Par$cle-core coupling in deformed nuclei: odd-A and doubly even-A iden$cal bands John ... · 2018-06-15 · Par$cle-core coupling in deformed nuclei: odd-A and doubly even-A iden$cal

Par$cle-corecouplingindeformednuclei:odd-Aanddoublyeven-Aiden$calbands

JohnL.Wood,SchoolofPhysics,GeorgiaIns$tuteof

Technology

FundamentalsofNuclearModels:Founda$onalModels--DavidJ.RoweandJLW,WorldScien$fic,2010[R&W]+secondvolumenearingcomple$on

Iden$calbands23yearsago:recognizedbutnotunderstood(andabandoned)

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. 199

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. Sci

. 199

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:485

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. Dow

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“Inpar$cular,thereduc$onoftheBCSpairingcorrela$onsduetotheblockingofoneandtwoorbitalsimplieslargechanges(upto30%)inthemomentsofiner$aandcannotbereconciledwiththese[iden$calband]systema$cs.”

Symmetric-topmodel:quantumnumbers

R&WFig.1.46

the z-axis and the 3-axis are not in a rigidly oriented relationship

R:collec$veangularmomentumJ:intrinsicspinI:totalspin/angularmomentumM:laboratory-frame,z-componentof I K:body-frame(symmetryaxis),3-componentofI;K=Ω

“Coriolis”interac$on:R�R= (I – J)�(I –J) = I�I - 2 I�J + J�J

J

=Ω

February 4, 2010 12:18 WSPC/Book Trim Size for 9.75in x 6.5in rw-book975x65

1.8 Low-energy collective structure in odd nuclei

same energy. However, if the potential is allowed to become spheroidal, the statesof |≠| = 1/2, 3/2, 5/2, . . . , separate in energy. For a prolate spheroidal potential,the |≠| = 1/2 orbital lies lowest and the energy increases with increasing |≠|. Thisis due to the fact that the density distribution of a single-particle state of low |≠|fits more readily inside an equipotential surface of a prolate potential than in anequipotential surface of an oblate potential, and vice-versa. This is shown in Figure1.70 for a 1h11/2 single-particle state. For more than one available shell-model jorbital, mixing of configurations with the same ≠º value from diÆerent j orbitalsoccurs.

The Nilsson model is essentially a deformed version of the single-particle shellmodel. It replaces the spherical harmonic oscillator potential of Equation (1.13)with a spheroidal potential. Thus, for deformed nuclei, the spherically symmetricsingle-particle Hamiltonian of Equation (1.19) is replaced by an axially symmetricNilsson model Hamiltonian

h :=p2

2M+ 1

2M£

!2?(x

2 + y2) + !2z z

2§

+Dl2 + ª l · s, (1.58)

where the x, y and z coordinates are defined relative to the intrinsic axes of therotor.

An equipotential surface for this Hamiltonian is a spheroid with semi axes Rx =Ry = R?, Rz related by the equation

!2?R

2x = !2

?R2y = !2

zR2x. (1.59)

For consistency, !? and !z, should be chosen such that the shape of this equipo-tential surface approximates the shape of a corresponding equidensity surface ofthe deformed core. Thus, based on the constancy of the nuclear matter density inthe interior of a nucleus, it is appropriate to express the frequencies of the Nilssonmodel Hamiltonian in terms of a volume-conserving scale transformation

!? = !0e≤/3, !z = !0e

°2≤/3, (1.60)

for which !x!y!z = const. := !30 (cf. Exercises (1.34) and (1.35)). When ≤ > 0,

the equipotential surface is then a prolate spheroid and when ≤ < 0 it is oblate.The total Hamiltonian for a nucleon coupled to the states of a K = 0 ground-

state rotational band of an even-nucleus core is given by

H :=~2R2

2= + h, (1.61)

where R is the angular momentum of the rotor (cf. Equation (1.43)). If j denotesthe angular momentum of the odd nucleon and I := R + j is the total angularmomentum of the odd-mass nucleus, then

H =~2I2

2= + h+~2j2

2= ° ~2

= I · j. (1.62)

81

rot

R2 ^

Thekeyconceptformodelingdeformednuclei:thesymmetric-top+Nilssonmodel





h :=p2

2M+ 1

2M£

!2?(x

2 + y2) + !2z z

2§

+Dl2 + ª l · s, (1.58)



!2?R

2x = !2

?R2y = !2

zR2x. (1.59)


!? = !0e≤/3, !z = !0e

°2≤/3, (1.60)




H :=~2R2

2= + h, (1.61)


H =~2I2

2= + h+~2j2

2= ° ~2

= I · j. (1.62)

81





h :=p2

2M+ 1

2M£

!2?(x

2 + y2) + !2z z

2§

+Dl2 + ª l · s, (1.58)



!2?R

2x = !2

?R2y = !2

zR2x. (1.59)


!? = !0e≤/3, !z = !0e

°2≤/3, (1.60)




H :=~2R2

2= + h, (1.61)


H =~2I2

2= + h+~2j2

2= ° ~2

= I · j. (1.62)

81





h :=p2

2M+ 1

2M£

!2?(x

2 + y2) + !2z z

2§

+Dl2 + ª l · s, (1.58)



!2?R

2x = !2

?R2y = !2

zR2x. (1.59)


!? = !0e≤/3, !z = !0e

°2≤/3, (1.60)




H :=~2R2

2= + h, (1.61)


H =~2I2

2= + h+~2j2

2= ° ~2

= I · j. (1.62)

81





h :=p2

2M+ 1

2M£

!2?(x

2 + y2) + !2z z

2§

+Dl2 + ª l · s, (1.58)



!2?R

2x = !2

?R2y = !2

zR2x. (1.59)


!? = !0e≤/3, !z = !0e

°2≤/3, (1.60)




H :=~2R2

2= + h, (1.61)


H =~2I2

2= + h+~2j2

2= ° ~2

= I · j. (1.62)

81

“coupling”=addHamiltonians(wavefunc$onswillbedirectproducts)

“coupling”=addspins/angularmomenta





h :=p2

2M+ 1

2M£

!2?(x

2 + y2) + !2z z

2§

+Dl2 + ª l · s, (1.58)



!2?R

2x = !2

?R2y = !2

zR2x. (1.59)


!? = !0e≤/3, !z = !0e

°2≤/3, (1.60)




H :=~2R2

2= + h, (1.61)


H =~2I2

2= + h+~2j2

2= ° ~2

= I · j. (1.62)

81



Er isotopes show a loss of the strongly-coupled pattern. The reduction of strongcoupling is termed decoupling.

Decoupling is a consequence of the Coriolis interaction, given in Equation 1.39.Its expectation value reduces to °(~2/=)h~I ·~ji for a spheroidal (axially symmetric)nucleus. The Coriolis interaction cannot be neglected if = is small or if h~I · ~ji islarge. The systematic behaviour shown in Figure 1.72 is attributed to a decreasingdeformation (decreasing =) with decreasing mass and a consequent increase in theinfluence of the Coriolis interaction. The Coriolis interaction evidently favours thealignment of ~j with ~I. This “rotation alignment” eÆect of the Coriolis interactionopposes the “deformation alignment” of the spheroidal potential (cf. Figures 1.70and 1.71) which favours the alignment of the probability density distribution ofthe particle with that of the slowly rotating rotor core. (Fuller details of Coriolisdecoupling will be given in Volume 2.)

Coriolis decoupling provides an explanation for the irregularities of the rotationalbands built on the 371 and 627 keV states of 175Lu (cf. Figure 1.69). These twobands have |≠| = 1/2 ( equal to the lowest spin in each band). In fact, the Coriolisinteraction is invariably important for |≠| = 1/2 bands. This is because it makesdiagonal contributions to the energies of |≠| = 1/2 states; a fact that becomes evidentwhen h~I ·~ji is expressed in the form

h~I ·~ji = h12(I+j° + I°j+) + Iz jzi. (1.68)

One finds that the operator I+j° + I°j+ has non-zero matrix elements betweenthe ≠ = ±1/2 components of a |≠| = 1/2 rotational state. The modified rotationalenergy formula that results is

EI = E0 +A[I(I + 1) + (°1)I+1/2(I + 1/2) a ±K,1/

2

], (1.69)

where a, the so-called decoupling parameter, is characteristic of the intrinsic stateof the nucleus. (This will be discussed in more detail in Volume 2.)

Exercises

1.29 Obtain values of a that appear in Equation (1.69) by fitting this equation to thebands in 175Lu built on the 353 and 627 keV states shown in Figure 1.69.

1.30 Obtain = for the bands shown in Figure 1.69 and compare with 174Yb (Figure1.60).

1.31 Plot EI vs. I(I + 1) for the band shown in Figure 1.68.

1.32 For the nuclei with N = 91 in Figure 1.43, use the Nilsson model diagram, Figure1.71, to identify N , nZ , § quantum numbers.

1.33 Identify the Nilsson configurations, [N,nZ ,§], associated with the bands shownin Figure 1.69.

85






h~I ·~ji = h12(I+j° + I°j+) + Iz jzi. (1.68)


EI = E0 +A[I(I + 1) + (°1)I+1/2(I + 1/2) a ±K,1/

2

], (1.69)


Exercises






85

Nilssonmodelplusrota$ons


1.7 Low-energy collective structure in doubly-even nuclei

molecules and symmetric top molecules will be discussed in Volume 2). Thus,within the framework of the rotor model, one interprets the even-I spin sequence asimplying a reflection symmetry of the nuclear rotor in a plane perpendicular to itssymmetry axis. An implication of such a symmetry is that the states |KIMi and|°K, IMi appear in linear combination

|KIMi+ "(°1)I+K |°K, IMi, (1.50)

where " = ±1 according as the intrinsic wave function is symmetric or antisymmetricunder rotation through an angle º about an axis perpendicular to the symmetryaxis. One sees that, for a symmetric (" = 1) combination, the states of odd I vanishwhen K = 0; only even values of I survive. For " = °1, a K = 0 band has an odd-Ionly spin sequence; such bands are also seen.

In addition to their characteristic I(I +1) spectra, some distinguishing featuresof a rotational nucleus are the huge values of the quadrupole moments of its I > 1states. These are given in the rotor model by the product of an intrinsic quadrupolemoment, Q0(ÆK), characteristic of the rotor band, and a geometric factor whichdepends on the angular momentum of the particular state;

Q(ÆKI) =3K2 ° I(I + 1)

(I + 1)(2I + 3)eQ0(ÆK), (1.51)

where Æ distinguishes bands with the same K. (Note, that as I increases for a givenK, Q(ÆKI) will change sign for K > 1/2.) The intrinsic quadrupole moment of arotational state can also be determined from the E2 transition rates between thestates of a rotor band, which have B(E2) values given by

B(E2;ÆKIi ! ÆKIf ) =5

16º(IiK, 20|IfK)2e2|Q0(ÆK)|2, (1.52)

where (IiK, 20|IfK) is a Clebsch-Gordan coe±cient.

1.7.3 Low-energy vibrational states in doubly-even nuclei

Figure 1.47 shows states at low energy in the singly-closed shell nucleus 118Sn thatare strongly excited in inelastic electron scattering. From the strength of excitation,it is deduced that the first excited state at 1.23 MeV is a collective quadrupoleexcitation and the excited state at 2.33 MeV is a collective octupole excitation. TheIº = 4+ states at 2.28, 2.49, and 2.73 MeV indicate some hexadecapole collectivity,but it is fragmented.

To infer the characters of collective excitations, such as seen in Figure 1.47,it is necessary to consider them within the context of a larger pattern of states.Figure 1.48(a) shows the states in 114,116,118Sn that have strong electric quadrupoletransitions to the first 2+ states.

The pattern is approximately that of a harmonic vibrator in all three nuclei.Recall (Section 1.7.1) that, for harmonic vibrations, one expects a degenerate two-

55

ε=+1reflec$onsymmetricε=-1reflec$onasymmetric

δK,1/2 = 1, K = ½ δK,1/2 = 0 otherwise

I+ := Ix + iIy , I- = Ix - iIy j+ := jx + ijy, j- = jx - ijy





h :=p2

2M+ 1

2M£

!2?(x

2 + y2) + !2z z

2§

+Dl2 + ª l · s, (1.58)



!2?R

2x = !2

?R2y = !2

zR2x. (1.59)


!? = !0e≤/3, !z = !0e

°2≤/3, (1.60)




H :=~2R2

2= + h, (1.61)


H =~2I2

2= + h+~2j2

2= ° ~2

= I · j. (1.62)

81

hNilsson“Coriolis”orrota$onal-par$clecoupling

Quantummechanicsofpar$cle-rotorcoupling:K=½bandsand“Coriolis”decoupling

-5.0-10.0 0.0 +5.0 +10.0

-200

0

200

400

600

800

1000

E(keV)

a

3/2

5/2

7/2

9/2

11/213/2 15/217/2 19/2 21/2

1/2

1/2[541]--183Re






h~I ·~ji = h12(I+j° + I°j+) + Iz jzi. (1.68)


EI = E0 +A[I(I + 1) + (°1)I+1/2(I + 1/2) a ±K,1/

2

], (1.69)


Exercises






85

A = 10.0 keV

599(5/2)

701664

618

496

639

435

260

852

1000955

879

1085

1184

1003

861

1127

871

759829892

1023

5/2

(5/2)

1/2

(1/2)

(7/2)

7/2

(7/2)

9/2

(9/2)

(9/2)

9/2

11/2

(11/2)

11/2

13/2

13/2

(13/2)(3/2)

(3/2)

15/2

(17/2)

17/215/2

514541 402 404 400

183Re

115

0

NnzΛΣ éê êé é

“Strongcoupling”ofsinglenucleontodeformedcore:Nilssonmodelplussymmetric-topmodel

C.S.Purryetal./Nuclear

PhysicsA672

(2000)54–8865Fig. 1a. Partial level scheme for 183Re, showing 1-quasiparticle bands.

9/2-[514] 1/2-[541] 5/2+[402] 7/2+[404] 1/2+[660]

PurryNPA67254

Ωπ[NnzΛ]

183Re75

C.S.Purryetal./Nuclear

PhysicsA672

(2000)54–8865Fig. 1a. Partial level scheme for 183Re, showing 1-quasiparticle bands.

9/2-[514] 1/2-[541]

5/2+[402]

7/2+[404]

1/2+[660]

Ωπ[NnzΛ]



when ~!z°~!? ¿ hDl2+ª l·si, the mixing of diÆerent eigenstates of h≤ is small andthe single-particle energies are defined by the good (asymptotic) quantum numbersN , nz, § (the eigenvalue of lz), and ≠ (the eigenvalue of jz).

A Nilsson model energy level diagram for the 50 < Z < 82 deformed region isshown in Figure 1.71. As expected, each energy level is two-fold degenerate which

6.5

6.3

6.1

5.9

5.7

5.5

5.3

5.1

4.9

4.70 0.1 0.2 0.3 0.4 0.5 0.6

1/2[521]

5/2[402]

7/2[404]

5/2[752]

3/2[642]3/2[521]

9/2[514]5/2[523]

7/2[633]

3/2[761]

1/2[411]

1/2[651]

1/2[770]

9/2[404]

5/2[642]

7/2[523]

1/2[530]

5/2[413]

3/2[532]

3/2[651]

1/2[301]

5/2[532]

1/2[660]

3/2[301]

1/2[541]

5/2[303]

1/2[420]

3/2[422]

7/2[413]

3/2[411]

3/2

[54

1]

1/2

[55

0]

1/2

[43

1]

5/2

[30

3]

5/2

[42

2]

3/2

[30

1]

3/2

[43

1]

1/2

[44

0]

1/2

[30

1]

1/2

[66

0]

3/2

[65

1]

5/2

[64

2]

1/2

[53

0]

7/2

[63

3]

3/2

[52

1]

1/2

[77

0]

11

/2[5

05

]

1/2

[65

1]

3/2

[76

1]

1/2

[40

0]

3/2

[40

2]

5/2

[75

2]

3/2

[64

2]

1/2

[52

1]

1 /2[4

00]

3/2[4

02]

11/2[5

05]

1/2

[541]

9/2[514]

5/2[4

02]

7/2[4

04]

1/2[411]

7/2[523]

3/2[411]

5/2[413]

5/2[532]

3/2[541]

1/2[550]1/2[420]

3/2[422]1/2[431]

9/2[4

04]

p1/2

g9/2

g7/2

d5/2

h11/2

d3/2

s1/2

h9/2

82

50

∋

ε α /h

ωo

Figure 1.71: A Nilsson diagram for protons in nuclei with 50 ∑ Z ∑ 82. Energies, in units of ~!0, are plottedagainst the deformation parameter, ≤. Energy levels are labelled by their spherical shell model quantumnumbers, l and j, at ≤ = 0, and by the asymptotic quantum numbers ≠[Nnz§] for ≤ 6= 0. (The figure isadapted from Lamm I.-L. (1969), Nucl. Phys. A125, 504.)

83

“Strongcoupling”ofsinglenucleontodeformedcore:Nilssonmodelplussymmetric-topmodel

energy

deforma$onUsually,band-headspin=Ω

75183Re

�

�

�

�

�182W

184Os

402éd5/2

541êh9/2

183Re

400

300

200

100

E γ(keV

)

2 4 6 8 10 12 14Ii

Rota?onalignmentplots—Eγ(keV)vs.Ii--causedby“Coriolis”interac$on

Ialignment≈4

Ialignment≈0

ΔI=2transi$ons

75

�

�

�

�

�

183Re

182W

402d5/2100.1

260.1

351.0

379.2

463.9

435.7

229.3

2

100

200

300

400

4 6 8Ii

Eγ

320.8

435.2 -0.5

é

259.9

379.2

-0.2

-1.0

0.0

321.8

Simplicitydemandsaven$on—especiallyinamany-bodyproblem

�

�

�

�183Re

400

E γ(keV

)

300

200

100

2 4 6 8 10 12 14 Iini$al

404êg7/2

Ialignment≈-0.3

514éh11/2

Ialignment≈0.2

182W

Eγvs.Iini?alplotIden?calbands

184Os�

ΔI=2transi$ons

75

Constant“alignments”n

n

n

n

n

0

100

200

300

400

500

600

700

800

0 2 4 6 8 10 12 14

182Wcore

9/2[514]exp

9/[514]fullcoriolis

9/2[514]halfcoriolis

Linear(182Wcore)

183Re

WithA.Stuchbery,Canberra(Semmes-Ragnarssoncode)

E γ(keV

)

Ii

182Wcore9/2[514]expt.9/2[514]fullCoriolis×9/2[514]halfCoriolisLinear(182Wcore)

157Dy / 158Dy

1000

500

E γ (k

eV)

10 40 30 20

Ii

157Dy

158Dy 505 α = + 1/2 é

Iden$calbands(oddandevenneighbors)canpersistindependentlyof“backbending”--notrecognized

C.S. Purry et al. / Nuclear Physics A 672 (2000) 54–88 77

(a)

(b)

Fig. 8. Alignments in 183Re: (a) 1-quasiparticle bands; (b) multi-quasiparticle bands.

PurryNPA67254

183Re

“Standard”alignmentplots(for183Re):obscure

iHarris = (I 0 + ω2I 1) ω

Harrisformulafittoeven-evencore;thensubtractiHarrisfromItogetialign

w w w www w w w w

w w

i align

Recall:p = mv L =I ω

�

�

�

� � ��¢

¢

¢¢

¢

¢

¢

¢�

¢�

n n

n n n n n n n n

182W

402éd5/2const. = 0 keV

541êh9/2const. = - 230 keV

183Re

400

300

200

100

E γ(keV

)

2 4 6 8 10 12 14Ii

Eγvs.Iini?alasarota?onalignmentplot

EI = E0 + AI(I+1) - 2A<I�J> ΔEI,I-2 = A(4I – 2) – const. Δ2EI,I-2 = 8A

MISCONCEPTION:commonly,useA = ΔEI,I-2 / (4I – 2) --yieldsanodd-evenstaggeringandconclusionthattheunpairednucleonblockspairingwhichisimportantformagnitudesofmomentsofinertia.CORRECTION:useA=Δ2EI,I-2 / 8 CONCLUSION:noinfluenceofpairingonmomentsofiner$a

75

“Coriolis”contribu$ontoenergydifferences

K=Ω=1/2

J

I

I + ΔI

ΔI = ΔR

R

<ΔI �J> large

K=Ω=max

I + ΔI

ΔI = ΔR I

R

J

<ΔI �J> ≈ 0

Philosophicalpoint:simpleeffectsnecessitatesimpleexplana$ons

Avectorialdifferenceeffect

CONCLUSIONS•  Odd-massnucleiexhibitbandsthatdifferfromtheneighboringeven-even

nucleusground-statebandsbyan“alignment”termdescribedbyI�j.Exceptforthisdifference,thebandsareiden-cal*.CONSEQUENTLY,thereisnoevidencefor:•  odd-par$cle“blocking”ofcorrela$onsinvolvedintheeven-evencore

collec$vity;•  “deforma$on-driving”effectscausedbytheoddpar$cle;•  “Coriolis”alignmenteffects(whichshouldscalewithincreasingrota$onalfrequency).*Therearesmalldifferenceswhichcanprobablybeavributedtobandmixing.

Sowhyare“momentsofiner$a”varyingwithspin?

Q0(b)

RotorModelinodd-massnuclei:Novaria?onofintrinsicquadrupolemomentswithspin

<If ||M(E2)|| Ii> = (2Ii + 1)1/2 (5/16π)1/2 <Ii K 2 0 | If K> eQ0

B(E2; Ii à If) = <If ||M(E2)|| Ii>2 / (2Ii + 1)

¤

¤¤

¤ ¤

¤¤

¤

¤

¤¤

¤

¤

¤

¤

¤

�

��

��

��

� � ��

� � �

��

¤ ��

I + 1 I + 3 I + 5 I + 7 I + 9 I + 2 I + 4 I + 6 I + 8 I + 10

8.0

8.0

8.0

6.0

6.0

6.0

¤Δ I = 1 � Δ I = 2

175LuK=7/2

169TmK=1/2

159TbK=3/2

avg.7.36Qgs7.48(4)

avg.7.30Qgs7.16(4)

avg.7.67

Data:175LuSkensvedetal.,NPA3661251981169TmWardetal.,NPA2891651977159TbChapmanetal.,NPA3972961983

Failureofasimplemodel

•  Rotormodelsrequiremoment(s)ofiner$a•  “Momentsofiner$a”changewithspin--data•  Intrinsicquadrupolemomentsdonotchangewithspin--data

•  Pairingdoesnotplayaroleinmagnitudesofmomentsofiner$a—data(presentmessage)

•  SOWHATISCHANGING?

Ž .Nuclear Physics A 662 2000 125–147

Ž .SU 3 quasi-dynamical symmetry as an organizationalmechanism for generating nuclear rotational motions

C. Bahri, D.J. RoweDepartment of Physics, UniÕersity of Toronto, Toronto, Ontario M5S 1A7, Canada

Received 15 June 1999; received in revised form 3 September 1999; accepted 6 September 1999

Abstract

The phenomenological symplectic model with a Davidson potential is used to constructrotational states for a rare-earth nucleus with microscopic wave functions. The energy levels and

Ž .E2 transitions obtained are in remarkably close agreement to within a few percent with those ofthe rotor model with vibrational shape fluctations that are adiabatically decoupled from the

Ž .rotational degrees of freedom. An analysis of the states in terms of their SU 3 content shows thatŽ .SU 3 is a very poor dynamical symmetry but an excellent quasi-dynamical symmetry for the

model. It is argued that such quasi-dynamical symmetry can be expected for any Hamiltonian thatreproduces the observed low-energy properties of a well-deformed nucleus, whenever the latter arewell-described by the nuclear rotor model. q 2000 Elsevier Science B.V. All rights reserved.

PACS: 03.65.Fd; 05.70.Fh; 21.60.Fw; 21.60.EvŽ .Keywords: Dynamical symmetry; Quasi-dynamical symmetry; Symplectic model; SU 3 ; Shell

model; Nuclear Structure

1. Introduction

A microscopic theory of nuclear structure would be very incomplete without asatisfactory description of nuclear rotational states in terms of many-nucleon quantum

w xmechanics. However, while the states of a truly rigid rotor can be handled with ease 1 ,they do not have square-integrable wave functions in either a spherical vibrational-modelor many-nucleon Hilbert space. Moreover, the expansion of liquid-like, soft-rotor, wavefunctions on any spherical basis is slowly convergent. This means that many majorshells are required for a realistic shell-model theory of nuclear rotational states. It alsomeans that a realistic calculation of nuclear rotational states in terms of interactingnucleons, without a priori knowledge of the kinds of correlations to expect, is animpossibly difficult task. The fact remains that nuclear rotational bands are exceedinglysimple; they are essentially characterized by a few intrinsic quadrupole moments and

0375-9474r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0375-9474 99 00394-2

“Apar$cularlyinteres$ngchallengewastolearnhowamodel,withoutpaircorrela$ons,couldgivecorrectmomentsofiner$awhenitisknownthatthecrankingmodelisonlysuccessfulwhenpairingcorrela$onsareincluded.Theearlycalcula$onsofParketal.indicatedthatthedominantcontribu$ontorota$onalenergiescamefromthepoten$alenergypartoftheHamiltonian,thuscallingintoques$ontheveryconceptofthemomentofiner$aasaninversecoefficientoftheL2terminthekine$cenergy.Theresultsofthepresentcalcula$onindicatethattheinclusionofonlystretchedstates,asinthecalcula$onofParketal.,tendstoexaggeratethiseffect.Nevertheless,itconfirmsthatthedominantcomponentoftherota$onalenergiescomesfromthepoten$alenergy;fortheself-consistentvalueofxonlyabout20%oftherota$onalenergycomesfromthekine$cenergyinthepresentcalcula$on.”

#

#--1984

ModelswithI(I+1)spectra,butwithoutrota$onalkinema$cs

•  Elliovmodel:HElliov=Hshellmodel+κQ�Q•  Interac$ngbosonmodel:Hboson=Hone-bodyboson+Vboson-boson(=Q�Q)•  Symplec$cshellmodel:Hsymplec$c=HSU(3)+κQ�Q+VΔN=2(GMR;GQR)GMR=giantmonopoleresonanceGQR=giantquadrupoleresonance

Anewuniversalmodelperspec?ve:shell,SU(3)andmul?-shell,Sp(3,R)structure

model, or by mixing of ð!;"Þ irreps [see, e.g., Thiamova,Rowe, and Wood (2006)], or by mixing of Spð3; RÞ irreps[see, e.g., Rowe, Vassanji, and Carvalho (1989)]. Mixingwithin an Spð3; RÞ irrep is of particular interest because itretains Spð3; RÞ as a dynamical symmetry (with all of itsalgebraic structure available for the calculation of matrixelements), but can produce observed collective quadrupolestrength without the need for effective charges. Indeed, thisallows one to restrict the full microscopic shell-model spaceof the nucleus to the most important modes of collectivedynamics.

A doubly closed shell nucleus, such as 16O, possesses theground-state irrep ð!;"Þ ¼ ð0; 0Þ and its Spð3; RÞ collectivedegrees of freedom are restricted to just the giant monopoleand quadrupole resonances. However, the first-excited state isð!;"Þ ¼ ð8; 4Þ (Rowe, Thiamova, and Wood, 2006) and itpossesses a highly collective band.

A leading challenge for the symplectic collective modeland its SU(3) submodel is identifying the lowest-energyirreps in a given nucleus when the nucleon number is large.This is a trivial task in the rotor model, and in the interactingboson model it is dictated by a simple recipe [the number ofSU(6) bosons is given by counting the number of valencenucleon pairs from the nearest closed shells]. An effectiveway to address this challenge for strongly deformed struc-tures has been put forward by Jarrio, Wood, and Rowe(1991) and Carvalho and Rowe (1992), and ways to extendthis method to weakly deformed structures have been sug-gested by Hess et al. (2002). Following ideas by Cseh andScheid (1992), extension to cluster structures may also be inreach.

From the perspective of the Bohr model and its full alge-braic realization, the algebraic collective model (ACM)(Rowe, Welsh, and Caprio, 2009; Rowe and Wood, 2010),Spð3; RÞ provides the means to look beyond this foundationalmodel of nuclear structure.

We add a few more observations regarding where we seedevelopments occurring:

$ The rapid advances in achieving a unified perspective ofnuclear structure in low-A nuclei via no-core shell-model techniques and the prospect of carrying outcalculations in all nuclei with the symplectic no-coreshell model (Dytrych et al., 2008) promises an excitingfuture for nuclear structure theory. Indeed, such

theoretical developments will be highly demanding ofexperimental techniques for identifying such structures.

$ Mean-field techniques are reaching unimagined levelsof sophistication from the perspective of our earlierreviews. These techniques can straightforwardly suggestsome of the shapes expected in mass regions far fromstability.

$ The nuclear shell model has reached an extraordinarilyhigh level of sophistication, combining the constructionof highly efficient algorithms with increased computingpower, to obtain the energies and wave functions of thelowest-lying excited states, even going up to high-spinvalues. There is clearly room for exploring new trunca-tion methods to the nuclear eigenvalue problem.

$ A topic for future investigation of nuclear structuresinvolving different shapes is the correlations involved.Pairing is not naturally incorporated into the symplecticmodels. Mean-field techniques emphasize independent-particle degrees of freedom in their use of Slater deter-minants. Indeed, we have undertaken in this review topoint to correlations (cf. Figs. 29, 30, 36, and 37) thatmay well be indicators of important components ofcoexisting structures.

$ To carry out symplectic model calculations for compari-son with data in heavy nuclei requires the identificationof the SU(3) irreps that dominate low-lying collectivestructures in heavy nuclei and the important interactionsthat mix these irreps. The application of such a programto shape coexistence in heavy nuclei is a leading re-search challenge for nuclear structure. There is an im-portant role to be played by phenomenological bandmixing applied to data, e.g., in the analysis of interbandE0 and E2 transition strengths in the first steps of such aprogram to reveal details of the underlying coexistingstructures.

IV. CONCLUSIONS AND OUTLOOK

There has been a shift in perspective on shape coexistencesince %30 years ago (the first review was in 1983) from anexotic phenomenon occurring in just a few mass regions to itspresence in almost all nuclei. The balance between shell andsubshell energy gaps (an independent-particle effect) and

FIG. 52. The vertical shells of the symplectic collective model, labeled by the number of oscillator quanta and the quantum numbers of theSU(3) subgroup of the model. Some details are discussed in the text. Adapted from Rowe (1985) and Carvalho et al. (1986).

1512 Kris Heyde and John L. Wood: Shape coexistence in atomic nuclei

Rev. Mod. Phys., Vol. 83, No. 4, October–December 2011

WorkinaCartesianharmonicoscillatorbasis:{nxnynz}N=nx+ny+nz,λ~2nz-nx-ny,μ~nx–ny--SU(3)shell“cores”.Allow2ħωadmixturesofL=0,2[GMR,GQR]configura$ons.Thespin-orbitandpairinginterac$onsareperturba$ons.

D.J.Rowe,A.E.McCoy,andM.A.Caprio,Phys.Scr.91(2016)033003

CONCLUSION

•  Weareattheendofthebeginning•  Nowbeginsthemiddle(ofthestudyofthenuclearmany-bodyproblem)

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