Lectures in Milano University Hiroyuki Sagawa, Univeristy of Aizu March, 2008
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Lectures in Milano UniversityHiroyuki Sagawa, Univeristy of Aizu
March, 2008
• 1. Pairing correlations in Nuclei
General aspects of HFB HFB and Quadrupole Respons in weakly bound states
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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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Hartree-Fock Bogoliubov approximation
, 1
1 = exp 0
2
n
Z a a
Trial Wave Function
,
' ',
1 = exp ' , ' ' 0
2 r rdrdr Z r r a a
��������������������������������������� ���
* ,ra r a
* *
,
, ' ' , ', 'Z r r r Z r
Coordinate Space Representation
( , ) 1ˆ, ( , )
( , )
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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Hartree-Fock Bogoliubov Equations in the coordinate space
0)()(2
)()V(E2)1(
d
d
0)()(2
)()V(E2)1(
d
d
lj2ljqp222
2
lj2ljqp222
2
rurm
rvrm
r
ll
r
rvrm
rurm
r
ll
r
)(r
Coupled differential HFB equations
Pairing Potential
Mean Field Potential V(r)
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bound
continuum
resonance
resonance
Mean field and HFB single particle energy
i
0
HFB
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Features of HFB solutions0 ,0Eqp
0E 1) case
0)( 0)V(
0E
qp
qp
rrr
qp22
qp2
21
qp
1qp
E m2
E m2
)()( )(
)(),(E
)(),(E
zinzjizh
rrhrv
rrhru
lll
llj
llj
2
0
)( rvdr lj
normalization occupation probability
1 )()( 0
22
drrvru ljlj
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22
qp2
22
qp
22qp
qp
m21
E m2
)(),(E
)()sin()()cos(),(E
0E 2) case
C
rrhrv
rrnrrjCru
llj
lljlljlj
)EE(),E( ),E( ''
0
rurudr ljlj
normalization occupation probability/MeV
2
qp
0
( ,E )ljdr v r
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Quasi-particle wave functions of weakly bound states
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Model
1. is fixed to be
eff =E qp(lj)
2. The depth of Woods-Saxon potential is changed to adjust the eigenenergy WS .
3. Mass number is fixed to be A~84 .
4. Average pairing strength is given for a fixed .
central spin-orbit CoulombV( ) V ( ) V ( ) V ( )r r r r
WS 1/2 WS 5/2= (3s ) or (2d )
Mean field of cooper pairs
Density dependent pairing interaction
Pair potential
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Volume-type and Surface-type Pairing Correlations
)()( rfr
dr
rdfrr
)()(
aRr
rfexp1
1)(
Volume-type
Surface-type
Average strength of pair field
drrfrdrrfrr )(/)()(0
2
0
2
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3s1/2orbit
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3s1/2orbit
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asymptotic behavior of (r) or v(r)
r
0i
0
HF
HFB
Pairing correlation may give a quenching on the halo effect. On the contrary, more states around the Fermi sursface will be weakly-bound states due to the pairing.
2 2qpE
2 (E )( ) exp( )
qpmv r r
2( ) exp( )imr r
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Effective Pair Gap of A=80
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Quasi-particle energy
HFBHHFBHFBHHFBHFBE kkk'')(
BCSHBCSBCSHBCSBCSE kkk'')(
22)()( kkk BCSE
Quasi-particle energy of HFB is very different from BCS for weakly-bound low-l orbits.
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bound
continuum
resonance
resonance
2qp excitations
i
0
HFB
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a) Ei, Ej both discrete
b) Ei discrete Ej continuum
Multipole Response Function
i j( E ) <0 , ( E ) 0
max
i j
2
i j
2
i j j i
0
( E ) 0 , ( E ) 0
B( E E ) (ij)J O 0
i O j (E , ) (E , ) (E , ) (E , )r
i j j idr u r v r u r v r
max
2
i j j i j
2
i j j i
0
B( E E ) E (ij)J O 0 (E E )
i O j (E , ) (E , ) (E , ) (E , )r
i j j i
d
dr u r v r u r v r
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c) Ei ,Ej both continuum
rmax =64fm max=10MeV
Sum Rule NEWSR
EWSR
max
)()(0
SdBm n
n
max
)( )(1
SdBm n
nn
max
i j
2
i j i j i j
2
i i i i i
0
( E ) 0 , ( E ) 0
B( E E ) E E (ij)J O 0 (E E )
E i O j (E , ) ( E , ) ( E , ) (E , )r
i j j i
d d
d dr u r v r u r v r
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Quadrupole Response
=1MeV
Volume pairing
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Volume pairing
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1. We solved a simplified HFB equations in the coordinate space with the correct asymptotic boundary conditions.
WS(d5/2) 0
(a) The peak energy becomes lower and the widths gets broader while the total strength increases dramatically.
(c) The continuum 2qp excitations involving weakly bound S ½
neutrons enhances NEWSR value compared with the results of BCS.
(d) HFB rms radius is slightly smaller than BCS one .
Summary
(b) HFB continuum effect plays an important role in low-energy quadrupole excitations.
(e) The 2qp excitations without S ½ neutrons show only enhancement.
2. Quadrupole Response in the limit of