Effects of high order deformation on superheavy high-K isomers H.L. Liu,1 F.R. Xu,2 P.M. Walker,3 and C.A. Bertulani1 1Department of Physics and Astronomy, Texas A&M University-Commerce, Commerce, Texas 75429-3011, USA 2School of Physics, Peking University, Beijing 100871, China 3Department of Physics, University of Surrey, Guildford, Surrey GU2 7XH, UK (Dated: January 21, 2011) Using, for thefirst time, configuration-constrained potential-energy-surface calculations with the inclusionofβ6 deformation,wefindremarkableeffectsofthehighorderdeformationonthehigh-K isomersin254No,thefocusofrecentspectroscopyexperimentsonsuperheavynuclei. Forshapeswith multipolaritysix,theisomersaremoretightlyboundand,microscopically,haveenhanceddeformed shell gaps at N = 152 and Z = 100. The inclusion of β6 deformation significantly improves the description of the veryheavy high-K isomers. 1 1 PACSnumbers: 21.10.-k,21.60.-n,23.20.Lv,27.90.+b 0 2 n By overcomingthe strong Coulomb repulsionbetween clei [13, 14]. The inclusion of β6 deformation can give a the large number of protons, shell effects can lead to extrabinding energyinexcessof1 MeV,resultinginim- J the so-called “island of stability” centered on a doubly proved reproduction of experimental masses [13]. The 0 magic nucleus beyond 208Pb that has yet to be identi- 254No moment of inertia calculated with the addition of 2 fied. On the way to the predicted island, new chem- β6 deformation is 17% larger than the calculation with ical elements up to Z = 118 [1, 2] have been synthe- only β2 and β4 deformations [15]. Remarkable β6 defor- ] h sized, while the transfermium nuclei have been studied mationswerepredictedintheA≈250massregion,with t in detail through spectroscopy experiments [3]. Of spe- thelargestmagnitude(β6 ≈−0.05)in254No[15]. Inthis - l cial note in spectroscopy studies are multi-quasiparticle work, we investigate the high order deformation effects c (multi-qp) high-K (K is the total angular momentum on 254No high-K isomers. u n projectiononto the symmetry axis)isomerswhosedecay Configuration-constrained potential-energy-surface [ to low-K states is inhibited due to K forbiddenness [4]. (PES) calculations [16] have been applied to the three- They provide a probe into the underlying single-particle dimensional deformationspace (β2, β4, β6) to determine 2 structurearoundtheFermisurface. Forexample,thesys- the deformations and excitation energies of multi-qp v tematic observation of Kπ =8− isomers in A≈ 250 nu- states. Other frequently-used deformation degrees of 1 1 clei demonstrates the existence of N =152 and Z =100 freedom such as γ and β3 are excluded as they are 2 deformed shell gaps [5]. Such informationis vital for de- calculated to be negligible in 254No. The observation of 4 termining the nuclear potential that can then be used large hindrance in K-forbidden γ-ray transitions (that 1. to predict properties of superheavy nuclei. Furthermore, indicates approximately good K quantum numbers) in 1 superheavy high-K isomers can have enhanced stability 254No has confirmed that the nucleus is well deformed 0 againstα decayand spontaneousfissiondue to unpaired and axially symmetric [11]. Reflection asymmetry can 1 nucleons [6], perhaps serving as stepping stones towards significantly reduce the outer barrier beyond the second v: the “island of stability”. potential well of a prolate superheavy nucleus, but does i Among the A ≈ 250 nuclei in which high-K isomers not affect the first well [17]. In addition, 254No has no X have been discovered,254No has been the focus of recent indication of β8 deformation [15]. Deformations with r experiments due to its relatively high production rate. multipolarity higher than eight have been demonstrated a Two-qpandfour-qphigh-Kisomerswerefirstestablished to be negligible in calculations [14]. Therefore, it is by Herzberg et al. [7], Tandel et al. [8] and Kondev et justified for us to limit the calculations to the (β2, β4, al. [9]. Later these isomers were extensively studied by β6) deformation space. Heßbergeret al. [10] andClarket al. [11], with emphasis We employ the axially deformed Woods-Saxon poten- onthespectrumabovethetwo-qpisomer. Alltheexper- tial with the set of universal parameters [18] to pro- iments agree on the existence of a four-qp isomer with a vide single-particle levels. In order to reduce the un- half-lifeintheregionof200µs,butthesuggestedconfigu- physicalfluctuationoftheweakenedpairingfield(dueto rationsarecontroversial. Heßbergeret al. [10]andClark theblockingeffectofunpairednucleons)anapproximate etal.[11]deriveddifferentlevelsbridgingthefour-qpand particle-number projection has been used by means of two-qpisomers. More workis required,both experimen- the Lipkin-Nogami method [19], with pairing strengths tal and theoretical, to confirm the 254No high-spin level determined by the average gap method [20]. In the structure. configuration-constrainedPES calculation, it is required Theoretical descriptions of superheavy nuclei have to adiabatically block the unpaired nucleon orbits that made continuous progress [12] along with experiments. specify a given configuration. This has been achievedby One important finding is that high order deformation, calculating and identifying the averageNilsson quantum especially β6, is significant in modeling very heavy nu- numbers for every orbit involved in a configuration [16]. 2 0.00 (a) (b) 8 Kπ=16+ 6 {ν9/2[734]⊗ν7/2[613]⊗ π7/2[514]⊗π9/2[624]} β6 4 –0.04 2 ) V 0.24 0.28 0.24 0.28 Me 0 ( β β E 2 2 -2 g.s. FIG.1: CalculatedPESsfor254Nogroundstate(a)andKπ = -4 16+{ν9/2−[734]⊗ν7/2+[613]⊗π7/2−[514]⊗π9/2+[624]}state (b). At each point (β2,β6), the energy is minimized with re- -6 Kπ=8−{ν9/2[734]⊗ν7/2[613]} spect to β4. The energy interval between neighboring con- Kπ=8−{π7/2[514]⊗π9/2[624]} tours is 200 keV. -8 0.2 0.3 0.4 0.5 0.6 β 2 The good quantum numbers of parity and Ω (the indi- vidualangularmomentumprojectionontothesymmetry FIG. 2: (Color online) 254No potential energy curves calcu- axis)facilitatetheconfigurationconstraintin(β2,β4,β6) latedwith(solidlines)andwithout(dashedlines)β6deforma- deformation space. The total energy of a state consists tion. The energy for each β2 point isminimized with respect of a macroscopic part that is obtained with the stan- to deformations β4 and β6. dard liquid-drop model [21] and a microscopic part that iscalculatedbytheStrutinskyshell-correctionapproach, includingblockingeffects. Theconfiguration-constrained the observed very small spontaneous fission branch of PEScalculationcanproperlytreattheshapepolarization ≈10−4 for the two isomers in 254No [10]. due to unpaired nucleons. Themulti-qpstatescalculatedwithandwithoutβ6de- In Fig. 1, we display the calculated PESs for 254No formationarecomparedwithexperimentaldatainFig.3. ground state (g.s.) and four-qp high-K state relevant (Note: since the excitation energy data for the Kπ = to experiments (see below). The PESs show that the 3+,8− states from different experiments [7–11] are simi- states have remarkable β6 deformations. The g.s. β6 lar, we adopt the earliest accurate data [7]; the detailed deformation -0.029 is smaller in magnitude than -0.05 data from the most recent experiment [11] are used for that was calculated by Muntian et al. [15]. This is be- the other states.) The Kπ = 3+ state is firmly assigned − − cause we employ the standard liquid-drop model with a theprotontwo-qpconfigurationπ1/2 [521]⊗π7/2 [514] sharp surface for the macroscopicenergy, while Muntian through g factor measurement [7, 8, 11]. The K = 3 et al. [15] used the Yukawa-plus-exponential model with coupling is energetically favored over the K = 4 cou- a diffuse surface that is relatively soft against deforma- pling due to the residual spin-spin interaction between tion. Sincethelattertreatmentseemsmorerealistic,our the quasiparticles [22, 23]. According to the Gallagher- calculations may slightly underestimate the magnitude Moszkowski(GM)rule[22,23],thespin-antiparallelcou- of the β6 deformation and hence its effects. Fig. 1 also plingisenergeticallyfavoredfortwoquasineutronsortwo showsthat the shape of254No is robustagainstmulti-qp quasiprotons, while the spin-parallel coupling is lower excitations, which verifies that the increase in moment in energy for the combination of a quasineutron and a of inertia of the high-K bands with respect to the g.s. quasiproton. The splitting energies for the A ≈ 180 nu- band is due to the reduction of pairing rather than a cleiarefoundto be inthe rangeof≈100−400keV[24]. change of deformation [11]. The influence of the high Theenergyistoosmalltosubstantiallychangethecalcu- order deformation on the stability is significant. The lation of a multi-qp state. Our model in its present ver- g.s. obtains an extra binding energy of 0.8 MeV due sion does not include the residual spin-spin interaction. toβ6 deformation. Themulti-qphigh-K statesalsohave The calculationsusually wellreproducethe energetically deeper potential wells than those calculated without β6 favored coupling (see e.g. Refs. [6, 16]). − − deformation, as shown in Fig. 2. The depth increase for Ourcalculationoftheπ1/2 [521]⊗π7/2 [514]config- the Kπ = 8−{π7/2−[514] ⊗ π9/2+[624]} state reaches uration with β6 deformation gives an excitation energy 0.856 MeV. Importantly, our calculations indicate that of 0.965 MeV, in very good agreement with the experi- the β6 deformationhas no influence onthe barrierpeaks mental data 0.988 MeV [7]. The low excitation energy − − (see Fig.2), so thatthe extrabinding energyresultsin a implies that the π1/2 [521] and π7/2 [514] orbits must net increase in fission barrier height. It is seen in Fig. 2 becloseinenergy. InFig.4,wepresentthesingle-particle that the multi-qp states have wider and higher fission levels calculated with and without β6 deformation. One barriersthantheg.s.,implyingenhancedstabilityagainst can see in Fig. 4 that the two orbits become nearly de- fission due to unpaired nucleons. This is consistent with generatedue to β6 deformationso that we obtainanim- 3 4.0 -5.5 -2.0 β=0.247, β=0.011 β=0.247, β=0.011 π 2 4 2 4 K -6.0 3/2[622] -2.5 9/2[624] 3.5 Kπ 16+ {πν79//22[[571344]]⊗⊗πν79//22[[662244]]⊗} -6.5 711//122/[[266[127302]]5] -3.0 1/2[521] 3.0 16+ 14+ {πν79//22[[571344]]⊗⊗πν39//22[[662224]]⊗} V)-7.0 N=152 -3.5 Z=100 7/2[514] 16+ Me 9/2[734] 7/2[633] 2.5 {πν79//22[[571344]]⊗⊗πν79//22[[661234]]⊗} E (-7.5 7/2[624] -4.0 3/2[521] 5/2[622] -8.0 -4.5 ) 2.0 10+ 8- 1/2[631] 5/2[642] eV {ν9/2[734]⊗ν7/2[624]} -8.5 -5.0 E (M 1.5 8- 1088+-- {{{νπν997/2//22[7[[75331444]⊗]]⊗⊗ννπ1791///222[[[667122345]]]}}} -9.0 β6=−0.029 β6=0.000 (a) -5.5 β6=−0.029β6=0.000 (b) 1.0 3+ 3+ {π1/2[521]⊗π7/2[514]} FIG.4: 254Noneutron(a)andproton(b)single-particlelevels calculated using the Woods-Saxon potential with theuniver- sal parameter set. 0.5 0.0 0+ 0+ g.s. were systematically observed in N = 150 isotones [3]. Exp. Cal. Cal. For the other Kπ = 8− neutron two-qp configuration, -0.5 with β6 without β6 ν9/2−[734]⊗ν7/2+[613], the energy calculated with β6 deformation is very similar to that of the proton two- FIG.3: Calculations of254Nomulti-qpstates withandwith- qp configuration π7/2−[514]⊗π9/2+[624] (see Fig. 3). Both the calculated Kπ =8− states are in better agree- outβ6 deformation,comparedwithexperimentaldata[7,11]. mentwith experiments thanthose calculatedwithout β6 deformation. This is attributed to the β6 deformation that enhances the N = 152 and Z = 100 deformed shell provedreproductionofthestatewiththeinclusionofthe gaps, leading to increased separation of the ν9/2−[734] high order deformation. It is worth noting that β6 de- and ν7/2+[613] orbits and decreased separation of the formation leads to an enlarged Z = 100 deformed shell π7/2−[514] and π9/2+[624] orbits. It should be noted gap, consistent with that predicted in Ref. [13]. Experi- that the K = 8 coupling for the neutron two-qp config- ment [5] has confirmed the existence of the gap together uration is not the energetically favored one of the GM with the stronger N = 152 gap. The Kπ = 3+ state doublet. When considering the residual spin-spin inter- − is of special interest because the π1/2 [521] orbit origi- action, the proton two-qp state, instead of the neutron nates from the spherical orbit 2f5/2 whose position rela- two-qpstate, couldbe the lowestKπ =8− state. Never- tive to the spin-orbit partner 2f7/2 determines whether theless, they remain close to each other because the GM Z =114 is a magic number for the “island of stability”. splitting energy is small. Experimentalinformation such Thegoodagreementbetweenexperimentsandourcalcu- as the g factor is needed to distinguish between the two lationswithβ6deformationdemonstratestheimportance configurations for the Kπ =8− isomer. of the high order deformation in very heavy nuclei and The two low-energy Kπ = 8− configurations can cou- the validity of the Woods-Saxon potential in this mass ple to form a four-qp Kπ = 16+ state, analogous to region. the well-known Kπ = 16+ isomer in 178Hf [4]. In- UnliketheKπ =3+ statewithitsconfigurationunam- deed, a four-qp 184 µs isomer has been observed. How- biguouslyassigned,theobserved266msKπ =8− isomer ever, its configuration is less clear than those of the hasits configurationcontroversiallyassignedinthe liter- two-qp states. Two possible configurations, Kπ = ature. The proton two-qp configuration π7/2−[514] ⊗ 16+{ν9/2−[734]⊗ν7/2+[624]⊗π7/2−[514]⊗π9/2+[624]} π9/2+[624]is suggestedfor the isomer in Refs. [7, 8, 10], and Kπ =14+{ν9/2−[734]⊗ν3/2+[622]⊗π7/2−[514]⊗ while the most recent experiment [11] favors a neutron π9/2+[624]}, were suggested in Ref. [7] and Refs. [8, 9], two-qp configuration. There are two possible Kπ = 8− respectively. The most recent experiment [11] preferred neutron two-qp configurations, ν9/2−[734]⊗ν7/2+[613] a spin-parity assignment of Kπ =16+. Our calculations andν9/2−[734]⊗ν7/2+[624]. Ourcalculationofthe lat- showninFig.3indicatethattheconfigurationsuggested terindicatesthatthestateistoohighinenergytobethe inRef.[7]ismuchhigherthanthefour-qpKπ =16+con- isomer. The high energyis because both orbits lie below figuration involving the ν7/2+[613] orbit. This is due to the large N = 152 shell gap. Therefore, it requires two thehighenergyoftheν9/2−[734]⊗ν7/2+[624]coupling, neutrons to cross the gap to form the state. The con- as discussed above. The Kπ =14+ configuration with a figuration favors the formation of an isomer in N = 150 low-Ω orbit ν3/2+[622] involved is calculated to be also nuclei where the Fermi surface is between the two or- higherthan the ν9/2−[734]⊗ν7/2+[613]⊗π7/2−[514]⊗ bits. Indeed, low energy isomers with this configuration π9/2+[624] configuration. Consequently, the calculated 4 TABLE I: Theoretical deformations and excitation energies a two-qp Kπ = 10+ state was observed in the most re- of multi-qp states in 254No. centexperiment[11],withtheconfigurationν9/2−[734]⊗ − ν11/2 [725] suggested. Fig. 3 shows that the calculated Kπ Configuration† β2 β4 β6 Ex(keV) excitationenergyis 1.479MeV, muchlowerthan the ex- 0+ g.s. 0.247 0.011 -0.029 0 perimentaldata2.013MeV[11]. However,theKπ =10+ 3+ ab 0.247 0.011 -0.030 965 state has unfavored spin-spin coupling that would in- 8− AB 0.241 0.012 -0.024 1357 creasetheexcitationenergy. Theenergyincrementcould 8− bc 0.245 0.009 -0.028 1378 reach ≈400 keV as our calculated excitation energy can be taken as the value for the favored coupling. 6− AE 0.247 0.010 -0.029 1427 10+ AD 0.244 0.010 -0.027 1479 As shown in Fig. 4, there exist several high-Ω orbits 7− bd 0.246 0.010 -0.028 1481 aroundthe 254NoFermisurfacethat cancouple to many otherhigh-K states. TableIsummarizesthecalculations 8+ cd 0.244 0.009 -0.027 1658 7+ BC 0.242 0.014 -0.026 1774 wcititahtitohneeinnecrlugsyioonfotfhβe6sdixe-fqoprmKatπio=n. 2T5h−e cstaalctuelaiste4d.5e2x2- 9− CD 0.246 0.012 -0.028 1881 MeV,comparableto3.942MeV,the excitationenergyof 8− AC 0.243 0.014 -0.025 2032 the observed 24+ g.s. band member [7]. The Kπ = 25− 9− BD 0.238 0.010 -0.022 2237 state could be close to the yrast line (where the state 16+ ABbc 0.240 0.010 -0.024 2722 has the lowest energy among the states with the same 14+ AEbc 0.245 0.008 -0.028 2803 angularmomentum), possibly forming anisomericstate. 18− ADbc 0.242 0.008 -0.026 2845 In summary, the effects of the high order deforma- 17+ ABCD 0.239 0.013 -0.023 3158 tion, β6,onthehigh-K isomersin254Noareinvestigated 16+ ACbc 0.241 0.012 -0.025 3407 by applying configuration-constrained PES calculations 25− ABCDbc 0.238 0.011 -0.023 4522 in (β2,β4,β6) deformation space. The isomers gain ex- 24− ABCDbd 0.239 0.013 -0.023 4631 tra binding energy due to the β6 deformation, imply- ing enhanced stability against fission. The high order 25+ ABCDcd 0.236 0.011 -0.021 4774 deformation rearrangesthe single-particle levels, leading † Neutron orbits 9/2−[734], 7/2+[613], 7/2+[624], 11/2−[725], to strengthened deformed shell gaps at N = 152 and and 3/2+[622] are represented by A, B, C, D, and E respec- Z =100, which influences the properties of the multi-qp tively. Proton orbits 1/2−[521], 7/2−[514], 9/2+[624], and states. These effects are found to be significant. All the 7/2+[633]arerepresentedbya,b,c,anddrespectively. observed multi-qp states in 254No are better reproduced by the calculations with β6 deformation. This indicates lowest-lying Kπ = 16+ state is likely the 184 µs isomer the importance of the high order deformation in calcu- lating multi-qp states in very heavy nuclei. due to its low energyandhigh K value, compatible with the experimentalevidence ofa Kπ =16+ spin-parityas- We are grateful to T.L. Khoo and F.G. Kondev signment [11]. The excitation energy calculated with β6 for suggesting the present work. This work was sup- deformationis2.722MeV,whichisclosetothemeasured ported in part by the US DOE under Grants DE- value of 2.928 MeV [11]. In Fig. 3, it can be seen that FG02-08ER41533 and DE-FC02-07ER41457 (UNEDF, the inclusion of β6 deformation increases the calculated SciDAC-2), and the Research Corporation; the Chinese energy, making it closer to the experimental value. Fur- Major State Basic Research Development Program un- thermore, the neutron component of unfavored residual der Grant 2007CB815000; the National Natural Sci- interaction is expected to further increase the energy. ence Foundation of China under Grants 10735010 and In addition to all the multi-qp states observed before, 10975006;and STFC and AWE plc (UK). [1] S. Hofmann and G. Mu¨nzenberg, Rev. Mod. Phys. 72, ConferenceonNuclearDataforScienceandTechnology, 733 (2000). Nice, France, 2007, edited by O. Bersillon et al. 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