lectures in istanbul hiroyuki sagawa, univeristy of aizu june 30-july 4, 2008
DESCRIPTION
Lectures in Istanbul Hiroyuki 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) pair BEC (Bose-Einstein condensation). - PowerPoint PPT PresentationTRANSCRIPT
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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)
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--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
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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 1401/3 = 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
2kv
2ku
i
kkkkk
kkkkk
avau
avau
0
0
exp1
BCS
kkk
k
k
kkkkk
aauv
N
aavu
Bogoliubov transformation
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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
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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
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Single-particle energy
Quasi-particle energy
Excitation energy
Positive energy
Negative energy
Pairing gap energy
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Quasi-particle excitations
2
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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
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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:
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Toward Universal Pairing Energy Density Functionals
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cnn kr /4 range effective mv /2 220 nnalimit unitary
mke MC /22..
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-
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Stable Nuclei Unstable Nuclei
Excitations to the continuum states in drip line nuclei
Breakdown of BCS approximation
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bound
continuum
resonance
resonance
Mean field and HFB single particle energy
i
0
HFB
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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
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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
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Odd-even mass difference
),1(),(2),1(2
),()3( ZNBZNBZNBZN N
NN )(
N=odd is recommended.
)()3( ),( HFBBCSZN
-B
N
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24O skin nucleus
16C
Borromian Nuclei
(any two body systems are not bound, but three body system is bound)
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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
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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
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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
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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
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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
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Two-particle density for 11LiTotal
S=0 S=1 or
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Two-particle density for 6HeTotal
S=0 S=1 or
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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
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Two-particle density for 24OTotal
S=0 S=1 or
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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
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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)
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Dipole Excitations
Response to the dipole field:
Smearing:
Experimental proof of di-neutron and/or cigar configurations
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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
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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)
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BCS-BEC crossover behavior in infinite nuclear matter
Neutron-rich nuclei are characterized by
Weakly bound levelsdilute density around surface (halo/skin)
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Probing the behaviorat several densities
Coexistence of BCS-BEC like behaviour of Cooper Pair in 11Li
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BCS
Crossover region
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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
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Di-neutron wave function in Borromean nuclei
(00)
Sum = 0.603PS=0 = 0.606
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: di-neutron: cigar-like
configurations
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Nulcear Matter Calc.11Ligood correspondence
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M. Matsuo, PRC73(’06)044309
Matter Calc.
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Two-particle density
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Gogny HFB calculations
N. Pillet, N. Sandulescu,and P. Schuck,e-print: nucl-th/0701086
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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
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Constraining the size of 11Li by various experiments
R
r
H. Esbensen et al., Phys. Rev. C (2007).
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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
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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
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Spatial structure of neutron Cooper pair in infinite matter
M. Matsuo, PRC73(’06)044309
BCS
Crossover region
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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
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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
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22)()( nnnn kkE
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22)()( nnnn kkE *22 2/)( nn mkk nnn U
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11Li
Di-neutron wave function in Borromean nuclei
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76, 064316(2007)
Messages from Nuclear Matter Calculations