shell model a uniﬁed view of nuclear...

Shell Model

A Unified View of Nuclear Structure

Frederic Nowacki1

18th STFC UK Postgraduate Summer School

Lancaster, August 24th-September 5th-2015

1Strasbourg-Madrid Shell-Model collaboration

Bibliography

Basic ideas and concepts in nuclear physicsan introductory approachHeyde K.IOP Publishing 1994

Shell model applications in nuclear spectroscopyBrussaard P.J., Glaudemans P.W.M.North-Holland 1977

The nuclear shell modelHeyde K.Springer-Verlag 1994

The nuclear shell modelA. Poves and F. NowackiLecture Notes in Physics 581 (2001) 70ff

The shell model as a unified view of nuclear structureE. Caurier, G. Martinez-Pinedo, F. Nowacki, A. Poves, A. P. ZukerRev. Mod. Phys. 77, 427 (2005)

Shell structure evolution and effective in-medium NN interactionN. SmirnovaEcole Joliot-Curie 2009

Outline

Lecture 1: Introduction, basic notions, shell model codes

and calculations

Lecture 2: Lanczos structure functions, Effective

Interactions

Lecture 3: Shell model applications to nuclear

spectroscopy

monopole and Multipole Hamiltonians

Hmonopole: spherical mean-field ( saturation and evolution of the spherical singleparticle levels).Hmultipole: correlator (pairing, quadrupole, octupole...)

Important property: 〈CS ± 1|H|CS ± 1〉 = 〈CS ± 1|Hmonopole|CS ± 1〉ZAdjust monopole part only on CS±1 particle nuclei

0

2.5

5

8 14 16

Ee

xc(

2+)

(Me

V)

Neutron number

O, Z=8

THE EXP 0

2.5

5

20 28 32

Neutron number

Ca, Z=20

THE EXP

-10

-5

0

5

8 14 16

ES

PE

(M

eV

) ds

0d5/21s1/2

-10

-5

0

5

20 28 32

fp0f7/21p3/2

All realistic NN interactionsprovide similar monopoleand multipole parts

The link to the phase shiftsis independent on thedetails of the potential andrenormalization procedures.

They all share the sameproblems, in particular, thelack of spin-orbit shellclosures.

Phys. Rev. C85 (2011) 051301

Shell evolution mechanism

Hmonopole =∑

i

niǫi +∑

i≤j

ni .nj Vij

Π ν Π ν

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

Shell gapDefinition

The shell gap ∆ is defined as the difference of binding energies

(positively defined)

∆ = [BE(N)− BE(N − 1)]− [BE(N + 1)− BE(N)]

= 2BE(N)− BE(N + 1)− BE(N − 1)

ZThe uncorrelated shell gap is therefore equal to the difference

of the corresponding ESPE and fully given by Hmonopole.

Landscape of medium mass nuclei

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

68 Ni 78 Ni

16 O

14 C

48 Ni 56 Ni

36 S34 Si32 Mg

52 Ca48 Ca40 Ca

22 O

42 Si

64 Cr

f7/2 g 9/2

f7/2

f5/2

g 9/2f5/2p 1/2p 3/2f7/2

p 3/2

90 Zr80 Zr

p 1/2g 9/2

d5/2

p 1/2f5/2p 3/2

12 Be

f7/2

40 Mg

74 Cr

8 8

20

28

20

28

32

40Z

28

2820

N

40

82014

Sn100 50

50

Landscape of medium mass nuclei

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

68 Ni 78 Ni

16 O

14 C

48 Ni 56 Ni

36 S34 Si32 Mg

52 Ca48 Ca40 Ca

22 O

42 Si

64 Cr

f7/2 g 9/2

f7/2

f5/2

g 9/2f5/2p 1/2p 3/2f7/2

p 3/2

90 Zr80 Zr

p 1/2g 9/2

d5/2

p 1/2f5/2p 3/2

12 Be

f7/2

40 Mg

74 Cr

8 8

20

28

20

28

32

40Z

28

2820

N

40

82014

Sn100 50

50

UNDERSTANDING REGULARITIESfor both SPHERICAL and DEFORMED systems

Magic Numbers: 24O, 48Ni, 54Ca, 78Ni, 100Sn

Islands of Deformation: 12Be, 32Mg, 42Si, 64Cr, 80Zr ...

Variety of phenomena dictated by shell structure

Close connection between collective behaviour and underlyingshell structure

H = Hm +HM

Interplay between• Monopole field (spherical mean field)• Multipole correlations (pairing, Q.Q, ...)

Effective Single Particle Energies: Trends

28O

34Si

40Ca

-20

-10

0

10

8 14 16 20

ES

PE

(M

eV

)

Proton number

d5/2s1/2d3/2f7/2

p3/2

p1/2f5/2

At the neutron drip line, the ESPE’s of28O are completely at variance with

those of 40Ca at the stability valley. Thechange from the standard ESPE’s of16O to the anomalous ones in 28O istotally due to the interactions of sd shellneutrons among themselves

Notice that the sd shell orbits remainalways below th pf shell with the ν0f 7

2

and ν0p 3

2

orbitals DO get inverted

The monopole part of theneutron-proton interaction restores theN=20 shell gap when the valley ofstability is approached

Spin-Tensor decomposition shows it ismainly a Central and Tensor effect

Magic numbers are associated to enegy gaps in the spherical meanfield. Therefore, to promote particles above the Fermi levels costsenergy

However some intruders configurations can overwhelm their loss ofmonopole energy with their huge gain in correlation energy

Several examples of this phenomenon exist in stable magic nuclei(as in 40Ca nucleus) in the form of coexisting spherical, deformedand superdeformed states in a very narrow energy range

At the very neutron rich or very proton rich edges, the T=0 and T=1channels of the effective nuclear interaction weight very differentlythan they do at the stability line. Therefore the effective singleparticle structure may suffer important changes, leading in somecases to the vanishing of established shell closures or to theappearance of new ones


28O

34Si

40Ca

-20

-10

0

10

8 14 16 20

ES

PE

(M

eV

)

Proton number

d5/2s1/2d3/2f7/2

p3/2

p1/2f5/2




2

and ν0p 3

2





28O

34Si

40Ca

-20

-10

0

10

8 14 16 20

ES

PE

(M

eV

)

Proton number

d5/2s1/2d3/2f7/2

p3/2

p1/2f5/2




2

and ν0p 3

2




0

1s (2) 2

1p(8)(6)

8

1g7/2

1f2p

1d2s 2

3s

(38)(40)(50)

1g

1h

1h11/2

2d

3p

2d3/2

2p1/2

2s

1p3/21p1/2

1s

2p3/2

1h9/2

3s1/2

2d5/2

2f7/22f5/2 3p3/23p1/2

(64)

(82)

2f

1g9/2

1f5/2

1f7/2

1d3/2

1d5/2

(8)

(6)(4)(2)

(10)

(8)(6)

(4)(2)

(12)

(2)(6)(4)

(12)

(8)

(6)(2)(4)

(2)

(2)

(4)

(126)

(20)(14)

1i13/2 (14)

N=5

N=4

N=3

N=2

N=1

N=0

(28) 28

82

126

50


28O

32Mg

34Si

40Ca

1.9

5.1

7.3

N=20

-20

-10

0

10

8 14 16 20

ES

PE

(M

eV

)

Proton number

d5/2

s1/2

d3/2f7/2

p3/2p1/2

f5/2




2

and ν0p 3

2





28O

32Mg

34Si

40Ca

1.9

5.1

7.3

N=20

-20

-10

0

10

8 14 16 20

ES

PE

(M

eV

)

Proton number

d5/2s1/2d3/2f7/2

p3/2

p1/2f5/2




2

and ν0p 3

2





28O

32Mg

34Si

40Ca

1.9

5.1

7.3

N=20

-20

-10

0

10

8 14 16 20

ES

PE

(M

eV

)

Proton number

d5/2s1/2d3/2f7/2

p3/2

p1/2f5/2




2

and ν0p 3

2




Spin-Tensor decomposition

V =∑

Vk =∑

Uk .Sk

k S S′ spin-tensor components

0 0 0 C=Central1 1

1 0 1 ALS=antisymmetric spin-orbit1 01 1 LS=spin-orbit

2 1 1 T=Tensor


28O

32Mg

34Si

40Ca

1.9

5.1

7.3

N=20

-20

-10

0

10

8 14 16 20

ES

PE

(M

eV

)

Proton number

d5/2s1/2d3/2f7/2

p3/2

p1/2f5/2




2

and ν0p 3

2




∆ (f 72-d 3

2) filling d 5

2between 32Mg and 34Si

G matrix SDPF-U diff.Tot 1.57 1.17 -0.40

Central 1.11 0.70 -0.41Vector -0.159 -0.155 0.004

LS -0.049 -0.12 -0.071ALS -0.11 -0.035 0.075

Tensor 0.61 0.63 0.02Nuclear shell evolution and in-medium NN interaction

N. Smirnova, K. Heyde, B. Bailly, F. Nowacki, K. Sieja

Phys. Rev. C86, 034314 (2012)


1d3/2

νΠ

1f7/22p3/2

Si35

2s1/2

1d5/2

∆ (p 3

2

-p 1

2

) filling s 1

2

between 34Si and 36S

KLS N3LO N3LO(bare) (4~ω)

Tot 0.58 0.58 0.60

Central 0.00 0.00 0.00Vector 0.58 0.58 0.60Tensor 0.00 0.00 0.00

G. Burgunder, PhD ThesisGANIL, December 2011


1d3/2

νΠ

1f7/22p3/2

2s1/2

1d5/2

37 S

∆ (p 3

2

-p 1

2

) filling s 1

2

between 34Si and 36S

KLS N3LO N3LO(bare) (4~ω)

Tot 0.58 0.58 0.60

Central 0.00 0.00 0.00Vector 0.58 0.58 0.60Tensor 0.00 0.00 0.00

G. Burgunder, PhD ThesisGANIL, December 2011

Correlation Energies

Let us consider the configurations with closed N=20

[sd ]12,Z (normal filling) and the ones with two neutrons

blocked in the pf shell [sd ]10,Z [pf ]2,0 (intruder)

And calculate the energy of the ground states at fixed

configuration, with and without correlations

Correlations energies: normal vs 2p-2h intruder

8 9 10 11 12 13 14 15 16 17 18

Z-14

-12

-10

-8

-6

-4

-2

0

0p-0h 2p-2h

F. Nowacki and A. PovesPhys. Rev. C90, 014302 (2014)


8 9 10 11 12 13 14 15 16 17 18

Z-14

-12

-10

-8

-6

-4

-2

0

0p-0h 2p-2h

28O



8 9 10 11 12 13 14 15 16 17 18

Z-14

-12

-10

-8

-6

-4

-2

0

0p-0h 2p-2h

28O

30Ne



8 9 10 11 12 13 14 15 16 17 18

Z-14

-12

-10

-8

-6

-4

-2

0

0p-0h 2p-2h

28O

30Ne

32Mg



8 9 10 11 12 13 14 15 16 17 18

Z-14

-12

-10

-8

-6

-4

-2

0

0p-0h 2p-2h

28O

30Ne

32Mg34Si


Gaps: normal vs 2p-2h intruder

8 9 10 11 12 13 14 15 16 17 18

Z

-2

0

2

4

6

8

10

12

14M

eVE(2p-2h)-E(0p-0h) : without correlationsE(2p-2h)-E(0p-0h) : with correlations

ISLAND OF INVERSION

30Ne

32Mg

34Si


Merging of the islands of inversion at N=20 and

N=28

8 10 12 14 16 18 20 22 24 26 28 30 32

N0

1

2

3

4

2+ e

xcita

tion

ener

gy in

MeV

EXPTH

0

20

40

60

80

100

34 36 38 40 42 44

BE

2(e2 .fm

4 )

Si

BE2,SMEXP

8 10 12 14 16 18 20 22 24 26 28 30 32

N

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2+ e

xcita

tion

ener

gy in

MeV

EXPTH

30 32 34 36 38 400

50

100

150

e2 fm4

B(E2) expB(E2) th

Si

Mg

Island of inversion at N=40, an old story: 1996

A. Poves

Island of inversion at N=40, an old story: 2003

8 10 12 14 16 18 20

Proton number

−20

−15

−10

−5

0

5

effe

ctiv

e S

PE

(M

eV)

N=20

1f7/2

1d3/2

2p3/2

20 22 24 26 28

Proton number

−2

0

2

4

6

effe

ctiv

e S

PE

(M

eV)

N=40

1g9/2

1f5/2

2d5/2

GANIL

More recent experimental information

RAPID COMMUNICATIONS

PHYSICAL REVIEW C 81, 051304(R) (2010)

Collectivity at N = 40 in neutron-rich 64Cr

A. Gade,1,2 R. V. F. Janssens,3 T. Baugher,1,2 D. Bazin,1 B. A. Brown,1,2 M. P. Carpenter,3 C. J. Chiara,3,4 A. N. Deacon,5

S. J. Freeman,5 G. F. Grinyer,1 C. R. Hoffman,3 B. P. Kay,3 F. G. Kondev,6 T. Lauritsen,3 S. McDaniel,1,2 K. Meierbachtol,1,7

A. Ratkiewicz,1,2 S. R. Stroberg,1,2 K. A. Walsh,1,2 D. Weisshaar,1 R. Winkler,1 and S. Zhu3

1National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824, USA2Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA

3Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA4Department of Chemistry and Biochemistry, University of Maryland, College Park, Maryland 20742, USA

RAPID COMMUNICATIONS

PHYSICAL REVIEW C 81, 061301(R) (2010)

Onset of collectivity in neutron-rich Fe isotopes: Toward a new island of inversion?

J. Ljungvall,1,2,3 A. Gorgen,1 A. Obertelli,1 W. Korten,1 E. Clement,2 G. de France,2 A. Burger,4 J.-P. Delaroche,5 A. Dewald,6

A. Gadea,7 L. Gaudefroy,5 M. Girod,5 M. Hackstein,6 J. Libert,8 D. Mengoni,9 F. Nowacki,10 T. Pissulla,6 A. Poves,11

F. Recchia,12 M. Rejmund,2 W. Rother,6 E. Sahin,12 C. Schmitt,2 A. Shrivastava,2 K. Sieja,10 J. J. Valiente-Dobon,12

K. O. Zell,6 and M. Zielinska13

1CEA Saclay, IRFU, Service de Physique Nucleaire, F-91191 Gif-sur-Yvette, France2GANIL, CEA/DSM-CNRS/IN2P3, Bd Henri Becquerel, BP 55027, F-14076 Caen, France

3CSNSM, CNRS/IN2P3, F-91405 Orsay, France

MSU

GANIL

SM framework

Island of inversion around 64Cr

S. Lenzi, F. Nowacki, A. Poves and K. Sieja

Phys. Rev. C 82, 054301, 2010.

f7/2

p1/2

f5/2

p3/2

Π

g9/2d5/2

Ca

ν

48

LNPS interaction:

based on realistic TBME

new fit of the pf shell (KB3GR, E. Caurier)

monopole corrections

g9/2-d5/2 gap set to 1.5 Mev in 68Ni

Calculations:

up to 14p-14h excitations across Z=28and N=40 gaps

up to 1010

m-scheme code ANTOINE (non publicversion)

Shape transition at N=40

68Ni

68Ni

0

0.5

1

1.5

2

2.5

3

20 22 24 26 28

E(2

+)

(MeV

)

Z

N=40

(a)SMEXP

0

100

200

300

400

500

20 22 24 26 28

B(E

2;2+

->

0+)

(e2 fm

4 )

Z

(b)

SMEXP

Nucleus νg9/2 νd5/2 configuration68Ni 0.98 0.10 0p0h(51%)66Fe 3.17 0.46 4p4h(26%)64Cr 3.41 0.76 6p6h(23%)62Ti 3.17 1.09 4p4h(48%)


66Fe

66Fe

68Ni

68Ni

0

0.5

1

1.5

2

2.5

3

20 22 24 26 28

E(2

+)

(MeV

)

Z

N=40

(a)SMEXP

0

100

200

300

400

500

20 22 24 26 28

B(E

2;2+

->

0+)

(e2 fm

4 )

Z

(b)

SMEXP



64Cr

64Cr

66Fe

66Fe

68Ni

68Ni

0

0.5

1

1.5

2

2.5

3

20 22 24 26 28

E(2

+)

(MeV

)

Z

N=40

(a)SMEXP

0

100

200

300

400

500

20 22 24 26 28

B(E

2;2+

->

0+)

(e2 fm

4 )

Z

(b)

SMEXP


Neutron effective single particle energies

-25

-20

-15

-10

-5

0

5

20 28 32

ES

PE

(M

eV)

Z

(a)

N=40

f7/2p3/2f5/2p1/2g9/2d5/2

reduction of the νf5/2-g9/2 gap withremoving f7/2 protons

proximity of the quasi-SU3 partnerd5/2

inversion of d5/2 and g9/2 orbitalssame ordering as CC calculations

60Ca 64Cr 68Ni

-25

-20

-15

-10

-5

0

5

8 14 16

ES

PE

(M

eV)

Z

(b)

N=20

d5/2s1/2d3/2f7/2p3/2

reduction of the νd3/2-f7/2 gap withremoving d5/2 protons

proximity of the quasi-SU3 partnerp3/2

inversion of p3/2 and f7/2 orbitals

28O32Mg

34Si

Neutron effective single particle energies

-25

-20

-15

-10

-5

0

5

20 28 32

ES

PE

(M

eV)

Z

(a)

N=40

f7/2p3/2f5/2p1/2g9/2d5/2

reduction of the νf5/2-g9/2 gap withremoving f7/2 protons

proximity of the quasi-SU3 partnerd5/2

inversion of d5/2 and g9/2 orbitalssame ordering as CC calculations

60Ca 64Cr 68Ni

-25

-20

-15

-10

-5

0

5

8 14 16

ES

PE

(M

eV)

Z

(b)

N=20

d5/2s1/2d3/2f7/2p3/2

reduction of the νd3/2-f7/2 gap withremoving d5/2 protons

proximity of the quasi-SU3 partnerp3/2

inversion of p3/2 and f7/2 orbitals

28O32Mg

34Si

G. Hagen et al.

Phys. Rev. Lett. 109, 032502 (2012)

Extension of collectivity N=40 towards N=50

0

1

2

34 36 38 40 42 44 46 48

En

erg

y (

MeV

)

Neutron number

2+

4+

LNPS-mLiteratureThis work

0

1

2

34 36 38 40 42 44 46 48

En

erg

y (

MeV

)

Neutron number

2+

4+


2

3

34 36 38 40 42 44 46 48R4

2=

E(4

+)/

E(2

+)

Neutron number


2

3

34 36 38 40 42 44 46 48R4

2=

E(4

+)/

E(2

+)

Neutron number


Extension of collectivity N=40 towards N=50

0

1

2

34 36 38 40 42 44 46 48

En

erg

y (

MeV

)

Neutron number

2+

4+


0

1

2

34 36 38 40 42 44 46 48

En

erg

y (

MeV

)

Neutron number

2+

4+


2

3

34 36 38 40 42 44 46 48R4

2=

E(4

+)/

E(2

+)

Neutron number


2

3

34 36 38 40 42 44 46 48R4

2=

E(4

+)/

E(2

+)

Neutron number


C. Santamaria et al., submitted for publication in Phys. Rev. Lett.

Spin-orbit shell closure far from stability

H. O.

H. O.

Π+

Π−

sd-pf: 42Si deformed

pf-sdg: 78Ni ???

sdg-phf: 132Sn

doubly magic

Evolution of Z=28 from N=40 to N=50

Evolution of N=50 from Z=40 to Z=28

Three-body forces in medium mass nuclei

16O 24O 40Ca 48Ca 68Ni 78Ni

-10

-5

0

5

8 14 16

ds

0d5/21s1/2

-10

-5

0

5

20 28 32

fp0f7/21p3/2

-10

-5

0

5

40 50 56

gd0g9/21d5/2

Evolution of the neutron effective single-particle energies

with neutron filling in ds, fp, and gd shells

“Universal” mechanism for the generation of T=1 spin-orbit

shell closures

Connection with 3N forces and ab-initio calculations:

• “works” for Coupled-Cluster and to be checked in nickels

• problems for “ab-initio” core shell-model

Three body forces and persistence of spin-orbit shell gaps in medium-mass nuclei: Towards the doubly magic 78NiK. Sieja, F. NowackiPhys. Rev. C85, 051301(R) (2012)

Physics around 78Ni

H. O.

H. O.

Π+/−

Π−/+

PFSDG-U interaction:


pf shell for protons and gds shell forneutrons


proton and neutrons gap 78Ni fixed tophenomenological derived values

Calculations:

excitations across Z=28 and N=50gaps

up to 1010 Slater Determinant basisstates

m-scheme code ANTOINE (nonpublic version)

Physics around 78Ni

��

��

��

��

��

��

��

��

��

��

��

��

H. O.

H. O.

Π+/−

Π−/+

PFSDG-U interaction:


pf shell for protons and gds shell forneutrons


proton and neutrons gap 78Ni fixed tophenomenological derived values

Calculations:

excitations across Z=28 and N=50gaps

up to 1010 Slater Determinant basisstates

m-scheme code ANTOINE (nonpublic version)

Shape coexistence in 78Ni

At first approximation, 78Nihas a double closed shellstructure for GS

But very low-lying 4p4hstructures

The first excited state 2+1 in

78Ni predicted at 2.8 MeVand to be a deformedintruder !!!

Necessity to go beyond(fpg 9

2d 5

2) LNPS space

Constrained deformed HF in the SM basis

(B. Bounthong, Ph D Thesis, Strasbourg)

Island of Deformation below 78Ni

Schematic SU3 predictions: arXiv 1404.0224

Nilsson-SU3 selfconsistency:

Quadrupole dominance in heavy N=Z nuclei

Ni78 Fe76 Cr74-5

-4

-3

-2

-1

0

1

2

Ene

rgie

s in

MeV

rea

ltive

to th

e 0p

0h s

tate

s

0p-0h2p-2h4p-4h

monopole + quadrupole model

proton gap (5MeV) and neutron gap (5MeV) estimates

Quasi-SU3 (protons) and Pseudo-SU3(neutrons) estimates

Qs = (〈2q20〉 + 3.)b2)2/3.5

E = ǫi〈ni〉 − ~ωκ(〈2q20(3)〉

15+

〈2q20(4)〉

23)

Prediction of Island of strong collectivity

below 78Ni !!!





Ni78 Fe76 Cr74-5

-4

-3

-2

-1

0

1

2

Ene

rgie

s in

MeV

rea

ltive

to th

e 0p

0h s

tate

s

0p-0h2p-2h4p-4h




Qs = (〈2q20〉 + 3.)b2)2/3.5

E = ǫi〈ni〉 − ~ωκ(〈2q20(3)〉

15+

〈2q20(4)〉

23)


below 78Ni !!!

E*(2+1 ) Qs BE2↓ Qi from Qi from β

(MeV) (e.fm2) (e2.fm4) Qs from BE2

72Cr 0.24 -48 551 167 166 0.3774Cr 0.29 -46 515 162 161 0.3676Cr 0.41 -29 239 100 108 0.22

76Fe 0.35 -39 346 135 132 0.26





Ni78 Fe76 Cr74-5

-4

-3

-2

-1

0

1

2

Ene

rgie

s in

MeV

rea

ltive

to th

e 0p

0h s

tate

s

0p-0h2p-2h4p-4h




Qs = (〈2q20〉 + 3.)b2)2/3.5

E = ǫi〈ni〉 − ~ωκ(〈2q20(3)〉

15+

〈2q20(4)〉

23)


below 78Ni !!!

0

1

2

34 36 38 40 42 44 46 48 50

En

erg

y (

MeV

)

Neutron number

2+

4+

PFSDG-ULiteratureThis work

Extension of Island of deformation to N=50 !!!

Spin-orbit shell closure far from stability

H. O.

H. O.

Π+

Π−

sd-pf: 42Si deformed

pf-sdg: 78Ni ???

sdg-phf: 132Sn

doubly magic

Evolution of Z=28 from N=40 to N=50

Evolution of N=50 from Z=40 to Z=28


34Si

42Si

7.5

5.1

-40

-35

-30

-25

-20

-15

-10

14 16 20 28 32 34 40

ES

PE

(M

eV

)

Neutron number

Silicium chain

d5/2s1/2d3/2


34Si

42Si

7.5

5.1

-40

-35

-30

-25

-20

-15

-10

14 16 20 28 32 34 40

ES

PE

(M

eV

)

Neutron number

Silicium chain

d5/2s1/2d3/2

∆ (d 52-d 3

2) filling f 7

2between 34Si and 42Si

G matrix SDPF-U diff.Tot -3.15 -2.38 +0.77

Central 0.24 -0.11 -0.35Vector -0.27 0.55 0.82

LS -0.11 0.11 0.22ALS -0.16 0.44 0.60

Tensor -2.65 -2.77 0.12


68Ni

78Ni

8.8

5.2

-30

-20

-10

0

20 28 32 38 40 50

ES

PE

(M

eV

)

Neutron number

f5/2p3/2p1/2

g9/2f7/2


68Ni

78Ni

8.8

5.2

-30

-20

-10

0

20 28 32 38 40 50

ES

PE

(M

eV

)

Neutron number

f5/2p3/2p1/2

g9/2f7/2

∆ (f 72-f 5

2) filling g 9

2between 68Ni and 78Ni

G matrix PFSDG-U diff.Tot -3.21 -3.64 -0.43

Central -0.28 -0.12 +0.16Vector -0.081 -1.70 -1.62

LS -0.081 -1.12 -1.04ALS 0.0 -0.58 -0.58

Tensor -2.84 -1.82 +1.02


120Sn

132Sn

5.2

5.9

-20

-10

0

50 56 64 66 70 82

ES

PE

(M

eV

)

Neutron number

g9/2

d5/2g7/2s1/2

d3/2h11/2


120Sn

132Sn

5.2

5.9

-20

-10

0

50 56 64 66 70 82

ES

PE

(M

eV

)

Neutron number

g9/2

d5/2g7/2s1/2

d3/2h11/2

∆ (g 92-g 7

2) filling h 11

2between 120Sn and 132Sn

Vlowk NNSP diff.Tot -2.15 +0.89 +3.04

Central -0.034 +0.30 +0.33Vector +0.12 +1.55 +1.43

LS -0.038 -0.06 -0.022ALS +0.49 +1.61 +1.12

Tensor -2.30 -0.96 +1.34

Summary

Monopole drift develops in all regions but the Interplay betweencorrelations (pairing + quadrupole) and spherical mean-field(monopole field) determines the physics.It can vary from :- island of inversion at N=20 and N=40- deformation at Z=14, N=28 for 42Si and shell weakening atZ=28, N=50 for 78Ni- deformation extending from N=40 to N=50 for Z=24-26 for 74Crand 76FeThe “islands of inversion” appear due to the effect of thecorrelations, hence they could also be called “islands ofenhanced collectivity”. As quadrupole correlations are dominantin this region, most of thei inhabitants are deformed rotors.Shape transitions and coexistence show up everywhere

Quadrupole energies can be huge and understood in terms ofsymmetries

Spin-Tensor Analysis show competing trends but varyingsignificantly from light to middle mass nuclei

Summary

Special Thanks to:

B. Bounthong, E. Caurier, H. Naidja, K. Sieja, A. Zuker

A. Poves

M. Hjorth-Jensen

H. Grawe, S. Lenzi, O. Sorlin

J. Herzfeld

shell model a uniﬁed view of nuclear...

Documents