Pairing correlations in Exotic Nuclei

September 22, 2011, Kyoto, Japan Hiroyuki Sagawa Univesity of Aizu

1.  Introduction 2.  IS+IV pairing interactions 3.  BEC-BCS crossover in Borromian Nuclei 4.  Di-neutron correlations and Dipole response 5.  Summary　　

Isospin  (IS+IV)  dependent  Pairing  Interac5on,  BCS-­‐BEC

Carlos Bertulani, Texas A&M University at Commerce Hong Feng Lu, China Agricultural University/UoA Hongliang Lui, Peking University

J. Margeuron, Orsay K. Hagino, Tohoku University, Japan P. Schuck, Orsay J. Carbonell , Grenoble

Systematic Study

Odd-Even mass staggering !3 =(")A

2B(A)-2xB(A+1)+B(A+2)( )

Isospin  dependent  Pairing  Interac5on  

Vpair (1,2) =1! P!2V0g! ",#! z!" #$!(

!r1 %!r2 )

[ ] [ ] [ ]1 2, , ,z z zg g gτ τ τρ βτ ρ βτ ρ βτ= +

[ ]1s n

0 0

, 1 f ( ) f ( )s n

z z s z ngα α

τ

ρ ρρ βτ βτ η βτ η

ρ ρ⎛ ⎞ ⎛ ⎞

= − −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

J.Margueron,  HS  and  K.  Hagino  PRC76,064316(2007)  and  PRC77,054309(2008)

s n n

( ) ( )f ( ) 1 f ( ), f ( )

( )n p

z z z z z

r rr

ρ ρβτ βτ βτ βτ τ

ρ

−= − = =

No  free  parameters!  

0

s n

nn scattering length, , , pairing gaps in nuclear and neutron matters n

Vη α η α

→

→

kkkkk

a

kkkkkkk

c

c

nn

c

cc

+−

−−=

⎥⎦

⎤⎢⎣

⎡

+−

++−=

ln1

ln2

12cot

π

αα

απδ

nn  scaPering  length  and  pairing  strength

( )nncnn aka 2/2 −= πα

T-­‐matrix  theory  provides  the  formulas

G. F. Bertsch and H. Esbensen

m/2V 220 απ=

Bare:Argonne V18 +3body

cnn kr π/4 range effective = mv /2 220 απ= ∞→nnalimit unitary

mke MC /22.. =

( )nncnn aka 2/2 −= πα

l Weakly interacting fermions l Correlation in p space (large coherence length)

l Interacting “diatomic molecules” l Correlation in r space (small coherence length)

M. Greiner et al., Nature 426(’04)537

cf. BEC of molecules in 40K

Nuclear  MaPer

76, 064316(2007)

Messages from Nuclear Matter Calculations

BCS-­‐BEC  crossover  phenomena

r

 density

np

V(r) aPrac5ve

             IS+IV    Pairing  in  Finite  Nuclei

HFB  calcula5ons  with  canonical  basis SLy4+isospin  dependent    pairing  interac5onsCa,  Ni,  Sn,  Pb  isotopes  

J.Margueron,  HS  and  K.  Hagino、 PRC77,054309(2008)

Deformed HF+BCS with odd particle blocking (EV8-odd)

SkP    +isospin  dependent    pairing  SLy4+isospin  dependent    pairing

     PRC80,  027303(2009)

19

Deformed HF+BCS with blocking EV8-ODD

EV8 (P. Bonche, H. Flocard and P.H. Heenen, CPC 171 (2005) 49) 1 – code has only 6,500 lines

2 - HF+BCS, 3-d coordinate mesh, axial symmetry, small continuum space

EV8 ODD:

3 – Blocking implemented by Bertsch (odd N) and Bertulani (odd Z).

700 800 900 1000 1100 1200 1300 1400 1500 1600 1700Pairing strength (MeVfm3)

0.0

0.5

1.0

! (M

eV)

IS isotopesIS isotonesIS+IV isotopesIS+IV isotones

A systematic study of pairing gaps varying the coupling strength C. Bertulani, Honglinag Lui, HS et al. (2011).

0

20

40

60

80

100

0 20 40 60 80 100 120 140 1600

20

40

60

80

100

20 40 60 80 100 120 140 160

|Sexpp -SIS

p |

Z

0

0.2

0.4

0.6

0.8

1

|Sexpn -SIS

n |δS (MeV)

|Sexpn -SIS+IV

n |

Z

N

|Sexpp -SIS+IV

p |

N

50 60 70N

5

10

15

S n (MeV

)

90 100 110 120N

120 130 140N

expISIS+IV

Z=46 Z=72 Z=90

30 40 50Z

0

5

10

15

S p (MeV

)

60 70 80Z

90Z

expISIS+IV

N=50 N=90 N=136

86 88 90 92 94 96Z

2

4

6

8

S p (MeV

)expISIS+IV

N=136

0

20

40

60

80

100

0 20 40 60 80 100 120 140 1600

20

40

60

80

100

20 40 60 80 100 120 140 160

|Δexpp -ΔIS

p |Z

0

0.2

0.4

0.6

0.8

1

|Δexpn -ΔIS

n |δΔ (MeV)

|Δexpn -ΔIS+IV

n |

Z

N

|Δexpp -ΔIS+IV

p |

N

50 60 70 80N

0.0

0.5

1.0

1.5

2.0!

n (MeV

)

90 100 110 120N

130 140 150N

expISIS+IV4.66/A0.31

Z=52 Z=78 Z=92

48 52 56 60 64Z

0.0

0.5

1.0

1.5

2.0

!p (M

eV)

64 68 72 76 80 84Z

72 76 80 84 88Z

expISIS+IV4.31/A0.31

N=76 N=102 N=112

1

TABLE I:

Data set Low isospin High isospin Di!erence

Neutrons Z = 52 Exp 1.36 1.08 -0.28

IS 1.52 1.41 -0.11

IS+IV 1.40 1.19 -0.21

Z = 78 Exp 1.13 0.99 -0.14

IS 0.96 1.16 0.20

IS+IV 0.87 0.91 0.04

Z = 92 Exp 0.77 0.56 -0.21

IS 0.90 0.80 -0.10

IS+IV 0.70 0.55 -0.15

Protons N = 76 Exp 1.19 0.93 -0.26

IS 1.13 0.87 -0.26

IS+IV 1.13 0.98 -0.15

N = 102 Exp 0.96 0.63 -0.33

IS 0.79 0.39 -0.40

IS+IV 0.92 0.59 -0.33

Z = 112 Exp 0.87 0.66 -0.21

IS 0.58 0.61 0.03

IS+IV 0.67 0.70 0.03

Average Gaps for low and high isospin

27

Perspectives •  IS+IV pairing interaction looks promising for the fitting purpose. •  Extension to include ((N-Z)/A)2 term + Two-body Coulomb. (M. Yamagami ) (M. Yamagami and H. Nakada) • Better EDF with pairing correlations: Pairing correlation properties should be isolated from the full functional. •  Are we much better than LDM for pairing gap residuals as a systematics?

yes !

l Weakly interacting fermions l Correlation in p space (large coherence length)

l Interacting “diatomic molecules” l Correlation in r space (small coherence length)

M. Greiner et al., Nature 426(’04)537

cf. BEC of molecules in 40K

16C

Borromian Nuclei

(any two body systems are not bound, but three body system is bound)

18C

Three-body model

Density-dependent delta-force

core n

n r1

r2

VWS

VWS

H. Esbensen, G.F. Bertsch, K. Hencken, Phys. Rev. C56(’99)3054

(note) recoil kinetic energy of the core nucleus

v0 ann α S2n

Ø Hamitonian diagonalization with WS basis Ø Continuum can be included by solving Green’s functions Ø Pauli blocking is properly taken into account. Important for

dipole excitation

Application to 11Li, 6He

Two-particle wave functions (J=0 pairs)

Hamiltonian diagonalization

l Continuum: box discretization l Energy cut-off:

Application to 11Li, and 6He

11Li, 6He: Typical Borromean nuclei 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-α elastic scattering

fmsa Hen 12.097.4)(4 ±=−

(Efimov states ?)

fmsa Lin )31/12(30)(9 −++−=−

Probing the behavior at several densities

Coexistence of BCS-BEC like behaviour of Cooper Pair in 11Li

: di-neutron : cigar-like

configurations

M. Matsuo, PRC73(’06)044309

Matter Calc.

Two-particle density

Dipole Excitations

Response to the dipole field:

Smearing:

Experimental proof of di-neutron

Comparison with expt. data (11Li)

Epeak=0.66 MeV B(E1) = 1.31 e2fm2 (E < 3.3 MeV)

T. Nakamura et al., PRL96,252502(2006)

Epeak ~ 0.6 MeV B(E1) = (1.42 +/- 0.18) e2fm2 (E < 3.3 MeV)

T. Aumann et al., PRC59, 1259(1999)

Geometry of Borromean nuclei 11Li

6He

B(E1)

matter radius

(11Li) (6He)

K.Hagino and H. S.,PRC76(’07)047302

“experimental” mean opening angle

C.A. Bertulani and M.S. Hussein, PRC76(’07)051602

θ12

11 e n

1θ

2e2θ

Li9

１１Li three-body break-up cross sections

2n

K. Hagino, H.S.,T.Nakamura and S.Shimoura, PRC80,031301(R)(2009)

Dalitz Plot of Triple coincidence experiments

Exp. T. Nakamura et al., to be published

Nakamura-san’s slides

Double differential strength function for 11Li full calculation (with FSI)

（without FSI)

(with FSI) No Vn-core

(No Vn-core

virtual s-state!

full calculation No Vn-core

(with FSI)

（without FSI)

Summary I Ø Application of three-body model to Borromean nuclei

11Li

Ø Di-neutron wave function for each R l  Concentration of a Cooper pair on the nuclear surface l  Relation to BCS-BEC crossover phenomenon

Ø E1 response and geometry of Borromean nuclei

Ø  n-n coincidence cross sections from 11Li* and 6He* l  importance of n-core interaction l  clear evidence of virtual s-state in n-9Li system l  correlation angle is determined experimentally.

l  strong pair correlations in di-neutrons θ12

l  Di-neutron correlations in 11Li and 6He Ø K.Hagino and H. S., PRC72(’05) 044321. Ø K.H.agino and H. S., PRC75(’07)021301(R). Ø K.Hagino, H. S., J. Carbonell, and P. Schuck, PRL99(’07)022506. Ø H. Esbensen, K.Hagino, P. Mueller, and H. S., PRC76(’07)024302. Ø K.Hagino and H. S., PRC76(’07) 047302.

l  energy and angular n-n coincidence cross sections

Recent publications:

Ø K.Hagino, H.S. ,T. Nakamura and S. Shimoura, PRC80,031301 (R)(2009).

l  Di-proton correlations in 17Ne Ø T.Oishi, K. Hagino and HS, PRC82,024315 (2010).

Ø Strong di-proton correlations is found in 17Ne Ø Two body Coulomb interaction decreases 13% of di-proton correlations.

Summary II