Lectures in Istanbul Hiroyuki Sagawa, Univeristy of Aizu June 30-July 4, 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) BCS （ Bardeen-Cooper-Schrieffer) pair BEC (Bose-Einstein condensation)

--Coexistence of BCS and BEC-like pairs in Infinite Matter and Nuclei-- Hiroyuki Sagawa Center for Mathematics and Physics, University of Aizu 1.Introduction 2.Pairing gaps in nuclear matter 3.Three-body model for borromian nuclei 4.BEC-BCS crossover in finite nuclei 5.Summary Pairing Correlations in Exotic Nuclei

Pairing correlations in nuclei Coherence length of a Cooper pair: much larger than the nuclear size (note)  = 55.6 fm R = 1.2 x 140 1/3 = 6.23 fm (for A=140) K.Hagino, H. Sagawa, J. Carbonell, and P. Schuck, PRL99,022506(2007).

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

BCS state Bogoliubov transformation

Two-body Hamiltonian Constrained Hamiltonian Gap equation

BCS formulas with seniority pairing interaction Seniority pairing Gap Equation  QP energy condition gives is obtained.  GS + S 

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

Quasi-particle excitations

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

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

Toward Universal Pairing Energy Density Functionals

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Stable NucleiUnstable Nuclei Excitations to the continuum states in drip line nuclei Breakdown of BCS approximation

 bound continuum resonance Mean field and HFB single particle energy ii 0 HFB

Hartree-Fock Bogoliubov approximation Trial Wave Function Coordinate Space Representation

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

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

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

Three-body model Density-dependent delta-force core n n r1r1 r2r2 V WS H. Esbensen, G.F. Bertsch, K. Hencken, Phys. Rev. C56(’99)3054 (note) recoil kinetic energy of the core nucleus v 0 a nn  S 2n  Hamitonian diagonalization with WS basis  Continuum: box discretization Important for dipole excitation Application to 11 Li, 6 He, 24 O

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 S 2n can be reproduced

Application to 11 Li, 6 He, 24 O 11 Li, 6 He: Typical Borromean nuclei Esbensen et al. 24 O: Another drip-line nuclei 11 Li: WS: adjusted to p 3/2 energy in 8 Li & n- 9 Li elastic scattering Parity-dependence to increase the s-wave component 6 He: WS: adjusted to n-  elastic scattering 24 O: WS: adjusted to s 1/2 in 23 O (-2.74 MeV) & d 5/2 in 21 O A.Ozawa et al., NPA691(’01)599

Two-body Density One-body density as a function of angle r1r1 r2r2  12

Two-particle density for 11 Li r1r1 9 Li n n r2r2  12 Set r 1 =r 2 =r, and plot  2 as a function of r and  12  two-peaked structure  Long tail for “di-neutron” di-neutroncigar-type) S=0S=0 or 1

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

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

Comparison among three nuclei 11 Li 6 He (p 3/2 ) 2 :83.0 % (d 5/2 ) 2 :6.11 %, (p 1/2 ) 2 :4.85 % (s 1/2 ) 2 :3.04 %, (d 3/2 ) 2 :1.47 % (p 1/2 ) 2 :59.1% (s 1/2 ) 2 :22.7% (d 5/2 ) 2 :11.5% for (p 1/2 ) 2 or (p 3/2 ) 2 for (s 1/2 ) 2

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

24 O (s 1/2 ) 2 :93.6% (d 3/2 ) 2 :3.61% (f 7/2 ) 2 :1.02% for (s 1/2 ) d 5/2 2s 1/2 22 O 24 O

Ground State Properties S 2n is still controversial in 11 Li. C. Bachelet et al., S 2n =376+/-5keV (ENAM,2004)

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

Peak positionSimple two-body cluster model: ScSc peak at E=8 S c /5=1.6 S c 6 He: E peak =1.55 MeV 1.6 S 2n =1.6 X 0.975= 1.56 MeV 11 Li: E peak =0.66 MeV 1.6 S 2n =1.6 X 0.295= 0.47 MeV 24 O: E peak =4.78 MeV 1.6 S 2s1/2 =1.6 X 2.74= 4.38 MeV 1.6 S 2n =1.6 X 6.45= MeV 6 He, 11 Li: dineutron-like excitation 24 O: s.p.-like excitation

Comparison with expt. data ( 11 Li) E peak =0.66 MeV B(E1) = 1.31 e 2 fm 2 (E < 3.3 MeV) New experiment :T. Nakamura et al., PRL96,252502(2006) E peak ~ 0.6 MeV B(E1) = (1.42 +/- 0.18) e 2 fm 2 (E < 3.3 MeV) T. Aumann et al., PRC59, 1259(1999)

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

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

BCS Crossover region

Two-particle density for 11 Li r 9 Li n n r  12 Total S=0 “di-neutron” configuration “cigar-like” configuration K.Hagino and H. Sagawa, PRC72(’05)044321

Di-neutron wave function in Borromean nuclei (00) Sum = P S=0 = 0.606

: di-neutron : cigar-like configurations

Nulcear Matter Calc. 11 Li good correspondence

M. Matsuo, PRC73(’06) Matter Calc.

Two-particle density

Gogny HFB calculations N. Pillet, N. Sandulescu, and P. Schuck, e-print: nucl-th/

Summary The three-body model is suceessfully applied to describe both the g.s. and excited states of drip line nuclei. Dipole excitations show strong threshold effect in the borromeans, while there is no clear sign of the continuum coupling in the skin nucleus 11 Li  Di-neutron wave function at different R coexistence of BCS/BEC like behavior of Cooper pair Further experimental evidence could be obtained by 2n correlation measurements of break-up reactions (Dalitz plot) and 2n transfer reactions. r 1 =r 2

Constraining the size of 11 Li by various experiments R r H. Esbensen et al., Phys. Rev. C (2007).

1.We have studied a role of di-neutron correlations in weakly bound nuclei on the neutron drip line by a three-body model. 2. Two peak structure in the g.s. density is found in the borromean nuclei: One peak with small open angle -> di-neutron Another peak with large open angle -> cigar- type correlation. 3. Di-neutron configuration is dominated by S=0, while the cigar depends on the nuclei having either S=1 or S=0. 4. Dipole excitations show strong threshold effect in the borromeans, while there is no clear sign of the continuum coupling in the skin nucleus. 5. Further experimental evidence can be obtained by 2n correlatioms measurements of Coulomb break-up reactions, 2n transfer ---. Summary 2

Motivation Spatial correlation of valence neutrons Analysis of Coul. Dissociation for 11 Li K. Ieki et al., PRL70(’93)730. S. Shimoura et al., PLB348(’95) 29. M.Zinser et al.,Nucl. Phys.A619(’97)151 K.Nakamura et al.,PRL96(’06) r R 9 Li n n Correlation angle? ? Di-neutron correlation

Spatial structure of neutron Cooper pair in infinite matter M. Matsuo, PRC73(’06) BCS Crossover region

BCS-BEC crossover behavior in infinite nuclear matter Neutron-rich nuclei Weakly bound levels dilute density around surface (halo/skin) pairing gap in infinite nuclear matter M. Matsuo, PRC73(’06)044309

BCS vacuum Quasi-particle energy

11 Li Di-neutron wave function in Borromean nuclei

76, (2007) Messages from Nuclear Matter Calculations