microball [13], a 4p arrayof 95 CsI(Tl) scintillators. A total of 775 3 106 g-g-gand higher-fold coincidences were recorded during theexperiment. The Hevimet collimators were removed fromthe HPGe detectors to enable g-ray multiplicity and sum-energy measurements for each event [14], and the 36Arevaporation channel was cleanly selected by applyingenergy conservation requirements [15] to events in whichtwo alpha particles were detected.Here we focus on the identification of a superdeformedband in 36Ar. The g-ray spectrum obtained by summingcoincidence gates set on all members of this band is© 2000 The American Physical Society 2693VOLUME 85, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 25 SEPTEMBER 2000shown in Fig. 1(a), and a partial decay scheme for 36Ar ispresented in Fig. 2. Transitions denoted by diamonds inFig. 1(a) firmly link the band to known low-spin states. Forexample, Fig. 1(b) shows a spectrum of the band obtainedwith a single gate set on the 4950-keV linking transition.The spin and parity assignments in Fig. 2 are based on an-gular distribution measurements which establish stretched-E2 character for all of the in-band and high-energy linkingtransitions. Two examples are shown in Figs. 1(c) and1(d). Legendre polynomial fits to these data (solid curves)yield (a2, a4) coefficients of (0.31 6 0.02, 20.09 6 0.03)and (0.31 6 0.02, 20.08 6 0.02) for the 3352 and 4166-keV g rays and establish the 61 and 41 states of theband. The 4951 keV 21 state was known from earlier35Cl p, g36Ar studies [16]. These studies also identifieda 01 state at 4329 keV which, based on the regularrotational spacing, is presumed to be the SD bandhead.The g-ray branchings from the 41 and 61 states of theband yield in-band to decay-out BE2 ratios of 148 6 6and 86 6 4, consistent with strongly enhanced in-bandtransitions. Assuming a similar ratio of BE2’s for the21 state, a 622-keV 21 ! 01 in-band transition wouldbe expected to carry 0.3% of the decay intensity. Thepresent experiment yields an upper limit of 1.0% for thisunobserved branch.0500010000150001000 2000 3000 4000 500002004006000 60 120 1805006007000 60 120 180250300350Eγ (keV)CountsAngle (degrees)Relative Yield Eγ = 3352 keV Eγ = 4166 keV4950 keVGate36ArSuperdeformed Band11861630 1970 + 19752160169724442602282129482981 335234484166 - 511406641664950 - 5114950(a)(b)(c) (d)FIG. 1. Gamma-ray spectra obtained by (a) summing coinci-dence gates set on all transitions in the 36Ar superdeformed band(circles), and (b) setting a single gate on the 4950-keV transition.Diamonds in (a) indicate linking transitions. Angular distribu-tions measured relative to the beam axis are shown for the (c)3352-keV and (d) 4166-keV transitions.26940 1 2 3 4051015Eγ (MeV)Angular Momentum (h)48Cr36ArExperimentShell Model(s1/2d3/2)4(pf)4FIG. 2. Partial decay scheme for 36Ar showing the superde-formed band ( left). Transition and level energies are given tothe nearest keV and arrow widths are proportional to transitionintensities. The inset compares the experimental and shell model“backbending” plots for the SD band in 36Ar, as well as experi-mental values for the ground band of 48Cr [2–4].The near-perfect rotational behavior of the 36Ar bandup to I  10h̄ is illustrated by the linear increase in an-gular momentum with rotational frequency (g-ray energy)shown in the inset in Fig. 2. The data for the ground bandof 48Cr [2–4], with low-spin deformation ´2  0.25, arealso shown for comparison. We note that (i) up to thebackbend, the rotational behavior in the 36Ar band is evenbetter than that of 48Cr, and (ii) despite the considerablechange of mass, the low-spin moments of inertia are com-parable, implying a larger deformation and/or quenching ofpairing correlations for the 36Ar band. The maximum spinavailable in the s12d324 pf4 configuration in 36Ar isIp  161, and the high-energy g ray from the 161 statesuggests a band termination similar to that at I  161 inthe  f728 configuration of 48Cr [7].A highly deformed band in 36Ar, in which four pf-shellorbitals are occupied, can readily be inferred from the gapin the energy levels of the Nilsson diagram at deformation´2  0.4 for particle number N , Z  18. This configu-ration is obtained from the harmonic oscillator SD shellgap at N , Z  16, ´2  0.6 [17] by adding two protonsand two neutrons to the upsloping 20252 Nilsson or-bital, in a manner analogous to SD shell gaps in heaviernuclei obtained by adding particles to flat or upsloping or-bitals above the harmonic oscillator gaps. In this sense,we refer to the 36Ar band as superdeformed. To formalizethese ideas, we have performed configuration-dependent[18] cranked Nilsson-Strutinsky (CNS) calculations forVOLUME 85, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 25 SEPTEMBER 200036Ar. The sd-shell Nilsson parameters of Ref. [19] havebeen adopted and pairing correlations have been ne-glected. The calculated energies of favored configurationsin 36Ar are compared with the experimental data in Fig. 3.The p pf2 ≠ n pf2 configuration, labeled 2, 2 inFig. 3(b), clearly provides the best description of theobserved band. Although pairing correlations are knownto affect the moment of inertia, and their absence in thesecalculations leads to the expected discrepancy betweenthe experimental and CNS low-spin slopes seen in Fig. 3,they do not strongly modify the structure of the wavefunction [6,8]. The CNS calculations are thus expected toprovide a reliable description of quadrupole properties [7].At low spin, a prolate shape with deformation ´2  0.40(b2  0.45) is calculated, and this intrinsic shape re-mains approximately constant up to I  8h̄. At higherspins the nucleus is predicted to change shape smoothly(increasing g at roughly constant ´2  0.4), terminatingat Ip  161 in a noncollective oblate (g  60±) state.We have carried out large-scale sd-pf SM calculationsfor 36Ar with the diagonalization code ANTOINE [20] andan effective interaction based on that of Ref. [21]. The en-tire pf shell was included and, following the argumentsof Ref. [10], separate calculations were performed withn  0, 2, and 4 particles promoted to this shell. The re-sulting yrast lines are shown in Fig. 3(c). For n  0, the-202468-2024680 2 4 6 8 10 12 14 16-202468Angular Momentum (h)E - 0.08234 I(I+1) (MeV)(a) Experiment(b) Cranked Nilsson(c) Shell ModelGround BandSuperdeformed Band10+1[0,0] [1,1]--[1,1]++[2,2](d5/2s1/2d3/2)20(s1/2d3/2)6(pf)2(s1/2d3/2)4(pf)4FIG. 3. Energies of positive parity, even spin states in 36Arrelative to a rigid rotor reference from (a) experiment, (b)cranked Nilsson-Strutinsky (CNS), and (c) spherical shell modelcalculations. The CNS configurations are labeled by  p, n,where p (n) is the number of proton (neutron) pf-shell orbitalsoccupied and superscripts indicate the signatures of odd sd-shellprotons and neutrons.complete sd20 space is readily diagonalized and accu-rately reproduces the observed ground-state band, includ-ing the 61 state identified in this work. For n  2 and 4,the model space becomes prohibitively large, and for thesecalculations a closed d52 orbital was assumed. To accountfor this truncation, the effective single-particle energies in17O were adjusted to give the proper level spacings in 29Siand the cross-shell monopole interaction was modified toobtain the correct spectrum for 41Ca. As can be seen inFig. 3 and in the inset of Fig. 2, these calculations sup-port the  pf4 assignment for the 36Ar band and providea truly remarkable description of the data. Even the slightdiscrepancy for the 101 level can, at least partially, be at-tributed to an interaction between this state and the nearbyyrast 101 state at 11 902 keV. The latter is a strong can-didate for the low-lying 101 state belonging to the n  2configuration which is predicted by both the SM and CNScalculations.Given the success of both models in reproducing theexperimental data, a detailed comparison of these comple-mentary descriptions is of considerable interest. Figure 4shows the calculated occupancies of spherical j shells forthe SD band. The CNS values have been obtained ac-cording to the formalism of Ref. [22]. We note that theapproximate constancy of the subshell occupancies at lowspin validates the concept of a rotational band built on afixed intrinsic state. At higher spins, the increasingly rapidvariations in the occupancies are associated with the pre-dicted change in the intrinsic deformation from prolate tooblate. For the pf shell, the occupancy trends from the01230 4 8 12 160123Angular Momentum (h)Orbital Occupation (π + ν)(a) Cranked Nilsson(b) Shell Model: (s1/2d3/2)4(pf)4d5/2 holess1/2d3/2f7/2p3/2f5/2p1/2s1/2d3/2f7/2p3/2f5/2p1/2FIG. 4. Occupancies of spherical j shells in the 36Ar superde-formed band from (a) cranked Nilsson-Strutinsky and (b) spheri-cal shell model calculations. In the Nilsson calculations thereare also small occupancies in higher major shells.2695VOLUME 85, NUMBER 13 P H Y S I C A L R E V I E W L E T T E R S 25 SEPTEMBER 2000two approaches are very similar, indicating the dominanceof quadrupole deformation in determining the structure ofthe wave function. The full shell-model treatment of allresidual interactions leads to only a slight increase in theupper pf-shell occupancies relative to the deformation-related mixing included in the CNS calculations. It hasbeen noted [23] that with a small number of particles in thepf shell, quadrupole properties can be largely accountedfor considering only the favored Dj  2 orbitals f72 andp32. The f52 and p12 orbitals do, however, lead to asignificant decrease of the moment of inertia and their in-clusion in the SM space is essential to obtain the quantita-tive agreement with the experimental data seen in Figs. 2and 3.In the sd shell, the truncation of the SM space tos12d32 leads to different occupancy predictions in thetwo methods. As shown in Fig. 4(a), the CNS calcu-lations, which provide a full description of the j-shellmixing associated with deformation, predict a substantialadmixture of d52 holes in the 36Ar SD band. The near-perfect agreement between experiment and the SM calcu-lation, which does not include the d52 orbital, suggeststhat its influence has been largely accounted for by the 1-MeV lowering of the effective s12 energy in 17O requiredto reproduce the 29Si spectrum in the modified space. It isalso noteworthy that the quadrupole collectivity predictedby the SM calculation is in good agreement with thedeformation predicted by the CNS calculation. AssumingK  0, an approximately constant intrinsic quadrupolemoment Q0  113 e fm2 (corresponding to b2  0.45)is obtained from both the SM spectroscopic quadrupolemoments and the calculated BE2’s connecting low-spinmembers of the band. This quadrupole moment repre-sents 80% of the maximum available in the sd16 pf4space [as attained in the unrealistic SU(3) limit], andwe therefore expect that including the d52 orbital in theSM space would not increase the quadrupole moment bymore than 5%–10%. A complete study of the role ofthe d52 orbital in the microscopic structure of the 36ArSD band, perhaps through a full sd-pf diagonalizationby Monte Carlo SM techniques (cf. Ref. [24]), would,of course, be of great interest. Such a calculation wouldalso provide information regarding the mixing of configu-rations with different numbers of pf-shell excitationsnecessary for a complete understanding of the decay outof the band and the interaction of the two 101 states inFig. 2. From the deformed mean field perspective, a fulltreatment of pairing correlations would provide additionalinsight into the relative contributions of deformation andpairing to the j-shell occupancies and, conversely, the SMoccupancies could be used to test, and improve upon, thestandard pairing approximations. Experimentally, lifetimemeasurements for states in the 36Ar SD band will providestringent tests of the quadrupole properties predicted byboth models.In summary, we have identified a superdeformed ro-tational band in the N  Z nucleus 36Ar. The band is2696linked to low-spin levels and is observed to its terminationat Ip  161. Cranked Nilsson-Strutinsky and large-scaleshell model calculations lead to a configuration assignmentinvolving the occupation of four pf-shell orbitals and pre-dict a low-spin quadrupole deformation b2  0.45. Withtwo major shells active for both protons and neutrons, yeta valence space dimension small enough to be approachedfrom the shell model perspective, this band provides anideal test case for further studies of the microscopic de-scription of rotational motion in nuclei.We thank W. E. Ormand for helpful discussions. Thiswork has been partially supported by the DOE underContracts No. DE-AC03-76SF00098, No. W-31-109-ENG-38, No. DE-AC05-96OR22464, and No. DE-FG05-88ER40406, the Swedish Institute, NFR (Sweden), DGES(Spain) under Grant No. PB96-53, NSERC (Canada), andIN2P3 (France).[1] J. P. Elliott, Proc. R. Soc. London 245, 128 (1958); 245,562 (1958).[2] J. A. Cameron et al., Phys. Lett. B 387, 266 (1996).[3] S. M. Lenzi et al., Z. Phys. A 354, 117 (1996).[4] F. Brandolini et al., Nucl. Phys. A642, 387 (1998).[5] E. Caurier et al., Phys. Rev. C 50, 225 (1994).[6] E. Caurier et al., Phys. Rev. Lett. 75, 2466 (1995).[7] A. Juodagalvis and S. Åberg, Phys. Lett. B 428, 227(1998); A. Juodagalvis, I. Ragnarsson, and S. Åberg, Phys.Lett. B 477, 66 (2000).[8] A. Poves, J. Phys. G 25, 589 (1999).[9] K. Hara, Y. Sun, and T. Mizusaki, Phys. Rev. Lett. 83, 1922(1999).[10] E. K. Warburton, J. A. Becker, and B. A. Brown, Phys. Rev.C 41, 1147 (1990).[11] T. Motobayashi et al., Phys. Lett. B 346, 9 (1995);E. Caurier et al., Phys. Rev. C 58, 2033 (1998); D. J. Deanet al., Phys. Rev. C 59, 2474 (1999).[12] I.-Y. Lee, Nucl. Phys. A520, 641c (1990).[13] D. G. Sarantites et al., Nucl. Instrum. Methods Phys. Res.,Sect. A 381, 418 (1996).[14] M. Devlin et al., Nucl. Instrum. Methods Phys. Res., Sect.A 383, 506 (1996).[15] C. E. Svensson et al., Nucl. Instrum. Methods Phys. Res.,Sect. A 396, 228 (1997).[16] P. M. Johnson, M. A. Meyer, and D. Reitmann, Nucl. Phys.A218, 333 (1974).[17] R. K. Sheline, I. Ragnarsson, and S. G. Nilsson, Phys. Lett.41B, 115 (1972).[18] T. Bengtsson and I. Ragnarsson, Nucl. Phys. A436, 14(1985); A. V. Afanasjev and I. Ragnarsson, Nucl. Phys.A591, 387 (1995).[19] D. M. Headly, R. K. Sheline, and I. Ragnarsson, Phys. Rev.C 49, 222 (1994).[20] E. Caurier, computer code ANTOINE, CRN, Strasbourg,1989.[21] J. Retamosa et al., Phys. Rev. C 55, 1266 (1997).[22] R. K. Sheline et al., J. Phys. G 14, 1201 (1988).[23] A. P. Zuker et al., Phys. Rev. C 52, R1741 (1995).[24] T. Otsuka, M. Honma, and T. Mizusaki, Phys. Rev. Lett.81, 1588 (1998).

Superdeformation in the N = z Nucleus 36Ar: Experimental, deformed mean field, and spherical shell model descriptions

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Superdeformation in the N = Z Nucleus $^{36}$ Ar: experimental, Deformed Mean Field, and Spherical Shell Model Descriptions

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Superdeformation in the N = Z Nucleus 36^{36}36Ar: experimental, Deformed Mean Field, and Spherical Shell Model Descriptions

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Superdeformation in the N = Z Nucleus $^{36}$ Ar: experimental, Deformed Mean Field, and Spherical Shell Model Descriptions