J.R.StoneTennessee/Oxford/RIKEN/Adelaide

jirina.stone@physics.ox.ac.uk

Quark-Meson-CouplingModelforFiniteNucleiNuclearMatterandbeyond

ColloquiumatTexasA&MUniversity,Commerce.31January2019

Nuclearforceisnotfundamental!!Twonucleons:nucleon-nucleonscatteringtractablewithmanyparametersnouniquemodelasyet

Manynucleons:forcedependsondensityandmomentum–strong,electro-weakinteractionsplayroleaswellassub-nucleonstructure

Nuclei

Hadronsareobjectsmadeofquarks

BaryonsareNucleons,andstrangeHyperons,Λ ,Σ,Ξ

ddensitynormalizedtonucleardensityd0=0.16fm-3

RHICLHC

NUCLEON-NUCLEONINTERACTIONINNUCLEARMEDIUM:

NUCLEARMANY-BODYPROBLEM(NMBP)ForcesandpotentialsderivedfromfreenucleonsystemsdonotworkdirectlyinfinitenucleiNoexactdirectsolutionoftheNMBPpossible– approximationhavetobesought

Dowehaveenoughconstraintstouniquelydetermineparametersofexistingtheories?

Dowebelievethisisarelevantquestion?

Nuclearmatter:Coldandhot:SymmetricAsymmetricPureneutron

NucleiandparticlesNuclearstructureNuclearreactionsFusionandfissionHeavyioncollisions:Rangeofenergies

AstrophysicsNeutronstarsSupernovaeNuclearsynthesisStarevolutionEarlyUniverseGravitywaves

Outlineofthetalk:

Twoapproachestotheproblem:

1Constructionofmodelswithenoughvariableparameterstoreproduceexperimentwithlessregardtophysicalcontentofthemodel

Examples: densitydependentenergyfunctional,realisticpotentials,ab-initiomodels,one-boson-exchangerelativisticmodels

2Constructionofmodelswithlimitednumberofparametersstronglyconstrainedbytheoryandseekagreementwithexperimentbyphysicalcontent

Example: Quarkmesoncouplingmodel

Infinitedensematter:

Systemofaninfinite numberofinteractingparticlesinaninfinitevolumewithafinite ratioofanumberofparticlesperunitvolume.

NoCoulombforcepresent– nosurfaceeffects–- translationalinvariance

interiorofneutronstars,core-collapsesupernovae,possiblylargeheavynuclei

Testingtheoriesundersimplifiedconditions

Astrophysicsandnuclearmatter

Phasesofnuclearmatter:

Symmetric (equalnumberofprotonsandneutrons)Asymmetric(unequal)Pureneutron

Symmetricmatterhasatdensityρ0 =0.16fm-3 :

Boundstate(E/A) =-16MeVSymmetryenergyE/A(sym)-E/A(pn)=30- 32MeVSlopeofthesymmetryenergyL =40-100MeVIncompressibility:traditionalK =240+/-30MeV

NEWVALUE 250– 315MeVStoneatal.,PRC89,044316(2014)

LudwigBoltzmann1844- 1906

TheEquationofState(EoS):Idealgas:

Nuclearmatter:

P = ε(ρ,T ) ε(ρ,T ) = EA

ρ,T( )ρ⎛⎝⎜

⎞⎠⎟f

∑f

µB = (P + ε ) / ρ

Twokeypoints:

I.TheEoS isdependentoncompositionCONSTITUENTS+INTERACTIONS

IIE/AandITSDENSITYDEPENDENCEmustbedeterminedbynuclearand/orparticlemodels.

Realisticbarenucleon

Phenomenologicaldensitydependence

20-60adjustableparametersSeveralthousandsofdatapoints:Freenucleon-nucleonscatteringandpropertiesofdeuteron

10-15adjustableparametersSeveraltensofdatapoints:Symmetricnuclearmatteratsaturation,Groundstatepropertiesoffinitenuclei

Usedinnuclearmattercalculations,shellmodel,ab-initiotheories

Usedinmean-fieldmodelsnon-relativisticHartree-Fock

NODENSITYDEPENDENTFURTHERTREATMENTNEEDEDTOUSEINNUCLEARENVIRONMENT

DENSITYDEPENDENCEINCLUDEDINANEMPIRICALWAYTHROUGHPARAMETERS

Ab-initiotechniques

BRIEF REPORTS PHYSICAL REVIEW C 74, 047304 (2006)

and single-particle energies in the Bethe-Goldstone equationhas been shown to introduce errors well below 1 MeV for thebinding energy at saturation [19].

Concerning the inclusion of three-body forces in the BHFapproach, we use the formalism developed in Refs. [5–7],namely a microscopic model based on meson exchange withintermediate excitation of nucleon resonances (Delta, Roper,and nucleon-antinucleon). The meson parameters in thismodel are constrained to be compatible with the two-nucleonpotential, where possible.

For the use in BHF calculations, this TBF is reduced toan effective, density-dependent, two-body force by averagingover the third nucleon in the medium, the average beingweighted by the BHF defect function g, which takes accountof the nucleon-nucleon in-medium correlations [6,8,20]:

Vij (r) = ρ

!d3rk

"

σk ,τk

[1 − g(rik)]2[1 − g(rjk)]2Vijk. (5)

The resulting effective two-nucleon potential has the operatorstructure

Vij (r) = (τ i ·τ j )(σ i ·σ j )V τσC (r) + (σ i ·σ j )V σ

C (r) + VC(r)

+ Sij (r)#(τ i ·τ j )V τ

T (r) + VT (r)$

(6)

and the components V τσC , V σ

C , VC, V τT , VT are density depen-

dent. They are added to the bare potential in the Bethe-Goldstone equation (1) and are recalculated together withthe defect function in every iteration step until convergenceis reached. This approach has so far been followed with theParis [6], the V14, and the V18 [7] potentials and the resultswill be shown in the following presentation of our results. Forcomplete details, the reader is refered to Refs. [5–7].

We begin in Fig. 1 with the saturation curves obtained withour set of NN potentials. On the standard BHF level (blackcurves) one obtains in general too strong binding, varyingbetween the results with the Paris, V18, and Bonn C potentials(less binding), and those with the Bonn A, N3LO, and IS(very strong binding). Including TBF (with the Paris, V14,and V18 potentials; red curves) adds considerable repulsionand yields results slightly less repulsive than the DBHF oneswith the Bonn potentials [16] (green curves). This is notsurprising, because it is well known that the major effect of theDBHF approach amounts to including the TBF correspondingto nucleon-antinucleon excitation by 2σ exchange within theBHF calculation [6,7]. This is illustrated for the case of the V18potential (open stars) by the dashed (red) curve in thefigure, which includes only the 2σ -exchange “Z-diagram”TBF contribution. The remaining TBF components are overallattractive and produce the final solid (red) curve in thefigure.

Figure 2 shows the saturation points of symmetric matterextracted from the previous results. Indeed there is a stronglinear correlation between saturation density and energy,confirming the concept of the Coester line. One can roughlyidentify three groups of results: The DBHF results with theBonn potentials as well as the BHF+TBF results with the Paris,V14, and V18 potentials lie in close vicinity of the empiricalvalue. The BHF results with Paris, V14, V18, and Bonn C forma group with about 1–2 MeV too-large binding and saturation

FIG. 1. (Color online) Energy per nucleon of symmetric nuclearmatter obtained with different potentials and theoretical approaches.For details see text.

at about 0.27 fm−3. The remaining potentials, in particular themost recent CD-Bonn, N3LO, and IS, yield strong overbindingat larger density, more than twice saturation density in thelatter cases. From a practical point of view, it would thereforeappear convenient to use the potentials of the former groupfor approximate many-body calculations, because the requiredcorrections are smaller, at least for Brueckner-type approaches.

Historically, there is the observation that the position ofa saturation point on the Coester line seems to be strongly

FIG. 2. (Color online) Saturation points obtained with differentpotentials and theoretical approaches. The (online blue) squareindicates the empirical region.

047304-2

BINDINGENERGYPERPARTICLEINSYMMETRICNUCLEARMATTER

Lietal.,PRC74,047304(2006)

“Realistic”modelsobtainedfromfreenucleonscatteringandrenormalizedtonuclearmedium.

NOTCALIBRATEDTOSYMMETRICNUCLEARMATTERPARAMETERS.

E/AMeV

J.Carlsonetal.Rev.Mod.Phys.87,1065(2015)

Akmaletal.,PRC58,1084(1998)

EoSofpureneutronmatterascalculatedindifferentmodels

Mariseta

l.,PRC

87,054318(2013)

Gezerlis etal.,PRL111,032501(2013)

Effectof3BF

Neutronstars:

Massesaremeasuredwell– 2solarmassRadiiaredifficultanduncertainMomentsofinertianotavailableasyetGravitationalwavessignalsnotfullyconstraining

ModelsyieldonlymassasfunctionofradiusCompositionofthestaruncertainbutcrucialfordeterminationoftheEoS

EOSCRITICALLYDEPENDENTONNUCLEAR/PARTICLEMODELS

Steineretal.,ApJ774:17(2013) J.Carlsonetal.Rev.Mod.Phys.87,1065(2015)

3BFuncertainty

AV8’+UIX

Empirical approach – polytropic EoS: Combination of models and observation dataAssumptions: There is only one EoS of high density matter

Steiner et al., ApJ Letters 765, L5 (2013)

P(nB)=KnBΓ

Latestmodelsmodels(rightpanel)oftheEoS calibratedondatafromthegravitationalwaveobservationoftheGW170817 NSmerger

(ZhaoandLattimer,Phys.Rev.D98,063020(2018))

Binary tidal deformability Λ ,

Finitenuclei

QuantumMonteCarloCoupledClusterMethodNoCoreShellModelGreen'sFunctionApproachesChiraleffectivefieldTheory

ShellmodelLocallyadjusted

DensityfunctionalmodelsGlobalorlocallyadjustedEmpiricalmassmodels

FigureadaptedfromSCIDACreport

Example:

Skyrme non-relativisticenergyfunctional(totalenergyofthesystemincludesdensitydependence

Designedtomodel:

Groundstatepropertiesoffinitenuclei:bindingenergies,radii,shapes,electromagneticmoments,lowamplitudecollectiveexcitations(giantresonances),single-particlestatesinnucleargroundstate

Derivedproperties:neutronskin,particleseparationenergies,particledriplines

Propertiesofnuclearmatter

Tony Hilton Royle Skyrme1922 - 1987

PHENOMENOLOGICALDENSITYDEPENDENTPOTENTIAL:THESKYRMEINTERACTION

P.-G.Reinhard,Phys.Scr.91,023002(2016).

Parametersadjustabletoexperiment:t0,t1,t2,t3,t4,t5,x0,x1,x2,x3,x4,x5,α,β,γ

Infinitenumberofcombinationsoftheparametersthatgivegoodagreementwithexperiment

NUCLEARMATTERPROPERTIESFROMMEANFIELDMODELSWITHDENSITYDEPENDENTSKYRMEEFFECTIVEINTERACTION:

1. 240non-relativisticmodelsbasedontheSkyrmeinteraction-densitydependenteffectivenucleon-nucleonforcedependentonupto15adjustableparameterswererecentlytestedagainstthemostup-to-dateconstraintsonpropertiesofnuclearmatter:

5satisfiedalltheconstraintsM.Dutraetal.,PRC85,035201(2012)BUTONLY2OFTHEM(SQMC(700)andKDEv1)ALSOWORKSATISFACTORILY)INFINITENUCLEI!!!P.Stevensonetal.,arXiv:1210.1592(2012)

EXAMPLESOFRESULTSOFTHESYMMETRYENERGYOFNUCLEARMATTERUSINGDIFFERENTSKYRMEMODELS

Santosetal.,arXiv:1507.05856v1(2015)

S(ρ) = S(ρ0 )− L (ρ0 − ρ)3ρ0

S(ρ0 ) ≈ aSym ≈ 30 MeVSymmetricnuclearmatter:

Energyperparticle

S.Bogner,R.Furnstahl,andA.Schwenk,Prog.Part.Nucl.Phys.65(2010)94R.Machleidt andD.R.Entem,Phys.Rept.503,(2011)1H.Hergert etal.,Phys.Rept.621(2016)165G.R.Jansenetal.,Phys.Rev.Lett.113(2014)142502

AB-INITIO:(eg.MR-iM-SRG,Coupledclusters,No-coreshellmodel,Green’sfunctionmodels,QuantumMonteCarloetc.

TechniquesbasedonEffectivefieldtheory– includeonlynucleon-pioninteraction.

Advantage:Provideground-stateandexcitedstatesproperties(includecorrelationsinthemeanfield)calculateinteraction+operators,estimateerrors

Disadvantages:Dependenceoncut-offparameters,uncertaintyin3-bodyforces,computationallyexpensivetoincludehigherordersintheoriesandtoapplytoheaviernuclei.ALREADYMULTIPLEMODELSEXIST

CourtesyofHeiko Hergert,NS2016,KnoxvilleTN

HeavyIonCollisions

Measurement: Beamenergy35AMeV– 5.5ATeVCollisions(Au,Au),(Sn,Sn),(Cu,Cu)butalso(p,p)foracomparisonTransverseandEllipticalparticleflow

Calculation:Transportmodels-- empiricalmeanfieldpotentialsFittodataà energydensityà P(ε)à theEoS(extrapolationtoequilibrium,zerotemperature,infinitematter)(e.g Danielewicz etal.,Science298,2002,Bao-AnLietal.,Phys.Rep.464,2008)

QuantumMolecularDynamics(e.g.Yingxun Zhang,Zhuxia Li,AkiraOno)

HeavyIoncollisions:

GSI,MSU,TexasA&M,RHIC,LHCexistingFAIR(GSI),NICA(Dubna,Russia)planned

er deflections.) The open and solid points inFig. 2 show measured values for the directedtransverse flow in collisions of 197Au projec-tile and target nuclei at incident kinetic ener-gies Ebeam/A, ranging from about 0.15 to 10GeV per nucleon (29.6 to 1970 GeV totalbeam kinetic energies) and at impact param-eters of b ! 5 to 7 fm (5 " 10#13 to 7 "10#13 cm) (13–16). The scale at the top ofthis figure provides theoretical estimates forthe maximum densities achieved at selectedincident energies. The maximum density in-creases with incident energy; the flow dataare most strongly influenced by pressurescorresponding to densities that are somewhatless than these maximum values.

The data in Fig. 2 display a broad maxi-mum centered at an incident energy of about2 GeV per nucleon. The short dashed curvelabeled “cascade” shows results for the trans-verse flow predicted by Eq. 1, in which themean field is neglected. The disagreement ofthis curve with the data shows that a repulsivemean field at high density is needed to repro-duce these experimental results. The othercurves correspond to predictions using Eq. 1and mean field potentials of the form

U ! $a% " b%&)/[1'(0.4%/%0)&–1] ' (Up

(5)

Here, the constants a, b, and & are chosen toreproduce the binding energy and the satura-tion density of normal nuclear matter whileproviding different dependencies on densityat much higher density values, and (Up de-scribes the momentum dependence of themean field potential (28, 33, 34) (see SOMtext). These curves are labeled by the curva-

ture K § 9 dp/d%)s/% of each EOS about thesaturation density %0. Calculations with largervalues of K, for the mean fields above, gen-erate larger transverse flows, because thosemean fields generate higher pressures at highdensity. The precise values for the pressure athigh density depend on the exact form chosenfor U. To illustrate the dependence of pres-sure on K for these EOSs, we show thepressure for zero temperature symmetricmatter predicted by the EOSs with K ! 210and 300 MeV in Fig. 3. The EOS with K !300 MeV generates about 60% more pres-sure than the one with K ! 210 MeV atdensities of 2 to 5 %0 (Fig. 3).

Complementary information can be ob-tained from the elliptic flow or azimuthalanisotropy (in-plane versus out-of-planeemission) for protons (24, 25, 36). This isquantified by measuring the average value*cos2+,, where + is the azimuthal angle ofthe proton momentum relative to the x axisdefined in Fig. 1. (Here, tan+ ! py/px , wherepx and py are the in-plane and out-of-planecomponents of the momentum perpendicularto the beam.) Experimental determinations of*cos2+, include particles that, in the cen-ter-of-mass frame, have small values for therapidity y and move mainly in directionsperpendicular to the beam axis. Negative val-ues for *cos2+, indicate that more protonsare emitted out of plane (+ - 90°or + -270°) than in plane (+ - 0°or + - 180°), andpositive values for *cos2+, indicate thereverse situation.

Experimental values for *cos2+, for in-cident kinetic energies Ebeam/A ranging from0.4 to 10 GeV per nucleon (78.8 to 1970 GeVtotal beam kinetic energies) and impact pa-rameters of b ! 5 to 7 fm (5 x 10#13 to 7 "10#13 cm) (17–19) are shown in Fig. 4. Neg-ative values for *cos2+,, reflecting a pref-erential out-of-plane emission, are observedat energies below 4 GeV/A, indicating thatthe compressed region expands while the

spectator matter is present and blocks thein-plane emission. Positive values for*cos2+,, reflecting a preferential in-planeemission, are observed at higher incident en-ergies, indicating that the expansion occursafter the spectator matter has passed the com-pressed zone. The curves in Fig. 4 indicatepredictions for several different EOSs. Cal-culations without a mean field, labeled “cas-cade,” provide the most positive values for*cos2+,. More repulsive, higher-pressureEOSs with larger values of K provide morenegative values for *cos2+, at incident en-ergies below 5 GeV per nucleon, reflecting afaster expansion and more blocking by thespectator matter while it is present.

Transverse and elliptic flows are also in-fluenced by the momentum dependencies(Up of the nuclear mean fields and the scat-tering by the residual interaction within thecollision term I indicated in Eq. 1. Experi-mental observables such as the values for*cos2+, measured for peripheral collisions,where matter is compressed only weakly andis far from equilibrated (28), now providesignificant constraints on the momentum de-pendence of the mean fields (21, 28). This isdiscussed further in the SOM (see SOM text).The available data (30) constrain the mean-field momentum dependence up to a densityof about 2 %0. For the calculated resultsshown in Figs. 2 to 4, we use the momentumdependence characterized by an effectivemass m* ! 0.7 mN, where mN is the freenucleon mass, and we extrapolate this depen-dence to still higher densities. We also makedensity-dependent in-medium modificationsto the free nucleon cross-sections followingDanielewicz (28, 32) and constrain these

Fig. 2. Transverse flow results. The solid andopen points show experimental values for thetransverse flow as a function of the incidentenergy per nucleon. The labels “Plastic Ball,”“EOS,” “E877,” and “E895” denote data takenfrom Gustafsson et al. (13), Partlan et al. (14),Barrette et al. (15), and Liu et al. (16), respec-tively. The various lines are the transport the-ory predictions for the transverse flow dis-cussed in the text. %max is the typical maximumdensity achieved in simulations at the respec-tive energy.

Fig. 3. Zero-temperature EOS for symmetricnuclear matter. The shaded region correspondsto the region of pressures consistent with theexperimental flow data. The various curves andlines show predictions for different symmetricmatter EOSs discussed in the text.

Fig. 4. Elliptical flow results. The solid and openpoints show experimental values for the ellip-tical flow as a function of the incident energyper nucleon. The labels “Plastic Ball,” “EOS,”“E895,” and “E877” denote the data of Gutbrodet al. (17), Pinkenburg et al. (18), Pinkenburg etal. (18), and Braun-Munzinger and Stachel (19),respectively. The various lines are the transporttheory predictions for the elliptical flow dis-cussed in the text.

R E S E A R C H A R T I C L E S

22 NOVEMBER 2002 VOL 298 SCIENCE www.sciencemag.org1594

on

Mar

ch 1

7, 2

012

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fro

m

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etal.,M

ol.Biol.

Cell12,3016

(2001).19.A.Fire

etal.,N

ature391,806

(1998).20.X.Wang,C

.Yang,D.Xue,unpublished

results.21.G.M.Stanfield,H

.R.Horvitz,M

ol.Cell5,423

(2000).22.Y.C.Wu,G.M.Stanfield,

H.R.Horvitz,

Genes

Dev.

14,536(2000).

23.E.M.Hedgecock,J.E.Sulston,J.N

.Thomson,Science

220,1277(1983).

24.F.Chenetal.,Science

287,1485(2000).

25.H.Li,H

.Zhu,C.J.Xu,J.Yuan,C

ell94,491

(1998).26.X.Luo,I.Budihardjo,H

.Zou,C.Slaughter,X

.Wang,

Cell94,481

(1998).

27.A.Grossetal.,J.

Biol.Chem.274,1156

(1999).28.X.Wang,G

enesDev.15,2922

(2001).29.B.Conradt,H

.R.Horvitz,C

ell93,519

(1998).30.!!!!

,Cell98,317

(1999).31.J.Cai,Y.Shi,unpublished

results.32.C.Savage

etal.,J.

CellSci.107,2165

(1994).33.Z.Song,H

.Steller,TrendsCellBiol.

9,M49(1999).

34.C.C.Mello,J.M

.Krame,D.Stinchcom

b,V.Ambros,

EMBOJ.10,3959

(1992).35.Wethank

A.Fire,Y.Kohara,and

M.Driscollforreagents

andstrains;

J.Parrish

fordiscussion;

andY.C.Wu,B.

Boswell,andX.D.Liuforhelpful

comments

onthis

manuscript.This

workwassupported

inpartbySearle

ScholarAwards

(D.X.andY.S.),

aRita

AllenScholar

Award(Y.S.),

theBurroughs

Wellcom

eFund

CareerAward(D.X.),and

grantsfrom

theU.S.Departm

entof

Defense

(D.X.)andNIH(D.X.and

Y.S.).

SupportingOnline

Material

www.sciencem

ag.org/cgi/content/full/298/5598/1587/DC1

Materials

andMethods

ReferencesandNotes

Figs.S1andS2

16July2002;accepted

27Septem

ber2002

Determ

inationof

theEquation

ofState

ofDense

Matter

Paweł

Danielew

icz, 1,2Roy

Lacey, 3William

G.Lynch

1*

Nuclear

collisionscancompress

nuclearmatter

todensities

achievedwithin

neutronstars

andwithin

core-collapsesupernovae.

Thesedense

statesof

matter

existmomentarily

beforeexpanding.W

eanalyzed

theflowofmatter

toextract

pressuresinexcess

of1034pascals,

thehighest

recordedunder

laboratory-controlledconditions.

Using

theseanalyses,

weruleoutstrongly

repulsivenuclear

equationsofstate

fromrelativistic

meanfieldtheory

andweakly

repulsiveequations

ofstatewithphase

transitionsatdensities

lessthan

threetimesthat

ofstable

nuclei,butnotequations

ofstate

softenedathigher

densitiesbecause

ofatransform

ationtoquark

matter.

The

nucleon-nucleoninteraction

isgenerally

attractiveat

nucleon-nucleonseparations

of(r)

!1

to2

fm(1

"10

#13

cmto

2"

10#

13

cm)

butbecom

esrepulsive

atsm

allsepara-

tions($

0.5fm

),m

akingnuclear

matter

difficulttocom

press.As

aconsequence,m

oststable

nucleiare

atapproxim

atelythe

same

“saturation”density,%

0&

2.7"

1014

g/cm3,

intheir

interiors,andhigher

densitiesdo

notoccur

naturallyon

Earth.

Matter

atdensities

ofup

to%

!9

%0

may

bepresent

inthe

interiorsof

neutronstars

(1),and

matter

atdensities

upto

about%!

4%

0m

aybe

presentin

thecore

collapseof

typeII

supernovae(2).

The

relationshipbetw

eenpressure,

density,and

temperature

describedby

theequation

ofstate

(EO

S)of

densem

attergoverns

thecom

-pression

achievedin

supernovaeand

neutronstars,

asw

ellas

theirinternal

structureand

many

otherbasic

properties(1–5).

Models

thatextrapolate

theE

OS

fromthe

propertiesof

nucleinear

theirnorm

aldensity

andfrom

nucleon-nucleonscattering

arecom

monly

ex-ploited

tostudy

suchdense

systems

(1,3–9).C

onsequently,it

isim

portantto

testthese

extrapolationsw

ithlaboratory

measurem

entsof

high-densitym

atter.

Nuclear

collisionsprovide

theonly

means

tocom

pressnuclear

matter

tohigh

densityw

ithina

laboratoryenvironm

ent.T

hepres-

suresthat

resultfrom

thehigh

densitiesachieved

duringsuch

collisionsstrongly

in-fluence

them

otionof

ejectedm

atterand

pro-vide

thesensitivity

tothe

EO

Sthatis

neededfor

itsdeterm

ination(10–19).

Fullequilibri-

umis

oftennotachieved

innuclearcollisions.

Therefore,itis

necessaryto

studyexperim

en-tal

observablesthat

areassociated

with

them

otionsof

theejected

matter

andto

describethem

theoreticallyw

itha

dynamical

theory(20–27

).T

orelate

theexperim

entalobservables

tothe

EO

Sand

theotherm

icroscopicsources

ofpressure,w

eapply

am

odelformulated

within

relativisticL

andautheory,

which

includesboth

stableand

excited(delta,

N*)

nucleons(thatis,baryons)

asw

ellaspions

(20,28).Itdescribes

them

otionof

theseparticles

bypredicting

thetim

eevolution

ofthe

(Wigner)

one-bodyphase

spacedistribution

functionsf(r,p,t)

forthese

particles,using

aset

ofB

oltzmann

equationsof

theform

'f't!

()p !*

"()

r f*#

()r !*

"()

pf*$

I

(1)

Inthis

expression,f(r,p,t)

canbe

viewed

semi-classically

asthe

probabilityof

findinga

particle,at

time

t,w

ithm

omentum

pat

positionr.

The

single-particleenergies

!in

Eq.

1are

givenin

alocal

frame

by

!$

KE

+U

(2)

where

KE

isthe

kineticenergy

andU

isthe

average(m

eanfield)

potential,w

hichde-

pendson

theposition

andthe

mom

entumof

theparticle

andis

computed

self-consistentlyusing

thedistribution

functionsf(r,p,t)

thatsatisfy

Eq.1

(20,28).The

particledensity

is%(r,t)

!,

dp"

f(r,p,t);theenergy

densitye

canbe

similarly

computed

from!

andf(r,p,t)

bycarefully

avoidingan

overcountingof

po-tential

energycontributions.

The

collisionintegral

Ion

theright-hand

sideof

Eq.

1governs

them

odificationsof

f(r,p,t)by

elasticand

inelastictw

o-bodycol-

lisionscaused

byshort-range

residualinter-

actions(20,

28).T

hem

otionsof

particlesreflect

acom

plexinterplay

between

suchcollisions

andthe

densityand

mom

entumdependence

ofthe

mean

fields.Experim

entalm

easurements

(12–19,29–31),theoreticalin-novations,

anddetailed

analyses(10,

20–29,32–34

)have

allprovided

important

insightsinto

thesensitivity

ofvarious

observablesto

two-body

collisions(29,

32)and

thedensity

andm

omentum

dependence(28,

33,34

)of

them

eanfields.

The

presentw

orkbuilds

onthese

earlierpioneering

efforts.Compression

andexpansion

dynamics

inenergetic

nucleus-nucleuscollisions.

Collision

dynamics

playan

important

rolein

studiesof

theE

OS.

Severalaspects

ofthese

dynamics

areillustrated

inFig.1

fora

colli-sion

between

two

Au

nucleiat

anincident

kineticenergy

of2

GeV

pernucleon

(394G

eV).

The

observablessensitive

tothe

EO

Sare

chieflyrelated

tothe

flowof

particlesfrom

thehigh-density

regionin

directionsperpendicular

(transverse)to

thebeam

axis.T

hisflow

isinitially

zerobutgrow

sw

ithtim

eas

thedensity

grows

andpressure

gradientsdevelop

indirections

transverseto

thebeam

axis.T

hepressure

canbe

calculatedin

theequilibrium

limit

bytaking

thepartial

deriv-ative

ofthe

energydensity

ew

ithrespect

tothe

baryon(prim

arilynucleon)

density%

P$

%2

"! '(e/%*"'% # $

s/%(3)

atconstant

entropyper

nucleons/%

inthe

collidingsystem

.T

hepressure

developedin

thesim

ulatedcollisions

(Fig.1)is

computed

microscopically

fromthe

pressure-stressten-

sorT

ij,which

isthe

nonequilibriumanalog

ofthe

pressure[see

supportingonline

material

1NationalSuperconducting

Cyclotron

Laboratoryand

Departm

entofPhysics

andAstronom

y,Michigan

StateUniversity,East

Lansing,MI48824

–1321,USA.

2Gesellschaft

furSchw

erionenforschung,64291

Darmstadt,

Germany.

3Departm

entofChemistry,

StateUniversity

ofNewYork,

StonyBrook,

NY

11794–3400,U

SA.

*Towhom

correspondenceshould

beaddressed.

E-mail:lynch@

nscl.msu.edu

RESEA

RCH

ARTICLES

22NOVEMBER2002

VOL298

SCIENCEwww.sciencem

ag.org1592

on March 17, 2012www.sciencemag.orgDownloaded from

Transportmodelswithparametersfittedtodataonellipticalandtransverseflow

Science298,1592(2002)

MatterinHICandcompactobjectshavedifferentEoS:CentralA-Acollision:StronglybeamenergydependentBeamenergy<1GeV/A:

Temperature:<50MeVEnergydensity:~1-2GeV/fm3

Baryondensity<ρ0Timescaletocool-down:10-22-24sNoneutrinos

Strong Interaction:(S,BandLconserved)Timescale10-24s

NEARLYSYMMETRICMATTER

InelasticNNscatterings,N,N*,Δ’sLOTSofPIONSstrangenesslessimportant(kaons)

?(Local)EQUILIBRIUM?

Proto-neutronstar:(progenitormassdependent)~8– 20solarmass

Temperature:<50MeVEnergydensity:~1GeV/fm3Baryondensity~2-3ρsTimescaletocool-down:1-10sNeutrinorichmatter

Strong+Weak Interaction:(BandLcon)Timescale10-10 s(ρ andTdependent)

HIGHLYASYMMETRICMATTER

HigherT:strangenessproducedininweakprocesses

LowerT:freeze-outN,strangebaryonsandmesons,NOPIONS,leptons

EQUILIBRIUM

0.6

0.8

1

1.2

1.4

ρ n,p/ρ

0

n 120120n 100120n 100100p 120120p 100120p 100100

0 200 400 600 800Ebeam [MeV/nucl]

0

0.05

0.1

0.15

δ

120120100120100100

200 400 600 800Ebeam [MeV/nucl]

200 400 600 800Ebeam [MeV/nucl]

200 400 600 800Ebeam [MeV/nucl]

S SM SMS SSM

Transportmodelresultsformaximumprotonandneutrondensityandisisospinasymmetryδ inacollisionodSn nucleiasafunctionofthelabbeamenergyupto800MeV/nucl

JRS,P.Danielewicz andY.Iwata,PRC96,014612(2017)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4ρ [fm-3]

0

20

40

60

S [M

eV]

S SM SMSSSMSV-bas

FourmodelsfordensitydependenceofthesymmetryenergyS,SM,SMS,SSM

Threeobservations:(i)Themaximumdensitydoesexceed1.4ρ0(ii)Theisospinasymmetryhasdoesnotincreasewithbeamenergy- thesystemretainsmemoryoftheoriginalasymmetry

(iii)TheobservablesareonlyweaklydependentonS(ρ)

NewPhysics(stillonaphenomenologicallevel)?

Quark-Meson-CouplinginsteadNucleon-MesonCoupling?

FUNDAMENTALQUESTIONS:

1.Isthenucleonimmutable?

2.Whenimmersedtoanuclearmediumwithappliedscalarfieldwithstrengthoforderofhalfofitsmassisitreallyunchangeable?

3.Isthiseffectrelevanttonuclearstructure?

Energysc

alesinhad

ronan

dNuclei

Replaceinteractionbetweennucleons

Byinteractionbetweenvalencequarksinindividualnon-overlappingnucleons

LookforthemodificationofthequarkdynamicsinanucleonduetopresenceofothernucleonsACCOUNTFORTHEMEDIUMEFFECT

History:Original:PierreGuichon(Saclay),PLB200,235(1988)SeveralvariantsdevelopedinJapan,Europe,Brazil,Korea,ChinaLatest:JRS,Guichon,ReinhardandThomas,PRL116,092501(2016)

Quark-Meson-CouplingModel(many-bodyeffectiveHamiltonian)

6

FIG. 10: Isosurface and surface plot of C(y) for a 10-sweepsmeared T-shape source with quark positions as in the seventhconfiguration of Table I. The maximum expulsion is 8.3% andthe isosurface is set to 4.4%. Further details are described inthe caption of Fig. 6.

FIG. 11: Isosurface and surface plot of C(y) for a 10-sweepsmeared Y-shape source with quark positions as in the seventhconfiguration of Table I. The maximum expulsion is 8.3% andthe isosurface is set to 4.4%. Further details are described inthe caption of Fig. 6.

FIG. 12: Isosurface and surface plot of C(y) for a 10-sweepsmeared L-shape source with quark separations of ℓ = 10.The maximum expulsion is 8.8% and the isosurface is set to4.4%. Further details are described in the caption of Fig. 6.

tive three-quark potential for the various quark positions,source shapes and Euclidean time evolutions. The vac-uum expectation value for W3Q is

⟨W3Q(τ)⟩ =∞!

n=0

Cn exp(−a Vn τ), (4)

where Vn is the potential energy of the n-th excited stateand Cn describes the overlap of the source with the n-th state. The effective potential is extracted from theWilson loop via the standard ratio

a V (r, τ) = ln

"

W3Q(r, τ)

W3Q(r, τ + 1)

#

. (5)

If the ground state is indeed dominant, plotting V as afunction of τ will show a plateau and any curvature canbe associated with excited state contributions. Statisticaluncertainties are estimated via the jackknife method [16].

Our results for the various quark positions and sourceshapes are shown in Fig. 16. All small shapes are stableagainst noise over a long period of time evolution andeven some of the largest shapes show some stability be-fore being lost into the noise.

Robust plateaus are revealed for the first four quarkpositions of Table I for the T and Y shape sources. Thissuggests the ground state has been isolated and indeedthe four lowest effective potentials of the T- and Y-shapesources agree. This result was foreseen in the qualita-tive analysis where Figs. 6 and 7 for the T- and Y-shapesources respectively displayed the same correlations be-tween the action density and the quark positions.

Conversely, the disagreement between Figs. 10 and 11indicates the ground state has not been isolated in oneor possibly both cases. Indeed the nontrivial slopes ofthe seventh effective potentials of Fig. 16 for the Y- andT-shape sources confirm this. On the other hand, thecurves are sufficiently flat to estimate an effective poten-tial at small values of τ , and given knowledge of the nodeposition from our qualitative analysis, one can make con-tact with models for the effective potential.

The expected r dependence of the baryonic potentialis [2, 4]

V3Q =3

2V0 −

1

2

!

j<k

g2CF

4πrjk+ σL , (6)

where CF = 4/3, σ is the string tension of the qq poten-tial and L is a length linking the quarks. There are twomodels which predominate the discussion of L; namelythe ∆ and Y ansatze.

In the ∆-ansatz, the potential is expressed by a sumof two body potentials [4]. In this case L = L∆/2 =3⟨dqq⟩/2 where L∆ is the sum of the inter-quark dis-tances. In the Y-ansatz [2, 6], L = LY = 3⟨rs⟩ is thesum of the distances of the quarks to the Fermat point.

LatticeQCDsimulationsofthestructureofanucleon

Schematicillustrationofamulti-baryonsystem:baryonscanappearcloseenoughtotoexchangemesonsbetweenquarksthroughdisturbanceintheQCDvacuum(Guichon)

Bissey,PRD76,114512(2007)

Exchangeσ,ω,ρ

http://www.physics.adelaide.edu.au/theory/staff/leinweber/VisualQCD/Nobel/index.html

Typicalstructureofgluon-fieldconfigurations,averagedovertime,describingthethevacuumpropertiesofQCD (actiondensity– left),chargedensity(right)

QCDvacuumfluctuationsareexpelledfromtheinteriorregionofabaryonliketheprotonisanimatedatleftandaquark-antiquarkpairattheright

Guichon,JRS,ThomasProg.Part.Nucl.Phys.2018

1.TakeabaryoninmediumasanMITbag(withoneqluon exchange)immersedinameandensitydependentscalarfield

2Approximatethesolutionofthebagequationsbyadynamicalnucleoneffectivemas

Thelasttermrepresentstheresponseofa nucleontothescalarfield withdbeingthescalar polarizability – theoriginofMANY- BODYforcesinQMC(noadditionalparameters– discalculated)

3.Thenucleonmesoncouplingconstantsrelatetothequarkmesoncouplingsas

whereqisthevalencequarkwavefunctionforafreebag.

OUTLINEOFTHEQMCMODEL

Guichonetal.,Nucl.Phys.A772,1,(2006),GuichonandThomasPRL93,132502(2004)

d = 0.0044 + 0.211*RB − 0.0357RB2

MN* (σ ) = MN − gσNσ + d

2(gσNσ )

2

σ

gσN = 3gσ

q drBag∫ qq(r ), gωN = 3gω

q , gρN = gρq

Guichon,Stone,Thomas:ProgressinParticleandNuclearPhysics100(2018)262–297

4. Solveself-consistentlyforthemesonfieldsusingthecondition

5.ConstructaquantizedHamiltonian/Lagrangianforagivensystem(non-rel)forfinitenucleiandrelativisticfornuclearmatter.

ThesedependonlyonnucleondynamicsandaresolvedbystandardHartree-Fockmethodstodetermineobservablesofinterest.

-densitydependenceoftheenergydensityfunctionalismicroscopically- calculated- multi-bodyforcesareautomaticallyincluded-exchangetermsarealwaysincluded- spin-orbittermappearsnaturallyinbothNRandRmodels- protonandneutrons.p.potentialsarecalculated– noneedforfitting.

∂E / ∂σ = 0 ∂E / ∂ω = 0 ∂E / ∂ρ = 0

Parameters(verylittlemaneuveringspace):

I.3nucleon-mesoncouplingconstantsinvacuumWedefine(forconvenience)

gσN ,gωN ,gρN

GσN = gσN2 /mσ

2 GωN = gωN2 /mω

2 GρN = gρN2 /mρ

2

II.Mesonmasses:ω,ρ,πkeeptheirphysicalvalues650MeV<Mσ <700MeV

III.Bagradius(freenucleonradius):1fm(limitedsensitivitywithinchange+/- 20%)

Allotherparameterseithercalculatedwithinthemodelorfixedbysymmetry

I. Nuclearmatterparametersusedinthefit:

-17MeV<E0 <-15MeV0.15fm<ρ0 <0.17fm-3

28MeV<J <34MeVL>22MeV250MeV<K0 <350MeV

+II.HF+BCSmodelincludingconstrainedcalculationatfixedquadrupolemomentQ20(twomoreparameters– protonandneutronpairingstrengths)

Calculated:groundstatebindingenergies,axialandoctupoledeformations,two-neutronseparationenergies,electromagneticproperties,single-particleenergies,pairinggaps,neutronskin,spin-orbitsplitting

DetailsinJRS,Guichon,ReinhardandThomas,PRL116,092501(2016)

TheparametersetisUNIQUEforeachmodelHamiltonian

Quadrupoledeformationofeven-evenGd isotopesacrosstheisotopicchain

Closed

shell

-0.2

-0.1

0

0.1

0.2

0.3

0.4

β 2

QMCSV-minFRDM

136 140 144 148 152 156 160 164Mass Number

-0.05

0

0.05

0.1

0.15

β 4

FIG. 4: (Color on-line) Ground state quadrupole deformation parameters β2 and β4 of even-even

Gd isotopes as calculated with QMC and SV-min, compared with FRDM.

11

SV-minKlupfel etal.(11parameters)Phys.Rev.C79,034310(2009)FRDMMolleretal.,ADNDT,59,185(1995)

108

Fig.6.2 Illustration of the multi pole deformations for A = 1, ... ,4. In each case the shapes are plotted for 0>-0 = na>-, with n = 0, ... ,3 and at = 0.15, a2 = 0.25, a3 = a4 = 0.15. The figures are not scaled correctly with respect to each other and the volume is also not kept constant with in-creasing deformation. Note the complicated shape con-stant with increasing defor-mation. Note the compli-cated shapes for larger de-formation.

6. Collective Models

preserve the invariance of the surface definition. As the spherical harmonics have a parity of (-1)\ so must a AW

6.2.2 Types of Multipole Deformations

The general expansion of the nuclear surface in (6.24) allows for arbitrary distor-tions. In this section the physical meaning of the various multi pole orders and their application will be examined for increasing values of A.

1. The monopole mode, A = O. The spherical harmonic Yoo(f?) is constant, so that a non vanishing value of aoo corresponds to a change of the radius of the sphere. The associated excitation is the so-called breathing mode of the nucleus. Because of the large amount of energy needed for compression of nuclear matter, however, this mode is far too high in energy to be important for the low-energy spectra discussed here. l The deformation parameter aoo can be used to cancel the overall density change present as a side effect in the other multipole deformations. Calculating the nuclear volume (see Exercise 6.1) shows that this requires

(6.31)

to second order.

2. Dipole deformations, A = I to lowest order, really do not correspond to a deformation of the nucleus but rather to a shift of the center of mass (see Exercise 6.1). Thus to lowest order A = 1 corresponds only to a translation of the nucleus, and should be disregarded for nuclear excitations.

3. Quadrupole deformations: the modes with A = 2 turn out to be the most important collective excitations of the nucleus. Most of the coming discussion of collective models will be devoted to this case, so a more detailed treatment is given in the next section.

3. The octupole deformations, A = 3, are the principal asymmetric modes of the nucleus associated with negative-parity bands. An octupole-deformed shape looks somewhat like a pear.

4. Hexadecupole deformations, A = 4: this is the highest angular momentum that has been of any importance in nuclear theory. While there is no evidence for pure hexadecupole excitation in spectra, it seems to play an important role as an admixture to quadrupole excitations and for the ground-state shape of heavy nuclei.

5. Modes with higher angular momentum are of no practical importance. It should be mentioned, though, that there is also a fundamental limitation in A, be-cause the range of the individual "bumps" on the nuclear surface described by

t The compression is seen experimentally and is quite important for determining the com-pression properties of nuclear matter. The assumption of a sharp surface, though, is not good enough for a quantitative description as its energy depends sensitively on how the density profile on the surface changes during the vibration.

106 6. Collective Models

we saw it is not quantitatively reliable, so we try to fit it to experimental data; in practice values between -I and % work well.

The last term is the density-functional version of the surface energy. It is ba-sically the energy due to a Yukawa interaction, but the energy for a homogeneous distribution has been subtracted (that is why e(r') - e(r) appears), so that it con-tributes only on the surface.

6.2 Nuclear Surface Deformations

6.2.1 General Parametrization

The excitation spectra of even-even nuclei in the energy range of up to 2 MeV show characteristic band structures that are interpreted as vibrations and rota-tions of the nuclear surface in the geometric collective model first proposed by Bohr and Mottelson [B052, B053a, B054] and elaborated by Faessler and Greiner [Fa62a,b, Fa64a,b, Fa65a-c]. The underlying physical picture is that of a nucleus modeled by a classical charged liquid drop, generalizing the concept that led to the Bethe-Weizsacker mass formula to dynamic excitation of the nucleus. For low-energy excitations the compression of nuclear matter is unimportant, and the thickness of the nuclear surface layer will also be neglected, so that we start with the model of a liquid drop of constant density and with a sharp surface. The in-terior structure, i.e., the existence of individual nucleons, is neglected in favor of the picture of homogeneous fluid-like nuclear matter.

The comparison of the results with experiment will sho'Y to what extent these approximations may be justified. However, it is already clear that the model of a liquid drop will be applicable only if the size of the nucleon can be neglected with respect to the size of the nucleus as a whole. This, as well as the neglect of the surface layer thickness, should work best in the case of heavy nuclei.

With these assumptions, the moving nuclear surface may be described quite generally by an expansion in spherical harmonics with time-dependent shape pa-rameters as coefficients:

R(B,¢,t)=RO(I+ f: t A=OIl=-A

(6.24)

where R(B,¢,t) denotes the nuclear radius in the direction (B,¢) at time t, and Ro is the radius of the spherical nucleus, which is realized when all the a AIl vanish.

In all the following formulas the ranges of summation ..\ = 0, ... , 00 and fl = -..\, ... ,..\ will be implied if not otherwise indicated.

The time-dependent amplitudes aAIl(t) describe the vibrations of the nucleus and thus serve as collective coordinates.

To complete the formulation of the collective model a Hamiltonian depending on the a AIl and their suitably defined conjugate momenta has to be set up. Before doing that, however, let us examine the physical meaning of the a AIl in more detail.

β2 quadrupole

β3 octupole

β4hexadecapole

Deformationparameters

146Gd(64,82)

P.A.M. Guichon et al. / Nuclear Physics A 772 (2006) 1–19 13

expect different values for nuclei with large isospin asymmetry. However, as can be seen fromTable 4, the differences are rather small. This is primarily because the spin–orbit splitting de-pends on the product of the spin–orbit form factor and the corresponding single-particle wavefunctions. Thus, if the wave functions are not strongly localised in the surface region, whereWτ (r) is effective, the influence of the isospin dependence of Wτ (r) upon the splitting need notbe so significant.

In Figs. 1, 2 we show the proton and neutron densities calculated with the QMC model andwith the Sly4 Skyrme force [19]. In the proton case we also show the experimental values [23].

Fig. 1. Proton densities of the QMC model compared with experiment and the prediction of the Skyrme Sly4 force.

Fig. 2. Neutron densities of the QMC model compared with the prediction of the Skyrme Sly4 force.

QMCprotondensitydistributioncomparedwithexperimentandSkyrme SLy4

40Ca

48Ca

16O

208Pb

-20

-15

-10

-5

0

-20

-15

-10

-5

0

0f7/2

0f5/2

1p3/2

1p1/2

0g9/2

1p3/2

0f5/2

1p1/2

0g9/2

1d5/2

protons

neutrons

78Ni (Z=28, N=50)QMC

exp cal

exp cal

neutrons

Skyrme SV-min

calexp

protonsprotons

exp cal

0f7/2

0f52

1p3/21p1/2

0g9/21p3/2

0f5/21p1/2

0g9/2

1d5/2

Spectrumofsingle-particleenergiesinthegroundstateof78Ni

DatatakenfromGrawe etal.,Rep.Prog.Phys.70,1525(2007)

GroundstatebindingenergyforselectedSHEnuclei

SkyrmeSV-minP.Klupfel etal.Phys.Rev.C79,034310(2009)

FiniteRangeDropletModel(FRDM)P.Molleretal.,ADNDT,59,185(1995)

Micro-macromodel(MM)I.Muntian etal.,AtomicNuclei,66,1015(2003)Acta Phys.Pol B34,2073(2003

Theory:

Experiment:H.J.Huang etal.,ChinesePhys.C41,030002and030003(2017)

Physicsofeven-evensuperheavy nucleiwith96<Z<110intheQuark-Meson-CouplingModelJ.R.Stone,K.Morita,P.A.M.Guichon,A.W.Thomashttps://arxiv.org/abs/1901.06064

8 10 12 14 16R [km}

0

0.5

1

1.5

2

2.5

Mg[s

olar

mas

s]

F-QMC700F-QMC700�4N-QMC700N-QMC�4

8 10 12 14 16R [km]

0

0.5

1

1.5

2

2.5

BJAPRSkM*

Mass-radiusofaneutronstarpredictionbytheQMCmodel(relativisticversion)

2Msolar massneutronstarwithfullhyperonoctetpredicted3yearsbeforeitsobservation– nohyperonpuzzle

0 0.2 0.4 0.6 0.8 1 1.20.001

0.01

0.1

1Po

pula

tion

0 0.2 0.4 0.6 0.8 1 1.2� [fm-3]

0.001

0.01

0.1

1

Popu

latio

n

QMC700

QMC700�4

np

n

e

�0�

�

�

µ

e

p

µ

�0

�

��

CompositionofmatterinaneutronstarcoreascalculatedintheQMCmodel(nucleon-hyperoninteractioncalculatedinameanfieldapproximation)

ThefirstevidenceofadeeplyboundstateofΞ--14Nsystem"K.Nakazawaetal.,Prog.Theor.Exp.Phys.(2015),033D02

Non-ExistenceofboundΣ hypernuclei

ExistenceofΛ - hypernuclei

Stone,Guichon,Matevosyan,Thomas,NPA792,341(2007)

Existenceofcascade-hypernucleus

Concludingremarks

1. Currentlowenergynuclearphysicsfacesanumberofchallenges

1. WeintroducetheQuark-Meson-MesonCouplingmodelbasedonstructureofthenucleon.Themodelhasasmallnumberofwellconstrainedparameters– UNIQUESETPERHAMILTONIAN

3.Theformalismworksforanybaryonswithoutincreasednumberofparameters

4.Themodelisconsistentwhenappliedtonuclearmatter,finitenuclei,hyperonicphysicsandcompactobjects

5. Isthereanindependentexperimentalconfirmationofamediumeffectadynamicalstructureofabaryon?EMC?

6.Whatisthewayforward?

LatticeQCD

increasinglysuccessfulnumericalexperimentforfreesystems

QuantumChromodynamics(QCD):CompletedescriptionofthestronginteractionParameters:Couplingconstantg andquarkmasses

FrankWilczek,PhysicsToday,August2000

CourtesyJamesZanotti

TWOPROBLEMS:THEEXPONENTIALWEIGHTINGFACTOROSCILLATESATFINITECHEMICALPOTENTIALS

OVERLAPPROBLEM(PLEASEGOOGLEFORDETAILS)

Thanksto:

Myco-authorsPierreGuichonandTonyThomas

P.-G.ReinhardforimplementationoftheQMCmodelintheSkyax 2Dcode

and

ChrisDűllmann forvaluablecommentsandsuggestionsconcerningexperimentaldataandfutureprospectsofSHE.