two-body correlations and pairing localization in open shell nuclei
DESCRIPTION
Two-Body Correlations and Pairing Localization in Open Shell Nuclei. Nicolae Sandulescu. Institute of Physics and Nuclear Engineering, Bucharest. G. Bertsch INT – Seattle N. Pillet CEA- Bruyeres-le-Chatel P. Schuck IPN -Orsay. Collaboration:. - PowerPoint PPT PresentationTRANSCRIPT
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