- Two categories of magic numbers : Harmonic Oscillator and Spin Orbit

- The role of proton-neutron interactions Disappearance of magic numbers Appearance of new magic numbers

- What do we mean by SO magic numbers ?

- Influence of binding energy on nuclear force ?

Note that

- Structural variations better seen in light nuclei-Extract general empirical rules / symmetries…

-> extrapolate to other regions

Mea

n-fi

eld

appr

oach

for

ato

mic

nuc

lei

L.S+

f7/2

d3/2

20

d5/2

p1/2

s1/2

p3/2

f5/2

28

40g9/2

50

14

H.O L2+

1d

1f

2s

2p

20N=2

N=3

1g

N=42d

40

20

N=1

88

40

Experimental progresses and challenges in the evolution of shell closuresO. Sorlin (GANIL)

ESNT 2010 - Saclay

82

The N=20 shell closure A prototypical case of HO shell number

N=204

3

2

1

0

E(2

+)

[MeV

]

12 16 20 24Neutron Number

16S

20Ca

12Mg

N/Z

38Ar 36S 34Si 32Mg 30Ne40Ca

200

400

B(E

2) [

e2 fm

4 ]

N=20

sd

sdfp

14Si

40

30

20

10

2016 24

S2N

(M

eV)

Neutron Number N

45Ca

36Ca 20

20

35Mg

27Mg

N=20 magic number Disappears !

40Ca

36S32Mg

2) Presence of intruder fp states, f and p reversed ?

3) New magic number at N=16 ?

ESPE in N=20 isotones and island of inversion

N=20

T. Utsuno et al. PRC (1999)

Isla

nd o

f in

vers

ion

0f7/2

d3/2s1/2d5/2

Role of the Vpnd5/2d3/2 and Vpnd5/2(fp) interactionsAttractive and repulsive tensor terms, respectively

Neu

tron

1) Reduction of the N=20 shell gap

Occupancy of fp states grows at N=20

occu

panc

yJ. R. Terry et al., PRC 77 (2008) 014316.

4030

3330

20301680

W. Catford et al., PRL 2010

25Ne0.80

0.150.44

0.75

0.73

(SF)

= 2, 3/2+

= 0, ½+

= 2, 5/2+ hole

= 1, 3/2-

( = 3),7/2-

J

= 1

= 3

= 2

= 2

= 1

- Proximity of f and p states to sd ones

- p and f states reversed, N=28 gap

24Ne(d,p)25Ne with TIARA+EXOGAM+VAMOS (GANIL)

Protons -> TIARAGammas -> ExogamNuclei -> Vamos

The ‘sizes’ of the N=20 and N=16 gaps in Oxygen (RIKEN)22O(d,p)23O reaction to probe the neutron N=16, 20 shell closures

23O

N=20 : 1.3 MeV

N=16 : 4.0 MeV

5/2+ observed PRL99 (2007)Elekes et al. PRL98 (2007) 102502

Gated on neutrons Gated on 4 MeV neutron peak

L=2

hole

34Si40Ca

d3/2s1/2d5/2

28O N=20

Utsuno, Otsuka et al.

16

Isla

nd o

f in

vers

ion

A ‘critical’ view :

Mechanism of inversion not proven :

0+2 not yet observed in 34Si and 32Mg

Hard to get 28O unbound using standard Vnn

No ‘direct’ (easy) determination of Vd5/2d3/2

due to deformation

So far monopole assumed constant

whatever neutron and proton binding

energy

True or not ? Can we check it ?-> Study of 26F

Nuclear interaction in the sd shell N

eutr

on

Empirical determination of Vd5/2d3/2

25F

Sp

183.46

25OSn

Sn+Sp

26Ffree

24O 168.38

BE(MeV)

24O

p

n

26F

Vpn (d3/2d5/2)

d5/2

d3/2

Exp monopole ~ 600keV weakerthan Shell Model !

continuum effects … ??Where is the 4+ ? Isomer ?

interact

J

Jnp

Jpn

pn J

jjvJV

)12(

),()12(

J

2/32/52/32/5 J

1

exp

2

exp1

?

Hoffman PRL 100(2008)Stanoiu thesis 2003,E(J=2)Jurado PLB 649 (2007)

41

32

J

US

Da

USDa

Generalization to other HO shell gaps

Same mechanism at play : -Drop in 2+ energy at N=8, 20 and 40

-Inversion between normal and intruder states at N=40

- Search for a (super)deformed 0+2 in 68Ni

-Prove the extreme deformation of 64Cr

Great similarity between the three cases of HO shell numbers

N=8 N=20 N=40

O. S. , MG Porquet PPNP (2008)

d5/2

d3/2

s1/2

d5/2

14

16

f7/2

Z=8

N~20 p3/2

Z=14

d5/2

d3/2s1/2

d5/2

14[ ]

f7/2

20

p3/2

Z=28

f7/2

p3/2

f5/2

f7/2

28

p1/2

[ ]

g9/2

40

f7/2

p3/2

f5/2

f7/2

28

p1/232

34

g9/2

Large N/Z

Z=20

N~40d5/2d5/2

p3/2

p1/2

p3/2

6

d5/2

Z=2

N~8 s1/2

p3/2

p1/2

p3/2

6[ ]

d5/2

8

Z=6

s1/2

Evolution of Harmonic Oscillator shell closures

Role of the p3/2- p1/2 interaction

Role of the d5/2- d3/2 interaction

Role of the f7/2- f5/2 interaction ?

SP

IN –

FL

IP

=0

IN

TE

RA

CT

ION N=14

N=28

N=50

Small gaps

The making of ‘SO’ magic numbers

Which physics ? Which interactions ?

20 28

Bin

ding

ene

rgy

Ene

rgy

[MeV

]

28

Neutrons

Evolution of neutron SPE in the Ca isotopic chain

20 28 20 28 Neutrons

2828

Courtesy M.G Porquet

No increase of the N=28 shell gap when f7/2 is filled Same with realistic VlowK interaction -> 3 body ?

> Same increase of the neutron shell gaps by about 3 MeV !> Same mechanism at play to create SO magic numbers -> empirical rule to be used to constraint these spacing for heavier nuclei

Building SO magic numbers by neutron-neutron interactions

O

20 2822 24 26

Ca

Neutron number

- 4

- 840 42 44 46 48 50

0

g9/2

d5/2

68Ni 78Ni

N=50?

From data around 90Zr

Ni

Neutron number

Extracted from BE’s, spectroscopy and SF’s

In collab with MG Porquet

Sn(23O)

Sn(22O)

Sn(17O)

E*(17O)

d3/2

p1/2

28

f5/2

p3/2

f7/2

14s1/2

d5/2 [ ] [ ]

> Role of nuclear forces :Modification of the N=28 shell gap ?SO and Tensor interaction ?

j=2 j=2

Enhanced E2 collectivity due to j=2

42Si

44S

N=28N=20

The study of the N=28 shell closure : a way to probe nuclear force

Ca, Z=20 48Ca

34Si

S, Z=16

Si, Z=14

36S

40Ca

46Ar

1- compression of proton orbits

neutron f7/2 filling

2- Evolution of neutron orbits due to pn interactions

proton sd removed

d3/2

s1/2

1.33

f5/2

f7/2

- 10

- 6

- 2

0

- 8

- 4

f7/2

p3/2

p1/2

f5/2

28

28

49Ca47Ar

ES

PE

(MeV

)

2018

Variation of single particle energies (SPE)

Tensor interaction (Otsuka) d3/2 –( f7/2-f5/2 )

+280keV per proton added in d3/2

-210keV

Evolution of SPE’s from tensor part of the proton-neutron interaction

Use of 46Ar (d,p) transfer reaction

Size of the N=28 shell gap

Reduction of SO splitting

L. Gaudefroy et al. PRL 97 (2006),

18d

p

p

28f7/2

p3/2

p1/2

f5/2

- 10

- 6

- 2

0

- 8

- 4

SP

E(M

eV) + 2

f7/2

p3/2

p1/2

f5/2

28

28

49Ca47Ar2018

Global trend of single particle energies between 49Ca and 43Si

derived from experimentally-constrained monopole variations

N=29

28

45S161443Si

28

f7/2

p3/2

p1/2

-A shrink of SPE’s due to two-body p-n interactions…-Favor particle-hole excitations and E2 collectivity

0

0

- Spherical, shape coexistence in 44S and deformation in 42Si

44S

Weak mixing between prolateand spherical shapes in the 0+

Qi ~ 55

Qi ~ 0

Electron spectroscopy to probe shape coexistence in 44S

Glasmacher et al., PLB 395 (97)

44S01

+

63(18)

2+

1330

BE2e2fm4

1365 2.62(2) s42(2)

02+

C. Force, S. Grévy et al. to be published

Ee- (keV)

1365 keV

e+ e-

e- conv

(E0) = 8.7(5) 10-3

BE2(0+2 → 2+

1)BE2(0+

1 → 2+1)

~1/7

f7/2d3/2s1/2

d5/2

14

[ ]

[ ] p3/2

42Si14

SP

IN- F

LI P

=

1 I

NT

ER

AC

TIO

N

N=28p3/2

f7/2

d3/2s1/2

d5/2

14

[ ]

[ ]

[ ]28[ ]

48Ca20

42Si

Collapse of the N=28 shell closure in 42Si

B. Bastin, S. Grévy et al., PRL 99 (2007)

Role of the d3/2 – f7/2 interaction

Decrease of the N=28 gap by ~1MeV for 6 protons

N=14 shell closure in 22O and 20C

Thirolf et al. PLB 485 (2000) M. Stanoiu et al. PRC 69 (2004) and (2009)

5

0

E(2

+)

(MeV

)

O

5 10 15 20Neutron Number

s1/2 N=14

d5/2p1/2

p3/2

6

[ ]

[ ]14[ ]

22O 8

20C

C

d5/2s1/2

p1/2

p3/2

6

[ ]

[ ]

20C 6 20C 6

Role of the p1/2 – d5/2 interaction

Decrease of the N=14 by ~1.6 MeV for 2 protons

f7/2d3/2s1/2

d5/2

14

[ ]

[ ] p3/2

42Si14

SP

IN- F

LI P

=

1 I

NT

ER

AC

TIO

N

s1/2 N=14

d5/2p1/2

p3/2

6

[ ]

[ ]14[ ]

22O 8

d5/2s1/2

p1/2

p3/2

6

[ ]

[ ]

20C 6 20C 6

N=28p3/2

f7/2

d3/2s1/2

d5/2

14

[ ]

[ ]

[ ]28[ ]

48Ca20

N=14

N=28

50

90Zr

N=50d5/2

g9/2

f5/2p3/2

f7/2

28

[ ]

[ ]

[ ][ ]

40

g9/2

f5/2p3/2

d5/2

28

[ ]

d5/2

78Ni50

50

N=50 ??

152Gd

N=82f7/2

h11/2

d5/2g7/2

g9/2

50

[ ]

[ ]

[ ]82[ ]

h11/2

d5/2g7/2

g9/2

28

[ ]

f7/2

64 132Sn50

[ ]82

N=82 strong

42Si

68Ni

48Ca40Ca

1

2

3

4

Occupation probability0 0.5 1

2+ e

nerg

y (M

eV)

34Si

?

The N=50 shell closure at 78Ni50

« Monopole propose, quadrupole dispose »A. Zuker

Some Conclusions

Robust effect of NN inteactions :

Proton Neutron interaction L=0 plays an essential role to modify HO shell gaps

Proton Neutron interaction L=1 plays an certain role to modify SO shell gaps->Perhaps not strong enough to supress the magicity in 78Ni50

Role of Vnn to create SO magic numbers -> Same increase of neutron shell gap (3MeV) for all SO magic numbers !

Modification of Vpn due to the presence of continuum ? Vpn d5/2d3/2 (26F) ~ 60% of canonical value only ! -> Other candidates YES !!!

Special thanks : S. Grévy, L. Gaudefroy, D. Sohler, Z. Dombradi, M. Stanoiu, M. G. Porquet, F. Nowacki and F. Azaiez

28

V lowk NN

No N=28 shell gap formationwith realistic interactions !

The N=28 shell gap and the role of 3 body forces

Holt, Otsuka, Schwenk, Suzuki

p3/2

p1/2

f7/2

f5/2

47Ar

: reduced by 330keV

Use of 46Ar (d,p) transfer reaction

Size of the N=28 shell gap

Reduction of SO splitting

L. Gaudefroy et al. PRL 97 (2006)

20

18d

p

p

28f7/2

p3/2

p1/2

f5/2

Evolution of the neutron SPE below 48Ca

46Ar

(2J+1)C2S=1.7(2J+1)C2S=2.44

(2J+1)C2S=1.36C2Sf=0.64

C2Sg=0.34

p

f

fp

f5/2

g9/2

neutrons

f7/2

p3/2

p1/2

28

protons

(jp<)

(jp>)

(jn>)

50

d5/2

78Ni

42Si and 78Ni are ‘mirror’ systems

Development of collectivity in 42Si

Doubly magic numbers originating from spin-orbit interaction

Mutual reductions proton and neutron gaps depends on the strength the tensor force

The proton and neutron gaps are connected by ℓ=2 connections with valence states

d3/2

f7/2

neutrons

d5/2

s1/2

14

protons

(jp<)

(jp>)

(jn>)

28p3/2

42Siℓ

=2

ℓ=

2

p1/2

f5/2

Role of proton-neutron forces in the N=28 region

E(1

/2+)

– E

(3/2

+)

(keV

)

Neutron Number

16s1/2

d3/2f7/2

1000

0

40 5042 44 46 48

Neutron Number

32p3/2

f5/2

g9/2

Cu (Z=29)exp Around 78Ni

f7/2

28

E(5

/2- )

– E

(3/2

- ) (k

eV)

??

in the N=50 region

- 10

- 6

- 2

0

- 8

- 4

f7/2

p3/2

p1/2

f5/2

28

28

49Ca47Ar

SP

E(M

eV)

2018

Change of SO splitting for p orbits

p1/2

Central density dependence (Piekarewicz)

p3/2

s1/20.66

+170keVper proton

-85keV

-No change of p1/2-p3/2 splitting between 41Ca and 37S after removal of 4 protons from d3/2

-Reduction of splitting due to s1/2

Gaudefroy et al. PRL 2007

Probe the density dependence of the SO interaction in 36S and 34Si

RMF calculations using NL3 interactionReduction of the SO splitting by 70%

MF / Skyrme or Gogny forces Reduction of the SO splitting by 40%

SM calculations spdf-NRReduction of SO splitting by 30%Bare forces VlowK reduction by 7% only

SO reduced

N=16 disappears !

B.G Todd Rutel et al. PRC 69 (2004) 1301(R)M. Grasso et al. NPA 2009

Analysis GANIL in progress

36S 34Si

34Si36S

Insert here one or two slides on the effect of continuum…

Part I :Properties of shell closures of ‘HO’ origin

The N=8 shell closure

Quadrupole excitations favored in BeFirst ‘Island of inversion’ ?12Be g.s. strongly mixed (Navin et al PRL85; Pain et al. 96)

= 2

14C

12Be

[

1- ]

12Be : Iwasaki et al., PLB 481 (2000) 7

12Be

14C

16O

Evolution of the N=8 shell closure

15O

15O

13C

8 6 4 2Z

-1-2-3-4-5

[1

/2- -

1/2

+]

-6

-7

d5/2

p1/2

p3/26

8s1/2

p1/2 p3/2

p1/2

p3/2

6

d5/2s1/2

11Be

11Be

Role of the p3/2-p1/2 interaction

Magic Numbers are a four-piece rock band from England comprising two pairs of brother and sister who previously went to The Cardinal Wiseman Roman Catholic High School in Greenford. The group was formed in 2002, releasing their critically acclaimed album titled The Magic Numbers in June 2005….

The Magic NumbersFrom Wikipedia, the free encyclopedia

Summary- Two classes of shell closures (magic numbers) : HO and SO- Proton-neutron interactions usually act to destroy them- Takes root in NN bare forces – link in progress- Forces be strong enough to destroy shell closures in heavy nuclei ? - Astrophysical consequences expected- Extrapolation to superheavies or unknown regions ?

g9/2

g7/2

d5/2

h11/2s1/2

d3/2

f7/2

p3/2h9/2

i13/2

f5/2

p1/2

g9/2

g7/2

d5/2

h11/2

s1/2

d3/2

f7/2

p3/2

f5/2p1/2

h9/2

Around 132Sn

126

82

50

82

N>>Z, drip-line

Nuclear Shell Structure Evolution

Mean field near stabilityStrong spin-orbit interaction

Reduced spin-orbit Tensor forcesMean field for N>>Z ?Effect of continuum ?

?

Adapted from J Dobaczewski

Major consequences :

1

1 : Reduction/disappearence of shell gaps -> modify the shape of r abundance peaks

2

2 : Change of g7/2 energy, increase the g7/2 → g9/2 GT transition, shorten -decay lifetimes

3

3 : The valence p states appear at weak excitation energy, favor neutron capure with n =0

No bound excited state in 23O and 24O

Size of N=16 > 4 MeV

Searching for a new N=16 shell closure

In-beam -ray spectroscopy using double step fragmentation

M. Stanoiu et al. PRC 69 (2004)

After this point the talk is finished…

Extra slides only !

Evidence of intruder configurations in neutron-rich Ne isotopes

Reduction of the N=20 shell gap ?

A. Obertelli Phys. Lett. B633 (2006)33

26Ne(d,p)27Ne in thick CD2 target 2 states at 765 and 885keVInclusive for 765keV, compatible with intruder

1/2+

L=0L=1L=2

p// (Gev/c)

Cro

ss s

ecti

on

L1

28Ne(-1n)27Netransition between 765 and 885keVIntruder state (765keV) has L1 from momentum distrib.

3/2-

J.R. Terry, Phys. Lett. B 640 (2006) 86

CD240Ar

22O

gammas23O

neutrons

d p

14d5/2

s1/2

d3/2

22O14

protons

f7/2

16

RIKEN

22O(d,p)23O reaction to probe the neutron N=16, 20 shell closures

Elekes et al. PRL98 (2007) 102502

42Si

Collapse of the N=28 shell closure in 42Si

B. Bastin, S. Grévy et al., PRL 99 (2007)

5

0

CE(2

+)

(MeV

)

O

5 10 15 20Neutron Number

M. Stanoiu submitted

20C

Knock-out reaction 12Be(-1n) to probe g.s. composition of 12Be

Sn0.3

11Be1/2 +1/2 -

=1

=0

1/2+

1/2-

Navin et al., PRL 85 (2000) 266

12Be

0.

E*(MeV) J

Confirms that the N=8 gap has collapsed

1.8

2.7

(5/2 +)

(3/2 -)

Almost equal SF values

Admixtures of s, p and d statesN=8 shell closure no longer present

Pain et al., PRL 96 (2006) 032502

11Be unbound1.8

Erel (MeV)

Large quadrupole deformation in the N=20 isotones below Z=14

Proton inelastic scatteringthick Liquid H target

Y. Yanagisa et al., PLB 566 (2003) 84

Isla

nd o

f in

vers

ion

sdfpsd SM predictions

20

8

f7/2

p3/2

d3/2s1/2

d5/2

p1/2

fp

14sd

at Z=14

20

8

14

16

at Z=12

2p-2h excitations

Known T1/2

130Cd

g9/

2

h11/2

d3/2

g7/2

d5/2

s1/2

h9/2p3/2f7/2

p1/2

82

neutrons

g9/2

p1/2

50

protons

Need for good extrapolations far from known regionsUnderstand bulk evolution of nucleusAlways protons removed in the same g9/2 shellProton()-neutron() interactions involving the g9/2 orbit, e.g. g9/2 - g7/2

Evolution of the N=20 shell closure

d3/2s1/2d5/2

!

Onset of deformation around 32Mg

Specific role of the d5/2 – d3/2 and d5/2 – f7/2 interac.

No longer determine the size of the spherical N=20 gap

Some consequences …

28O

Evolution of BE shows that :

N=20 gap remains large and constant as long as protons occupy d3/2 and s1/2 orbits

pn interactions involved have similar strength

Vpn(d3/2f7/2) Vpn(d3/2d3/2)

Vpn(s1/2f7/2) Vpn(s1/2d3/2)

40Ca

7/2-

34Si

f7/2

d3/2

s1/2

g9/2p3/2

f5/2

h11/2

g7/2

d5/2d3/2

d5/2

f7/2

g9/2

14

28

50 s1/2

f7/2

d3/2 s1/2

d5/2

14[ ]

g9/2

p3/2f5/2

f7/2

28[ ]

h11/2

g7/2d5/2 d3/2

g9/2

50 s1/2

N=20

N=44

N=70

SP

IN- F

LI P

=

1 I

NT

ER

AC

TIO

N

[ ]

19K

N=28

29Cu

N=40

51Sb

N≤64

Large N/Z

-5

0

5

5

Eff

ecti

ve S

ingl

e P

arti

cle

Ene

rgy

(MeV

)

5

5

0

10 15 20Neutron Number

d5/2s1/2

d3/2

C

E(2

+)

(MeV

)

10 15 20Neutron Number

O

14

16d5/2

s1/2

d3/2

16

How will proton-neutron interactions (np=0,1)change this picture ? For large N/Z ratios, the L2 and L.S terms are expected to be reduced

Simplified mean-field approach for atomic nuclei

L.S+

f7/2

d3/2

20

d5/2

p1/2

s1/2

p3/2

f5/2

28

40g9/2

50

14

H.O L2+

1d

1f

2s

2p

20N=2

N=3

1g

N=42d

40

20

N=1

88

40

Z=14

d5/2

d3/2s1/2

d5/2

14[ ]

f7/2

20

d5/2

d3/2

s1/2

d5/2

14

16

f7/2

Z=8

N~20

N=20

T. Otsuka EPJA (2004) 69

ESPE in N=20 isotones and island of inversion

Isla

nd o

f in

vers

ion

Vpn(d3/2d5/2) >> Vpn(d3/2d3/2)

0f7/2

d5/2 d3/2s1/2

Z=20

d5/2

d3/2s1/2

d5/2

14[ ]

20

d5/2

d3/2

s1/2

d5/2

14

16

f7/2

Z=8

s1/2

d3/2 [ ]

Z=14

d5/2

d3/2s1/2

d5/2

14[ ]

20s1/2

d3/2

f7/2

p3/2f7/2

p3/2p3/2 = 2

unbound

occu

panc

y

J. R. Terry et al., PRC 77 (2008) 014316.

Ground state composition of Mg isotopes at N=18, 20

60NaI detectors, = 20%

At N=20Constancy of B(E2) and E(2+) for Z=14-20Sudden drop of E(2+)Sudden rise of B(E2) at 32Mg Excitations to the neutron fp shells are required

4

3

2

1

012 16 20 24

E(2

+)

[MeV

]

Neutrons

12Mg16S

20Ca

N=20

N/Z

40Ca 38Ar 36S 34Si 32Mg 30Ne

200

400

B(E

2) [

e2 fm

4 ]

N=20

sd

sd+fp

14Si

From 14C to 12Be or 10He, the removal of p3/2 protons

provoke the breaking of the N=8 shell gap, inferred from -energy of the 1/2-, 1/2+ states

-1-, 2+ systematics,-SF’s derived from –1n neutron knock-out reaction

Role of the proton-neutron interaction p3/2-p1/2

p3/2

p1/2

p3/2

6[ ]

d5/2

8

p3/2

p1/2

p3/2

6

d5/2

Z=6

Z=2

s1/2 s1/2

Summary for the N=8 shell closure

= 2

Sn= 2.7(1) MeV

23000 nuclei

No bound excited state in 23O and 24O

Monte Carlo20%feeding

exp

Doppler corrected

23O

23O

Raw spectra

22O

6671 nuclei

Sn=4.19(10) MeV

Monte Carlo20%feedingexp

24O 24O

4180

0+

3+

1+

30Mg12

94%

335(17)ms

log ft I

18

2+

30Al

GT dominated by d3/2 d5/2

Large occupancy of d3/2 orbital

688

13 17

Beta-decay of Mg isotopes0+

1+

2+

32Mg12

55%

735

86(5)ms

2765

3202

1+

1+

(4-)(4+)

1179

25%

11%

4.16

4.4

log ft I

20

32Al

GT strength to g.s. much weaker

Missing occupancy of the d3/2 orbital

few %>7? (1-)

13 19

data S. Grévy (GANIL)

2 neutrons in d3/2

4 neutrons in d3/2

28

20

f7/2

p3/2

d3/2

s1/2

d5/214

28

20

8

f7/2

p3/2

d3/2s1/2

d5/2

p1/2

14

GT

GT

41P2643P28

2+

2+

42Si

43P

41P

Collapse of the N=28 shell closure in 42Si