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

Stable Nuclei Unstable Nuclei

Excitations to the continuum states in drip line nuclei

Breakdown of BCS approximation

Hartree-Fock Bogoliubov approximation

1 = exp 0

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

u E rr Y r

v E r r

ˆ| |0

New quasi-particle picture different to BCS quasi-particle!!

wave function will be

non-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(

0)()(2

)()V(E2)1(

lj2ljqp222

　 Coupled differential HFB equations

Pairing Potential

Mean Field Potential V(r)

continuum

resonance

Mean field and HFB single particle energy

Features of HFB solutions0 ,0Eqp

0E 1) case

0)( 0)V(

)()( )(

)(),(E

zinzjizh

)( rvdr lj

normalization occupation probability

1 )()( 0

drrvru ljlj

)(),(E

)()sin()()cos(),(E

0E 2) case

rrnrrjCru

lljlljlj

)EE(),E( ),E( ''

rurudr ljlj

normalization occupation probability/MeV

( ,E )ljdr v r

Quasi-particle wave functions of weakly bound states

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

rfexp1

Volume-type

Surface-type

Average strength of pair field

drrfrdrrfrr )(/)()(0

3s1/2orbit

asymptotic behavior of (r) or v(r)

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.

continuum

resonance

2qp excitations

a) Ei, Ej both discrete

b) Ei discrete Ej continuum

Multipole Response Function

i j( E ) <0 , ( E ) 0

i j j i

( 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

i j j i j

i j j i

B( E E ) E (ij)J O 0 (E E )

i O j (E , ) (E , ) (E , ) (E , )r

i j j i

dr u r v r u r v r

c) Ei ,Ej both continuum

rmax =64fm max=10MeV

Sum Rule NEWSR

SdBm n

)( )(1

SdBm n

i j i j i j

i i i i i

( 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 dr u r v r u r v r

Quadrupole Response

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

Lectures in Milano University Hiroyuki Sagawa, Univeristy of Aizu March, 2008

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