Lectures in Istanbul Hiroyuki Sagawa, Univeristy of Aizu June 30-July 4, 2008

1. Lectures in IstanbulHiroyuki Sagawa, Univeristy of AizuJune 30-July 4, 2008 • 1. Giant Resonances and Nuclear Equation of States • 2. Pairing correlations in Exotic nuclei -- BEC-BCS crossover -- BCS（Bardeen-Cooper-Schrieffer)pair BEC (Bose-Einstein condensation)

2. Pairing Correlations in Exotic Nuclei --Coexistence of BCS and BEC-like pairs in Infinite Matter and Nuclei-- Hiroyuki Sagawa Center for Mathematics and Physics, University of Aizu • Introduction • Pairing gaps in nuclear matter • Three-body model for borromian nuclei • BEC-BCScrossover in finite nuclei • Summary

4. Pairing correlations in nuclei Coherence length of a Cooper pair: much larger than the nuclear size (note) x = 55.6 fm (for A=140) R = 1.2 x 1401/3 = 6.23 fm K.Hagino, H. Sagawa, J. Carbonell, and P. Schuck, PRL99,022506(2007).

5. Pairing Phase Transition(second order phase transition) Particles Fermi energy Holes order parameter

6. BCS state Bogoliubov transformation

7. Two-body Hamiltonian Constrained Hamiltonian Gap equation

8. BCS formulas with seniority pairing interaction -GS+S- Seniority pairing D Gap Equation QP energy is obtained. condition gives

9. Quasi-particle energy Excitation energy Positive energy Pairing gap energy Single-particle energy Negative energy

10. Quasi-particle excitations

11. Weakly interacting • fermions • Correlation in p space • (large coherence length) • Interacting • “diatomic molecules” • Correlation in r space • (small coherence length) cf. BEC of molecules in 40K M. Greiner et al., Nature 426(’04)537

12. BCS-BEC crossover Cooper pair wave function: crossover BEC (strong coupling) BCS (weak coupling) Correlation in r space (small coherence length) Correlation in p space (large coherence length) x x

16. Toward Universal Pairing Energy Density Functionals

21. -

23. Stable Nuclei Unstable Nuclei Excitations to the continuum states in drip line nuclei Breakdown of BCS approximation

24. Mean field and HFB single particle energy ei continuum HFB resonance 0 bound 2l l resonance

25. Hartree-Fock Bogoliubov approximation Trial Wave Function Coordinate Space Representation

26. New quasi-particle picture different to BCS quasi-particle!! wave function will be upper comp. non-local lower comp local Pair potential Pair potential goes beyond HF potential

28. Odd-even mass difference N=odd is recommended. -B N

30. 24O skin nucleus 16C Borromian Nuclei (any two body systems are not bound, but three body system is bound)

31. H. Esbensen, G.F. Bertsch, K. Hencken, Phys. Rev. C56(’99)3054 Three-body model n Density-dependent delta-force r1 VWS VWS v0 ann a S2n r2 core n (note) recoil kinetic energy of the core nucleus • Hamitonian diagonalization with WS basis • Continuum: box discretization Important for dipole excitation Application to 11Li, 6He, 24O

32. Density-dependent delta interaction H. Esbensen, G.F. Bertsch, K. Hencken, Phys. Rev. C56(’99)3054 • Two neutron system in the vacuum: • Two neutron system in the medium: : adjust so that S2n can be reproduced

33. Application to 11Li, 6He, 24O 11Li, 6He: Typical Borromean nuclei Esbensen et al. 24O: Another drip-line nuclei A.Ozawa et al., NPA691(’01)599 11Li: WS: adjusted to p3/2 energy in 8Li & n-9Li elastic scattering Parity-dependence to increase the s-wave component 6He: WS: adjusted to n-a elastic scattering 24O: WS: adjusted to s1/2in 23O (-2.74 MeV) & d5/2 in 21O

34. Two-body Density r1 q12 r2 One-body density as a function of angle

35. Two-particle density for 11Li n r1 q12 9Li r2 n Set r1=r2=r, and plot r2 as a function of r and q12 • two-peaked structure • Long tail for “di-neutron” S=0 S=0 or 1 di-neutron cigar-type)

36. Two-particle density for 11Li Total S=1 or S=0

37. Two-particle density for 6He Total S=1 or S=0

38. Comparison among three nuclei 11Li 6He (p1/2)2 :59.1% (s1/2)2 :22.7% (d5/2)2 :11.5% (p3/2)2 :83.0 % (d5/2)2 :6.11 %, (p1/2)2 :4.85 % (s1/2)2 :3.04 %, (d3/2)2 :1.47 % for (p1/2)2 or(p3/2)2 for (s1/2)2

39. Two-particle density for 24O Total S=1 or S=0

40. 24O 0 2s1/2 -2.74 22O -3.80 1d5/2 24O (s1/2)2 :93.6% (d3/2)2 :3.61% (f7/2)2 :1.02% for (s1/2)2

41. Ground State Properties S2n is still controversial in 11Li. C. Bachelet et al., S2n=376+/-5keV (ENAM,2004)

42. Experimental proof of di-neutron and/or cigar configurations Dipole Excitations Response to the dipole field: Smearing:

43. Peak position Simple two-body cluster model: Sc peak at E=8 Sc /5=1.6 Sc 6He: Epeak=1.55 MeV 1.6 S2n=1.6 X 0.975= 1.56 MeV 11Li: Epeak=0.66 MeV 1.6 S2n=1.6 X 0.295= 0.47 MeV 24O: Epeak=4.78 MeV 1.6 S2s1/2=1.6 X 2.74= 4.38 MeV 1.6 S2n =1.6 X 6.45= 10.32 MeV 6He, 11Li: dineutron-like excitation 24O: s.p.-like excitation

44. Comparison with expt. data (11Li) Epeak=0.66 MeV B(E1) = 1.31 e2fm2 (E < 3.3 MeV) T. Aumann et al., PRC59, 1259(1999) New experiment :T. Nakamura et al., PRL96,252502(2006) Epeak ~ 0.6 MeV B(E1) = (1.42 +/- 0.18) e2fm2 (E < 3.3 MeV)

45. BCS-BEC crossover behavior in infinite nuclear matter • Weakly bound levels • dilute density around surface (halo/skin) Neutron-rich nuclei are characterized by

46. Coexistence of BCS-BEC like behaviour of Cooper Pair in 11Li Probing the behavior at several densities

47. BCS Crossover region

48. Two-particle density for 11Li K.Hagino and H. Sagawa, PRC72(’05)044321 Total n r “cigar-like” configuration q12 9Li r S=0 n “di-neutron” configuration

49. Di-neutron wave function in Borromean nuclei (00) Sum = 0.603 PS=0 = 0.606

50. : di-neutron configurations : cigar-like