Critical-Point Structure in Finite Nuclei

Properties of quantum shape-phase transitions in finite nuclei are considered in the framework of the interacting boson model. Special emphasis is paid to the dynamics at the critical-point of a general first-order phase transition.

6

002 ceD 4 1v5102160/th-lcu:nviXraCritical-PointStructureinFiniteNuclei

A.Leviatan

RacahInstituteofPhysics,TheHebrewUniversity,Jerusalem91904,Israel

Abstract.Propertiesofquantumshape-phasetransitionsin nitenucleiareconsideredintheframeworkoftheinteractingbosonmodel.Specialemphasisispaidtothedynamicsatthecritical-pointofageneral rst-orderphasetransition.

Keywords:Quantumshape-phasetransitions,critical-points,interactingbosonmodelofnucleiPACS:21.60Fw,21.10Re,05.70.Fh

Phasetransitionsassociatedwithachangeofshapeareknowntooccurindynamicalsystemssuchasnuclei.Recently,ithasbeenrecognizedthatsuchquantumshape-phasetransitionsareamenabletoanalyticdescriptionsatthecriticalpoints[1,2].Fornu-cleitheseanalyticbenchmarksofcriticalitywereobtainedinthegeometricframeworkofaBohrHamiltonianformacroscopicquadrupoleshapes.Inparticular,theE(5)[1](X(5)[2])benchmarkisapplicabletoasecond-( rst-)ordershape-phasetransitionbetweensphericalanddeformedγ-unstable(axially-symmetric)nuclei.Empiricalevi-denceofthesebenchmarkshavebeenpresented[3,4].Animportantissueconcerningphasetransitionsinrealnucleiistheroleofa nitenumberofnucleons.Thisaspectcanbeaddressedinthealgebraicframeworkoftheinteractingbosonmodel(IBM)[5]whichdescribeslow-lyingquadrupolecollectivestatesinnucleiintermsofasystemofNmonopole(s)andquadrupole(d)bosonsrepresentingvalencenucleonpairs.Thethreedynamicalsymmetrylimitsofthemodel:U(5),SU(3),andO(6),describethedy-namicsofstablenuclearshapes:spherical,axially-deformed,andγ-unstabledeformed.Ageometricvisualizationofthemodelisobtainedbyanintrinsicenergysurfacede nedbytheexpectationvalueoftheHamiltonianinthecoherent(intrinsic)state[6,7]

|β,γ;N =(N!) 1/2(b c)N

|0 ,

(1)

whereb c=(1+β2) 1/2[βcosγd 0+βsinγ(d 2+d 2

)/√

Properties of quantum shape-phase transitions in finite nuclei are considered in the framework of the interacting boson model. Special emphasis is paid to the dynamics at the critical-point of a general first-order phase transition.

0.2

f(β)

0.1

h

β+ β0

β

FIGURE1.

TheIBMenergysurface,Eq.(3),transition.The√atthecriticalpointofa rst-orderphase 2 1+12positionandheightofthebarrierareβ+=1+β0respectively.andh=f(β+)=β

forsmallβ.ThisisthesituationencounteredintheU(5)-O(6)phasetransition.Ina

rst-orderphasetransitiontheenergysurfacehastwocoexistingminimawhichbecomedegenerateatthecriticalpoint.TheU(5)-SU(3)phasetransitionisaspecialcaseofthisclass,forwhichthebarrierseparatingthesphericalandaxially-deformedminimaisex-tremelysmall,andhencethecriticalenergysurfaceisrather at.Numerical[3,10]and(approximate)analytic[11,12]studieswithintheIBM,showthattheU(5)-O(6)[U(5)-SU(3)]criticalHamiltonianscapturetheessentialfeaturesoftheE(5)[X(5)]modelswhichemployin nitesquare-wellpotentials.Inthepresentcontributionweconsiderthepropertiesofageneral rst-orderphasetransitionwithanarbitrarybarrier.Inthiscasethecriticalenergysurfacesatis esb>0andb2=4ac,andforγ=0hastheform

Ecri(β)=cN(N 1)f(β)

f(β)=β2(1+β2) 2(β β0)2.

(3)

AsshowninFig.1,Ecri(β)exhibitsdegeneratesphericalanddeformedminima,atβ=0andβ=β0=2a

Properties of quantum shape-phase transitions in finite nuclei are considered in the framework of the interacting boson model. Special emphasis is paid to the dynamics at the critical-point of a general first-order phase transition.

"deformed"

543210

"spherical"

31

25

32

Probability (%)

100500100500100500100500

32232101

31252202

E

23

22

02

64320

0121

02345678910L

012345678910012345678910

ndnd

FIGURE2.Leftpotion:spectrumofHint,Eq.(5),withh2=0.1,β0=1.3andN=10.Rightportion:thenumberofdbosons(nd)probabilitydistributionforselectedeigenstatesofHint.

d (2).ByconstructionHhasasetofsolvabledeformeddint2µ

eigenstateswithenergyE=0,whicharethestates|β=β0;N,L withangularmomen-tumL=0,2,4,...,2Nprojectedfromtheintrinsicstate|β=β0,γ=0;N .Ithasalso

0 ≡|sN and|N,nd=τ=3,L=3 solvablesphericaleigenstates:|Nd=τ=L= ,n 2(N 3)+5respectively.ForlargeNthespectrumwithenergyE=0andE=3h2β0

ofHintisharmonic,involving5-dimensionalquadrupolevibrationsaboutthesphericalminimumwithfrequencyε,andbothβandγvibrationsaboutthedeformedminimum,withfrequenciesεβandεγgivenby

withP2,µ

=β0s dµ+

2

N,εγ=ε=εβ=h2β0

9

O(6) 5N

,(7)

=nwheren dandN d+n sarethed-bosonandtotal-bosonnumberoperatorsrespectively.

GdenotesthequadraticCasimiroperatorofthegroupGasde nedin[9].InHereC

Properties of quantum shape-phase transitions in finite nuclei are considered in the framework of the interacting boson model. Special emphasis is paid to the dynamics at the critical-point of a general first-order phase transition.

18.26

137

10

+

18.3310

169

+

C3

8

+

Rotor

8

+

11.95

145

12.00

165

6

+

+

6.98

148

6

+

6.31

0.03

+

7.00

157

3.32

140

4

100

+

3.33

143

4

100

+

1.00.0

2+0

+

1.00.0

20

++

16.58

137

10

+

C5

8

+

11.23

145

6.74

148

6

+

6.01

+

0.03

+

3.28

140

4

100

1.00.0

20

+

+

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