Lectures in IstanbulHiroyuki Sagawa, Univeristy of Aizu

June 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) 　pairBEC (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 1401/3 = 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

2kv

2ku

i

kkkkk

kkkkk

avau

avau

0

0

exp1

BCS

kkk

k

k

kkkkk

aauv

N

aavu

Bogoliubov transformation

3421

4321

432121

21

21 41

kkkkkkkk

kkkkkkkk

kk aaaaVaatH

NHH

'

220

''0

''''

'

'''

' )(kk

k

kkkkkkk

kkkkkk VvuV

2'''' '

'21

kk

kkkkkkkkkkkkk vVVtt

22

2

22

2

'

'

)(1

21

)(1

21

kk

kk

kk

kk

u

v

Two-body Hamiltonian

Constrained Hamiltonian

Gap equation

0' BCSHBCS

NvNk

k 0

22BCSBCS

BCS formulas with seniority pairing interaction

Seniority pairing

Gap Equation

22qp )(E i

i

22

2

22

2

)(1

21

)(1

21

i

ii

i

ii

u

v

QP energy

i condition gives

is obtained.

iqpE

2122 ii uv

GS+S

i

J

ii aa)0(

S

2 2

2 1G ( )i

i

Single-particle energy

Quasi-particle energy 　　

Excitation energy

Positive energy

Negative energy

Pairing gap energy

Quasi-particle excitations

2

Weakly interacting fermionsCorrelation 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 40K

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

cnn kr /4 range effective mv /2 220 nnalimit unitary

mke MC /22..

-

Stable Nuclei Unstable Nuclei

Excitations to the continuum states in drip line nuclei

Breakdown of BCS approximation

bound

continuum

resonance

resonance

Mean field and HFB single particle energy

i

0

HFB

Hartree-Fock Bogoliubov approximation

, 1

1 = exp 02

n

Z a a

Trial Wave Function

,

' ',

1 = exp ' , ' ' 02 r r

drdr Z r r a a

* ,ra r a

* *

,

, ' ' , ', 'Z r r r Z r

Coordinate Space Representation

( , ) 1 ˆ, ( , )( , )

lj mlj

lj

u E rr Y r

v E r r

ˆ| |0

H N

Z

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

),1(),(2),1(2

),()3( ZNBZNBZNBZN N

NN )(

N=odd is recommended.

)()3( ),( HFBBCSZN

-B

N

24O skin nucleus

16C

Borromian Nuclei

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

Three-body model

Density-dependent delta-force

coren

nr1

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

S2n

Hamitonian diagonalization with WS basisContinuum: box discretization Important for

dipole excitationApplication to 11Li, 6He, 24O

Density-dependent delta interactionH. 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

Application to 11Li, 6He, 24O

11Li, 6He: Typical Borromean nuclei Esbensen et al.

24O: Another drip-line 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- elastic scattering

24O:

WS: adjusted to s1/2 in 23O (-2.74 MeV) & d5/2 in 21O

A.Ozawa et al., NPA691(’01)599

2 1 2 12 1 2 1 2

0 12 1 2 12 2 1 2 12

( , , ) ( , ) ( , )

= ( , , ) ( , , )

gs gs

S S

r r r r r r

r r r r

Two-body Density

2 212 1 1 2 2 2 1 2 12

0 0

( ) 4 ( , , )r dr r dr r r

One-body density as a function of angle

2

0 212 2 12 12 12

1 312 2 12 12

For (p) J=0 configuration, S=0 and S=1 contributions;

sin( ) ( ) sin( )con ( )

sin( ) ( ) sin ( )

S

S

2For (s) J=0 configuration, only S=0 is possible.

r1

r2

12

Two-particle density for 11Li

r1

9Li

n

n

r2

12

Set r1=r2=r, and plot 2 as a function of r and 12

two-peaked structureLong tail for “di-neutron”

di-neutron cigar-type)

S=0 S=0 or 1

Two-particle density for 11LiTotal

S=0 S=1 or

Two-particle density for 6HeTotal

S=0 S=1 or

Comparison among three nuclei11Li 6He

(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 %

(p1/2)2 :59.1%(s1/2)2 :22.7% (d5/2)2 :11.5%

for (p1/2)2 or (p3/2)2 for (s1/2)2

Two-particle density for 24OTotal

S=0 S=1 or

24O

(s1/2)2 :93.6%(d3/2)2 :3.61% (f7/2)2 :1.02%

for (s1/2)2

0

-2.74

-3.801d5/2

2s1/2

22O24O

6 23/2

11 21/2

He 0.975 21.3 13.2 (p ) 83.0

Li 0.295 41.4 26.3 (p ) 59.1

21/2

24 21/2

(s ) 22.7

O 6.452 35.2 10.97 (s ) 93.6 ( s1/2 d5/2 = 2.739MeV, = 3.806MeV)

2 2 2 22n nn c-2nnucleus S (MeV) r (fm ) r (fm ) configuration (%)

Ground State Properties

S2n is still controversial in 11Li.

C. Bachelet et al., S2n=376+/-5keV (ENAM,2004)

Dipole Excitations

Response to the dipole field:

Smearing:

Experimental proof of di-neutron and/or cigar configurations

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 MeV11Li: Epeak=0.66 MeV 1.6 S2n=1.6 X 0.295= 0.47 MeV24O: 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 excitation24O: s.p.-like excitation

Comparison with expt. data (11Li)

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

New experiment :T. Nakamura et al., PRL96,252502(2006)

Epeak ~ 0.6 MeVB(E1) = (1.42 +/- 0.18) e2fm2 (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 levelsdilute density around surface (halo/skin)

Probing the behaviorat several densities

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

BCS

Crossover region

Two-particle density for 11Li

r

9Li

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 = 0.603PS=0 = 0.606

: di-neutron: cigar-like

configurations

Nulcear Matter Calc.11Ligood correspondence

M. Matsuo, PRC73(’06)044309

Matter Calc.

Two-particle density

Gogny HFB calculations

N. Pillet, N. Sandulescu,and P. Schuck,e-print: nucl-th/0701086

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

11Li

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.

r1 =r2

Constraining the size of 11Li 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 11Li

K. Ieki et al., PRL70(’93)730.S. Shimoura et al., PLB348(’95) 29.M.Zinser et al.,Nucl. Phys.A619(’97)151K.Nakamura et al.,PRL96(’06)252502.

rR

9Lin

n

Correlation angle?

?

Di-neutron correlation

Spatial structure of neutron Cooper pair in infinite matter

M. Matsuo, PRC73(’06)044309

BCS

Crossover region

BCS-BEC crossover behavior in infinite nuclear matterNeutron-rich nuclei Weakly bound levels

dilute density around surface (halo/skin)

pairing gap in infinite nuclear matter

M. Matsuo, PRC73(’06)044309

kkk

kkk

k

k

kkkkk

N

aauv

N

aavu

0,0

0

0

'1BCS

exp1

BCS

0BCS

},{ 0},{ '''

k

kkkkkk

BCSHBCSBCSHBCSE kkk''

BCS vacuum

Quasi-particle energy

22)()( nnnn kkE

22)()( nnnn kkE *22 2/)( nn mkk nnn U

11Li

Di-neutron wave function in Borromean nuclei

76, 064316(2007)

Messages from Nuclear Matter Calculations