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Physics
Letters
B
www.elsevier.com/locate/physletb
S
-shaped
heat
capacity
in
an
odd–odd
deformed
nucleus
Balaram Dey
a,
∗
,
N. Quang Hung
b,
Deepak Pandit
c,
Srijit Bhattacharya
d,
N. Dinh Dang
e,
L.T. Quynh Huong
f,
g,
Debasish Mondal
c,
S. Mukhopadhyay
c,
Surajit Pal
c,
A. De
h,
S.R. Banerjee
iaSahaInstituteofNuclearPhysics,1/AF-Bidhannagar,Kolkata,700064,India
bInstituteofFundamentalandAppliedSciences,DuyTanUniversity,HoChiMinhCity700000,VietNam cVariableEnergyCyclotronCentre,1/AF-Bidhannagar,Kolkata,700064,India
dDepartmentofPhysics,BarasatGovt.College,Barasat,N24Pgs,Kolkata,700124,India
eQuantumHadronPhysicsLaboratory,RIKENNishinaCenterforAccelerator-BasedScience,RIKEN,2-1Hirosawa,Wakocity,Saitama351-0198,Japan fDepartmentofNaturalScienceandTechnology,UniversityofKhanhHoa,NhaTrangCity,KhanhHoaProvince652124,VietNam
gFacultyofPhysicsandEngineeringPhysics,VietnamNationalUniversityHoChiMinhCity–UniversityofScience,HoChiMinhCity748355,VietNam hDepartmentofPhysics,RaniganjGirls’College,Raniganj,713358,India
i(Ex)VariableEnergyCyclotronCentre,1/AF-Bidhannagar,Kolkata,700064,India
a
r
t
i
c
l
e
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f
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Articlehistory:
Received30October2018
Receivedinrevisedform30November2018 Accepted4December2018
Availableonline6December2018 Editor:V.Metag
Keywords:
Nuclearthermodynamics Statisticalmodelcalculation Pairingphasetransition
We examine the thermodynamic propertiesof mass A∼200 nucleiutilizing angularmomentum (J) gatednuclearleveldensities(NLDs)extractedintheexcitationenergyrangeof2–15 MeV.Interestingly, the experimental NLDsare ingood agreement withthe results ofa microscopicapproach, whichis derivedbasedontheexactpairingplustheindependent-particlemodelatfinitetemperature(EP+IPM), whereas the conventionalHartree–Fock BCS(HFBCS) and Hartree–Fock–Bogoliubovplus combinatorial method(HFBC)failtodescribethesedata.Consequently,thethermodynamicpropertiesofthosenuclei atfinite angularmomentumhavebeen extracted usingtheEP+IPM NLDs. Whiletheheatcapacities of200Tl,211Poand212At(nearsphericalnuclei)followthetrendasexpectedinodd–oddandeven–odd
masses,surprisinglyan
S
-shapedheatcapacityisfoundinodd–odddeformednucleus184Re.Ithasbeenshownthatthis
S
-shapedheatcapacityobservedin184Reiscausedbynotonlythebreakingofnucleon Cooperpairsbutalsothechangeofpairinginducedbydeformation.©2018TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.
Along-standingprobleminnuclearphysicsistheexperimental observationofthepairingphasetransitioninatomicnuclei. Ther-modynamicpropertieslikesuperfluidityandpairing phase transi-tion are well-established facts in infinite nuclear matter [1], e.g. the core of neutron star. The recent observation of rapid cool-ingofpulsars, e.g. Cassiopeia A,hasbeeninterpretedinterms of superfluidityandneutrontripletpairing [2].However,these prop-erties digress from infinite nuclear matter to finite nucleus due to the statistical fluctuations in the order parameter. Therefore, thegradualtransitionfromstronglycorrelatedpairedstatesto un-pairedonesinatomicnucleimaynot beasevident asin infinite matters [1]. Thisinduces a highdegree of interests in the study
*
Correspondingauthor.E-mailaddresses:[email protected](B. Dey),[email protected] (N. Quang Hung).
of nuclear thermodynamics, especially in the energy domain of neutron binding. On the other hand,in finite nuclei it has been seen that the neutron pairing gap depends upon shell and sub-shellclosures [3].Again,newshellclosuresemergeandolderones disappear in exotic nuclei. Therefore, it will not be an exagger-ation to say the role of shell effects on the nuclear pairing is highly important [4].Althoughsuchtheoreticalstudiesweredone in the past [5], yet experimental investigations are very rare in the existingliterature. Similarly,therelationship betweennuclear deformation andpairing has not been studied in much detail at higherexcitationenergies.Calculationswithintherelativisticmean field (RMF)[6],finite-temperatureHartree–Fock(FTHF) [7],Dirac– Hartree–Fock–Bogoliubov(DHFB)[8,9],etc.,havebeencarriedout in the past to understand the effect of deformation on pairing andnuclearthermodynamics. Butexperimental dataarestill very scarceintermsofthedependenceofangularmomentumand de-formationonnuclearpairing.
https://doi.org/10.1016/j.physletb.2018.12.007
Thestudyofthermodynamicsoffinitenucleiinvolvesthe reli-ableextractionofexperimentalnuclearleveldensity(NLD). Exper-imentaldetermination ofNLDshasbeen difficultinthe pastdue tothe absence ofsuitable experimental techniques.Recently, the thermal propertiesof some nuclei (56,57Fe [10], 91–98Mo [11,12], 116,117Sn [13], 148,149Sm [14], 161,162Dy [15,16], 166,167Er [16,17], 172,173Yb[18])weremeasuredbytheOslogroupusinganew tech-nique, calledthe Oslo method,and the S-shaped canonical heat capacity has been extracted, signalizing the pairing phase tran-sition around the critical temperature of Tc
=
0.5–1.0 MeV. The NLDsandthermodynamicalpropertiesofPtandAuisotopeswere alsostudiedinRef. [19].IntheOslotechnique,theNLD hasbeen extractedfromtheprimaryγ
-rayspectrameasuredincoincidence with the particle using inelastic and/or transfer reactions [20]. However, this method is limited only up to low excitation en-ergy E∗ (belowtheparticlethreshold) andangularmomentum J (onlyafew¯
h).ThislimitationinE∗ and Jhasrecentlybeen over-comebymeasuringthe NLDfromtheevaporatedparticlespectra in the compound nuclear reactions [21–23]. Although the major problemconnectedwiththisparticleevaporationtechniqueisthe possiblecontributionfromthemultistepanddirectreactions,the useoflow-energylight-ionbeamhasprovidedan excellentscope forextractingtheNLDfromaparticularnucleardecaychanneland thecontributions from the directreaction could be ruled out by measuringtheparticlespectraatbackwardangles [23].Fusion re-actionswithalphaparticlespopulatethecompoundnucleusinthe sameisospinstatesasthat ofthetarget nucleus.However, ithas been shown earlier by A. Voinov et al., [22] that the extracted level densities from the experimental neutron evaporation spec-trumwithdeuteronbeam(Tz=
0)givesthesameleveldensityas thatobtainedbyusing3Hebeam(Tz
=
1/2)andthediscretelevel densitiesasobtainedfromRIPL.On the other hand, the measured NLD (up to the particle threshold)intheOslotechniquewascomparedwiththe Constant-Temperatureformula[24] andthenextrapolateduptothehigher energyusingtheFermi-gas(FG)model [25] toestimate the ther-modynamic quantities. However, the functional form of the NLD is till date not adequately known due to the lack of availability of experimental data at higher E∗ and J. Therefore the use of a single consistent theoretical calculation, instead of adding two differentphenomenological models, to investigate the thermody-namic properties of atomic nuclei based on the measured NLDs belowandabove the particlethresholdcould be a better choice. Veryrecently, theNLD hasbeen measured ina 96Tcnucleus be-low and much above the particle threshold energy at different J-values using the particle evaporation technique and compared withtheaidofaconsistenttheoreticalcalculation,namelythe ex-actpairing plus independent-particlemodelatfinitetemperature (EP
+
IPM) [23]. Thus, the angular momentum gated thermody-namicparameterscouldbeestimated.Inaddition,tillnowmostof thethermalpropertieshavebeenstudiedinthenucleiwherethe shelleffect(δS) isverysmall(lessthan∼
1 MeV). Thelargeshell effect(δS∼
10 MeV)massregionhasrarelybeenstudiedandthus itisvery importanttoinvestigatehowthe thermodynamic prop-ertiesbehavebecauseofthelargeeffectofnuclearshells.In this Letter, the NLDs and, consequently, the nuclear ther-modynamic quantities(TQ) forfour differentnuclei(184Re, 200Tl, 211Po, and212At)are estimated. Thosenucleiare chosen soasto
understandthe shellanddeformation effects onthe TQ.184Re is deformedwithsmallshelleffect(quadrupoledeformation param-eter
β2
=
0.23 withδS
= −
2.06),whereasother threenucleihave largershelleffects(δS= −
7.32,−
9.56 and−
8.42 for200Tl,211Po, and212At,respectively)butwithalmostzerodeformation [26].In addition,theeffectofeven–odd(211Po)andodd–odd(212At)pair-Fig. 1.Experimental neutronenergyspectra(symbols)alongwiththecascade cal-culations(solidlines)takenfromRefs.[27,28].Thedashedanddashed-dottedlines representthecontributionsfrom1nand2nchannels,respectively.
ingcorrelationsonthethermodynamicpropertiesofatomicnuclei hasalsobeeninvestigated.
In the present work the NLD have been extracted using the previously reportedneutronevaporationspectraatlowexcitation energies by K. Banerjee et al., [27] and Pratap Roy et al., [28], wherethe experiments wereperformed usingthe
α
-beamsfrom the K-130 cyclotron at VECC. The compound nuclei 185Re, 201Tl, 212Po,and213Atwerepopulatedatdifferentinitialexcitationener-gies(E∗
∼
18–50 MeV)inthereactions4He+
181Ta,4He+
197Au, 4He+
208Pband4He+
209Bi,respectivelytoinvestigatetheshellandcollectiveeffectsinNLD.Thelowestexcitationenergydataof 23.2, 23.9, 18.5 and 18.2MeV for 185Re, 201Tl, 212Po, and 213At, respectivelyhavebeenusedtoextracttheNLDstostudythe ther-modynamicproperties.
Itshouldbepointedoutthattheneutronenergyspectrashould haveanegligiblecontaminationfromthenon-compoundreactions toobtaintheNLDusingparticlespectra.Hence,theneutron spec-traatthemostbackwardangle(150◦)wereusedtoextractthe in-verseleveldensityparameterearlier[27,28].Thepreviousstudies [27,28] alsoconfirmedthatthe neutronenergyspectrameasured atbackwardanglesfromalphainducedreactionsindifferentmass regionsaremainlydominatedbythecompoundnuclear(CN) reac-tions. Inorderto identifythepre-dominantcontributor, the neu-tronevaporationspectrafromdifferentnucleiinthedecaychainof theCNwereextractedfromthecascadecode [29] andareshown inFig.1 alongwiththeexperimental data.It wasfound thatthe contribution of the neutron energy En
>
3 MeV is mainly domi-nated by 1n channel andenables usto extract the experimental NLDusingtheapproachpresentedinRef. [23].TheexperimentalNLDwas extractedusingtherelation follow-ingtheapproachpresentedinRefs. [21,30,31],namely
ρ
exp(
EX)
=
ρ
fit(
EX)
(
dσ
/
dE)
exp(
dσ
/
dE)
fit.
(1)Here,
(
dσ
/
dE)exp
and(
dσ
/
dE)fit
are the experimental neutron evaporation andbest-fit theoretical spectra taken fromRefs. [27, 28],respectively.Thequantityρ
fit(EX)
isthebest-fitleveldensity takenfromthecascadecalculation [29],whichisgivenbyρ
(
EX,
J)
=
2J
+
1 12θ
3/2√
aexp
(
2√
aU
)
U2
.
(2)Here,
θ
=
2Ieff¯
h2 , Ieff is the effectiverigid-body moment of inertia,
Fig. 2.Angular momentumgatedNLDsalongwiththeresultsofdifferenttheoretical calculationsfor J=12h¯.FilledsymbolsaretheextractedNLDsinthepresentwork andopensymbolsaretakenfromRef. [37] weightedoverour J-distribution.
istheneutronenergyinthecenter-of-massframe, Sn isthe neu-tron separationenergy; U
=
E∗−
J(Jθ+1)−
P,where P isthe pairing energy term, and E∗ is the initial excitation energy. The leveldensityparameter“a”wasdeducedfromtheexperimentally determined asymptotic NLD parametera by using the Ignatyuk’s parameterization [32] givenbya
= ˜
a[
1+
δ
SU
[
1−
exp(
−
γ
U)
]]
,
(3)where,
γ
=
0.4A4/3 ˜a istherateatwhichtheshelleffectisdamped withincreasing the excitation energy. The values of the ground-state shellcorrection
δ
S are definedasthedifferencesofthe ex-perimental and theoretical (liquid drop) masses, and mentioned previouslyforthefourselectednuclei.Theextractedvaluesfor in-verselevel densityparameter (k=
A/
a˜
) are 9.7±
0.7,10.0±
0.7, 7.8±
0.4 and8.3±
0.3 MeV for184Re, 200Tl, 211Po and212At, re-spectively [27,28]. In this way, the NLDs at different excitation energiesforfiniteangularmomentumare extracted(asshownin Fig. 2) and further used to investigate the thermal properties of atomicnuclei.Anatomicnucleuscanbeassumedasastatisticalensemble be-causeofaveryhighdensityofstatesevenatlowexcitationenergy (few MeV). The NLD is directly related to the partition function of the statistical ensemble,which is a fundamental quantity de-scribing the statistical properties of a system in thermodynamic equilibrium. Mostofother TQscanbe expressedinterms ofthis partition function or its derivative. In principle, the thermody-namicpropertiesofnuclearsystemcanbedescribedbasedonthe grandcanonical ensemble(GCE),canonical ensemble (CE),or mi-crocanonical ensemble(MCE) [33].In thiswork,the CEhas been usedinordertoinvestigatethethermodynamicalpropertiesofall selectednuclei.TheexpressionoftheCEpartitionfunctionisgiven by
Z
(
T,
J)
=
En
En=0
ρ
(
En,
J)
e−En
T
δ
En,
(4)where
ρ
(
En,
J)
isthe NLDat EX=
En andtotalangular momen-tum J, andδ
En is the energy interval. The free energy F, aver-age energy E, entropy S, andheat capacity C can be expressed basedontheCEpartitionfunction(4) asF= −
TlnZ,S= −
∂
F/∂
T, E=
F+
T S, andC=
∂
E/∂
T. Formally,the calculationusing eqn.Fig. 3.Angular-momentum-gated thermodynamicquantitiesasfunctionsof temper-atureobtainedbyusingtheEP+IPM leveldensitiesfor J=12¯h.
(4) requiresaninfinitesummation.However,theextractedNLDsin Fig.2onlycovertheexcitationenergyupto15 MeV.Therefore,the extracted NLD has been compared with the different theoretical calculationssuch asthe EP
+
IPM [34],HFBCS[35], andHFBCfor positive andnegative parities[36]. Finally,thebestmatched the-oretical NLD has beenused to investigatethe thermalproperties ofatomicnuclei.Detailsofthetheoreticalcalculationemployedin thepresentworkarediscussedinRef. [23].The experimental angular momentum gated NLDs compared with the results of different microscopic calculations (EP
+
IPM, HFBCS, and HFBC) are shown in Fig. 2. The level density pa-rameter and angular momentum are measured in the experi-ment andused in thestatistical modelcalculation to extract the NLD. Theuncertainty intheNLD duetothestatisticalmodel pa-rameters has been checked and found to be∼
5–10%. The error due to J-distribution (±
4 h¯
) has been taken care of by using theextractedNLD weightedovertheexperimental J-distribution. The low-energy NLD data (∼
1 MeV, represented by the open circles in Fig. 2) are obtained from the total NLD taken from Ref. [37]. The latter is determined by counting the numbers of experimental discrete levelsinthelow-energyregion. These low-energy data are then multipliedby the spin distribution f(
J)
=
(2
J+
1)exp[−
J(
J+
1)/2σ
2]
/(2
σ
3√
2π
)
withσ
2 beingthe spin cut-off factortaken fromthesystematics [38] and weighted over the experimental angular-momentum distribution for a proper comparison withthehigh-energy data.An overall agreement be-tween theexperimental dataandthepredictionsbytheEP+
IPM calculationsisclearlyobserved,whiletheHFBCSandHFBCcannot explain thedata.Itshouldbe mentionedherethatno normaliza-tionisrequiredwithintheEP+
IPM calculationswhencomparing withtheextractedNLD.As theEP+
IPM resultsmatchwellwith the data, this model is further used to calculate the thermody-namicquantitiesoffourselectednucleiat J=
12¯
h.Fig. 4.Exact pairinggaps(withoutangularmomentumor J=0)asfunctionsof temperatureforfournucleiunderconsideration[a,d],atdifferentdeformationsfor 184Re[b,e],andusingdifferentvaluesofpairinggapsatzerotemperaturefor184Re [c,f].
they show smooth variation at higher T (above T
=
1.6 MeV), wherethe shellstructuresare predictedto be melted [32,39,40]. Mostnoteworthyisthenatureofheatcapacityfor184Re,whichis completelydifferent fromthe other threenuclei. There is a pro-nouncedbump(S-shape)inthe heatcapacityof184Re ataround T∼
0.8 MeV,whichisclosetothecriticaltemperatureTc,where the pairing gap collapses as predicted by the conventional BCS theory.This S-shape in the heatcapacity isinterpreted as a fin-gerprintforpairingtransitioninnuclei. Theheatcapacities211Po, 212At,and200Tldonotshowthepronounced S-shapestructureasexpectedforodd–oddandodd–evennuclei.Itisalsointerestingto notethattheheatcapacitiesoftheisotones211Poand212Atshow verysimilarbehaviorandaddingone protononlyslightlyreduces theheatcapacityinthismassregion.
In order to understand the difference in the heat capacities, the exact neutron and proton pairing gaps have been calculated asfunctionsof T as showninFigs. 4(a) and(d). Althoughthese exact gaps are obtained only for the non-rotating case (without angularmomentumor J
=
0),theyarepredictedtodecreasewith increasing J.However, thedifference betweenthegaps for J=
0 and12h¯
,asshown inFig.4, shouldbe very small[41–43].It is seenthattheexactpairinggapschangedifferentlywithT.Thegap at T<
1 MeV for184Re is quite larger than that of other nuclei andit alsohas the steepest slope.As 184Re nucleusis deformed (β2=
0.23),its pairinggaps havebeencalculatedatdifferent de-formations as shown in Figs. 4(b) and (e). As can be seen from thesefigures,thepairing gapswithβ2
=
0 (no deformation)have thesteepestslope.Consequently,theheatcapacityforβ2
=
0 (and J=
12¯
h)showsaclearS-shapeasseeninFig.5(a).However,the NLD forβ2
=
0 does not matchthe experimental data asshown inFig.5(c). Next,theheatcapacityfor184Re hasbeencalculated bykeepingβ2
=
0.23 andchangingthepairinggapssuch ashigh gap,bestgap,lowgap, andnearlyzerogapasshowninFigs.4(c) and (f).Theheatcapacitiesatlow andzerogapsdonotshowthe S-shape and the corresponding NLDs do not fit the data, while those at high gap and best gap show a clear S-shape as seen in Fig. 5(b). It should be mentioned herethat the NLD for best gapmatches experimental data well and was considered for theFig. 5. [a, b] Heat capacities as functions oftemperature at different deforma-tionsand pairing gaps for 184Renucleus. [c, d]Extracted NLDsalong withthe EP+IPM calculationsatdifferentdeformationsandpairinggapsfor184Renucleus. Thedashed-dottedlinesin[a]standfortheresultsobtainedwithintheEP+IPM withoutthecollectiveenhancementfactor.
thermodynamic calculation. To see ifthe collective enhancement contributesincausingtheS-shapeheatcapacityfor184Re[27,44], theNLDcalculationswithintheEP
+
IPM werecarriedoutby set-ting the collective enhancement factors krot and kvib [34] equal to 1. The obtained results (dash-dotted lines in Fig. 5(a)) show thattheS-shapeoftheheatcapacityremains,thatisthecollective enhancementplays noroleinthisissue.Therefore,itcanbe con-cludedthattheS-shapeofheatcapacityseenin184Reat J=
12¯
h isonlyduetothechangeofpairinggapscausedbythechangeof deformation.Insummary,nuclear leveldensitiesofodd–odd andodd–even nuclei in A
∼
200 have been extracted by utilizing the neutron evaporation energy spectra of Refs. [27,28] measured using the low-energyα
beam. The microscopic EP+
IPM calculation ex-plains well all the experimental data in this mass region and, consequently, is used to study the thermodynamic properties of 184Re, 200Tl, 211Po,and212At nuclei.While theheat capacities of200Tl,211Po,and212At,whicharenearlyspherical,displaythe
cor-rectbehaviorasexpectedincaseofeven–odd/odd–oddsystems,a pronounced S-shaped heat capacity is observed for an odd–odd deformed 184Re nucleus. It has been shown that this observed S-shapeintheheatcapacityof184Reisduetothechangeof pair-ing gapscausedby thechange ofdeformation.Thus, itwouldbe veryinterestingtoextracttheNLDsandstudythethermodynamic propertiesofdifferentdeformednucleiinasystematicwayinnear future.
Acknowledgements
The authors are very grateful to Dr. Pratap Roy for providing the experimental neutron energy spectra. The numerical calcula-tionswithintheEP
+
IPM,whichwerecarriedoutusingthe FOR-TRAN IMSL Library by Visual Numerics on the RIKEN supercom-puterHOKUSAI-GreatWaveSystem, aresupportedby theNational Foundation for Science and Technology Development (NAFOS-TED) of Vietnam through Grant No. 103-04.2017.323. Balaram Dey acknowledges financial support from theSERB-NPDF (India), SERB/PDF/2017/000246.References
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![Fig. 5. [a, b] Heat capacities as functions of temperature at different deforma- deforma-tions and pairing gaps for 184 Re nucleus](https://thumb-us.123doks.com/thumbv2/123dok_us/8200958.2174328/4.918.67.450.101.446/capacities-functions-temperature-different-deforma-deforma-pairing-nucleus.webp)
