Two-Body Correlations and Pairing Localization in Open Shell Nuclei

Nicolae Sandulescu

Institute of Physics and Nuclear Engineering, Bucharest

G. Bertsch INT – SeattleN. Pillet CEA- Bruyeres-le-ChatelP. Schuck IPN -Orsay

Collaboration:

T. Nakamura et al, PRL96(2006)252502

degrees 48 1418-12

K. Hagino, H.Sagawa,J.Carbonell,P.Schuk, PRL 99, 2007

12

Two-particle correlations in halo nuclei: 11Li ..

0.35.01 22, ncr

Two-body correlations in open shell nuclei: general definitions

• two-body density

• two-body correlations

• two-body correlations in configuration space

0||00||00||0|| 2jjiijjiiij cccccccck 2

i22 viii uk

• two-body correlations in BCS

describes correlations between two generic nucleons

comnmonly associtated to pair transfer amplitude

NNN drdrrrrrrN

...|),......,(|),( 32

221...

122112

2

]1)[()(),(|),(| 12212122

21 Prrrrrrk

)(r )(rv 0|)()(|0),( 2i1ii1221 i

iurararrk

k is not the wave function of the collective Cooper pair

ii cci

i

u

v)'()(

u

v )',(

i

i rrrr ii

-influence of pair fluctuations - structure of pairs in BCS and in exact models

Two-Body Correlations and Pairing Localization in Open Shell Nuclei

Main Issues

• Pairing correlations: surface/bulk localization ?

• What is the size of correlated pairs ?

m

kFP

2

• Dependence of correlations on pairing tratment ?

• What are the effects of (strong) two-body correlations ?

- enhancement of pair transfer - soft dipole/octupole modes ? next talk by Matsuo

“In nuclei, the pairs cannot be localized within dimensions smaller than the nuclear radius R ”. (A. Bohr et B. R. Mottelson, Nuclear Structure, vol II)

)( )]([)( rrVr eff )(r )(rv ),(iii

iiurrk

k

kk

k

k

V

UE

V

U

h

h

)()(

)()(

0|),()(|0 ),( 11222211 rararr

1r

2r

r

R

θ

0021 ),( ),( SS Rrkrr

• pairing tensor in coordinate representation :

• HFB equations

Localization of pairing correlations in open shell nuclei

2/122

2/124

sin,,

sin,,)(

ddrrRr

ddrrRrR• coherence length

calculations with Gogny force D1S

N. Pillet, N. S, P. Schuck; PRC76, 2007

Localization of pairing correlations: Sn isotopes

mixed

volume

surface

N. Pillet, N. S, P. Schuck; PRC76, 2007

N. S, P. Schuck, X. Vinas, PRC 71 (2005) 054303

Pairing localization: Skyrme-HFB calculations

Vpair =V0[1-r-r’)

Single- particle wave functions: Sn isotopes

2/5d

2/7g

2/1s

2/3d2/11h

)(vu )( 2

kkk rrk k

Pairing field in Sn isotopes

)( )]([)( rrVr eff

N. S, P. Schuck, X. Vinas, PRC 71 (2005) 054303

Coherence length

Pairing localization : generic features

N. Pillet, N. S, P. Schuck; PRC76, 2007

2),( rR

Coherence length in Ca and Sn isotopes

N. Pillet, N.S., P. Schuck, PRC76, 2007

222 )r,R(Rr)r,R(P

Probability distribution

Uncorrelated Probability Distribution

The Effect of Parity Mixing

N. P

illet, N

.S., P

. Schuck, P

RC

76, 2007

)(cos )()(v)12(4

1),( 21nlj21

1

lnlnljn

nljBCS PrRrRujrr

)(cos )()(v)12(4

1),( 21nlj21

1

lnlnljn

nljBCS PrRrRujrr

Parity mixing : generic features

21r case The r

1)1(P );()()(P : note ll uPu ll

) 0for weight (maximum coherently add termsall ,2

If

sign opposite haveparity different with terms/2, If

2

torelative symmetric ison distributi the

parity,given a with termsonly the keeps one If

Particular case of 36Ca

Comparison with 22O

Results and analysis for 3 isotopic chains

Importance of choosing a large configuration space, not restricted to the major shell

Parity mixing in deformed nuclei

9015262 Sm

http://www-phynu.cea.fr

Pairing localization in 152Sm

N. Pillet, N.S, P. Schuck, J.F. Berger, in preparation

Preliminary results

Pairing localization in 152Sm

Preliminary results

How much depend the correlations on pairing treatment ?

Correlation Energies in Sm isotopes

one can reduce the error by restricting to a smaller space !

G. Dussel, S. Pittel, J. Dukelsky, P. Sariguren, PRC 2007

Solutions of pairing Hamiltonian

• BCS

•PBCS

• Exact solution

R. W. Richardson and N. Sherman, Nucl. Phys. 52 (1964)221

(identical collective pairs)

(non-identical collective pairs)

Correlation Energies in BCS and PBCS

BCS PBCS

N. S , G. Bertsch, arXiv 2008

Two-body correlations in configuration space

Npair =8

N. S , G. Bertsch, arXiv 2008

2i

222 v|)(||)(||| iiiii uNBCSccNBCSk

weak coupling intermediate coupling strong coupling

Pair transfer amplitudes in BCS and PBCS

Npair =8

g=0.32 g=0.42 g=0.87

weak coupling intermediate coupling strong coupling

2i

222 v|)(||)(||| iiiii uNBCSccNBCSk

22 |)(||)2(||| NPBCSccNPBCSk iiPBCSii

i

iikgT 22 )()

2(

7.1)( 2 BCS

PBCS

T

T2.1)( 2

BCS

PBCS

T

T

Npair =8

Two-body correlations and pair wave function

N. S , G. Bertsch, arXiv 2008

)'()(v)',( i rrurrk iii

i

pairing tensor

pair wave function

)'()(2

1 )',(

rrE

rr iii

E

i

ixK 4/1

)'()(u

v )',(

i

i rrrr ii

measure of correlations

BCS, PBCS

exact model

L. Cooper, 2007 (Meeting on 50 years of BCS)

Probability distribution inside nuclei

for 120Sn

N. P

illet, N

.S., P

. Schuck, P

RC

76, 2007

BCS-BEC transition in the surface of nuclei ?

II) analyse the action on PBCS state:

bosonic behaviour:

Bosonic character of two-body correlations

testing the bosonic character

N=8, g=0.87

ii cci

i

u

v

1

u

v

2|],[|

2i

2i

NBCSBCSI) analyse the average

!)(

1|

2/1 NN

N

N

NN NNN |1||1

N

NN

1 1

N

N

0.3 1

N

N

two-body operator :

Summary and Conclusions

• small coherence length (2-3 fm) in the nuclear surface

• localization properties (surface/volume) depends essentially on s.p.states closer to chemical potential and less on pairing force

proper description for loosely bound nuclei !?

• two-body correlations associated to the pair wave function look similar in BCS and PBCS

• in the exact model all pairs are different and their collectivity depends on the position of their energies with respect to the chemical potential

• pair transfer amplitudes are underestimated by BCS

Thanks for your attention !

2j

22

j

0 |)2(v)(| | )()()0(| ..... AAudrrrRrRjjBd

djjj

G. Ortiz, J. Dukelsky, Phys Rev A72(2005)043611

BCS-to-BEC crossover from the exact BCS solution

L. Cooper, 2007 (Meeting on 50 years of BCS)

Wave function of Cooper pairs: BCS

A. Legget, 2007 (Meeting on 50 years of BCS)

“Wave function of Cooper pairs”: Legget

Condensation Fraction in Sm Isotopes

G. Dussel, S. Pittel, J. Dukelsky, P. Sariguren, PRC 2007

Pairing Gaps

N. S , G. Bertsch, arXiv 2008

Uncorrelated Probability Distribution

Particular case of 36Ca

Comparison with 22O

Results and analysis for 3 isotopic chains

Importance of choosing a large configuration space, not restricted to the major shell

Condensation Fraction for N=8 pairs