Extreme Correlations; Deformation and …webs.um.es/albaladejo/aguilas2011/umar_2011_3.pdfExtreme Correlations; Deformation and Superdeformation ALFREDO POVES Departamento de F´ısica

Extreme Correlations; Deformation andSuperdeformation

ALFREDO POVESDepartamento de Fısica Teorica and IFT, UAM-CSIC

Universidad Autonoma de Madrid (Spain)

Universidad Internacional del MarAguilas, July 25-28, 2011

Alfredo Poves Extreme Correlations; Deformation and Superdeformation

Outline

Deformation and Superdeformation at N=Z

Triaxiality; The light Xenon isotopes

Correlated pairs, the case of 92Pd

Quadrupole Collectivity; SU3 and its variants

Summary


Collectivity at N=Z

Four protons and four neutrons on top of doubly magic40Ca, suffice to produce a well behaved rotor. In N=Znuclei, a proper treatment of the neutron proton pairing,isovector and isoscalar is compulsory in order toreproduce the experimental moment of inertia.


ISM dialogues with BFM: the 48Cr case

0.0 1.0 2.0 3.0 4.0Eγ (MeV)

2

4

6

8

10

12

14

16

J

Exp

SM-KB3CHFBSM-GOGNY


ISM dialogues with BMF: the 48Cr case

Intrinsic states in the laboratory frame wave-functions.

J B(E2)exp B(E2)th Q0(B(E2))2 321(41) 228 1074 330(100) 312 1056 300(80) 311 1008 220(60) 285 9310 185(40) 201 7712 170(25) 146 6514 100(16) 115 5516 37(6) 60 40


ISM dialogues with BMF: the 48Cr case

The intrinsic state in the intrinsic description:


Coexistence: Spherical, Deformed andSuperdeformed states in 40Ca

In the valence space of two major shells1f5/2

2p1/2

2p3/2

1f7/2

pf -shell1d3/2

2s1/2

1d5/2

sd-shell

The relevant configurations are:[sd]24 0p-0h in 40Ca, SPHERICAL[sd]20 [pf]4 4p-4h in 40Ca, DEFORMED[sd]16 [pf]8 8p-8h in 40Ca, SUPERDEFORMED


The correlation energies

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10

Ene

rgy

(MeV

)

np-nh configuration

Uncorrelated

np-nh 0+

r2pf 0+


The correlation energies

In the 8p-8h configuration the correlations amount to18.5 MeV. 5.5 MeV are due to T=1 pairing and 0.5 MeV toT=0 pairing, thus the neutron-proton pairing contribution is2.33 MeV. The remaining 12.5 MeV are most likely ofquadrupole origin.

In the 4p-4h configuration, the pairing contributions are thesame, but the quadrupole is just 3.5 MeV.

The physical gound state gains 5 MeV of pairing energy bymixing with the other np-nh states, the ND bandhead2 MeV, and the SD bandhead nothing


The np-nh energies as a function of J

5

10

15

20

0 2 4 6 8 10

Ene

rgy

(MeV

)

np-nh configuration

16+

14+

12+

10+

8+

6+

4+

2+

0+


The superdeformed band: 8p-8h

0 2 4 6 8

10 12 14 16 18 20 22 24

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5Eγ (MeV)

J

Exp.8p-8h


The Superdeformed band: Mixed calculation

0

2

4

6

8

10

12

14

16

0 0.5 1 1.5 2 2.5 3 3.5 4Eγ (MeV)

J

Exp.r2pf


The triaxial deformed band: Mixed calculation

0

2

4

6

8

10

12

14

16

0 0.5 1 1.5 2 2.5 3 3.5 4Eγ (MeV)

J

Exp.r2pf


Transition Quadrupole Moments

100

120

140

160

180

200

220

240

2 4 6 8 10 12 14 16

Q0(

t) (

e fm

2 )

J


Comparing with experiment

0

02

4

6

8

10

0

2

4

6

8

10

12

14

16

3

5

7

9

11

13

2

4

12

14

16

0

02

4

6

8

10

0

2

4

6

8

10

12

14

16

3

5

7

9

11

13

2

4

12

14

16

812

1014

1062

1090

263

502

210

179

184

188

190

192

181

1110

1124

1134

1094176

18

1.7

116

21

13521

49

3227179

10

1.0

579

813

874

906

844

292

397

346

227

161

75

112

49

214

211

175

110

80

133

546

557

429

427 2.7

12

7.8

43

1.80.1

6.8

19

583.0

49

202891

16

0.2

284

Exp. Th.

40CaE

xcita

tion

Ene

rgy

(MeV

)

0

2

4

6

8

10

12

14

16

18

20

22

24

26


Collectivity in the light Xenon isotopes

110Xe

0+ 0

2+ 350

4+ 920

6+ 1710

8+ 2640

10+ 3730

12+ 4950

14+ 5980

2+ 11003+ 13304+ 1560

5+ 1880

6+ 2210

0+ 0

2+ 470

4+ 1110

(6+) 1890

shell model exp.

1005

1450

1568

1591

1530

1431

0.05

17741395938

600


Collectivity in light Xenon isotopes

J E* Eγ BE2 Qsp Q0 Q0 β(BE2) (Qsp)

2+ 0.35 0.35 1005 -62 225 217 0.164+ 0.92 0.57 1450 -78 226 215 0.166+ 1.71 0.79 1568 -83 224 208 0.168+ 2.64 0.94 1591 -87 220 207 0.1610+ 3.73 1.09 1530 -86 213 198 0.1512+ 4.95 1.22 1431 -85 204 191 0.1514+ 5.98 0.99 0.05 -126 1 27916+ 6.63 0.69 111 -125 56 27318+ 7.51 0.88 1184 -130 183 28220+ 8.51 1.00 1043 -134 172 288



J E* Eγ BE2 Qsp Q0 Q0 β(BE2) (Qsp)

2+2 1.10 +61

3+ 1.33 0.23 1774 -1.34+

2 1.56 0.23 1395 -38 219 261 0.185+ 1.88 0.32 938 -54 217 234 0.176+

2 2.21 0.33 600 -74 209 259 0.17

From the ratioBE2(2+

γ→2+

y )

BE2(2+γ→0+

y )γ=20o.

Q(2+γ )=-Q(2+

y ) and Q(3+)≈0suggest that the γ band can be labeled by K=2.



The valence space r4h contains a pseudo-SU3 triplet plus theintruder orbit 0h11/2. What happens if we remove the intruderorbit from the space?

The moments of inertia of the bands get reduced by 30%

The backbending is suppressed

The triaxiality is reduced to γ = 12o

The magnetic moments are fully consistent with therotational model up to J=20.


The Xenon isotopes

0

0.5

1

1.5

2

2.5

3

50 60 70 80

Ene

rgy

(MeV

)

N

2+ sm4+ sm

2+ exp4+ exp


Aligned T=0 pairs in 92Pd

The proper counter of correlated pairs, which must haveexpectation value zero in a completely filled orbit andexpectation value 1 in the two particle (two hole) states withJ0T0, has the following form:

NJ0T0p = βJ0

∑

J

αJ,J0 [(a†a†)JT0(aa)JT0 ]00

αJ,J0 = −2J0 + 1

∑

J(2J + 1)for J 6= J0

αJ0,J0=

∑

J 6=J0(2J + 1)

∑

J(2J + 1)

βJ0=

∑

J(2J + 1)∑

J 6=J0(2J + 1)


Aligned T=0 pairs in 92Pd

Using it we obtain the following values for the number ofcorrelated JT=90 pairs in the yrast band of 92Pd.

Jπ r3g space g9/2 space0+ 1.34 2.262+ 1.48 2.324+ 1.65 2.356+ 1.69 2.38

24+ 3.87 3.89

Had we used the operator without removing the monopole, wewould have obtained 8.73 pairs for the 0+ computed in the g9/2space, instead of 2.26. In view of the numbers in the table, thecalculation in the realistic valence space softens somehow thecondensate interpretation.


The mechanism of deformation andsuperdeformation in the laboratory frame

Consider the quadrupole force alone, taken to act in the p-thoscillator shell. It will tend to maximize the quadrupole moment,which means filling the lowest orbits obtained by diagonalizingthe operator Q0 = 2q20 = 2z2 − x2 − y2. Using the cartesianrepresentation, 2q20 = 2nz − nx − ny , we find eigenvalues 2p,2p − 3,. . . , etc. By filling the orbits orderly we obtain theintrinsic states for the lowest SU(3) representations: (λ,0) if allstates are occupied up to a given level and (λ, µ) otherwise.For instance: putting two neutrons and two protons in theK = 1/2 level leads to the (4p,0) representation. For fourneutrons and four protons, the filling is not complete and wehave the (triaxial) (8(p − 1),4) representation for which weexpect a low lying γ band.


SU3 and Quasi-SU3

0

3

6

9

12

1/2

3/2

5/2

7/2

9/2

1/2

3/2

5/2

7/2

9/2

Q + k0SU3

quasi.SU3

Nilsson orbits for SU(3) (k = 2p) and quasi-SU(3)(k = 2p − 1/2)


SU3 and Quasi-SU3

In jj coupling the angular part of the quadrupole operatorq20 = r2C20 has matrix elements

〈j m|C2|j + 2 m〉 ≈3[(j + 3/2)2 − m2]

2(2j + 3)2 ,

〈j m|C2|j + 1 m〉 = −3m[(j + 1)2 − m2]1/2

2j(2j + 2)(2j + 4)

The ∆j = 2 numbers are—within the approximation made—identicalto those in LS scheme, obtained by replacing j by l. The ∆j = 1matrix elements are small, both for large and small m, correspondingto the lowest oblate and prolate deformed orbits respectively. If thespherical j-orbits are degenerate, the ∆j = 1 couplings, though small,will mix strongly the two ∆j = 2 sequences (e.g., (f7/2p3/2) and(f5/2p1/2)). The spin-orbit splittings will break the degeneracies andfavour the decoupling of the two sequences. Hence the idea ofneglecting the ∆j = 1 matrix elements and exploit thecorrespondence

l −→ j = l + 1/2 m −→ m + 1/2 × sign(m).Alfredo Poves Extreme Correlations; Deformation and Superdeformation

SU3 and Quasi-SU3

The resulting “quasi SU(3)” quadrupole operator respectsSU(3) relationships, except for m = 0, where thecorrespondence breaks down. The resulting spectrum forquasi-2q20 is shown together with the SU3 one. The result isnot exact for the K = 1/2 orbits but a very good approximation.To check the validity of the decoupling, a Hartree calculationcan be done for H = εHsp + Hq, where Hsp is the observedsingle particle spectrum in 41Ca (essentially equidistant orbitswith 2MeV spacings) and Hq is the quadrupole force in with aproperly renormalized coupling. The result is exactly a Nilssoncalculation,

Hmq0 = ~ω

(

εHsp −δ

32q20

)

,

whereδ

3=

14〈2q20〉

〈r2〉=

〈2q20〉

(p + 3/2)4 .


Quasi-SU3

Nilsson diagrams in the pf shell. Energy vs. single particlesplitting ε (left panel), energy vs. deformation δ (right panel)

0.0 0.5 1.0 1.5 2.0ε

−10.0

−5.0

0.0

5.0

10.0∆E

(M

eV)

1/2[330]

1/2[321]

1/2[310]

1/2[301]

3/2[321]

3/2[312]

3/2[301]

5/2[312]

5/2[303]

7/2[303]

−0.30 −0.10 0.10 0.30δ

1/2[330]

1/2[321]

1/2[310]

1/2[301]

3/2[321]

3/2[312]

3/2[301]

5/2[312]

5/2[303]

7/2[303]

In the right panel of the figure the results are given in the usualform. In the left panel we have turned the representationaround: since we are interested in rotors, we start from perfectones (SU(3)) and let ε increase. At a value of ≈ 0.8 the fourlowest orbits are in the same sequence as the right side of thefigure. The real situation corresponds to ε ≈ 1.0.Alfredo Poves Extreme Correlations; Deformation and Superdeformation

Quasi-SU3

Quasi-SU3 is confirmed by an analysis of the wavefunctions:For the lowest two orbits, the overlaps between the purequasi-SU(3) wavefunctions calculated in the restricted ∆j = 2,fp-space and the ones in the full pf shell exceed 0.95throughout the interval 0.5 < ε < 1. More interesting still: thecontributions to the quadrupole moments from these two orbitsvary very little, and remain close to the values obtained at ε = 0(i.e., from the Quasi-SU3 Nilsson orbits).


The quasi-SU3 spherical configurations

The calculations in the restricted (fp)n spaces accountremarkably well for the results in the full major shell ((pf )n). Thesame happens in larger spaces. The intrinsic quadrupolemoments Q0 remain constant to within 5% up to a critical Jvalue at which the bands backbend.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

J

E(J+2)-E(J) fp8fp4.gds4

gds8gds4.hfp4

Yrast transition energies Eγ = E(J + 2)− E(J) for theconfigurations pf 8, pf 4 gds 4, gds 8, gds 4 hfp 4


Heavier nuclei: Quasi+Pseudo SU(3)

0

3

6

9

12

[ [] ]hg

50 82

r4 5r

hfp

igds

MeVPositive

Negative

Quasi-SU(3) is a variant of SU(3) that obtains for moderate spin-orbitsplittings. For other forms of single particle spacings, thepseudo-SU(3) scheme will be favoured (in which case we have to usethe SU3 Nilsson levels with pseudo-p = p − 1). The figure gives aschematic view of the single particle energies above 132Sn. Thespace consists of two contiguous major shells—in protons (π) andneutrons (ν)—adequate for a SM description of the rare earth region.rp is the set of orbits in shell p excluding the largest. For the uppershells the label l is used for j l 1 2


Heavier nuclei: Quasi+Pseudo SU(3)

We can estimate the quadrupole moments for nuclei at theonset of deformation. We shall assume quasi-SU(3) operates inthe upper shells, and pseudo-SU(3) in the lower ones. Thenumber of particles in each shell for which the energy will belowest will depend on a balance of monopole and quadrupoleeffects, but Nilsson diagrams suggest that when nuclei acquirestable deformation, two orbits K =1/2 and 3/2—originating in theupper shells of the figure—become occupied, i.e., the upperblocks are precisely the 8-particle configurations. Theircontribution to the electric quadrupole moment is then

Q0 = 8[eπ(pπ − 1) + eν(pν − 1)],

with pπ = 5, pν = 6; eπ and eν are the effective charges.Consider even-even nuclei with Z=60-66 and N=92-98,corresponding to 6 to 10 protons with pseudo-p = 3, and 6 to10 neutrons with pseudo-p = 4 in the lower shells. From theSU3 Nilsson diagrams we obtain easily their contribution to Q0,which added to the Quasi-SU3 contributions yields a totalAlfredo Poves Extreme Correlations; Deformation and Superdeformation

Heavier nuclei: Quasi+Pseudo SU(3)At fixed n, the value is constant in the four cases because the orbitsof the triplet K=1/2, 3/2, 5/2 have zero contribution for p=3. Theresults, using effective charges of eπ = 1.4, eν = 0.6 calculated arecompared in the table with the available experimental values . Theagreement is quite remarkable and no free parameters are involved.Note in particular the quality of the prediction of constancy (or ratherA2/3 dependence) at fixed n, which does not depend on the choice ofeffective charges. The discrepancy in 152Nd is likely to be ofexperimental origin, since systematics indicate, with no exception,much larger rates for a 2+ state at such low energy (72.6 keV).

N Nd Sm Gd Dy92 4.47 4.51 4.55 4.58

2.6(7) 4.36(5) 4.64(5) 4.66(5)94 4.68 4.72 4.76 4.80

5.02(5) 5.06(4)96 4.90 4.95 4.99 5.03

5.25(6) 5.28(15)98 5.13 5.18 5.22 5.26

5.60(5)Alfredo Poves Extreme Correlations; Deformation and Superdeformation

The region around the N=Z=40 nucleus 80Zr

The pseudo+quasi SU3 valence space comprises orbits fromtwo major shells

3s1/2

2d5/2 QUASI p=41g9/2

sdg-shell1p1/2

1f5/2 PSEUDO p=21p3/2

pf -shell

The relevant configurations for 80Zr are:[pf]40 0p-0h, doubly magic, SPHERICAL

[pf]36 [sdg]4 4p-4h[pf]32 [sdg]8 8p-8h


The region around N=Z=40

The 4p-4h and 8p-8h configurations contain oblate and prolatestates. The quadrupole moments can be easily computed withthe help of the SU3 Nilsson diagrams. The results are:

4p-4h: Q0=44 b2 e fm2 , Q0=-44 b2 e fm2

8p-8h, Q0=70 b2 e fm2 , Q0=-70 b2 e fm2

The experimental values in the region are consistent with the8p-8h prolate choice, but the model exhibits extreme oblateprolate coexistence, as it has been found in other nuclei in theregionRecent SMMC calculations in the pf+sdg valence space byLanganke et al.(NPA728(2003)109) also reach the sameconclusion, giving an occupation number of the 1g9/2 orbit of3.6 protons and 3.6 neutrons, fully consistent with thePseudo+Quasi-SU3 8p-8h surmise. The Vampir calculations ofPetrovici et al. (NPA710(2002)246) rather favor 12p-12hconfigurationsThe BE2↑ in 80Zr deduced from the Q0 value of the prolateAlfredo Poves Extreme Correlations; Deformation and Superdeformation

Coexistence near N=Z=20

In the 4p-4h intrinsic state of 36Ar, the two neutrons and twoprotons in the pf -shell can be placed in the lowest K=1/2quasi-SU3 level of the p=3 shell. This gives a contributionQ0=25 b2. In the pseudo-sd shell. p=1 we are left with fourparticles, that contribute with Q0=11 b2.In the 4p-4h state of 40Ca these values are Q0=25 b2 and Q0=7b2, while in the 8p-8h the values are Q0=35 b2 and Q0=11 b2

Using the proper values of the oscillator length it obtains:36Ar 4p-4h band Q0=136 e fm2 (Q0=173 e fm2)40Ca 4p-4h band Q0=125 e fm2 (Q0=148 e fm2)40Ca 8p-8h band Q0=180 e fm2 (Q0=226 e fm2)In very good accord with the data. The values in blue assumestrict SU3 symmetry in both shells. The SM results almostsaturate the quasi-SU3 bounds. The SU3 values are a 25%larger.


Conclusions

The geometry of the spherical mean field orbits giving rise todeformed rotors pertains to variants of Elliott’s SU3(pseudo-SU3, quasi-SU3).

In the well deformed limit, the effect of pairing is mainly tomodify the moment of inertia. Neutron-proton pairing isresponsible for about 50% of the total effect in N=Z nuclei.

np-nh configurations across N=Z=20 produce superdeformedshapes that can be explained in the pseudo-SU3+quasi-SU3scheme. This scheme applies also to other mass regions,either proton rich as in 80Zr (N=40, Z=40), or neutron rich as in32Mg (N=20), or 40Mg and 42Si (N=28).


Documents

Extreme Correlations; Deformation and …webs.um.es/albaladejo/aguilas2011/umar_2011_3.pdfExtreme Correlations; Deformation and Superdeformation ALFREDO POVES Departamento de F´ısica