J.R.StoneTennessee/Oxford/RIKEN/Adelaide
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
18.D.Arnoult
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.