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Shells Evolution and Core Excitations in Semi-Magic
Nuclei
F. Nowacki
To cite this version:
Fred
eri NOWACKI Institut Pluridis iplinaireHubertCurien
23rueduLoess BP28
F67037StrasbourgCedex,Fran e
Re entadvan es in LargeShell Model al ulationsallownowto treat extended valen e spa es and more omplete des riptionsof (semi-)magi nu lei an be a hieved with in lusion of ore ex itations. The interplay betweenshell evolution and ore ex itations in semi-magi nu lei will be illustratedfor Tin isotopi hains in the framework of Large Shell Model al ulations. pnandnnmonopolerelativein uen ewillbetra edba kon E e tiveSingleParti leEnergiesandB(E2)'s.
1. Introdu tion
Like other nite quantum many-body systems, atomi nu lei are har-a terized by shell stru ture but also residual orrelations among its on-stituents. Closetostability,shellstru turehasbeenetablishedforlongnow andshowupinidenti edmagi numbers(2,8,20,28,40,50...) orrespond-ing to spheri al losed shell nu lei. On theopposite, when a tive nu leons an intera t throughresidual intera tion, orrelations an develop leading to olle tive deformation orsuper uidity. But orrelations alsomanifestin low-lying spe trum of magi nu leiand thepresen e of deformed states of manyparti les-holesnature havebeenobserved indoublemagi nu leilike 16 O, 40 Caor 56 Ni.
One of the main interest a tually in nu lear stru ture is to study how this ompetitionbetweenmagi spheri al on gurationsanddeformedones evolve farfrom stabilitywith neutron ex ess and ifshell losures ormagi numbers observed around the stability linestill persist. The ase of
32 Mg and thedisapperan eofN=20shell losure??wasapioneering aseof this new domainof nu lear stru tureinvestigations.
hasbeenshownto be ru ialto thedes riptionofmanynu learproperties. Starting from as hemati tensor oftheform
V T =(~ 1 :~ 2 )([s~ 1 ~ s 2 ℄) 2 :Y 2 )f(r)
they deriveda losure relationforthemonopolepart ofthisintera tion: (2j > +1)V T j>;j 0 +(2j < +1)V T j<;j 0 =0; Here j > and j < denote for l+ 1 2 and l 1 2
respe tively,lbeing the orbital angular momenta. From thisrelationone an follow theevolution of spin-orbitpartners j
> and j
<
of a given proton orneutron uidwith the lling oftheother uidj
0 : itisseenthatV T j>;j 0 andV T j<;j 0
haveoppositesignsand thatalsothetensorintera tionbeingofdipoletype,itisattra tivebetween shellsj
< /j
0 >
and repulsive betweenshellsj > /j 0 > . 2. Tin isotopes
Inthe parti ular aseof the tinregion, the tensormonopolee e ts de-velopand an beputineviden e,forexampleinthespe tros opyevolution between
91 Zrand
101
Sn. In gure 1, the experimental low-lying spe trum of
91
Zris shown. The orresponding spe trum of 101
Sn was obtained by a tting pro edure of single parti le energies and two-body matrix elements in the(g7 2 ;d5 2 ;d3 2 ;s1 2 ;h11 2
) (r4h) valen espa e on top ofa 100
Sn ore. The leare e t of the llingof theg
9 2
protonorbital auses thedropof the 7 2 + state from 1.88 to 0.2 MeV. This neutronstate is of single parti le nature and risesdowndueto thestrongpnattra tionbetweenspin-orbitpartners V7
2 ;
9 2
. As quoted in [2 ℄, the tensor me anism predi ts also repulsive inter-a tion V11
2 ;
9 2
as ould be inferedfrom the riseof the 11 2 between 91 Zrand 101
Sn. While the state in 101
Sn is evidently the h11 2
single parti le state, theone in
91
Zris arryingonlya fra tionalpartofthespe tros opi fa tor strength [3 ℄ and has probably strong admixture of [(p1
2 ) 1 (g9 2 ) 1 ℄ (d5 2 ) on guration (infered from Gross-Frenkel interation [4 ℄ in
89
Zr for exam-ple). The experimental determinationofthe lo ationofthe h11
2
single par-ti le entroid should therefore be ru ialto determine the nature and the strength of V pn h 11 2 g 9 2
monopoleintera tion. At thispoint, it isinterestingto noti e thatthe spe tros opi properties all along thetin hain,from
101 Sn to
131
Sn(see gure?? ),willbe governed bythesame intera tingorbitalsas theonesintheprevious asefrom
91 Zrand
101
Sn. Inparti ular,depending on thenature(l1=2)of thesu essive lledneutronorbitals,the intera -tionwithg9
2
Zr
91
0.0
5/2+
1/2+
1.20
2.17
2.04
1.88
3/2+
11/2−
7/2+
Sn
101
0.0
5/2+
7/2+
0.20
1.54
1.69
11/2−
2.40
3/2+
1/2+
Fig.1. Energyevolutionoflow-lyingstatesbetween 91 Zrand 101 Sn.
0
100
200
300
400
500
600
50
60
70
80
B(E2) (e
2
fm
4
)
Neutron number
exp
t=0
t=2
t=4 g9g7d5
t=4 gds
-15
-10
-5
0
50
60
70
80
ESPE (MeV)
Neutron number
g9/2
d5/2
g7/2
s1/2
d3/2
h11/2
Fig.2. left: BE2(2 +
!0 +
)fortinisoptopes;right: protonESPEfortinisotopes.
In left panel of gure 2 are represented the B(E2) valuesalong all the isotopi tin hain. The dis retepointsare theexperimentalmeasurements whilethe ontinuous urves orrespondtoshellmodel al ulationswith dif-ferent proton trun ationlevels (0p0h, 2p2h,trun ated and full4p4h). The al ulations are performed in the gds valen e spa e for protons and (r4h) forneutrons. Several pointsareto be ommentedfrom these urves: - theexperimental urveshows anleft/right asymmetrymeaning enhan ed transitions for the lightest tin isotopes. This is on rmed by the re ent Rising measurement in
108 Sn [5 ℄.
- shell model al ulations presented here are able to reprodu e the ampli-tudes of the transitions with standard polarization e e tive harge of 0.5. Theneedof oreex itation is ru ialand demonstratedbythelarge ampli-tudegain betweent=0 (0p0h) andt=4 (4p4h) results.
ti ular due to neutron ore ex itation and non onvergen e of the 4p4h al ulations. Su ha gainshouldbe suÆ ientto disymmetrizethe urve,in agreement withtheexperimentaltrend.
It isinterestingto putin orresponden e previouspanel withtherightone of gure2wheretheunderlyingprotonstru tureisillustratedwiththe evo-lution ofthe E e tive SingleParti leEnergies [6 ℄ along thetin hain. The redu tion of the proton gap at mid-shell explains the orresponding large amplitudesof theBE(2)'s. Onthe otherhand, theV
pn h 11 2 g 9 2 monopole inter-a tion(whi h ould notbedeterminedfrom
91
Zrspe trum)isneededtobe suÆ entlyattra tive (of theorder of the V
pn h 11 2 g 7 2
) to hinder thetransitions at the end of the shell and lose suÆ iently
132
Sn. This appear to be in ontradi tion withthepredi tion ofthetensor behaviour suggestedin [2℄.
3.Con lusion
Large ShellModel al ulations forthe tin isotopi hainare performed forthe al ulationsofele tromagneti E2strength. In lusionof ore ex ita-tionsallowto des ribe theamplitudeofexperimentalvalueswithstandard polarizatione e tive harges. In ontradi tion to the behaviour of a pure tensorfor e,theproton-neutronmonopoleintera tionV
h 11 2 g 9 2 needto beas attra tive asV h 11 2 g 7 2 one. REFERENCES [1℄ C.Thibaultet al. Phys.Rev.C12,644(1975) [2℄ T.Otsukaet al.
Phys.Rev.Lett.95,232502(2005) [3℄ C.R.BinghamandM.L.Halbert
Phys.Rev.C2,2297(1970) [4℄ R.GrossandA. Frenkel,
Nu .Phys.A267,85(1976)
[5℄ A.Banuetal.,Phys.Rev.C72,061305(R)(2005) [6℄ Y.Utsuno, T.Otsuka,T.MizusakiandM.Honma
