PHYSICS REPORTS

EI.SEVIER PhysicsReports242 (1994) 139—148

Microscopictheory of thermodynamicpropertiesof finite nuclei

JamesP. Varya~*,Mahmoud A. Hasanba Departmentof PhysicsandAstronomy,Iowa State University,Ames,IA 50011, USA

b AppliedScienceUniversity, Amman,Jordan

Abstract

We developamethodto evaluatelargeamplitudetime-dependentHartree—Fock(TDHF) dynamicsin asinglemodeand in the presenceof a thermalbath. We first apply the method with realistic effective interactionsin the smalloscillation regimeto isothermalandisentropicbreathingmode oscillations.We presentthe coefficientsfor the T = 0incompressibilitiesandbreathingmode frequenciesfor 160 and40Ca.We obtain a significanteffect on the collectivemass due to the non-locality of the realistic effective interaction.The consequenceis that the interaction producesaremarkablylow valuefor theincompressibilitywhileproviding an adequatedescriptionof thebreathingmodeenergy.We concludethat nuclearmattermay havean incompressibilitysignificantly lower than 200MeV. We presentinitialresultsfor largeamplitudestaticcompressionof 40Caat T = 0 which include the excitationof nucleonsto deltas.

1. Introduction

The time evolutionof hot nucleihasbeeninvestigatedwithin avariety of many-bodymethods.We first reviewsomeearlierresults[1] thatextendtime-dependentHartree—Fock(TDHF) to treattime evolution with assumedglobal thermodynamicequilibrium [2—4]. We generalizethe ap-proachof Ref. [2] to arbitrarypairs of conjugateoperatorsleading to “mode-selected”TDHFsuitablefor large amplitudecollective motion. We implement the methodwith realistic effectiveHamiltonians which, in principle, have more predictive power than purely phenomenologicalHamiltonians.A numberof thermodynamiccoefficients are easily obtainedin studiesof smallamplitude motion andshould serveto broadenthe basis for comparisonsbetweentheory andexperimentandbetweendifferent theories.We thenconsiderthe largeamplitudecompressionoflight nucleiunderastatic loadat temperatureT = 0 to beginto investigatethepopulationof theA-degreeof freedom.

* Correspondingauthor.

0370-1573/94/S7.00 © 1994 ElsevierScienceB.V. All rights reserved.SSDI 03 70-1573 (94) 00020-4

140 J.P. Vary, MA. Hasan/PhysicsReports242 (1994) 139—148

2. TDHF dynamics in one mode at T = 0

The T = 0 time-dependentmean-fieldtheorycurrently providesthe bestdescriptionof nucleardynamicsat low energies[5, 6]. We considertwo single-particleoperators,Q, P interpretedasacollectivecoordinateandacollectivemomentum.Their naturedependson the kind of motionthat we want to focus upon.We require that Q—~Q andP —~ — P under time reversalandthat[P, Q] ~ 0. We then form a constrainedHartree—Fock(CHF) calculation on the many-bodyHamiltonian H by minimizing the functional

G[p] = H[p] — )~1Q[p] — 1~2P[p] . (1)

Theresultingdensitymatrixis afunctionof theconstrainingparametersp = p (‘~i, ).2). BecauseP is

time-oddP(~1,)~2) hasa time-oddpart representinga net collective current.If we interpret G, H, Q, P as functionsof 2~and )L2 throughp then,by the Feynman—Hellman

theorem

= —Q, ~G/ô22= —P. (2)

A Legendre transformation with respect to P and Q providesthe complementaryrelationships

= , ~H/?iP= ~.2 (3)

assumingthat Q(~1,~2)andP~1,)~2)can be inverted everywhere.The equationsof motion areobtainedby requiringthat the Heisenbergequationsfor P, Q be

satisfiedin meanvalue, i.e.,

11 (d/dt)trp~i= trpi [A,~] and h(d/dt) tr p,~h= tr pi [A, /~], (4)

wherethe lower casescript letterssignify matricesin the self-consistentsingle-particlespace.We canthendemonstratethe relations

0 = (d/dt)trp9 = )~2trpi[,~,9]/h= ~H/~P~(P,Q), (5)

= (d/dt)trp,~h= ).1trpi[9,,1h]/h =

Except for the commonfactor ~ thesearecanonicalequationsfor P, Q.Note that ~(P,Q) canbe scaledawayby a renormalizationof P. Q or by a rescalingof thetime

variable. Theseequationsare not restrictedto small amplitudesand constitute“mode-selectedTDHF”.

3. Finite temperature formalism

The extensionto finite temperatureof the resultsin the previoussectionfollows in astandardandstraightforwardmanner.Insteadof the energywe minimize the grand potentialQ with thetemperatureandthe chemicalpotentialconstrainingthe valuesof theentropyS and the particlenumber N. Thus

Q[p] = G[p] — TS[p] — ~tN[p] , (6)

J.P. Vary, M.A. Hasan/PhysicsReports242 (1994) 139—148 141

whereG[p] is definedin Eq. (1) and

S[p] = —tr{plnp + (1— p)ln(i — p)} and N[p] = trp. (7)

We then obtain H,N,S,QandP as functions of ~ T and p., through the dependenceof theself-consistentp on thesequantities.Fromthesewe arriveat the analoguesof Eq. (2)

8Q/8p= —N, ~3Q/aT= —S, ~Q/~1 = —Q, aQ/a..~.2= —P. (8)

A Legendre transformation yields the equivalent of Eq. (3)

= , aH/eP= ‘~2~ ?3H/0S= T, ~H/~N = p., (9)

with the correspondingassumptionsof invertability. Other partial Legendretransforms arerelevant in different situations. For isothermal processes at constant N we would considerthe freeenergyandaparalleldevelopmentwould follow [1]. Equationsof motion for ensembleaveragesareobtainedanalogousto thoseof Eq.(5). Within the assumptions of a mean field treatment, theabovetheory is valid for largeamplitudemotion.

4. Isentropic small oscillations

The study of large amplitude collectivemotion at finite temperature is now reduced to the studyof an energysurfaceobtainedfrom aCHF calculationwith four Lagrangeparameters.Two ofthem (‘ii, )L2) areof dynamicalnatureandrepresenttheforceandvelocity fieldsneededto describecollective motion in agiven mode. The other two (T, p.) embodythe thermalconstraintson themotion.The numericaldeterminationof thesesurfacesis quitefeasiblewith realisticHamiltonians.Oneresult is the functionH(P, Q, S,N) which, for constantS,N, yields thetrajectoryof isentropicmotion as the line of constantH in the phasespaceof P andQ. For smallamplitudesnearanequilibriumpoint with valuesP = 0, Q = q0 the surfacecanbe expandedup to quadratictermsas

H(P,Q,S,N)= iks(S,N)(Q— qo)2 + -~B~(S,N)P2, (10)

where

k5 = a

2H/aQ2IS.N = 02l/~QIs,N and B~= ~2H/~P2Is,N= a22/~PIS,N. (ii)

The equationsof motion from the analoguesof Eq. (5) are

d/dt(Q — qo) = +BP~ and dP/dt = —k~(Q— qo) (12)

for either isentropicor isothermalmotion. Here B andkhave the appropriate subscript and ~ iscalculatedat the equilibrium point. We obtain immediatelyharmonicmotion with frequency

cv=~/~.The calculationaboveis asimple exampleof how thermodynamiccoefficientscanbe obtained

from constrainedfinite temperatureHartree—Fock(FTHF) calculations.Thesecoefficients aredefinedin generalas secondderivativesof energy(or free energy)at an equilibrium point. HerekandB

1 havethe meaningof ageneralizedspringconstantandmassparameterwhoseprecisephysicalsignificancedependson the natureof theoperatorsP andQ.

142 J.P. Vary, M.A. Hasan/PhysicsReports242 (1994) 139—148

5. Ingredients of the calculations

The full Hamiltonian actingon all A nucleons is defined by

HTrei+V+Vc, (13)

wherewe usea relativekineticenergyoperatorTrei, the realisticnucleon—nucleonpotential V. andthe Coulomb potential V~.We then develop an effective Hamiltonian based on the majorassumptionthat the many-bodymethod we invoke is suitable to approximatethe results ofa shell-modeldiagonalizationin a very large but finite no-corebasis. We approximatethetemperature-independenteffective no-coreHamiltonian with Heff = Trei + VCff + V~.In this ap-proximationwe neglectrenormalizationeffects on Trei andon V~,andwe calculateonly leadingordercontributionsto V~.We alsointroducecertainphenomenologicaladjustmentswhich areaimedatcompensatingfor theseapproximationsandfor deficienciesin the Hartree—Fockapprox-imation. For full details refer to Ref. [1].

To describeisoscalarmonopolemotion we usefor P andQ:

Q ~ r~ and j3 rzrz_~(p..r.+ ~ (14)

where~, themeansquareradius,is thecollectivecoordinate,while fi, proportional to the radialflow, is the collective momentum.The commutatoris

= [iP,Q] /h = 2Q~/h. (15)

Having madeachoicefor ~ andfi we cannow assignaphysicalmeaningto the thermodynamiccoefficients defined in Eq. (ii). It is customary[7] to definethefinite nucleusincompressibilityas

KA = r~d2/dr2(E/A)Irro , (16)

wherer0 is the rms radius of the nucleus. In our case

r0 = [(1/A)trpr2]112 = (17)

andKA is relatedto k definedin Eq. (11) by

4<r2>2 ~ — k 18

A ~QAAdditional thermodynamiccoefficientsfor specificheatsandthermalexpansioncanbedefinedandevaluated[1].

ForHamiltonianswith local two-bodyinteractionsandfor ourchoicesof PandQ thecollectivemassparameteris [1]

B1 = öP/~i~2I52=o= mA/h

2Q . (19)

This result hasbeenshownto be true alsofor interactionsof Skyrmetypeby Bohigas[8]. In ourcase,theinteractionis non-localandthereforeEq. (19) will not hold in general.However,we candefinea ratio

= (h2/mAQ)(8P/~2) (20)

f.P. Vary, M.A. Hasan/PhysicsReports242 (1994)139—148 143

and therefore assign the non-locality effects of the interaction to an effective mass. It alsoincorporatesthe finite nucleuseffectsandthe dynamicsof the radial flow. Until amore detailedstudyof thisquantity is donewe mayjust think of ~ as aconvenientway of displayingthe resultsfor aP/a~2.

6. Results for small amplitude motion

A thoroughstudyof the temperaturedependenceof many thermodynamiccoefficientsin 160

and40Cahasbeencompleted[1] usingtheserealistic effectiveHamiltonians.The breathingmode

frequencieshw at T = 0 were found to be in reasonableaccord with experimentor systematictrendsasappropriate.For 160KA wasfoundto be98.2MeV atT = 0 and the corresponding ~was1.02. PhenomenologicalHamiltonianswith similar 160 values of hw andKA havevaluesof K fornuclearmatter(K~)of around 230MeV. On the other hand, for 40Ca, KA is only 91.1 MeV atT = 0 andthe corresponding~ is 0.75. Thisdrop in KA and in ~ with increasing A is significant anddeviatesremarkablyfrom resultspreviouslyreportedin the literature.

Due to thiseffect of non-locality it is within the realm of possibility thataconsistentdescriptionof breathingmodefrequenciesin finite nuclei may lead to K~,,values significantly lower than200MeV. Work to evaluateheaviernuclei is in progressto further investigatethis possibility.

This tendencyfor a low incompressibilityrelative to previousresultsmay provide abasis forunderstandingsupernovaexplosionsvia thepromptmechanismin certainmodels[9].

7. Large amplitude motion

As afirst steptowardsthestudyof largeamplitudemotion,werestrictourselvesto T = 0 andtoastatic constraint.That is, we solve the constrainedHF problemwith ~2 = 0, therebyretainingonly the constraint,)L1 Q which is againthe isoscalarmonopoleradial operatorof Eq. (14).

Sincewe addresslargeamplitudecompressionwe must studytwo majornew issues.First, forsufficiently large compressions,we expect that at least some nucleons will be converted toA resonancesin order to minimizethekineticenergywhich isbeingdrivenup by thePauliprinciplerestrictionson nucleonicstates.Second,we must addressthe issueof what is a sufficient modelspacefor agiven compressionto insureconvergence.

We haveearlierinvestigatedthe excitationsof theA-degreeof freedomin constrainedHF witha limited basisspace[10]. We employedthe samerealistic Heff for nucleons as described aboveexceptthe Coulombpotentialwas turnedoff. For the nucleon—deltatransitionpotentialsandthenucleon—deltaanddelta—deltainteractionsweemployedsemi-phenomenologicalcoupledchannelsinteractions[ii]. We found anucleonto deltaphasetransitionin 40Caat largecompressionbutnot in 160.The characterof the transitionwasreminiscentof afirst orderphasetransitionsincenosolutionsof the constrainedHF equationswereobtainedfor acertainrangeof rms values.

Wenowaddressthesecondissueof an adequatesizefor themodelspaceto treatafixed degreeofcompression.Clearly,the higherthe degreeof compressionwe wish to study,the largerthemodelspacewe needto include.To seethis relationshipquantitativelyweconsidernucleononly degreesof freedom under compression in a succession of enlarged model spaces.

144 f.P. Vary, MA. Hasan/PhysicsReports242 (1994) 139—148

—20 I I

—40

160

-60

shells X~ ~ 1~W

“6 .997 1.410 8.00 —

“7 .998 1.419 7.85”———

1::. 2.22.42.62.8 3

rriiis (fni)

Fig. 1. Constrained spherical HFenergy as a function of the point mass rms radius for 160, Thedifferentcurvesindicatethe behavior in different model spaces. Note that the Hamiltonian is adjusted within each model space to give theexperimental rms and binding energy at equilibrium. The Hamiltonian parameters aregivenin thelegendfor eachmodelspace.

Themethodsfor extendingHeff to largerspacesis describedin Ref. [1]. As additionalshells areincludedit is necessaryto adjust the parametersof Heff to give agreementbetweensphericalHartree—Fock (SHF) and experiment for the binding energy and rms radius with no radialconstraintbut with theCoulombpotentialturnedon. The bindingenergy(point rms massradius)thatwefit in eachmodelspaceis 127.0MeV(2.75fm), 342.1MeV (3.45fm), and483.7MeV(3.73fm)for 160 40Ca and 56Ni, respectively.

To accomplishtheabovefits, weadjustthe overallstrengthof Trei by multiplying with ~ andtheoverall strengthof V~by multiplying with )~.We alsoscalethematrix elementsto anewoscillatorbasischaracterizedby new valuesof hcv, theoscillatorenergyspacingas describedin Ref. [1]. Thenewvaluesof)L~,,~and hw required to obtain the SHFfits in each model spacefor eachnucleusarequotedin Figs. 1—3.

In Figs. 1—3 wedepict the constrained SHF results at T = 0for 160 40Ca and 56Ni, respectively.In each case we plot the HFenergy E(HF) versustherms radiusfor asequenceof modelspacesizesdefinedby thenumberof majoroscillatorshellsincluded.Foreachmodelspacethe insetprovidestheparametervaluesfor theHamiltonianthatgive goodagreementbetweenSHF andexperiment(including 17,, andsetting ~ = 0).

It is clearfrom theseresultsthat thereis goodconvergencein E(HF) versusrms nearequilibriumas a function of the model space size. Thus, the results presented above for the compressibilities aresupportedasconvergedresults.

f.P. Vary, M.A. Hasan/PhysicsReports242 (1994) 139—148 145

100 I I I I I I

—150 I 4OCa

“ shells Xt X,~ ~W117 ~995 1.310 7.25”—200 I 119 .996 1.344 6.55”———

26 2 8 3323436384

rr,iis (fin)

Fig. 2. ConstrainedsphericalHF energy as a functionof thepoint massrms radiusfor 40Ca.Consult the capitontoFig. 1.

—50 I I I I I I

-100 56Njshells ~ X~ 11W

—150 I “7 .985 1.294 753II —119 .990 1.299 7.25”~~—

1111 .990 1.309 6.0011 —“12 .990 1.311 6.941

—200

—250 ~“

3~ ‘~...,\

26 218 3 3234363844244

rrn~(fin)

Fig. 3. ConstrainedsphericalHF energyas a functionof thepoint massrms radiusfor 56Ni. Consult the captiontoFig. 1.

146 J.P. Vary, MA. Hasan/PhysicsReports242 (1994) 139—148

.300.1.,

-310.0 -

NucleonsbeI-

.32~~()‘ -

4

N Ca~ -330.0 N -

Nucleoiis + A’ s -~

-340.0 —_ — -

.350.0 I I Ill II !i3.200 3.250 3.300 3.350 3.400 3.450 3.500

FIllS

Fig. 4. ConstrainedsphericalHF energyas afunction of the point massrms radiusfor 40Ca.Themodelspaceincludessix majorshells for the nucleonsandfive orbitals for the deltas.Theuppercurveomitsthe deltaswhile thelower curveincludesthem.

Thenextmajor featureis that, aswe increasethemodelspacesize,we canincreasethe degreeofcompressionat which convergencein E(HF) is achieved.For example, in 56Ni seven shells areadequateto describethesystemcompressedto an rms of 3.6fm, nineshellsareadequatefor 3.4fm,and11 shellsareadequatefor about3.3fm. Fora radiusof 3.3fm the56Ni hasits volume(basedonthe rms) decreasedby 30% andE(HF) hasincreasedfrom —483.7to —433.9MeV or 49.8MeVincrease.

Finally, weshowin Figs.4 and 5 some initial large amplitude static compression results for 40Cafor nucleons only and for the case where we allow nucleons to be excited to A’s. Weevaluate thesevariational results for the case of six oscillator major shells for the nucleons and five orbitals for theA’s (Os

312,1P3/2,OP1/2,OP3/2,OP5/2).We usethe effectiveHamiltonianfor the nucleonsasdescribed

aboveandthe transitionpotentialsof Ref. [11] for the N—A and A—A interactions.Figure4 displaysthe E(HF) versusrmsradiusresultswith only nucleonsasasolid line andthe

results when A’s are addedas a dashedline. There is virtually no difference in the results atequilibrium but substantial differences emerge even for modest compression. Welimit the range ofcompression to the region where this model space may be considered a reasonable one in light ofthe resultsof Fig. 2.

Theexcitationof theA-degreeof freedomwith compressionin theseT= 0 resultsis displayedinFig. 5. At the maximum compression studied here, the results indicate the conversion of less than10% of a single nucleon to a A.

f.P. Vary, M.A. Hasan/PhysicsReports242 (1994)139—148 147

0.lc

0.08

~ 0.06

40Ca~ 0.04

0.02

o.o~ I I I I I3.240 3.280 3.320 3.360 3.400 3.440

r~1115(1m)

Fig. 5. Delta occupationnumber as a function of the point mass rms radiusunder compressionof40Ca within

constrainedsphericalHF. The calculationis the sameonedepictedas the lower curvein Fig. 4.

Overall theseresults for the populationof A’s under compressionindicate potentially majorconsequences for the nuclear equation of state and we will report on more extensive studies of thisphenomenonelsewhere[12].

Acknowledgements

This work wassupportedin part by NSF GrantNo. INT-9215017.Additional supportfor thiswork was provided by the US Departmentof Energyunder Grant No. DE-FGO2-87ER40371Division of High EnergyandNuclearPhysics.

References

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