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Lectures in Istanbul Hiroyuki Sagawa, Univeristy of Aizu June 30July 4, Giant Resonances and Nuclear Equation of States 2. Pairing correlations in Exotic nuclei  BECBCS crossover  BCS （ BardeenCooperSchrieffer) pair BEC (BoseEinstein condensation) BCS （ BardeenCooperSchrieffer) pair BEC (BoseEinstein condensation)
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Coexistence of BCS and BEClike 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.Threebody model for borromian nuclei 4.BECBCS crossover in finite nuclei 5.Summary Pairing Correlations in Exotic Nuclei
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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).
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Pairing Phase Transition (second order phase transition) order parameter Particles Fermi energy Holes
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BCS state Bogoliubov transformation
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Twobody Hamiltonian Constrained Hamiltonian Gap equation
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BCS formulas with seniority pairing interaction Seniority pairing Gap Equation QP energy condition gives is obtained. GS + S
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Singleparticle energy Quasiparticle energy Excitation energy Positive energy Negative energy Pairing gap energy
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Quasiparticle excitations
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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
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BCSBEC 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:
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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
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bound continuum resonance Mean field and HFB single particle energy ii 0 HFB
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HartreeFock Bogoliubov approximation Trial Wave Function Coordinate Space Representation
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New quasiparticle picture different to BCS quasiparticle!! wave function will be nonlocal local Pair potential goes beyond HF potential Pair potential upper comp. lower comp
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Oddeven mass difference N=odd is recommended. B N
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24 O skin nucleus 16 C Borromian Nuclei (any two body systems are not bound, but three body system is bound)
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Threebody model Densitydependent deltaforce 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
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Densitydependent 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
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Application to 11 Li, 6 He, 24 O 11 Li, 6 He: Typical Borromean nuclei Esbensen et al. 24 O: Another dripline nuclei 11 Li: WS: adjusted to p 3/2 energy in 8 Li & n 9 Li elastic scattering Paritydependence to increase the swave 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
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Twobody Density Onebody density as a function of angle r1r1 r2r2 12
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Twoparticle 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 twopeaked structure Long tail for “dineutron” dineutroncigartype) S=0S=0 or 1
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Twoparticle density for 11 Li Total S=0 S=1 or
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Twoparticle density for 6 He Total S=0 S=1 or
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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
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Twoparticle density for 24 O Total S=0 S=1 or
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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
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Ground State Properties S 2n is still controversial in 11 Li. C. Bachelet et al., S 2n =376+/5keV (ENAM,2004)
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Dipole Excitations Response to the dipole field: Smearing: Experimental proof of dineutron and/or cigar configurations
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Peak positionSimple twobody 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: dineutronlike excitation 24 O: s.p.like excitation
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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)
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BCSBEC crossover behavior in infinite nuclear matter Neutronrich nuclei are characterized by Weakly bound levels dilute density around surface (halo/skin) Weakly bound levels dilute density around surface (halo/skin)
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Probing the behavior at several densities Coexistence of BCSBEC like behaviour of Cooper Pair in 11 Li
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BCS Crossover region
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Twoparticle density for 11 Li r 9 Li n n r 12 Total S=0 “dineutron” configuration “cigarlike” configuration K.Hagino and H. Sagawa, PRC72(’05)044321
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Dineutron wave function in Borromean nuclei (00) Sum = P S=0 = 0.606
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: dineutron : cigarlike configurations
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Nulcear Matter Calc. 11 Li good correspondence
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M. Matsuo, PRC73(’06) Matter Calc.
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Twoparticle density
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Gogny HFB calculations N. Pillet, N. Sandulescu, and P. Schuck, eprint: nuclth/
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Summary The threebody 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 Dineutron 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 breakup reactions (Dalitz plot) and 2n transfer reactions. r 1 =r 2
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Constraining the size of 11 Li by various experiments R r H. Esbensen et al., Phys. Rev. C (2007).
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1.We have studied a role of dineutron correlations in weakly bound nuclei on the neutron drip line by a threebody model. 2. Two peak structure in the g.s. density is found in the borromean nuclei: One peak with small open angle > dineutron Another peak with large open angle > cigar type correlation. 3. Dineutron 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 breakup reactions, 2n transfer . Summary 2
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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? ? Dineutron correlation
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Spatial structure of neutron Cooper pair in infinite matter M. Matsuo, PRC73(’06) BCS Crossover region
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BCSBEC crossover behavior in infinite nuclear matter Neutronrich nuclei Weakly bound levels dilute density around surface (halo/skin) pairing gap in infinite nuclear matter M. Matsuo, PRC73(’06)044309
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BCS vacuum Quasiparticle energy
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11 Li Dineutron wave function in Borromean nuclei
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76, (2007) Messages from Nuclear Matter Calculations
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