Li Gang CaoIMP-CAS, Lanzhou

Collaborators: H. Sagawa(Aizu University) G. Colo(Milano University)

Microscopic study of ISGMR in Cd, Sn and Pb isotopes

From nucleon structure to nuclear structure and compact astrophysical objects , Beijing,

Kavli, 2012.6.11-29

The Nuclear equation of state

)6()()()(),( 44

22 OISISEI

A

ESNM

22 )()(),( ISEI

A

ESNM

)4(3632

)(3

0

00

2

0

000 O

QKEESNM

)( 00 SNMEE

0

2

2200

)(9

SNME

K

0

3

3300

)(27

SNME

Q

Nuclear structureHeavy ion collisionPhysics of neutron star

The incompressibility of nuclear matter K

The incompressibility of nuclear matter can not be measured directly, it can be deduced from the distribution Of ISGMR in heavy nuclei, such as 208Pb.

The incompressibility of nuclear matter K

The incompressibility of nuclear matter can not be measured directly, it can be deduced from the distribution Of ISGMR in heavy nuclei, such as 208Pb.

5 10 15 20 250.0

0.1

0.2

0.3

0.4

Fra

ctio

n E

0 E

WS

R/M

eV

E(MeV)

Exp. SKI3(258) SLy5(230) SKP(201)

208Pb

?K

MeVK 230

is found around 230 MeV by using non relativistic interaction, such as Skyrme and Gogny

In the case of relativistic interaction, It is around 250 MeV.

Both of them can produce very well the ISGMR energy in Pb208

Using different density dependent Skyrme force, Colo point out that it is possible to reproducing the ISGMR energy in Pb208 for some Skyrme interaction Which have 250 MeV of incompressibility.

Piekarewicz built non-linear relativistic Lagrangian with 230 MeV incompressibility, which can produce the ISGMR energy in Pb208. both non relativistic and relativistic.

Why Tin is so soft?

Or

Why pb is so hard?

New problem is appeared.

Some groups try to solve this problem by introducing a isospin dependent Incompressibility, they can get better description in Sn isotopes, but fails in Pb208. J. Piekarewicz, PRC79, 054311 (2009)

Since the Sn isotopes are open shell nuclei, the pairing shallPlay a role both in ground statesand excited states. Using the Constraint HF(HFB) and energy-weighted sum rule approach, one can get the constrained energy of ISGMR, it is foundthat pairing has an important effect in producing the ISGMR energies in Sn Isotopes. E. Khan PRC80, 011307(R)(2008).

Based on the HFB+QRPA calculation, the ISGMR energies in SnIsotopes are obtained using different Skyrme interaction, but There is No satisfied conclusion according to those calculationBecause the calculations are not fully self-consistent, such as The spin-orbit interaction is dropped . J. Li et.al.,PRC78,064304(2008)

Or the HF+BCS+QRPA(QTBA). The spin-orbit interaction is dropped. V. Tselyaev, PRC 79, 034309 (2009)

T. Sil, et.al., Phys. Rev. C73, 034316 (2006). The spin-orbit residual interaction produces an attractive effect on the ISGMR strength, the energies are pushed down by about 0.6MeV.

In this talk, I will present our recent work on ISGMR in Cd, Sn and Pb isotopes. It is based on the HF+BCS+QRPA.

)1(**

Y

XE

Y

X

AB

BA

mnijresj

mjinmnijim

resmnijimj

VijVmnB

V

inVmjA

min

min

)(

)(

Vmjin includes： full Skyrme force, spin-orbit, coulomb, and also the pairing in pp channel

The strength function is

n

nEEnFES )(ˆ0)(2

The various moments are defined as

dEESEm kk )(

1

3

0

1

1

1 ,,m

mE

m

mE

m

mE scencon

And various energies are defined as

equals to 1, 0.5,0 corresponding to surface, mixed, and volume Pairing.

MeVn 334.1

MeVn 321.1

MeVn 841.0

)()(

1),( 210

021 rrr

VrrVpair

Results for Pb isotopes

The so-called mutually-enhanced-magicity effect, which is proposed by Lunney and Zeldes.

E. Khan, Phys. Rev. C80, 057302 (2009).

MeVSKP

MeVSKM

MeVSLy

202

217

2305*

204 206 208

13.0

13.5

14.0

m1/

m0(

MeV

)

A

Exp. RCNP Exp. Lui. Exp. KVI filling approx. volume pairing surface pairing mixed pairing

Pb isotopes SLy5

(a)

204 206 208

13.0

13.5

14.0

m1/

m0(

MeV

)

A

Pb isotopes SkM*

(b)

204 206 208

13.0

13.5

14.0

m1/m

0(M

eV)

A

Pb isotopes SkP

(c)

Results for Pb isotopes

Results for Sn isotopes

MeVSKP

MeVSKM

MeVSLy

202

217

2305*

MeVSKP

MeVSKM

MeVSLy

202

217

2305*

110 112 114 116 118 120 122 124

15.0

15.5

16.0

16.5

m1/

m0(

MeV

)

A

Exp. RCNP Exp. T&M filling approx. volume pairing surface pairing mixed pairing

Sn isotopesSLy5

(a)

110 112 114 116 118 120 122 124

15.0

15.5

16.0

16.5

m1/

m0(

MeV

)

A

Sn isotopesSkM*

(b)

110 112 114 116 118 120 122 124

15.0

15.5

16.0

16.5

m1/

m0(

MeV

)

A

Sn isotopesSkP

(c)

MeVSKP

MeVSKM

MeVSLy

202

217

2305*

pairingmixed

QRPAHFBCS

Results for Cd isotopes

MeVSKP

MeVSKM

MeVSLy

202

217

2305*

104 106 108 110 112 114 116 11815.0

15.5

16.0

16.5

17.0m

1/m

0(M

eV

)

A

Exp. RCNP filling approx. volume pairing surface pairing mixed pairing

Cd isotopesSLy5

(a)

104 106 108 110 112 114 116 11815.0

15.5

16.0

16.5

17.0

m1/

m0(

MeV

)

A

Cd isotopesSkM*

(b)

104 106 108 110 112 114 116 11815.0

15.5

16.0

16.5

17.0

m1/

m0(

MeV

)

A

Cd isotopesSkP

(c)

We have studied the ISGMR in Cd, Sn and Pb isotopes based on the fully self-consistent HF+BCS plus QRPA calculations. The SLy5, SKM*, and SKP and different pairing interactions are used in our calculations.

We found that the pairing plays a role in producing the ISGMR properties.

The SLy5 interaction together with the effect of pairing can give better description on ISGMR both in Cd and Pb isotopes, but it still can not get better results in Sn isotopes, while SKM* can produce the experimental data in Cd and Sn isotopes, but fails in Pb isotopes, for SKP, it fails for all isotopes because the incompressibility is too low.

For future work, the calculations based on fully self-consistent HFB+QRPA may give more satisfied description, or other effect will be investigated, such as surface effect…….

Summary

Thank You!