1. Structure of exotic nuclei Takaharu Otsuka University of Tokyo / RIKEN / MSU 7 th CNS-EFES summer school Wako, Japan August 26 – September 1, 2008 A presentation supported by the JSPS Core-to-Core Program “International Research Network for Exotic Femto Systems (EFES)” Day 2

2. Section 1: Basics of shell model Section 2: Construction of effective interaction and an example in the pf shell Section 3: Does the gap change ? - N=20 problem - Section 4: Force behind Section 5: Is two-body force enough ? Section 6: More perspectives on exotic nuclei Outline

3. What is the shell model ? Why can it be useful ? Introduction to the shell model How can we make it run ? Day-1 lecture : Basis of shell model and magic numbers density saturation + short-range interaction + spin-orbit splitting  Mayer-Jensen’s magic number  Valence space (model space) For shell model calculations, we need also TBME (Two-Body Matrix Element) and SPE (Single Particle Energy)

4. Start from a realistic microscopic interaction M. Hjorth-Jensen, et al., Phys. Repts. 261 (1995) 125 –Bonn-C potential –3rd order Q-box + folded diagram 195 two-body matrix elements (TBME) and 4 single-particle energies (SPE) are calculated  Not completely good （ theory imperfect) Vary 70 Linear Combinations of 195 TBME and 4 SPE Fit to 699 experimental energy data of 87 nuclei M. Honma et al., PRC65 (2002) 061301(R) G-matrix + polarization correction + empirical refinement Microscopic Phenomenological An example from pf shell (f 7/2, f 5/2, p 3/2, p 1/2 ) GXPF1 interaction

5. G-matrix vs. GXPF1 T=0 … attractive T=1 … repulsive Relatively large modifications in V(abab ; J0 ) with large J V(aabb ; J1 ) pairing 7= f 7/2, 3= p 3/2, 5= f 5/2, 1= p 1/2 two-body matrix element input output

6. Systematics of 2 + 1 Shell gap N=28 N=32 for Ca, Ti, Cr N=34 for Ca ?? Deviations in E x Cr at N ≧ 36 Fe at N ≧ 38 Deviations in B(E2) Ca, Ti for N ≦ 26 Cr for N ≦ 24 40 Ca core excitations Zn, Ge g 9/2 is needed

7. GXPF1 vs. experiment 56 Ni th. exp. 57 Ni

8. Probability of closed-shell in the ground state Ni 48 Cr neutron proton total 56 Ni (Z=N=28) has been considered to be a doubly magic nucleus where proton and neutron f 7/2 are fully occupied. ⇒ Measure of breaking of this conventional idea doubly magic

9. 54 Fe yrast states 0p-2h configuration 0 +, 2 +, 4 +, 6 + …  (f 7/2 )  2 more than 40% prob. 1p-3h … 1 st gap One-proton excitation 3 +, 5 + 7 + ～ 11 + 2p-4h … 2 nd gap Two-protons excitation 12 + ～ States of different nature can be reproduced within a single framework p-h : excitation from f 7/2

10. 58 Ni yrast states 2p-0h configuration 0 +, 2 + … (p 3/2 ) 2 1 +, 3 +, 4 + … (p 3/2 ) 1 (f 5/2 ) 1 more than 40% prob. 3p-1h … 1 st gap One-proton excitation 5 + ～ 8 + 4p-2h … 2 nd gap One-proton & one-neutron excitation 10 + ～ 12 + p-h : excitation from f 7/2

11. N=32, 34 magic numbers ? Issues to be clarified by the next generation RIB machines

12. Monopole part of the NN interaction Angular averaged interaction Effective single particle energy Isotropic component is extracted from a general interaction. In the shell model, single-particle properties are considered by the following quantities …….

13. Effective single-particle energy (ESPE) ESPE : Total effect on single- particle energies due to interaction with other valence nucleons Monopole interaction, v m ESPE is changed by N v m N particles

14. Effective single-particle energies Lowering of f5/2 from Ca to Cr - weakening of N=34 - Rising of f5/2 from 48 Ca to 54 Ca - emerging of N=34 - Why ? n-n p-n f 5/2 p 3/2 p 1/2 34 32 new magic numbers ? Z=20 Z=22 Z=24

15. Exotic Ca Isotopes : N = 32 and 34 magic numbers ? exp. levels :Perrot et al. Phys. Rev. C (2006), and earlier papers 52 Ca 54 Ca 53 Ca 2+2+ 2+2+ ? 51 Ca

16. Exotic Ti Isotopes 55 Ti 56 Ti 54 Ti 2+2+ 2+2+ 53 Ti

17. G GXPF1 ESPE (Effectice Single- Particle Energy) of neutrons in pf shell f 5/2 Ca Ni Why is neutron f 5/2 lowered by filling protons into f 7/2 f 5/2

18. Changing magic numbers ? We shall come back to this problem after learning under-lying mechanism.

19. Section 1: Basics of shell model Section 2: Construction of effective interaction and an example in the pf shell Section 3: Does the gap change ? - N=20 problem - Section 4: Force behind Section 5: Is two-body force enough ? Section 6: More perspectives on exotic nuclei Outline

20. Left-lower part of the Nuclear Chart Studies on exotic nuclei in the 80~90’s リチウム１１ neutron halo nuclei (mass number) stable exotic -- with halo A 11 Li neutron skin proton halo Stability line and drip lines are not so far from each other  Physics of loosely bound neutrons, e.g., halo while other issues like 32 Mg Neutron number  Proton number 

21. Neutron halo 11 Li 208 Pb About same radius Strong tunneling of loosely bound excess neutrons Nakamura’s lecture

22. Drip line 中性子数 （同位元素の種類） Neutron number  Proton number  In the 21 st century, a wide frontier emerges between the stability and drip lines. Stability line A nuclei (mass number) stable exotic Riken’s work huge area

23. Also in the 1980’s, 32 Mg low-lying 2 +

24. Phys. Rev. C 41, 1147 (1990), Warburton, Becker and Brown 9 nuclei: Ne, Na, Mg with N=20-22 Basic picture was energy intruder ground state stable exotic sd shell pf shell N=20 gap ~ constant deformed 2p2h state Island of Inversion

25. One of the major issues over the millennium was to determine the territory of the Island of Inversion - Are there clear boundaries in all directions ? - Is the Island really like the square ? Shallow (diffuse & extended) Straight lines Steep (sharp) Which type of boundaries ?

26. N semi-magic intruder Small gap vs. Normal gap For larger gap, f must be larger  sharp boundary normal For smaller gap, f is smaller  diffuse boundary The difference  v is modest as compared to “semi-magic”. open-shell v=0  v=smaller  v=large v ~ Inversion occurs for semi-magic nuclei most easily Max pn force

27. Na isotopes : What happens in lighter ones with N < 20 Original Island of Inversion

28. Electro-magnetic moments and wave functions of Na isotopes ― normal dominant : N=16, 17 ― strongly mixed : N=18 ― intruder dominant : N=19, 20 Onset of intruder dominance before arriving at N=20 Q  Config. Monte Carlo Shell Model calculation with full configuration mixing : Phys. Rev. C 70, 044307 (2004), Utsuno et al. Exp.: Keim et al. Euro. Phys. J. A 8, 31 (2001)

29. Level scheme of Na isotopes by SDPF-M interaction compared to experiment N=16 N=17 N=18N=19

30. Major references on MCSM calculations for N~20 nuclei "Varying shell gap and deformation in N~20 unstable nuclei studied by the Monte Carlo shell model", Yutaka Utsuno, Takaharu Otsuka, Takahiro Mizusaki and Michio Honma, Phys. Rev. C60, 054315-1 - 054315-8 (1999) “Onset of intruder ground state in exotic Na isotopes and evolution of the N=20 shell gap”, Y. Utsuno, T. Otsuka, T. Glasmacher, T. Mizusaki and M. Honma, Phys. Rev. C70, (2004), 044307. Many experimental papers include MCSM results.

31. Monte Carlo Shell Model (MCSM) results have been obtained by the SDPF-M interaction for the full-sd + f7/2 + p3/2 space. Expansion of the territory Effective N=20 gap between sd and pf shells Ca O Ne Mg SDPF-M (1999) WBB (1990) ~2MeV ~5MeV Neyens et al. 2005 Mg Tripathi et al. 2005 Na Dombradi et al. 2006 Ne Terry et al. 2007 Ne

32. Phys. Rev. Lett. 94, 022501 (2005), G. Neyens, et al. Strasbourg unmixed USD (only sd shell) Tokyo MCSM 2.5 MeV 0.5 MeV 31 Mg 19

33. New picture energy intruder ground state stable exotic sd shell pf shell N=20 gap changing deformed 2p2h state Conventional picture energy intruder ground state stable exotic sd shell pf shell N=20 gap ~ constant deformed 2p2h state spherical normal state ?

34. Steep (sharp) Straight lines Shallow (diffuse & extended) Expansion of the territory Island of Inversion ? ? Ca O Ne Mg SDPF-M (1999) ~2MeV ~6MeV Effective N=20 gap between sd and pf shells Island of Inversion is like a paradise constant gap

35. Why ?

36. Section 1: Basics of shell model Section 2: Construction of effective interaction and an example in the pf shell Section 3: Does the gap change ? - N=20 problem - Section 4: Force behind Section 5: Is two-body force enough ? Section 6: More perspectives on exotic nuclei Outline

37. From undergraduate nuclear physics, density saturation + short-range NN interaction + spin-orbit splitting  Mayer-Jensen’s magic number with rather constant gaps (except for gradual A dependence) Robust mechanism - no way out -

38. One pion exchange ~ Tensor force Key to understand it : Tensor Force

39. Key to understand it : Tensor Force  meson (~  +  ) : minor (~1/4) cancellation  meson : primary source Ref: Osterfeld, Rev. Mod. Phys. 64, 491 (92)  Multiple pion exchanges  strong effective central forces in NN interaction (as represented by  meson, etc.)  nuclear binding One pion exchange  Tensor force This talk : First-order tensor-force effect (at medium and long ranges)

40. How does the tensor force work ? Spin of each nucleon is parallel, because the total spin must be S=1 The potential has the following dependence on the angle  with respect to the total spin S. V ~ Y 2,0 ~ 1 – 3 cos 2  attraction  =0 repulsion  =  /2  S relative coordinate

41. Deuteron : ground state J = 1 proton neutron Total spin S=1 Relative motion : S wave (L=0) + D wave (L=2) Without tensor force, deuteron is unbound. The tensor force is crucial to bind the deuteron. Tensor force does mix No S wave to S wave coupling by tensor force because of Y 2 spherical harmonics

42. Monopole part of the NN interaction Angular averaged interaction Effective single particle energy Isotropic component is extracted from a general interaction. In the shell model, single-particle properties are considered by the following quantities …….

43. Intuitive picture of monopole effect of tensor force wave function of relative motion large relative momentum small relative momentum attractive repulsive spin of nucleon TO et al., Phys. Rev. Lett. 95, 232502 (2005) j > = l + ½, j < = l – ½

44. j>j> j<j< j’ < proton neutron j’ > Monopole Interaction of the Tensor Force Identity for tensor monopole interaction (2j > +1) v m,T + (2j < +1) v m,T = 0 ( j’ j > ) ( j’ j < ) v m,T : monopole strength for isospin T T. Otsuka et al., Phys. Rev. Lett. 95, 232502 (2005)

45. Major features Opposite signs T=0 : T=1 = 3 : 1 (same sign) Only exchange terms (generally for spin-spin forces) tensor proton, j > neutron, j’ < spin-orbit splitting varied

46. j>j> j<j< j’ < proton neutron j’ > Tensor Monopole Interaction : total effects vanished for spin-saturated case Same Identity with different interpretation (2j > +1) v m,T + (2j < +1) v m,T = 0 ( j’ j > ) ( j’ j < ) v m,T : monopole strength for isospin T no change

47. j>j> j<j< s 1/2 proton neutron Tensor Monopole Interaction vanished for s orbit For s orbit, j > and j < are the same : (2j > +1) v m,T + (2j < +1) v m,T = 0 ( j’ j > ) ( j’ j < ) v m,T : monopole strength for isospin T

48. Monopole Interaction of the tensor force is considered to see the connection between the tensor force and the shell structure

49. Tensor potential tensor no s-wave to s-wave coupling differences in short distance : irrelevant

50. Proton effective single-particle levels (relative to d 3/2 ) f 7/2 neutrons in f 7/2 d 3/2 d 5/2 proton neutron  meson tensor exp. Cottle and Kemper, Phys. Rev. C58, 3761 (98) Tensor monopole

51. Spectroscopic factor for -1p from 48 Ca: probing proton shell gaps w/ tensorw/o tensor d 3/2 -s 1/2 gap d 5/2 -s 1/2 gap Kramer et al. (2001) Nucl PHys A679 NIKHEF exp.

52. Example : Dripline of F isotopes is 6 units away from O isotopes Sakurai et al., PLB 448 (1999) 180, … Tensor force d 5/2 d 3/2 N=16 gap : Ozawa, et al., PRL 84 (2000) 5493; Brown, Rev. Mex. Fis. 39 21 (1983) only exchange term

53. Monte Carlo Shell Model (MCSM) results have been obtained by the SDPF-M interaction for the full-sd + f7/2 + p3/2 space. Expansion of the territory Effective N=20 gap between sd and pf shells Ca O Ne Mg SDPF-M (1999) WBB (1990) ~2MeV ~5MeV Neyens et al. 2005 Mg Tripathi et al. 2005 Na Dombradi et al. 2006 Ne Terry et al. 2007 Ne

54. 1h 11/2 neutrons 1h 11/2 protons 1g 7/2 protons Exp. data from J.P. Schiffer et al., Phys. Rev. Lett. 92, 162501 (2004 ) + common effect (Woods-Saxon) Z=51 isotopes h 11/2 g 7/2 51 Sb case No mean field theory, (Skyrme, Gogny, RMF) explained this before. Opposite monopole effect from tensor force with neutrons in h 11/2. Tensor by  meson exchange

55. Single-particle levels of 132 Sn core Weakening of Z=64 submagic structure for N~90 64

56. Weakening of Z=64 submagic structure for N~90 8 neutrons in 2f 7/2 reduces the Z=64 gap to the half value 8 protons in 1g 7/2 pushes up 1h 9/2 by ~1 MeV 1h 9/2 64 2d 3/2 2d 5/2 Proton collectivity enhanced at Z~64

57. Mean-field models (Skyrme or Gogny) do not reproduce this reduction. Tensor force effect due to vacancies of proton d 3/2 in 47 18 Ar 29 : 650 (keV) by  +  meson exchange. f 5/2 f 7/2 Neutron single-particle energies

58. A city works its magic. … N.Y. RIKEN RESEARCH, Feb. 2007 Magic numbers do change, vanish and emerge. Conventional picture (since 1949) Today’s perspectives

59. Effect of tensor force on (spherical) superheavy magic numbers Woods-Saxon potential Tensor force added Occupation of neutron 1k17/2 and 2h11/2 1k17/2 2h11/2 Neutron N=184 Proton single particle levels Otsuka, Suzuki and Utsuno, Nucl. Phys. A805, 127c (2008)

60. Anatomy of shell-model interaction 解剖（学）

61. Start from a realistic microscopic interaction M. Hjorth-Jensen, et al., Phys. Repts. 261 (1995) 125 –Bonn-C potential –3rd order Q-box + folded diagram 195 two-body matrix elements (TBME) and 4 single-particle energies (SPE) are calculated  Not completely good （ theory imperfect) Vary 70 Linear Combinations of 195 TBME and 4 SPE Fit to 699 experimental energy data of 87 nuclei M. Honma et al., PRC65 (2002) 061301(R) G-matrix + polarization correction + empirical refinement Microscopic Phenomenological Shell evolution by realistic effective interaction : pf shell GXPF1 interaction

62. G-matrix vs. GXPF1 T=0 … attractive T=1 … repulsive Relatively large modifications in V(abab ; J0 ) with large J V(aabb ; J1 ) pairing 7= f 7/2, 3= p 3/2, 5= f 5/2, 1= p 1/2 two-body matrix element input output

63. T=0 monopole interactions in the pf shell GXPF1A G-matrix (H.-Jensen) Tensor force (  +  exchange) f-fp-p f-p “Local pattern”  tensor force

64. T=0 monopole interactions in the pf shell GXPF1A G-matrix (H.-Jensen) Tensor force (  +  exchange) Tensor component is subtracted

65. The central force is modeled by a Gaussian function V = V 0 exp( -(r/  ) 2 ) (S,T dependences) with V 0 = -166 MeV,  =1.0 fm, (S,T) factor (0,0) (1,0) (0,1) (1,1) -------------------------------------------------- relative strength 1 1 0.6 -0.8 Can we explain the difference between f-f/p-p and f-p ?

66. GXPF1 G-matrix (H.-Jensen) Central (Gaussian) - Reflecting radial overlap - Tensor force (  +  exchange) T=0 monopole interactions in the pf shell f-fp-p f-p

67. T=1 monopole interaction

68. T=1 monopole interactions in the pf shell j = j’ Tensor force (  +  exchange) G-matrix (H.-Jensen) GXPF1A Basic scale ~ 1/10 of T=0 Repulsive corrections to G-matrix

69. j = j’ Tensor force (  +  exchange) G-matrix (H.-Jensen) GXPF1A Central (Gaussian) - Reflecting radial overlap - T=1 monopole interactions in the pf shell

70. (Effective) single-particle energies Lowering of f5/2 from Ca to Cr : ~ 1.6 MeV = 1.1 MeV (tensor) + 0.5 MeV (central) KB3G n-n p-n KB interactions : Poves, Sanchez-Solano, Caurier and Nowacki, Nucl. Phys. A694, 157 (01) Rising of f5/2 from 48 Ca to 54 Ca : p3/2-p3/2 attraction p3/2-f5/2 repulsion

71. Major monopole components of GXPF1A interaction T=0 - simple central (range ~ 1fm) + tensor - strong (~ 2 MeV) - attractive modification from G-matrix T=1 - More complex central (range ~ 1fm) + tensor - weak ~ -0.3 MeV (pairing), +0.2 MeV (others) - repulsive modification from G-matrix even changing the signs Central force : strongly renormalized Tensor force : bare  +  meson exchange Also in sd shell….

72. T=0 monopole interactions in the sd shell SDPF-M (~USD) G-matrix (H.-Jensen) Tensor force (  +  exchange) Central (Gaussian) - Reflecting radial overlap -

73. T=1 monopole interactions in the sd shell SDPF-M (~USD) G-matrix (H.-Jensen) Tensor force (  +  exchange) j = j’ Basic scale ~ 1/10 of T=0 Repulsive corrections to G-matrix

74. This is not a very lonely idea  Chiral Perturbation of QCD S. Weinberg, PLB 251, 288 (1990) Tensor force is explicit Short range central forces have complicated origins and should be adjusted.

75. We discussed on how single-particle levels change (shell evolution) as functions of Z and N in exotic nuclei due to various components of the nuclear force. 1.Tensor force 2.Central force Single-particle properties (shell structure) are one of the most dominant elements of nuclear structure. Example : The deformation (of low-lying states) is a Jahn-Teller effect to a good extent. Summary of Day2-1

76. Summary of Day2 - 2 Some components of NN interactions show characteristic patterns of the shell evolution Tensor force : - variation of spin-orbit splitting - strong  unexpected oblate deformation of a doubly-magic 42 Si Central force : - differentiates different radial nodal structure of single-particle orbits - stronger (< tensor) Tensor + Central combined (typically in the ratio 2:1)  lowering of neutron f5/2 in Ca-Cr-Ni inversion of proton f5/2 and p3/2 in Cu for N~46 54 Ca, 78 Ni, 100 Sn Day-One experiments of RIBF