Giant and Pygmy Resonance in Relativistic Approach

The Sixth China-Japan Joint Nuclear PhysicsThe Sixth China-Japan Joint Nuclear Physics

May 16-20, 2006

Shanghai

Zhongyu Ma

China Institute of Atomic Energy, Beijing

Collaborators: Ligang Cao, Baoqiu Chen, Jun Liang

Introduction

Nucleus moving away from the valley of -stability

diffuse neutron: neutron skin, hallo structure,

new magic numbers, new modes of excitations, etc.

Significant interest on low-energy

excited states

GDR (Coulomb excitations)

restoring force proportional to the symmetry energy

Pygmy resonance

Loosely bound neutron coherently oscillate

against the p-n core

neutron density distribution, neutron radius,

skin et al. density dependence of symmetry energy

Astrophysical implications

Fully Consistent RRPA

RRPA -- Consistent in sense:

ph residual interaction determined from the same Lagrangian for g.s. RRPA polarization operator

i=,, i=1, , 3 for , , , respectively

consistent to RMF no sea approx.

Include both ph pairs and h pairs

Z. Y. Ma, et al., Nucl. Phys. A703(2002)222

RRPA TDRMF at small amplitude limit

TDRMF at each time no sea approximation is calculated in a stable complete set basis P.Ring et al., Nucl. Phys. A694(2001)249

2 3 30 1 2 0( , ; , '; ) ( , ; , ; ) ( , ; ) ( , ; , '; )i

i i ii

Q Q E g d k d k Q E D E Q E 1 1 2 2k k k k k k k k

Fig: ISGMR NL1,NL3,TM1,NLSH

Z.Y. Ma, et al., Nucl. Phys. A686(2001)173

M.E. of vector fields coupling h and ph----Largely reduced due to Dirac str.

Cancellation of the & fields --- not take place, Large M.E. coupling h and ph exist

Treatment of the continuum

Resonant states in the continuum

Metastable states in

the centrifugal & Coulomb Barrier

Discretization of the continuum

Expansion on Harmonic Oscillator basis

Box approximation: set a wall at a large distance

Exact treatment of the continuum

Set up a proper boundary condition

Single particle resonance with energy and width

Green’s function method

Scattering phase shift

Centrifugal & Coulomb

Nuclear Pot.

Total

Scattering phase shift

Boundary conditions:

Normalized by

phase shift: = /2 resonant state

For proton, Dirac Coulomb functions have to be solved

for ~1 Z large large diff. from norn Coulomb wf

Cao & Ma PRC66(02)024311

W. Grainer “Rel. Quantum Mach.”

( ) [ ( ) tan ( )], forl lG kr A j kr n kr r R

†( , ) ( ', ) ( ')E r E r dr E E

1cos

2

M EA

k

Example of resonance states

More resonant states

for p than those for n

due to the Coulomb

barrier

-10 -5 0 5 10

-80

-60

-40

-20

0

20

3p3/2

3p1/2

2g9/2

Vp(r)+V

c(r)

Vp(r)

Vn(r)

N P 1j15/2

1i11/2

1h9/2

1i11/2

1i13/21h

9/2 2f7/2

2f5/2

1i13/2

2f5/2

3s1/22d3/2

2d5/21g7/2

1g9/22p

1/2

2p3/2 1f

5/21f7/2

1g9/2

2p1/2

2p3/2

1f5/2

1f7/2

2s1/2

1d3/2

1d5/2

1p1/2

1p3/2

1s1/2

2s1/2

1d3/2

1d5/2

1p1/21p3/2

1s1/2

Ene

rgy(

MeV

)

R(fm)

120Sn

Resonant continuum in pairing correlations

Pairing correlations play a crucial role in MF models for open shell

HF+BCS and RMF+BCS simple

successful in nuclei when F not close to the continuum

HFB and RHB important in nuclei near the drip-line

HF eq. + gap eq. are solved simultaneously

states in continuum are discretized in both methods

Resonant states : HFB eq. are solved with exact boundary conditions

Grasso, Sandulescu, Nguyen, PRC64(2001)064321

Discretization of the continuum overestimates pairing corr.

Effect of the continuum on pairing --- mainly by a few resonant

states in the continuum

RMF+BCS with resonant states including widths

BCS with the continuum

Gap equation :

Nucleon densities:

Continuum level density

Pairing correlation energy

BCS are good in the

vicinity of the stable

line.

Width effects are

large for nuclei far

from the stability line.

60 70 80 900

1

2

3

4

5

6

ER

MF-E

RM

F+

BC

S(M

eV

)

A

RMF+BCS RMF+BCSR RMF+BCSRW

Ni-isotopes

N=50

Cao, Ma , Eur. Phys. J. A 22 (2004)189

Quasi-particle RRPA

2

0

2 1

(4 )( , ; , '; ) ( )

2 1 ( ) ( )

( ( ) ) (1 )

L L L Lj jR

L

P Q P QP Q k k E A

L E E E i E E E i

A u v v u

Response function

Unperturbed polarization operator

BCS occupation prob.

Outside the pairing active space

Positive unoccu. states

occu. states

Negative states

2v 22 2 21 , n nu v E

2 2, 0, 1nE v u

2 2, 1, 0nE v u

2 2, 0, 1nE v u

1( , ; , '; ) Im ( , ; , '; )RR Q Q E Q Q E

k k k k

Ni-isotopes

64 68 72 76 80 84 88 92 96 100

3.5

4.0

4.5

5.0Ni-isotopes

R(f

m)

A

neutron proton

64 68 72 76 80 84 88 92 96 100-8.8

-8.4

-8.0

-7.6

-7.2

-6.8

EB/A

(MeV

)

A

Theory Exp.

64 68 72 76 80 84 88 92 96 1000.0

0.2

0.4

0.6

0.8

1.0

1.2

Ni-Isotopes

Rn-R

p(fm

)

A

Extended RMF+BCS

s.p. resonant states

2d5/2,2d3/2,1g7/2,2f7/2,1h11/2,

G=20.5/A MeV

Cao, Ma, Modern Phys. Lett. A19(2004)2845

IVGDR Ni-isotopes

0

2

4 70Ni

C

84Ni

0

2

4

72Ni

86Ni

0

2

4

74Ni88Ni

(e2 fm

2 MeV

-1)

IVG

DR

0

2

4 76Ni

Re

spo

nse

Fu

nct

ion 90Ni

0

2

4 78Ni

C

92Ni

0

2

4

E(MeV)E(MeV)

80Ni

94Ni

0 5 10 15 20 25 300

2

4

82Ni

0 5 10 15 20 25 30

96Ni

58Ni – 64Ni are stable

vibration of p-n

Ni-isotopes A=70~96

The response functions

of IVGDR in QRRPA

Loosely bound neutron coherently oscillate against the p-n core

EH ~ 16 MeV

low-lying dipole <10 MeV

IVGDR in Ni-isotopes

0.2 0.4 0.6 0.8 1.05

6

7

8

9

10E

L(MeV

)

rn-r

p

68 72 76 80 84 88 92 96 1004

6

8

10

EL(M

eV)

A

68 72 76 80 84 88 92 96 10014

16

18

20

EH(M

eV)

m1/m

0 GDR

E(A)=31.2A-1/3+20.6A-1/6

68 72 76 80 84 88 92 965

10

15

20

25

30

35

Per

cent

age(

%)

Cao, Ma, Modern Phys. Lett. A19(2004)2845

GDR

restoring force proportional to the symmetry energy

Linear dep. on the neutron skin

Experiments on GDR

Gibelin and Beaumel (Orsay), exp. at RIKEN

inelastic scattering of 26Ne + 208Pb

60 MeV/u 26Ne secondary beam

Dominated by Coulomb excitations

selective for E1 transitions.

Thesis of J. Gibelin IPNO-T-05-11

Future work:

28Ne + 208Pb

Theoretical investigation – practical significance0 5 10 15

10-6

10-5

10-4

10-3

10-2

10-1

Den

sity

(fm

-3)

r(fm)

p(26Ne)

n(26Ne)

p(28Ne)

n(28Ne)

Cao, Ma, PRC71(05)034305

Properties of 26,28Ne

Extended RMF+BCS with NL3

GQR check the validity of spherical assumption

26Ne, 28Ne IVGDR

0 5 10 15 20 25 30 350.0

0.5

1.0

R(e

2 fm2 M

eV

-1)I

VG

DR RRPA

QRRPA

26Ne

Cao, Ma, PRC71(05)034305

0 5 10 15 20 25 30 350.0

0.5

1.0

1.5

R(e

2 fm2 M

eV-1

)IV

GD

RE(MeV)

RRPA QRRPA

28Ne

Sum rule

Low-lying GDR in 26Ne exhaust about 4.9% of TRK sum rule

28Ne 5.8%

Comparisons of Low-lying dipole state in 26Ne

Authors Methods Shape Result

Elias(Orsay) SHF+BCS+ spherical 11.7 not coll.

QRPA(RF)

Cao, Ma(CIAE) RMF+BCS(R.) spherical 8.4(5%) coll.

PRC71(2005)034305 +QRRPA(RF)

Peru(CEA) def. HFB(Gogny) Spherical 10.7 coll.

+QRPA(Matrix)

Ring(TUM) def. RHB +QRRPA Deformed 7.9 9.3 less coll.

(Matrix)

Exp.(Gibelin,Beaumel) measure ? ~9(5%)

IPNO-T-05-11

Preliminary

Symmetry Energy and GDR

Restoring force of GDR

Symmetry energy in NM

All parameters give very good

description of g.s. properties,

NM saturations

Centroid energy of GDR

Ecen=m1/m0

Linear dep on the symmetry

energy at saturation energy

May give constraint:

33 MeV< asym(0)<37 MeV

32 33 34 35 36 37 38 39 40 41 42 4310

11

12

13

14

15

16

17

18

19

90Zr

144Sm

E(M

eV)

IVG

DR

S (0) (MeV)

208Pb

NLSHNL3 NL-BA

NLENLZ2

NLVT

Density dep. of symmetry energy

Non linear - coupling

Todd, Pickarewicz, PRC67(03)

Modify the poorly known density dep. of symmetry energy

Without changing the agreement with existing NM, g.s. properties

Softening of the symmetry energy

NL3 B/A=16.24MeV

asym=37.3 MeV

0=.148fm-3(kF=1.3fm-1)

K=272 MeV

asym=25.67 MeV at =.1fm-3(kF=1.15fm-1)

int ( )v g g L=

0.00 0.05 0.10 0.15 0.200

10

20

30

40

50

60

asy

m (

Me

V)

(fm-3)

V=0.0

V=0.015

V=0.025

Ground state properties in 132Sn

v B/A(MeV)

rp(fm) rn(fm) rn-rp 0 asym(MeV) asym(=0.

1)

0 8.320 4.643 4.989 0.346 0.148 37.34 25.68

0.015 8.340 4.647 4.949 0.302 0.148 34.05 25.68

0.025 8.346 4.653 4.926 0.273 0.148 32.41 25.68

Exp. 8.355

B/A, rp slightly changed

asym softened

rn-rp becomes small

3.0 3.5 4.0 4.5 5.0 5.5 6.0

-60

-50

-40

-30

-20

-10

0

n (M

eV)

Neutron rms radius (fm)

V=0.0

V=0.025

NL3

132Sn

Pygmy Resonance & Symmetry Energy

20 21 22 23 24 25 26 27 287.0

7.5

8.0

8.5

9.0

9.5

10.0

10.5

11.0

E(M

eV

) P

DR

S () (MeV) (at =0.1 fm-3)

132Sn

0 5 10 15 20 25 300

2

4

6

R (

e2 fm

4 Me

V-1)

IVG

DR

E (MeV)

v=0.025

v=0.015

v=0.0

132Sn

NL3

Epeak(Pygmy)=8.0 MeV above one n separation energy

Epeak(GDR)= 13.8, 14.0, 14.2 MeV Adrich et al. PRL95(05)132501

GDR : peak energy is shifted dep. on the symmetry energy at 0

Pygmy resonance is kept unchanged at 8.0 MeV

It may set up a constraint on the density dep. of symmetry energy

GSI

Summary

Theoretical investigations on Pygmy resonance in quasi-particle RPA

non-relativistic QRPA

relativistic approaches QRRPA

Pairing correlation is important, coupling to the continuum

Extended RMF+BCS

the s.p. resonance in the continuum including widths

GDR -- restoring force is proportional to the symmetry energy

systematic study

33 MeV < asym(0) < 37 MeV

New excitation modes in exotic nuclei

Pygmy modes are related to neutron skin and density dependence of

symmetry energy

Thanks Thanks