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

Stable Nuclei Unstable Nuclei

Excitations to the continuum states in drip line nuclei

Breakdown of BCS approximation

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

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

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)

bound

continuum

resonance

resonance

Mean field and HFB single particle energy

i

0

HFB

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

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

Quasi-particle wave functions of weakly bound states

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

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

3s1/2orbit

3s1/2orbit

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

Effective Pair Gap of A=80

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.

bound

continuum

resonance

resonance

2qp excitations

i

0

HFB

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

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

Quadrupole Response

=1MeV

Volume pairing

Volume pairing

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