Compression-mode resonances in the calcium isotopes and implications for the asymmetry term in nuclear incompressibility

University of Groningen

Compression-mode resonances in the calcium isotopes and implications for the asymmetry

term in nuclear incompressibility

Howard, K. B.; Garg, U.; Itoh, M.; Akimune, H.; Bagchi, S.; Doi, T.; Fujikawa, Y.; Fujiwara, M.;

Furuno, T.; Harakeh, M. N.

Published in:

Physics Letters B

DOI:

10.1016/j.physletb.2019.135185

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Howard, K. B., Garg, U., Itoh, M., Akimune, H., Bagchi, S., Doi, T., Fujikawa, Y., Fujiwara, M., Furuno, T.,

Harakeh, M. N., Hijikata, Y., Inaba, K., Ishida, S., Kalantar-Nayestanaki, N., Kawabata, T., Kawashima, S.,

Kitamura, K., Kobayashi, N., Matsuda, Y., ... Yang, Z. (2020). Compression-mode resonances in the

calcium isotopes and implications for the asymmetry term in nuclear incompressibility. Physics Letters B,

801, [135185]. https://doi.org/10.1016/j.physletb.2019.135185

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Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Compression-mode

resonances

in

the

calcium

isotopes

and

implications

for

the

asymmetry

term

in

nuclear

incompressibility

K.B. Howard

a

,

U. Garg

a

,

∗

,

M. Itoh

b

,

H. Akimune

c

,

S. Bagchi

d

,

e

,

f

,

T. Doi

g

,

Y. Fujikawa

g

,

M. Fujiwara

h

,

T. Furuno

g

,

h

,

M.N. Harakeh

e

,

i

,

Y. Hijikata

g

,

K. Inaba

g

,

S. Ishida

b

,

N. Kalantar-Nayestanaki

i

,

T. Kawabata

j

,

S. Kawashima

c

,

K. Kitamura

c

,

N. Kobayashi

h

,

Y. Matsuda

b

,

A. Nakagawa

b

,

S. Nakamura

h

,

K. Nosaka

c

,

b

,

S. Okamoto

g

,

S. Ota

k

,

S. Weyhmiller

a

,

Z. Yang

h

a_{DepartmentofPhysics,UniversityofNotreDame,NotreDame,IN 46556,USA} b_{CyclotronandRadioisotopeCenter,TohokuUniversity,Sendai980-8578,Japan} c_{DepartmentofPhysics,KonanUniversity,Hyogo658-8501,Japan}

d_{AstronomyandPhysicsDepartment,SaintMary’sUniversity,Halifax,NSB3H3C3,Canada} e_{GSIHelmholtzzentrumfürSchwerionenforschungGmbH,D-64291Darmstadt,Germany} f_{Justus-LiebigUniversity,35392Giessen,Germany}

g_{DepartmentofPhysics,KyotoUniversity,KitashirakawaOiwake,Sakyo,Kyoto606-8502,Japan} h_{ResearchCenterforNuclearPhysics,OsakaUniversity,Osaka567-0047,Japan}

i_{KVI-CART,UniversityofGroningen,9747AAGroningen,theNetherlands} j_{DepartmentofPhysics,OsakaUniversity,Toyonaka,Osaka540-0043,Japan} k_{CenterforNuclearStudy,TheUniversityofTokyo,Wako,Saitama351-0198,Japan}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received12November2019

Receivedinrevisedform17December2019 Accepted19December2019

Availableonline27December2019 Editor:D.F.Geesaman Keywords: Collectivity Giantresonance Nuclearincompressibility Equationofstate

Recent data onisoscalar giant monopole resonance (ISGMR) in the calcium isotopes 40,44,48_{Ca have}

suggested thatKτ,theasymmetryterminthenuclearincompressibility,hasapositivevalue.Avalue

of Kτ >0 is entirely incompatible with present theoretical frameworks and, if correct, would have

far-reaching implications onour understanding ofmyriad nuclear and astrophysical phenomena. This paper presentsresultsofanindependent ISGMRmeasurementwiththe40,42,44,48_Ca(

_α

_,

_α

_)reactionat Eα=386 MeV. Theseresults conclusivelydiscountthe possibilityofapositive value for Kτ,and are

consistentwiththepreviously-obtainedvaluesforthisquantity.

©2020TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

Theisoscalargiantmonopoleresonance(ISGMR)hasbeen well-establishedasthemostdirectmeans bywhichone canconstrain theincompressibility ofnuclearmatter. Theincompressibilityofa nucleusisextractedfromtheresonanceenergy,EISGMR,suchthat:

EISGMR

= ¯

h



KA m



r2 0

,

(1)

where KA is theincompressibility ofthe nucleusof mass A

un-dergoingtheexcitation,m isthefree-nucleonmass,and



r₀2



isthe mean-square radius ofthe ground-state density. Withindifferent

*

Correspondingauthor.

E-mailaddress:garg@nd.edu(U. Garg).

modelframeworks,thevalueofEISGMR isassociatedwithdifferent

momentratiosoftheISGMRstrengthdistribution[1,2],the extrac-tionofwhichistheprimary goalofmanyexperimentsventuring to constrain theincompressibility, K_∞,and, thence, the Equation ofState(EoS)ofinfinitenuclearmatter[3].

Inasystemwithaproton-neutronimbalance,theEoSdepends additionallyontheasymmetryparameter

η

= (

N

−

Z

)/

A,andthe symmetryenergy,S

(

ρ

)

.InthesamewaythattheISGMRprovides a direct measurement of K_∞, the curvature of the EoS of sym-metric nuclearmatter, the trendof measurements ofnucleiwith varying values of

η

yields a direct constraintofthe curvature of S

(

ρ

)

.Foramorecompletediscussionofthemeansbywhich prop-erties of the giant resonances provide constraints on the EoS of suchasymmetricnuclearmatter,wereferthereadertoRefs. [3,4].

https://doi.org/10.1016/j.physletb.2019.135185

0370-2693/©2020TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

2 K.B. Howard et al. / Physics Letters B 801 (2020) 135185

Themicroscopic formalism forextractingproperties ofinfinite nuclearmatter frommeasurements on finitenucleiisdetailedin Ref. [5]. Themacroscopicleptodermousexpansion ofKA interms

ofpropertiesofinfinitenuclearmattergives: KA

≈

K∞

+

KsurfA−1/3

+

Kτ

η

2

+

KCoul

Z2

A4/3

.

(2)

Equation (2) is usefulin determining the value of Kτ for fi-nitenuclei,owinginparttotheisolated dependenceon

η

within the expression,aswell asthefairly minimal changes in the sur-facetermwithinanisotopicchain.Thegeneralprescriptionforthe sameisdetailedinRefs. [6,7],andinvolvesquadraticallyfittingthe dependenceofKA

−

KCoulZ2

/

A4/3 on

η

witha modelfunction of

theformKτ

η

2

₊

_{c,withc beingaconstant.The}

Kτ valuesso ex-tracted are consistent withone another: Kτ

= −

550

±

100 MeV andKτ

= −

555

±

75 MeV,fortheeven- A112−124_Snand106−116_Cd

isotopes,respectively[6–8].

The corresponding definition of K_τ∞ in terms ofproperties of theEoSforinfinitenuclearmatteris1 _[9]:

K_τ∞

=

Ksym

−

6L

−

Q_∞

K_∞L (3)

within which L and Ksym are, respectively, the slope and

curva-tureof S

(

ρ

)

,andQ_∞

/

K_∞ istheskewnessparameterfortheEoS ofsymmetricnuclearmatter. Theimplicationsofthisarethat ex-perimentalconstraintsonKτ arisingfrommeasurementsofKAon

finitenucleiarehelpfulindeterminingthedensitydependenceof thesymmetryenergy;thisargumentispredicatedonthe smooth-ness withwhich the valuesof KA vary acrossthe nuclear chart.

Indeed,ashasbeenarguedinRef. [10], anynuclear structure ef-fectsonpropertiesofthegiantresonanceswhichariseinanarrow regionof thechart ofnuclides woulddramaticallyaffectour un-derstanding of the collective model upon which has rested the understandingoftheseresonances.

Inlightofallthis,theresultsreportedrecentlyfor40,44,48_Caby

theTAMUgroup[11–13] wereverysurprising:themomentratios fortheISGMRand,therefore,theKAvaluesfor40,44,48Caincreased

withincreasing mass number. The most immediateconsequence ofthis, consideringEq.(2), isthat Kτ isapositive quantity,andit was shownin Ref. [12] that a large positive value of Kτ models the data well. In a test of hundreds of energy-density function-als currently in usein the literature, the valuesof Kτ extracted were consistently between

−

800 MeV

≤

Kτ

≤ −

100 MeV [14]. Examination of Eq. (3) also directly suggests that the symmetry energy would need to be extremely soft in order to accommo-date Kτ

>

0 [15]. Moreover, the hydrodynamical model predicts EISGMR

∼

A−1/3,whiletheresultsofRefs. [11–13] indicatedexactly

the opposite: the ISGMR energies increasing with mass number overtheisotopicchain.

Theseresultsclearlydemandedanindependentverification be-foresignificanttheoreticaleffortswereexpendedinunderstanding, and explaining, this unusual and unexpected phenomenon. This Letterpresentstheresultsofsuchanexperiment;itisfoundthat theISGMRsinthemeasuredcalciumisotopes( A

=

40,42,44,48) followthe “normal”patternof EISGMR decreasing withincreasing

mass,rulingout apositivevalueforKτ .

ThemeasurementswerecarriedoutattheResearchCenterfor Nuclear Physics (RCNP) at Osaka University, utilizing a beam of

1 _{ItshouldbenotedthatK}∞

τ isnot equaltothevalueofKτextractedfromfinite

nucleiutilizingthemethodologyofEq.(2),justasK∞=KA.However,throughthe sameself-consistentmechanismsbywhichmeasurementsofKAservetoconstrain

K∞ asdescribedbyBlaizot[5],determining valuesofKτ fromfinitenucleican

constraintheEoSforasymmetricinfinitenuclearmatter.

Fig. 1. Measureddouble-differentialcross-sectionspectrafrom 40,42,44,48_Ca(α,α₎ atθLab=0.7◦,afterparticleidentificationandsubtractionoftheinstrumental

back-ground.

386-MeV

α

-particleswhich was,forall practical purposes, “halo-free”. This beamimpinged onto enriched 40,42,44,48_{Ca target foils}

of areal densities

∼

1–3 mg/cm2. The scattered

α

-particles were momentum analyzed in the high-resolution magnetic spectrom-eter, Grand Raiden [16]. The focal-plane detection system was comprised of a pair of vertical and horizontal position-sensitive multiwire drift chambers in coincidence withplastic scintillators which providedthe particleidentificationsignal [17,18]. Arecent andcomprehensivedescriptionofthe procedureemployed inthe offline data analysis has been presented in Ref. [10]. Here, we briefly revisit the most salient points: in addition to the lateral dispersion of the spectrometer allowing for the scattered parti-cles to be distributed across the horizontal focal plane accord-ing to their momentum, the unique optical properties of Grand Raidenallowfor

α

-particleswhosemomentumtransfersoccurred only at the target to be coherently focused onto the median of the vertical focal plane distribution. Onthe other hand, particles that undergo scattering processes before or after the target are, correspondingly, over- or under-focused along the vertical axis. The latter events constitute the instrumental background, which in thepresent methodologycan be eliminated fromthe inelastic spectraprior tofurtheranalysis. Thisremovesany,andall, ambi-guities associated withmodeling theinstrumental background in the subsequentextractionof theISGMR strengthdistributions. In contrast,themeasurements reportedinRefs. [11–13] employeda phenomenologicalmodeling oftheinstrumental background,thus introducinganadditionaluncertaintyintheanalysis[19];this pro-cesswassuggestedasthemostlikelyreasonbehinddifferencesin the extractedISGMRstrength distributions notedrecentlyforthe

A

∼

90 nuclei[10,20–22].

Inelastic scattering data were obtained over a broad angu-lar range 0◦

≤ θ

Lab

≤

10◦, and the acceptance of the

spectrome-ter along the lateral dispersiveplane ranged fromapproximately 10

≤

Ex

≤

35 MeV.Foreachangularsettingofthespectrometer,a

precisemulti-pointenergycalibrationwasacquiredviathe analy-sisof24_Mg(α

_,

_α

_{)spectra.Theenergylossthroughthetargetfoils}

was accountedforwithinthemodelframework ofSRIM[23].The inelasticangulardistributionswereextractedin200-keV-widebins for40,42,44Ca;inordertoachievecomparablestatistical uncertain-ties, a wider 1-MeV-wide bin size was used for the analysis of

48_Ca.The“0◦_{” spectra,wheretheISGMRcrosssectionsare}

maxi-mal,arepresentedinFig.1.

To extract the strength distributions of the giant resonances, it is necessary to have an optical model parameter (OMP)-set to be employed inthe Distorted-Wave Born Approximation(DWBA)

Table 1

Optical-modelparametersextractedfromfitsto elas-ticscatteringangulardistributionsfor42,44,48_Ca.

Def-initionsoftheparametersareprovidedinRef. [10].

Vvol Wvol RI aI [MeV] [MeV] [fm] [fm] 40_{Ca 37.4 31.6 4.47 0.990} 42_{Ca 37.4 31.6 4.53 0.990} 44_{Ca 37.4 31.1 4.64 0.990} 48_{Ca 41.2 32.7 4.82 0.939}

calculations. To adequately constrain the OMP, elastic scattering data were measured for 42,44,48Ca over a broader angular range (5◦

−

25◦). Thenuclear reactions code

PTOLEMY

was utilized for a

χ

2 _{minimization of DWBA results from a single-folding,}

den-sitydependent,hybridopticalmodelpotential[24] relativetothis datawiththeempiricaldensitydistributionsreportedin[25].The extracted OMPs are presented in Table 1; further details of this procedurehavebeenprovidedelsewhere[10].Becauseof unavail-abilityofelasticscatteringdata on40Ca, OMPsextractedfor42Ca wereemployed forthat nucleus. TheuseofOMPs fromanearby nucleushasbeenshowntohavenegligibleeffectintheresultsof thegiant resonancestrength extraction [7], which isfurther evi-dencedby theminimalvariationinthe OMPsthemselvesasseen inTable1for42_Caand44_{Ca.We furthernotethat Refs. [12] and}

[13] hasalsoemployed the sameOMP-set inthe analyses ofthe

44_Caand48_{Cagiantresonancedatawhichoriginallymotivatedthis}

work.

The Multipole-Decomposition Analysis (MDA) of the inelastic spectrawas carriedoutemploying thenow“standard”procedure, described, for example, in Refs. [10,22,26,27]. The experimental double-differentialcrosssectionsovertheEx

=

10

−

31 MeVregion

weredecomposedintoalinearcombinationoftheDWBAangular distributionsforpureangularmomentumtransfers:

d2

σ

exp

_(θ

c.m.

,

Ex

)

d

dEx

=



λ Aλ

(E

x

)

d2

σ

_λDWBA

(θ

c.m.

,

Ex

)

d

dEx

.

(4)

The Aλ

(

Ex

)

coefficientscorrespond tothe fractionofthe

energy-weighted sum rule (EWSR) for the multipolarity

λ

exhausted within a particular energy bin [2]. DWBA cross sections for isoscalarmodes were included inthe MDA up to

λ

max

=

8, and

thecontributionfromtheisovectorgiantdipoleresonance(IVGDR) wasaccountedforusingtheGoldhaber-Tellermodelandthe avail-able photoneutron data for the calcium isotopes [28,29]. Typical resultsoftheMDAarepresentedinFig.2.

Fromthe Aλ coefficients,the strengthdistributions forthe

IS-GMRwerecalculatedusingthecorrespondingEWSRrelationships [2]. Shown in Fig. 3 are the extracted ISGMR strength distribu-tionsforeachofthecalciumnucleiinvestigatedinthiswork.From theseextractedstrengthdistributions, S

(

Ex

)

,the momentsofthe

strengthdistributionwereextractedintheusualway: mk

=



S(Ex

)E

kxdEx

.

(5)

The moment ratios

√

m1

/

m−1,m1

/

m0, and

√

m3

/

m1 that are

customarilyusedincharacterizingtheexcitationenergyofthe IS-GMR[30] arepresentedinTable2.Thequoteduncertaintieshave beenestimatedusingaMonteCarlosamplingfromtheprobability distributions of the individual Aλ

(

Ex

)

andconstitute a 68%

con-fidence interval. The pattern of moment ratios observed in the calcium isotopic chain(decreasing with A, asexpected fromthe A−1/3rule)iscontrarytothatreportedinRef. [12] viz. increasein themomentratioswithincreasing A.

Fig. 2. Multipole-decomposition analysesfor40,42,44,48_{Caforexcitationenergybins}

centeredat18MeV(leftpanels)and28MeV(rightpanels).Shownarethetotalfits (solidblacklines),aswellasthefittedcontributionsfromtheisoscalarmonopole (red),dipole(blue),quadrupole(green),andhighermultipolemodes(cyan).Also shownisthecontributionfromtheisovectorgiantdipoleresonance(dot-dashed line), basedonknown photoneutroncross-sectiondata andthe Goldhaber-Teller model.

Fig. 3. Extracted isoscalar monopole strength distributions for40,42,44,48_Ca. In addition to the momentratios exhibiting the expected be-havior over the isotopic chain, the demonstrated trend for the extracted finite incompressibilities, KA, is even more illustrative

(see Fig. 4):The agreement ofthe extracted KA valueswith the

behaviormodeledbytheleptodermousexpansionofEq.(2) using theacceptedvaluesfor Kτ and K_∞ israthergood,andstands in starkcontrasttotheresultsfromRef. [12].WhiletheextractedKA

for44Ca isconsistentwiththat whichwasmeasured inRef. [12], the KA forthe extrema of40Ca and 48Ca follow precisely

4 K.B. Howard et al. / Physics Letters B 801 (2020) 135185

Table 2

Percentages of the EWSR (m1)for the ISGMR strength

dis-tributions, as well as the corresponding moment ratios (in MeV) calculated over the energy range 10−31 MeV. The constrained-model(√m1/m−1),centroid(m1/m0),and

scaling-model(√m3/m1)energies arepresented.The corresponding

quantitieswhichwerereportedbythe TAMUgrouparealso shownforcomparison;thesewerecalculatedovertheenergy range9−40 MeV[11–13].Inallcases,thequoteduncertainties inthe%EWSR(m1)areonlystatistical;therecanbe15%–20%

additionaluncertaintyfromtheDWBAcalculationsthemselves (fromthechoiceoftheOMP,forexample).

RCNP 40_Ca 42_Ca 44_Ca 48_Ca m1% 102+₋34 89+ 3 −3 88+ 4 −4 78+ 4 −3  m1 m−1 19.5+0.1 −0.1 19.0+ 0.1 −0.1 18.9+ 0.1 −0.1 19.0+ 0.2 −0.2 m1 m0 20.2+₋00..11 19.7+ 0.1 −0.1 19.5+ 0.1 −0.1 19.5+ 0.1 −0.1  m3 m1 22.3 +0.1 −0.1 21.7+ 0.1 −0.1 21.5+ 0.1 −0.1 21.3+ 0.3 −0.3 TAMU 40_Ca 44_Ca 48_Ca m1% 97+₋1111 75+ 11 −11 95+ 11 −15  m1 m−1 18.3+0.30 −0.30 18.73+ 0.29 −0.29 19.0+ 0.1 −0.1 m1 m0 19.2+0.40 −0.40 19.50+ 0.35 −0.33 19.9+ 0.2 −0.2  m3 m1 20.6+₋00..4040 21.78+ 0.84 −0.72 22.6+ 0.3 −0.3

Fig. 4. The incompressibility,KA,forthecalciumisotopesinvestigatedinthiswork (bluesquares).These werecalculatedwithinthe scalingmodelfromthe experi-mentaldata(EISGMR=√m3/m1,forconsistencywiththepresentationofRef. [12];

seeTable 2).The expectedtrend forthesevaluesutilizing thepreviously docu-mentedcentralvalueforKτ= −550 MeV,andK∞=220 MeVasinputtoEq.(2) ispresented(bluedashedline),alongwiththesamecalculationbutwiththevalue

Kτ= +582 MeVreportedinRef. [12] (reddottedline).Afittothedataleadstoa

curvethatisnearlyidenticaltothatshownabove(bluedashedline)andleadstoa valueofKτ= −510±115 MeV.Forcomparisonpurposes,thedatafromRef. [12]

areshown(redcircles),aswellasthe KA valuescalculatedfromthe ISGMR re-sponsespredictedbytherelativisticFSUGarnetinteraction(greensquares)[15,31]. Thesolidlinesthroughthedatapointsaremerelytoguidetheeye.

additional datapoint for42_{Ca – which was absent in the TAMU}

analysis–that followsthesamegeneraltrendastheother three isotopesfound inthepresentworkinspires greater confidencein ourresults.Thesedata,thus,conclusivelyexcludethepossibilityof apositive Kτ valueforthecalcium nuclei.We alsonote thatthe

fitpresentedfortheTAMUdatainRef. [12] correspondedto K_∞= 200MeV,whichissignificantlylowerthanthecurrentlyaccepted valueof240

±

20 MeVforthisquantity [3,32].

Also presented in Fig. 4 are the KA values derived from the

40,42,44,48_{Ca strength distributions predicted by the FSUGarnet}

[15,31] relativistic interaction. The K_τ∞ has a moderate value of

−

247

.

3 MeV for this particular interaction and, accordingly, the KA valuesare observedtodecrease overtheisotopicchain,

qual-itatively similar to theexperimental results.This trendis indeed expected,andobserved,fortheoverwhelmingmajorityof interac-tionsandmodels[14].

Theseresults,ofcourse,begthequestionastotheoriginofthe differencesintheextractedISGMRresponsesfromthoseobtained bytheTAMUgroup.Themostobviousdifferencebetweenthe ex-perimental techniques liesin the accounting of theinstrumental backgroundandphysicalcontinuum.Whereasinthepresentwork theformerisalmost completelyeliminatedandthephysical con-tinuumisincludedwithintheextractedISGMRstrength,theTAMU group subtracts bothby approximatinga smoothbackground un-derlyingtheinelasticspectra.Asstatedearlier,thishasresultedin similar discrepancies intheextracted ISGMRstrengths, especially atthehigherexcitationenergies(Ex

>

20MeV),inthe A

∼

90

nu-clei[10,20–22].

In summary, motivated by the great cause for concern that would arise were Kτ

>

0 a reality, we have carried out a sys-tematic measurement ofthe ISGMR responseof 40,42,44,48Ca and extracted the nuclear incompressibilities, KA, therefrom. In

con-trast to prior results [11–13], the ISGMR strength distributions, and themetrics that are generallyused to characterize the exci-tation energy of the response, obey expected trends. It may be concluded, therefore, that there are no localstructure effects on the ISGMRstrength distribution inthecalcium region ofthe nu-clear chart and that a positive value for the asymmetry term of nuclearincompressibility,Kτ ,isruledout.

Acknowledgements

We thank Profs. W.G. Newton, J. Piekarewicz, and H. Sagawa for their comments on the implications of a positive Kτ in nu-clearstructureandnuclearastrophysicsapplications.Weare grate-ful, further, to Prof. J. Piekarewicz for providing the results of theFSUGarnet calculations.KBHacknowledges thesupportofthe ArthurJ.SchmittFoundation, aswell astheLiuInstitute forAsia andAsianStudies,andtheCollegeofScience, UniversityofNotre Dame.SWwouldliketothanktheGlynnFamily Honorsprogram at the University ofNotre Dame forfinancial support. This work has been supported in part by the National Science Foundation (GrantNo.PHY-1713857).

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