- 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 Mean-field approach for atomic nuclei L.S + f 7/2 d 3/2 20 d 5/2 p 1/2 s 1/2 p 3/2 f 5/ g 9/ H.O L2L2 + 1d 1f 2s 2p 20 N=2 N=3 1g N=4 2d N= Experimental progresses and challenges in the evolution of shell closures O. Sorlin (GANIL) ESNT Saclay 82

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

N= E(2 + ) [MeV] Neutron Number 16 S 20 Ca 12 Mg N/Z 38 Ar 36 S 34 Si 32 Mg 30 Ne 40 Ca B(E2) [e 2 fm 4 ] N=20 sd sdfp 14 Si S 2N (MeV) Neutron Number N 45 Ca 36 Ca Mg 27 Mg N=20 magic number Disappears ! 40 Ca 36 S 32 Mg

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) Island of inversion 0f 7/2 d 3/2 s 1/2 d 5/2 Role of the V pn d 5/2 d 3/2 and V pn d 5/2 (fp) interactions Attractive and repulsive tensor terms, respectively Neutron 1) Reduction of the N=20 shell gap Occupancy of fp states grows at N=20 occupancy J. R. Terry et al., PRC 77 (2008)

W. Catford et al., PRL Ne (SF) = 2, 3/2 + = 0, ½+ = 2, 5/2 + hole = 1, 3/2 - ( = 3),7/2 - JJ = 1 = 3 = 2 = 1 - Proximity of f and p states to sd ones - p and f states reversed, N=28 gap 24 Ne(d,p) 25 Ne with TIARA+EXOGAM+VAMOS (GANIL) Protons -> TIARA Gammas -> Exogam Nuclei -> Vamos

The ‘sizes’ of the N=20 and N=16 gaps in Oxygen (RIKEN) 22 O(d,p) 23 O reaction to probe the neutron N=16, 20 shell closures 23 O N=20 : 1.3 MeV N=16 : 4.0 MeV 5/2 + observed PRL99 (2007) Elekes et al. PRL98 (2007) Gated on neutronsGated on 4 MeV neutron peak L=2 hole

34 Si 40 Ca d 3/2 s 1/2 d 5/2 28 O N=20 Utsuno, Otsuka et al. 16 Island of inversion A ‘critical’ view : Mechanism of inversion not proven : not yet observed in 34 Si and 32 Mg Hard to get 28 O unbound using standard V nn No ‘direct’ (easy) determination of Vd 5/2 d 3/2 due to deformation So far monopole assumed constant whatever neutron and proton binding energy True or not ? Can we check it ? -> Study of 26 F Nuclear interaction in the sd shell Neutron

Empirical determination of Vd 5/2 d 3/2 25 F SpSp O SnSn SnSn +S p 26 F free 24 O BE(MeV) 24 O p n 26 F V pn (d 3/2 d 5/2 ) d 5/2 d 3/2 Exp monopole ~ 600keV weaker than Shell Model ! continuum effects … ?? Where is the 4 + ? Isomer ? interact J 1 exp 2 1 ? Hoffman PRL 100(2008) Stanoiu thesis 2003,E(J=2) Jurado PLB 649 (2007) J 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 in 68 Ni -Prove the extreme deformation of 64 Cr Great similarity between the three cases of HO shell numbers N=8 N=20 N=40 O. S., MG Porquet PPNP (2008)

d 5 /2  d 3/2 s 1/2 d 5/ f 7/2 Z=8 N~20 p 3/2 Z=14 d 5 /2  d 3/2 s 1/2 d 5/2 14 [ ] f 7/2 20 p 3/2 Z=28 f 7 /2  p 3/2 f 5/2 f 7/2 28 p 1/2 [ ] g 9/2 40 f 7 /2  p 3/2 f 5/2 f 7/2 28 p 1/ g 9/2 Large N/Z Z=20 N~40 d 5/2 p 3 /2 p 1/2 p 3/2 6 d 5/2  Z=2 N~8 s 1/2 p 3 /2  p 1/2 p 3/2 6 [ ] d 5/2 8 Z=6 s 1/2 Evolution of Harmonic Oscillator shell closures Role of the  p 3/2 - p 1/2 interaction Role of the  d 5/2 - d 3/2 interaction Role of the  f 7/2 - f 5/2 interaction ? SPIN –FLIP  =0 INTERACTION N=14 N=28 N=50 Small gaps

The making of ‘SO’ magic numbers Which physics ? Which interactions ?

2028 Binding energy Energy [MeV] 28 Neutrons Evolution of neutron SPE in the Ca isotopic chain Neutrons 28 Courtesy M.G Porquet No increase of the N=28 shell gap when f 7/2 is filled Same with realistic V lowK 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 Ca Neutron number g 9/2 d 5/2 68 Ni 78 Ni N=50 ? From data around 90 Zr Ni Neutron number Extracted from BE’s, spectroscopy and SF’s In collab with MG Porquet S n ( 23 O) S n ( 22 O) S n ( 17 O) E*( 17 O)

d 3/2 p 1/2 28 f 5/2 p 3/2 f 7/2 14 s 1/2 d 5/2 [ ]  > Role of nuclear forces : Modification of the N=28 shell gap ? SO and Tensor interaction ?  j=2  Enhanced E2 collectivity due to  j=2 42 Si 44 S N=28 N=20 The study of the N=28 shell closure : a way to probe nuclear force Ca, Z=20 48 Ca 34 Si S, Z=16 Si, Z=14 36 S 40 Ca 46 Ar 1- compression of proton orbits neutron f 7/2 filling 2- Evolution of neutron orbits due to pn interactions proton sd removed

d 3/2  s 1/ f 5/2 f 7/ f 7/2 p 3/2 p 1/2 f 5/ Ca 47 Ar ESPE(MeV) Variation of single particle energies (SPE) Tensor interaction ( Otsuka )  d 3/2 – ( f 7/2 -f 5/2 ) +280keV per proton added in d 3/2 -210keV Evolution of SPE’s from tensor part of the proton-neutron interaction Use of 46 Ar (d,p) transfer reaction  Size of the N=28 shell gap  Reduction of SO splitting L. Gaudefroy et al. PRL 97 (2006), 18 d p pp 28 f 7/2 p 3/2 p 1/2 f 5/2

SPE(MeV) + 2 f 7/2 p 3/2 p 1/2 f 5/ Ca 47 Ar Global trend of single particle energies between 49 Ca and 43 Si derived from experimentally-constrained monopole variations N= S Si 28 f 7/2 p 3/2 p 1/2 -A shrink of SPE’s due to two-body p-n interactions… -Favor particle-hole excitations and E2 collectivity Spherical, shape coexistence in 44 S and deformation in 42 Si

44 S Weak mixing between prolate and spherical shapes in the 0 + Q i ~ 55 Q i ~ 0 Electron spectroscopy to probe shape coexistence in 44 S Glasmacher et al., PLB 395 (97) 44 S (18) BE2 e 2 fm (2)  s 42(2) C. Force, S. Grévy et al. to be published E e- (keV) 1365 keV e + e - e - conv  (E0) = 8.7(5) BE2(0 + 2 → ) BE2(0 + 1 → ) ~1/7

 f 7 /2 d 3/2 s 1/2 d 5/2 14 [ ] p 3 /2 42 Si 14 SPIN-FLIP  =1 INTERACTION N=28 p 3 /2 f 7 /2 d 3/2 s 1/2  d 5/2 14 [ ] 28 [ ] 48 Ca Si Collapse of the N=28 shell closure in 42 Si B. Bastin, S. Grévy et al., PRL 99 (2007) Role of the  d 3/2 – f 7/2 interaction Decrease of the N=28 gap by ~1MeV for 6 protons

N=14 shell closure in 22 O and 20 C Thirolf et al. PLB 485 (2000) M. Stanoiu et al. PRC 69 (2004) and (2009) 5 0 E(2 + ) (MeV) O Neutron Number s 1 /2 N=14  d 5 /2 p 1/2 p 3/2 6 [ ] 14 [ ] 22 O 8 20 C C d 5 /2 s 1 /2 p 1/2 p 3/2 6 [ ]  20 C 6 6 Role of the  p 1/2 – d 5/2 interaction Decrease of the N=14 by ~1.6 MeV for 2 protons

 f 7 /2 d 3/2 s 1/2 d 5/2 14 [ ] p 3 /2 42 Si 14 SPIN-FLIP  =1 INTERACTION s 1 /2 N=14  d 5 /2 p 1/2 p 3/2 6 [ ] 14 [ ] 22 O 8 d 5 /2 s 1 /2 p 1/2 p 3/2 6 [ ]  20 C 6 6 N=28 p 3 /2 f 7 /2 d 3/2 s 1/2  d 5/2 14 [ ] 28 [ ] 48 Ca 20 N=14 N= Zr N=50 d 5 /2 g 9 /2 f 5/2 p 3/2  f 7/2 28 [ ] 40 g 9 /2 f 5/2 p 3/2  d 5/2 28 [ ] d 5 /2 78 Ni 50 N=50 ?? 152 Gd N=82 f 7 /2 h 11 /2 d 5/2 g 7/2  g 9/2 50 [ ] 82 [ ] h 11 /2 d 5/2 g 7/2  g 9/2 28 [ ] f 7 / Sn 50 [ ] 82 N=82 strong

42 Si 68 Ni 48 Ca 40 Ca Occupation probability energy (MeV) 34 Si ? The N=50 shell closure at 78 Ni 50 « 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 78 Ni 50 Role of V nn to create SO magic numbers -> Same increase of neutron shell gap (3MeV) for all SO magic numbers ! Modification of V pn due to the presence of continuum ? V pn d 5/2 d 3/2 ( 26 F) ~ 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 formation with realistic interactions ! The N=28 shell gap and the role of 3 body forces Holt, Otsuka, Schwenk, Suzuki

p 3/2 p 1/2 f 7/2 f 5/2 47 Ar : reduced by 330keV Use of 46 Ar (d,p) transfer reaction  Size of the N=28 shell gap  Reduction of SO splitting L. Gaudefroy et al. PRL 97 (2006) d p pp 28 f 7/2 p 3/2 p 1/2 f 5/2 Evolution of the neutron SPE below 48 Ca 46 Ar (2J+1)C 2 S=1.7 (2J+1)C 2 S=2.44 (2J+1)C 2 S=1.36 C 2 S f =0.64 C 2 Sg=0.34 p f f p

f 5/2 g 9/2 neutrons f 7/2 p 3/2 p 1/2 28 protons (j p <) (j p >) (j n >) 50 d 5/2 78 Ni 42 Si and 78 Ni are ‘mirror’ systems Development of collectivity in 42 Si 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 d 3/2 f 7/2 neutrons d 5/2 s 1/2 14 protons (j p <) (j p >) (j n >) 28 p 3/2 42 Si  ℓ=2 p 1/2 f 5/2

Role of proton-neutron forces in the N=28 region E(1/2 + ) – E(3/2 + ) (keV) Neutron Number 16 s 1/2 d 3/2 f 7/2  Neutron Number 32 p 3/2 f 5/2  g 9/2 Cu (Z=29) exp Around 78 Ni f 7/2 28 E(5/2 - ) – E(3/2 - ) (keV) ?? in the N=50 region

f 7/2 p 3/2 p 1/2 f 5/ Ca 47 Ar SPE(MeV) Change of SO splitting for p orbits p 1/2 Central density dependence (Piekarewicz) p 3/2  s 1/ keV per proton -85keV -No change of p 1/2 -p 3/2 splitting between 41 Ca and 37 S after removal of 4 protons from  d 3/2 -Reduction of splitting due to  s 1/2 Gaudefroy et al. PRL 2007

Probe the density dependence of the SO interaction in 36 S and 34 Si RMF calculations using NL3 interaction Reduction of the SO splitting by 70% MF / Skyrme or Gogny forces Reduction of the SO splitting by 40% SM calculations spdf-NR Reduction of SO splitting by 30% Bare forces V lowK 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 36 S 34 Si 36 S

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 Be First ‘Island of inversion’ ? 12 Be g.s. strongly mixed (Navin et al PRL85; Pain et al. 96)   = 2 14 C 12 Be  [1 - ] 12 Be : Iwasaki et al., PLB 481 (2000) 7 12 Be 14 C 16 O Evolution of the N=8 shell closure 15 O 13 C Z  [1/ /2 + ] d 5/2 p 1/2 p 3/2 6 8 s 1/2 p 1/2 p 3/2 p 1/2 p 3/2 6 d 5/2 s 1/2 11 Be Role of the  p 3/2 - p 1/2 interaction

The 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 Numbers From 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 ?

g 9/2 g 7/2 d 5/2 h 11/2 s 1/2 d 3/2 f 7/2 p 3/2 h 9/2 i 13/2 f 5/2 p 1/2 g 9/2 g 7/2 d 5/2 h 11/2 s 1/2 d 3/2 f 7/2 p 3/2 f 5/2 p 1/2 h 9/2 Around 132 Sn N>>Z, drip-line Nuclear Shell Structure Evolution Mean field near stability Strong spin-orbit interaction Reduced spin-orbit Tensor forces Mean 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 g 7/2 energy, increase the g 7/2 → g 9/2 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 23 O and 24 O 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 26 Ne(d,p) 27 Ne in thick CD 2 target 2 states at 765 and 885keV Inclusive  for 765keV, compatible with intruder 1/2 + L=0 L=1 L=2 p // (Gev/c) Cross section L1L1 28 Ne(-1n) 27 Ne transition between 765 and 885keV Intruder state (765keV) has L  1 from momentum distrib. 3/2 - J.R. Terry, Phys. Lett. B 640 (2006) 86

CD 2 40 Ar 22 O gammas 23 O neutrons d pp 14 d 5/2 s 1/2 d 3/2 22 O 14 protons f 7/2 16 RIKEN 22 O(d,p) 23 O reaction to probe the neutron N=16, 20 shell closures Elekes et al. PRL98 (2007)

42 Si Collapse of the N=28 shell closure in 42 Si B. Bastin, S. Grévy et al., PRL 99 (2007) 5 0 C E(2 + ) (MeV) O Neutron Number M. Stanoiu submitted 20 C

Knock-out reaction 12 Be(-1n) to probe g.s. composition of 12 Be SnSn Be 1/2 + 1/2 -  =1  =0 1/2 + 1/2 - Navin et al., PRL 85 (2000) 266  12 Be 0. E*(MeV) J Confirms that the N=8 gap has collapsed (5/2 + ) (3/2 - ) Almost equal SF values Admixtures of s, p and d states N=8 shell closure no longer present Pain et al., PRL 96 (2006) Be unbound 1.8 E rel (MeV)

Large quadrupole deformation in the N=20 isotones below Z=14 Proton inelastic scattering thick Liquid H target Y. Yanagisa et al., PLB 566 (2003) 84 Island of inversion sdfp sd SM predictions

20 8 f 7/2 p 3/2 d 3/2 s 1/2 d 5/2 p 1/2 fp 14 sd at Z= at Z=12 2p-2h excitations

Known T 1/2 130 Cd  g 9/2 h 11/2 d 3/2 g 7/2 d 5/2 s 1/2 h 9/2 p 3/2 f 7/2 p 1/2 82 neutrons g 9/2 p 1/2 50 protons Need for good extrapolations far from known regions Understand bulk evolution of nucleus Always protons removed in the same g 9/2 shell Proton(  )-neutron( ) interactions involving the g 9/2 orbit, e.g.  g 9/2 - g 7/2

Evolution of the N=20 shell closure d 3/2 s 1/2 d 5/2 !  Onset of deformation around 32 Mg  Specific role of the  d 5/2 – d 3/2 and  d 5/2 – f 7/2 interac.  No longer determine the size of the spherical N=20 gap  Some consequences … 28 O Evolution of BE shows that :  N=20 gap remains large and constant as long as protons occupy d 3/2 and s 1/2 orbits  pn interactions involved have similar strength V pn (d 3/2 f 7/2 )  V pn (d 3/2 d 3/2 ) V pn (s 1/2 f 7/2 )  V pn (s 1/2 d 3/2 ) 40 Ca 7/ Si

f 7 /2 d 3/2 s 1/2 g 9 /2 p 3/2 f 5/2 h 11 /2 g 7/2 d 5/2 d 3 /2    d 5/2 f 7/2 g 9/ s 1 /2 f 7 /2 d 3/2 s 1/2  d 5/2 14 [ ] g 9 /2 p 3/2 f 5/2  f 7/2 28 [ ] h 11 /2 g 7/2 d 5/2 d 3 /2  g 9/2 50 s 1 /2 N=20 N=44 N=70 SPIN-FLIP  =1 INTERACTION [ ] 19 K N=28 29 Cu N=40 51 Sb N≤64 Large N/Z

Effective Single Particle Energy (MeV) Neutron Number d 5/2 s 1/2 d 3/2 C E(2 + ) (MeV) Neutron Number O d 5/2 s 1/2 d 3/2 16

How will proton-neutron interactions (  np =0,1)  change this picture ? For large N/Z ratios, the L 2 and L.S terms are expected to be reduced Simplified mean-field approach for atomic nuclei L.S + f 7/2 d 3/2 20 d 5/2 p 1/2 s 1/2 p 3/2 f 5/ g 9/ H.O L2L2 + 1d 1f 2s 2p 20 N=2 N=3 1g N=4 2d N=

Z=14 d 5 /2  d 3/2 s 1/2 d 5/2 14 [ ] f 7/2 20 d 5 /2  d 3/2 s 1/2 d 5/ f 7/2 Z=8 N~20

N=20 T. Otsuka EPJA (2004) 69 ESPE in N=20 isotones and island of inversion Island of inversion Vpn(d 3/2 d 5/2 ) >> Vpn(d 3/2 d 3/2 ) 0f 7/2 d 5/2 d 3/2 s 1/2

Z=20 d 5 /2  d 3/2 s 1/2 d 5/2 14 [ ] 20 d 5 /2  d 3/2 s 1/2 d 5/ f 7/2 Z=8 s 1/2 d 3/2 [ ] Z=14 d 5 /2  d 3/2 s 1/2 d 5/2 14 [ ] 20 s 1/2 d 3/2 f 7/2 p 3/2 f 7/2 p 3/2   = 2 unbound

occupancy J. R. Terry et al., PRC 77 (2008) Ground state composition of Mg isotopes at N=18, 20

60NaI detectors,   = 20% At N=20  Constancy of B(E2) and E(2 + ) for Z=14-20  Sudden drop of E(2 + )  Sudden rise of B(E2) at 32 Mg  Excitations to the neutron fp shells are required E(2 + ) [MeV] Neutrons 12 Mg 16 S 20 Ca N=20 N/Z 40 Ca 38 Ar 36 S 34 Si 32 Mg 30 Ne B(E2) [e 2 fm 4 ] N=20 sd sd+fp 14 Si

From 14 C to 12 Be or 10 He, the removal of p 3/2 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 p 3/2 -p 1/2 p 3 /2  p 1/2 p 3/2 6 [ ] d 5/2 8 p 3 /2 p 1/2 p 3/2 6 d 5/2 Z=6  Z=2 s 1/2 Summary for the N=8 shell closure   = 2

S n = 2.7(1) MeV nuclei No bound excited state in 23 O and 24 O Monte Carlo 20%feeding exp Doppler corrected 23 O Raw spectra 22 O 6671 nuclei S n =4.19(10) MeV Monte Carlo 20%feeding exp 24 O 4180

Mg 12 94%  335(17)ms log ftII Al GT dominated by d 3/2   d 5/2 Large occupancy of d 3/2 orbital Beta-decay of Mg isotopes Mg 12 55%  (5)ms (4 - ) (4 + ) % 11% log ftII Al GT strength to g.s. much weaker Missing occupancy of the d 3/2 orbital few % >7 ? (1 - ) data S. Grévy (GANIL) 2 neutrons in d 3/2 4 neutrons in d 3/ f 7/2 p 3/2 d 3/2 s 1/2 d 5/ f 7/2 p 3/2 d 3/2 s 1/2 d 5/2 p 1/2 14  GT

41 P P Si 43 P 41 P Collapse of the N=28 shell closure in 42 Si