Strong enhancement of level densities in the crossover from spherical to deformed neodymium isotopes

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Physics Letters B

www.elsevier.com/locate/physletb

Strong enhancement of level densities in the crossover from spherical to deformed neodymium isotopes

M. Guttormsen

^a,∗

, Y. Alhassid

^b

, W. Ryssens

^b,1

, K.O. Ay

^c

, M. Ozgur

^c

, E. Algin

^c,d

, A.C. Larsen

^a

, F.L. Bello Garrote

^a

, L. Crespo Campo

^a

, T. Dahl-Jacobsen

^a

, A. Görgen

^a

,

T.W. Hagen

^a

, V.W. Ingeberg

^a

, B.V. Kheswa

^a,e

, M. Klintefjord

^a

, J.E. Midtbø

^a

, V. Modamio

^a

, T. Renstrøm

^a

, E. Sahin

^a

, S. Siem

^a

, G.M. Tveten

^a

, F. Zeiser

^a

aDepartmentofPhysics,UniversityofOslo,N-0316Oslo,Norway

bCenterforTheoreticalPhysics,SloanePhysicsLaboratory,YaleUniversity,NewHaven,CT 06520,USA

cDepartmentofPhysics,EskisehirOsmangaziUniversity,FacultyofScienceandLetters,TR-26040Eskisehir,Turkey dDepartmentofElectricalandElectronicsEngineering,AATScienceandTechnologyUniversity,01250Adana,Turkey eDepartmentofPhysics,UniversityofJohannesburg,P.O.Box524,AucklandPark2006,SouthAfrica

a rt i c l e i n f o a b s t r a c t

Articlehistory:

Received24November2020

Receivedinrevisedform11February2021 Accepted8March2021

Availableonline12March2021 Editor: B.Blank

Keywords:

Nuclearleveldensity Oslomethod

ShellmodelMonteCarlo Mean-fieldtheory Collectiveenhancement

Understandingtheevolutionofleveldensitiesinthecrossoverfromsphericaltowell-deformednuclei hasbeen along-standingprobleminnuclearphysics. Wemeasure nuclear leveldensities forachain ofneodymiumisotopes^{142,144−151}Nd whichexhibitsuchacrossover.Theseresultsrepresentthemost completedatasetofnuclearleveldensitiestodateforanisotopicchainbetweenneutronshell-closure and towards mid-shell. We observea strong increase ofthe level densities along the chain withan overallincreasebyafactorof≈^{150 atanexcitationenergyof6MeVandsaturationaroundmass150.}

LeveldensitiescalculatedbytheshellmodelMonteCarlo(SMMC)areinexcellentagreementwiththese experimentalresults.Basedonourexperimentalandtheoreticalfindings,weofferanexplanationofthe observedmassdependenceoftheleveldensitiesintermsoftheintrinsicsingle-particleleveldensityand thecollectiveenhancement.

©^{2021TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense} (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP³.

1. Introduction

Compound-nucleus reaction cross sections are indispensable in a variety of applications such as understanding stellar nucle- osynthesis [1],designing next-generationnuclearreactors [2],and optimizingtransmutationofnuclear waste [3].Such reactions are well understood by Hauser-Feshbach theory [4], but this theory requiresasinputstatisticalnuclearproperties,suchasthenuclear leveldensity(NLD).However,themicroscopiccalculationofNLDs in thepresence ofcorrelationsis a challenging many-bodyprob- lem. Furthermore, experimental data are usually limited to low excitation energies [5] and neutron resonance measurements at the neutron separation energy [6], making it difficult to bench- marktheoreticalmodels.

*

Correspondingauthor.

E-mailaddresses:[email protected](M. Guttormsen), [email protected](Y. Alhassid).

1 Currentaddress:Institutd’Astronomieetd’Astrophysique,UniversitéLibrede Bruxelles,CampusdelaPlaineCP226,BE-1050Brussels,Belgium.

UnderstandingtheeffectsofdeformationontheNLDisalong- standing open problem in nuclear physics; see,e.g., Ref. [7] and referencestherein.Intheabsenceofpairingcorrelations,thereare twomaincompetingeffectsthatinfluencetheNLDofadeformed nucleusascomparedtoasphericalnucleusofsimilarmass:(i)in mean-field theory the onset of deformation breaks the magnetic degeneracyofthesphericalsingle-particlelevels,leadingtoanef- fectivedecreaseofthe averagesingle-particlelevel densityatthe Fermienergy [8], and thus lowering the NLD;and(ii) rotational bandsbuiltontopofeachoftheseintrinsicmean-fieldconfigura- tionsleadtoenhancementoftheNLD [9,10].

Bjørnholmet al. [9] predicted acollective enhancementfactor

≈^{10 forvibrationsand}≈^{100 forrotations,altogethera factorof}

≈^{1000.In empiricalFermi gas models,effects of} collectivityare oftenabsorbedinaneffectiveenergy-dependentleveldensitypa- rameterthat wasintroducedtoaccountforshelleffectsandpair- ingcorrelations[11].Alternatively,variousphenomenologicalNLD models [12–17] weremodified to includeempiricalenhancement factors explicitly. Modern combinatorial and mean-field methods mustalsobeaugmentedbyphenomenologicalcollectiveenhance- mentfactors [18–20].

https://doi.org/10.1016/j.physletb.2021.136206

0370-2693/©^{2021TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense}(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP³.

Fig. 1.MatriceswithinitialexcitationenergyEversusγ-rayenergyEγ fromparticle-γcoincidencesobtainedbythe¹⁴⁴Nd(d,pγ)¹⁴⁵Ndreaction.ThefirststepsoftheOslo methodconsistofestablishingthea)raw,b)unfolded,andc)primaryγ-raymatrices.

Recent shell-model Monte Carlo (SMMC) calculations [21,22]

indicatedasignificantly smallercollectiveenhancement thansug- gestedbyBjørnholm etal. [9].However, asstatedbyJunghanset al. [23], experimental information on the NLD is mandatory for quantifying thiseffect.A reliabletheoretical modelshould repro- duce the NLD well not only at the low-lyingdiscrete levels and atthe neutronresonance energy,butalsofor abroadexcitation- energy region. The Oslo method used in this work provides the functionalformoftheNLDintheenergyrangebetweenthelow- lying discrete levels and the neutron separation energy, where oftenthereisnootherexperimentaldataavailable.Someprelimi- naryresultsofthepresentworkwerereportedinRef. [24].

We present a systematic study of the NLD for a chain of neodymiumisotopes,startingfrom¹⁴²NdattheN=^{82 shellclo-} sure and up to the well-deformed ¹⁵¹Nd, probing the effect of collectivity on the NLD in an unprecedented way. We have also performedmicroscopic SMMCcalculationsfor ¹⁴²Nd up to¹⁵²Nd and find them to be overall in excellent agreement with exper- iment. We show that the combined effect of a decrease in the single-particleleveldensitywithmassnumberandacollectiveen- hancementresultsinanincreaseoftheNLDwithdeformationthat saturatesaroundmass150.

2. Experiment

The light-ion reactions were performed at the Oslo Cyclotron Laboratory. The targets were metallic foils of ^{142,144,146,148,150}Nd with thicknesses of≈^{2 mg/cm2 and}enrichments of ≈^{97%. The} targetswere bombarded withproton anddeuteronbeams ofen- ergies 16.0 MeV and 13.5 MeV, respectively. The SiRi particle- telescopesystem [25] wasappliedtodeterminetheoutgoingpar- ticle type and energy. The 64 particle telescopes were located

≈ ^{5cm fromthe target in eight angles between126◦ and140◦} withrespecttothebeamdirection. Theparticleenergyresolution was ≈^{150keV(FWHM).The}

γ

raysfollowing thereactionswere measured withtheNaI(Tl)scintillator array CACTUS [26] and the LaBr3(Ce)scintillator arrayOSCAR.Additional detailsare provided intheSupplementalMaterial [27].

3. TheOslomethod

WeextracttheNLDfor^{142,144−151}NdapplyingtheOslomethod [28,29] forasetofparticle-

γ

raycoincidences.Fromthemeasured ejectile,weobtaininformationontheinitialexcitationenergyE of theresidualnucleus.The

γ

rays detectedincoincidencewiththe ejectilerevealthedecaypropertiesfromthisspecificexcitationen- ergy.Fig.1ashowshowthedataaresortedintoamatrixofinitial excitation energies E versus the

γ

-ray energy Eγ. This raw ma- trixisunfolded foreach excitationenergybin(Fig. 1b) usingthe known detectorresponse functions [30,32]. Finally,the first gen- eration (primary)

γ

-ray matrix P(Eγ,E) isobtained, see Fig.1c.

Thefirst-generationprocedureisbasedonaniterativesubtraction technique [31] whichseparatesthedistributionofthefirstemitted

γ

raysfromallavailable

γ

cascades.

Table 1

ParametersforextractingNLDandsystematicuncertaintiesinneodymiumisotopes.

Alsolistedarethequadrupoledeformationβ2andtemperatureTCTofEq. (3).

A β2 TCT Sn σ(Sn) D0 ρ(Sn)

(MeV) (MeV) RMI (eV) (10⁶MeV⁻¹) 142 0.092(2) 0.65(5) 9.828 6.6(7) 19(4)^c 1.23(35)^b 143 0.109(5)^a 0.61(3) 6.124 6.1(6) 1035(135) 0.07(2) 144 0.125(2) 0.63(3) 7.817 6.3(6) 37.6(21) 0.32(5) 145 0.138(5)^a 0.59(3) 5.755 5.9(6) 450(50) 0.16(4) 146 0.151(2) 0.62(3) 7.565 6.2(6) 17.8(7) 0.67(11) 147 0.176(5)^a 0.57(3) 5.292 5.8(6) 346(50) 0.20(5) 148 0.200(2) 0.59(3) 7.333 6.1(6) 5.9(11) 2.4(6) 149 0.242(5)^a 0.54(3) 5.039 5.8(6) 165(14) 0.42(9) 150 0.283(2) 0.61(4) 7.376 6.2(6) 3.0(10)^c 4.8(18)^b 151 0.314(10)^a 0.54(3) 5.335 6.0(6) 169(11) 0.43(9)

152 0.345(9) 7.278 6.3(6)

a Interpolatedbetweeneven-Aneighbors.

b Scaledfromsystematics [27].

c Adjustedtoreproduceρ(Sn).

The next step in the Oslo methodis to factorize the primary

γ

-raymatrixby

P

(

E_γ

,

E

) ∝

T

(

E_γ

) ρ (

E

−

^Eγ

).

(1) Here,wehaveappliedtheBrinkhypothesis [33]:the

γ

-raytrans- mission coefficient T ^is approximatelyindependent of excitation energy and spin/parity. The factorization is justified by Fermi’s goldenrule [34,35],whichstatesthatthedecayrateisproportional totheNLDatthefinalexcitationenergyafteremittingtheprimary

γ

ray. The fitting procedureperformed usingEq. (1) enablesthe simultaneousextractiontheNLDandthe

γ

-raytransmissioncoef- ficient.However, ithas beenshown [28] that anytransformation oftheform

ρ (

E

−

^Eγ

) →

^Aexp

[ α (

E

−

^Eγ

)] ρ (

E

−

^Eγ

),

(2) givesthesamefitto P(Eγ,E).Todeterminetheparameters A and

α

in (2), we useother experimental data.At low excitations, we normalizethe NLD toknown discrete levels [36]. At highexcita- tions, we used the measured average neutron s-wave resonance spacingD0 [37] attheneutronseparationenergySn.

Toconvertthe measured D₀ tototallevel density,weusethe spincutoffmodel [38,39].Thevalues

σ

(Sn)ofthespincutoffpa- rameterattheneutronseparationenergy Sn areestimatedbased on a rigid-body moment of inertia (RMI) [27] and are tabulated in Table 1. Table 1 also includes the quadrupole deformation β2 from Ref. [40] and the temperature T_CT extracted by fitting the constant-temperatureformula

ρ

CT

(

E

) = (

1

/

TCT

)

exp

[ (

E

−

^E0

)/

TCT

]

⁽³⁾

tothe high-energy datapoints, where E0 isa shift parameterto match

ρ

(S_n) [27].Equation(3) isusedto extrapolatethe experi- mentalNLDtohigherexcitationenergiesthanmeasured.

Fig. 2.ExperimentallyextractedNLDs(solidbluecircles)ofthe^{142,144−151}Ndisotopes.ThegrayhistogramsshowtheNLDofknowndiscretelevels.ThetotalNLDsevaluated fromneutroncaptureresonancespacingsD0aredisplayedasopenblacksquares.SMMCleveldensitiesforthe^142−150Ndisotopesareshownbysolidredsquares.

4. SMMCcalculations

TheSMMCmethod [41,42] enablestheexactcalculation(upto statisticalerrors) of NLDsin theframework ofthe configuration- interaction (CI)shellmodel.This methodallowsusto usemany- particle model spaces that are manyorders of magnitude larger thanthosethatcanbetreatedbyconventionalshellmodelmeth- ods [43].Incontrast tocombinatorialandmean-fieldapproaches, theSMMCapproach doesnot requireanyempiricalenhancement factors, andistherefore asuitable approach forstudying thede- formationdependenceoftheNLD.

We carried out SMMC calculationsin the proton-neutronfor- malism [44] for thechainofneodymiumisotopes ^142−152Nd.The CIshellmodelspaceincludesthecomplete50−^{82 shellplusthe} 1f_7/2 orbital for protons, and the complete 82−^{126 shell plus} the 0h_11/2 and 1g_9/2 orbitals forneutrons. The effective interac- tionparameters aregiveninRef. [21].Fortheodd-massisotopes, there isasignproblemassociatedwiththeprojection onan odd numberofneutronsatlowtemperaturesandtheground-stateen- ergiesweretakenfromRef. [45].Thelatterestimatedground-state energiesforalltheoddneodymiumisotopesinthechainwiththe exceptionof¹⁵¹Nd.

Incontrasttostate densitiesthatcountthe2J+1 degeneracy ofeachlevelwithspin J,themeasuredleveldensitiescounteach such level only once. In SMMC, the level densities are obtained by projection on M=^{0 (M}=¹/2)foreven-mass(odd-mass)nu- clei [46,47]. SMMC state densities for the neodymium isotopes werepresentedinRef. [48].WeprovidemoredetailsfortheSMMC calculationsintheSupplementalMaterial [27].

5. Results

In Fig. 2 we compare the experimentally extracted NLDs of 142,144−¹⁵¹NdwiththeSMMCresults.Above anexcitation energy of∼²−^{3 MeV,the}experimentalNLDsarealmostlinearinalog- arithmicscaleandarewell-describedbytheconstant-temperature formula (3). It was conjectured that this behavior emerges once the firstpairofnucleons isbroken [49–51], i.e., foran excitation energy E>2,whereisthepairing gap.Incontrasttorecent findingsin^167,168,169Tm [52],wedonotobserveanyexperimental ortheoreticalsignaturesofirregularbumpsintheNLDcurves.

Fig.3 showstheexperimental andSMMC NLDsfromFig.2at three excitation energies of 2.5,5 and 7.5 MeV asa function of deformationβ2.Thedeformationoftheeven-massisotopesisde- termined from the compilation of Pritychenko et al. [40], using

Fig. 3.Experimental(open squares)and SMMC(solidsquares)leveldensitiesfor 142−151NdatexcitationenergiesE=2.5,5.0 and7.5 MeV.Theexperimentaldata pointsatE=7.5 MeVareextrapolatedusingtheconstant-temperatureformula(3) withvaluesofTCTgiveninTable1.ThecurvesarecalculatedfromEq. (4);seetext.

themeasured B(E2)valuesbetweenthegroundstateandthefirst excited2⁺state.Fortheodd-massisotopes,weassumeadeforma- tionthatistheaverageoftheireven-massneighbors.Thesevalues ofβ2arelistedinTable1;seealsotheSupplementalMaterial [27].

Atexcitationenergiesof2.5 MeVand7.5 MeV,theNLDisde- termined,respectively, by known low-lying discrete levels which weassume tobe acompletesetandbythe averageneutronres- onancespacingD0,whileattheintermediateexcitationenergyof 5 MeV,theNLDis determinedby theOslomethod.We findthat the deformation dependence of the experimental NLDs at these threeexcitationenergiesfollowcloselytheempiricalform

ρ (β

2

) =

^Cexp

[− η (β

2

− β

₂^max

)],

⁽⁴⁾

whereC and

η

arefitparameters andβ₂^max=⁰.25.Theresulting fitsofEq. (4) totheexperimentaldataareshownbythecurvesin Fig.3.Weobtain similarvaluesoftheparameter

η

fortheeven- andodd-massisotopes with

η

=¹¹⁸,136 and166 at E=².5,5.0 and7.5MeV,respectively.Thereisastrongodd-eveneffectwhere the NLD of an odd-mass nucleus is higher than the NLDs of its even-massneighbors,whichcanbe attributedto theblockingef- fectoftheoddneutron [53].

Fig. 4. The totalenhancement ρ(A)/ρ(A=144)of theNLDs ofthe even-mass neodymiumisotopesrelativeto¹⁴⁴NdatE=6 MeV.Thisenhancement,asdeter- minedfromexperiment(openbluesquares),iscomparedwithSMMC(solidblue squares),meanfield(redcircles)andthemean-fieldcorrectedbythecollectiveen- hancementfactorK(blackcircles).K≈³.4,8.6,11.2 in^148,150,152Nd,respectively.

We observe overall excellent agreement between the experi- mentalandSMMCNLDs.TheincreaseintheNLDwithmassnum- beranditssaturationarewell reproducedforboththeeven- and odd-massisotopes upto mass150.Ingeneral,we findlargerde- viations for the odd-mass isotopes, in particular at the highest energy of E=⁷.5 MeV. We note, however, that the experimen- tal resultsat thisenergyare estimatedby extrapolating theOslo data beyond the neutron separation energy using the constant- temperatureformula(3), andthusthe comparisonisnot ascon- clusiveasfortheeven-massisotopes.

The systematics based on Eq. (4) describe a decrease in NLD beyondneutronnumber N=^{90. Whileno} experimentaldata are available for¹⁵²Ndandbeyond, such adecrease was recentlyre- portedintheneighboringsamariumisotopes,inwhichtheNLDof 155Sm is morethan two-fold lower than the NLD of¹⁵³Sm [54].

The SMMC calculationsin the neodymiumisotopes do not show suchadecreaseoftheNLDbetweenN=^{90 andN}=^92,ascanbe seeninFig.3.

6. Explanationofthemassdependence

Fig.4showsthetotalenhancementoftheNLDatE=^{6 MeVof} theeven-mass neodymiumisotopesrelative tothe NLDof¹⁴⁴Nd, thelightest isotopeinthe chainforwhichan experimentalvalue ofD0exists.Overall,weobserveexcellentagreementbetweenex- perimentandtheSMMCresults.We findalargeenhancement in the experimental NLDby a factorof≈^{150 for 150Nd (relativeto}

142Nd). Theobserved collective enhancement issubstantially less thanthatpredictedbythemodelBjørnholmetal. [9].Thismodel was implemented in Ref. [22] in the context of CI shell model Hamiltonians(seeSec. IV.B.6)andwasshowntooverestimate the rotational enhancementfactor whencompared withexact SMMC results.

We calculated the intrinsic mean-field state density

ρ

mf us- ing the methods of Refs. [22,55] and employing the same effec- tiveinteractionasintheSMMCcalculations.Thisintrinsicdensity is determined mostly by the average single-particle level density g_n( F)oftheneutronsattheFermienergy

F.Indeformednuclei, rotational bands built on top of intrinsic band heads lead to an enhancement of the total state density, described by a collective enhancement factor K =

ρ

state/

ρ

mf [21]. In Fig. 2 ofthe Supple- mental Material [27] we show the calculated K asa function of excitation energyfor¹⁵⁰Nd.Thiscollective enhancement of

ρ

state is also reflected in the total level density, since these two den- sities are related by a spin cutoff parameter that is only weakly dependentonmass.Thus,themassdependenceoftheNLDisde-

terminedby two factors:

ρ

mf andK.With82 neutrons,¹⁴²Ndis semi-magicandits neutronFermi energyisinthe middleofthe shellgapandisthus characterizedby a relativelylow gn( F).As we start filling the 82−^{126 major shell in 144Nd, the neutron} Fermi energyrises, so that it is closeto the 2f_7/2 orbital, lead- ingto a sharpincrease in gn( F) andthusin

ρ

_mf.The gn( F) of the sphericalmean-field solution continues to increase in ¹⁴⁶Nd, though at a more moderate rate. The increase in

ρ

mf is shown by thered circlesin Fig.4. Themean-field solution forourshell modelinteractionbecomesdeformedstarting in ¹⁴⁸Nd. Deforma- tionliftsthesphericaldegeneracyofthesingle-particlelevelsand decreases gn( F)in thedeformedisotopes, leading to adecrease in

ρ

mf compared to ¹⁴⁶Nd.This decrease is compensated by the rise of K with deformation. As a result, the total NLD enhance- mentincreases withmassbutsaturates around ¹⁵⁰Nd,forwhich theSMMCNLDisvery similartothat in¹⁵²Nd.Thisisshownby theblackcirclesinFig.4,describingtheproductof K andthein- creaseof

ρ

mf (relativeto ¹⁴⁴Nd),andfollowcloselytheobserved totalenhancementoftheNLDwithmassnumberA.

7. Conclusions

WeextractedexperimentalNLDsforalongchainofneodymium isotopes using the Oslo method. We observed a large total en- hancementoftheNLDinthecrossoverfromsphericaltodeformed isotopes, whichsaturates around A=^150.The availability of ex- perimental data for such a long isotopic chain makes theseiso- topes an excellent benchmark for testing the quality of current andfutureNLD models. We calculatedSMMC NLDsoftheseiso- topesintheframeworkoftheCIshellmodelandfoundthemtobe overallinexcellentagreementwiththeexperimentalNLDs.Weex- plainedthemassdependenceoftheNLDsbythecombinedeffects of the intrinsicsingle-particle level density andof the collective enhancement.

Declarationofcompetinginterest

Theauthorsdeclarethattheyhavenoknowncompetingfinan- cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

Acknowledgements

We thankJ.C. Müller, P. Sobas andJ. Wikne forproviding ex- cellent experimental conditions. This work is supported by The ScientificandTechnologicalResearchCouncilofTurkey(TUBITAK) withprojectnumber115F196.A.C.L.acknowledgesfundingofthis researchbytheEuropeanResearchCouncilthroughERC-STG-2014, grant agreement no. 637686. A.C.L. acknowledges support from the “ChETEC” COST Action (CA16117), supported by COST (Euro- peanCooperationinScienceandTechnology).Thisworkbenefited fromsupportbytheNationalScienceFoundation underGrantNo.

PHY-1430152(JINACenterfortheEvolutionoftheElements).This work was supported in part by the NationalScience Foundation underGrantNo.OISE-1927130(IReNA).TheworkofY.A.andW.R.

was supported in part by the U.S. DOEgrant No. DE-SC0019521.

TheSMMCcalculationsusedresourcesoftheNationalEnergyRe- searchScientificComputingCenter,whichissupportedbytheOf- fice of Science ofthe U.S. Departmentof Energy under Contract No. DE-AC02-05CH11231. We also thank the Yale Center for Re- searchComputingforguidanceanduseoftheresearchcomputing infrastructure.

This work was partially supported by projects 263030 and 262952 of the Norwegian Research Council. The OSCAR detector wasfundedbytheNorwegianResearchCouncilproject245882.

Appendix A. Supplementarymaterial

Supplementarymaterialrelatedtothisarticlecanbefoundon- lineathttps://doi.org/10.1016/j.physletb.2021.136206.

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