Lectures in IstanbulHiroyuki Sagawa, Univeristy of Aizu

June 30-July 4, 2008

• １ . Giant Resonances and Nuclear

Equation of States

• ２ . Pairing correlations in Nuclear Matter

　　 　 and 　 Nuclei

Giant Resonances and Nuclear Equation of States

Istanbul,Turkey, June 30, 2008

H. Sagawa, University of Aizu

1. Introduction

2. Incompressibility and ISGMR

3. Neutron Matter EOS and Neutron Skin Thickness

4. Isotope Dependence of ISGMR and symmetry term of Incompressibility

5. Summary

---nuclear structure from laboratory to stars----

IS mode IV(Isospin) mode

n p

Spin mode Spin-Isospin mode

pn

rYr ˆ22

Various excitation mode of finite nucleus (spin ｘ isospin x multipolarity)

IS monopole mode (compression mode)

pro

ton

s

neutrons

82

50

28

28

50

82

2082

28

20

126

Density F

unctional T

heory

self-

consistent M

ean Field

r-process

rp-p

roce

ss

Shell Model

Theory: roadmap

Ab

Initi

o

Three-body model

Nuclear matter

Neutron matter

A

Theoretical Mean Field Models

Skyrme HF model

Gogny HF model+tensor correlations

RMF model

RHF model

Many different parameter sets make possible to do systematic study of nuclear matter properties.

+pion-coupling, rho-tensor coupling

1 2 1 2 1 2

1 2

0 0 1 1

2 2 3 3

2 21 2

(effective zero-range force with density and momentum dependent terms) Skyrme interaction 11 12

11 16

sky r r r r

r r r

V r , t x t x

t x t

r P r r P k ' k

P k ' k x P

1 2 1 2

1 2

1 2 1 2

22

0 0 0 0

3 3 3 3

1 1 2

2

2 2 2

2

2 2

1 11 1

2 4 2

1 1 1 11

12 2 12 21

1 3 18

Hamiltonian density

n p n p n p n p

n p

k , k 'i i

r r

r r

t x t x

t x t x

t x t x

H ,m

-

i k ' kW

����

1 1 2 21 1 1

1 1+ derivative terms4 2 2

n n p p

n p p n Coult x t x H

Skyrme Hartree Fock (SHF) model

HFHF),( Skypn VTH

2

20 0

Hamiltonian density for infinite nuclear matter :

Isoscalar part Isovector part

1lim lim2

Physical properties of the infinite s

nm n p

nmnm

I I

H ,

Hh HI

0

0

0

Saturation density : Symmetry energy :

0

Saturation energy per nucleon : 1st derivative of :

( SHF ),

( RMF

ymmetric nuclear matter

sym

nm

nm

nm

nm

nm

nm

nm a

E

h

E

E

J

L

h

hM

J

222 2

2 2

)

Incompressibility : 2nd derivative of :

9 9

3

sym

nm

s

n

ym

nm m

K

K

K

Kh

L

n pI

0 1 2 2

0 3 1 2

22 3 5 3

22 3 1 5 3

22 3 1 5 3

2

22 3

0

3 1 2 2

Isoscalar part

3 10 3 5 4

5 8 163 1 3

3 5 410 8 16 80

3 9 31 3 5 4

5 16 8Isovector part

16 8

nmnm nm

nmnm n

/ /

/ /

/ /

/

m nm

nm nm nm

nm

c cm

c cm

c cm

cm

t t t x

t t t t x

t

tJ

t

E

K t x

0 0 3 3 1 1 2 2

0 0 3 3

1 1 2 2

3 3 1 1

1 5 3

22 3 1

5 3

22 5

23 1 3

2

1 11 2 1 2 3 4 5

48 243 1

1 2 1 2 16 8 165

3 4 524

3 51 2 1 3 4 5

3 16 12

/

/

/

/ /

nm nm nm

nmnm nm

nm

n nm nmm msy

x t x t x t x

t x t x

t x t x

t x t x t

L

c

c

K

c

x

m

c cm

Physical properties of the infinite nuclear matter by the parameters of Skyrme interaction

Lagrangian of RMF

1 SI 2 SIII 3 SIV

4 SVI 5 Skya 6 SkM

7 SkM* 8 SLy4 9 MSkA

10 SkI3 11 SkI4 12 SkX

13 SGII

Notation for the Skyrme interactions

14 NL3 15 NLC

16 NLSH 17 TM1

18 TM2 19 DD-ME1

20 DD-ME2

Notation for the RMF parameter sets

Parameter sets of SHF and relativistic mean field (RMF) model

SHF

RMF

Nuclear Matter

Nuclear Matter EOS

Incompressibility K

Isoscalar Giant Monopole Resonances

Isoscalar Compressional Dipole Resonances

Self consistent HF+RPA calculations

Self consistent RMF+RPA (TD Hatree) calculations,( , ) experimtent

Supernova Explosion

Self-consistent HF+RPA theory with Skyrme Interaction

1. Direct link between nuclear matter properties and collective

excitations 　　　　　　　

2. The coupling to the continuum is taken into account properly

by the Green’s function method.

3. The sum rule helps to know how much is the collectiveness 　 of obtained states.

4. Numerical accuracy will be checked also by the sum rules.

Tamm-Dancoff Approximation(TDA)Random Phase Approximation(RPA)

QQH ,

miim

miimim

mi aaYaaXQ ,,

0RPAQ

Y

X

Y

X

AB

BA

10

01**

p-h phonon operator

RPA equation

m

Fermi Energy

i

mnijnjmi

nijmijmnimnjmi

vB

vA~

~)(

(0) *

0 0

1 1G ( , '; ) ( ) ' ( ')i i

i occupied i i

r r r r r ri h i h

����������������������������������������������������������������� �����

where ( ) and are the HF single particle wave function and energyi ir

0 ( ) ( )i i ih r r

RPA Green’s Function Method

Unperturbed Green’s function

The inverse operator equation can be solved as

*

0

1' ( , ) ( ', ) ( , ')ljm ljm lj

ljmi

r r Y r Y r g r rh i

������������� �

where*

2

2 ( ) 1( , ') ( ) ( )

( , )lj

m rg r r u r v r

W u v

, ' if '

', if '

r r r r r r

r r r r r r

and

0

where ( ) is the regular (irregular) solution at the origin and

( ) behaves like a standing (outgoing) wave at infini .

) 0

t

(

y

i

uh

v

u v

u v

RPA (0) (0) RPA (0) 1 (0)(1 )v v

G G G G G G

Strength function2

0( ) | ( ) | 0 ( )nn

S E n f r E E E

*1 21 2

1( ) Im{G( , '; )} ( )dr dr f r r r f r

������������������������������������������������������������������������������������

RPA Green's function is then given by

IS monopole

)ˆ()( 02 rYrrf

(355MeV)

(217MeV)

(256MeV)

K=217MeV for SkM*

K=256MeV for SGI

K=355MeV for SIII

2 2 2AK ( ) /

mm r

(RPA)

Youngblood, Lui et al., （２００２）

(Gogny interaction)

What can we learn about neutron EOS from nuclear physics?

Nuclear Matter EOS

Incompressibility K

Isoscalar Monopole Giant Resonances

Isoscalar Compressional Dipole Resonances

Neutron surface thickness Pressure of neutron EOS

Size ~10fm Neutron star ~10km

size difference ~ 1810

K (230 10) MeV for Skyrme

(230 10) MeV for Gogny

(250 10) MeV for RMF

（ G. Colo ,2004 ）

(Lalazissis,2005)

H. Sagawa, University of Aizu

1. Introduction

2. Incompressibility and ISGMR

3. Neutron Matter EOS and Neutron Skin Thickness

4. Isotope Dependence of ISGMR and symmetry term of Incompressibility

5. Summary

---nuclear structure from laboratory to stars----

Giant Resonances and Nuclear Equation of States

Istanbul, 　 Turkey, 　 June 30, 2008

Lake Inawashiro

Mt. Bandai

UoA

Neutron Star MassesThe maximum mass and radii of neutron stars largely depend on the composition of the central core. Hyperons, as the strange members of the baryon octet, are likely to exist in high density nuclear matter. The presence of hyperons, as well as of a possible K-condensate, affects the limiting neutron star mass (maximum mass). Independent of the details, Glendenning found a maximum possible mass for neutron stars of only 1.5 solar masses (nucl-th/0009082; astro-ph/0106406).

                                                                                                                                                                                                                                                

Figure: Neutron stars are complex stellar objects with an interior                                                                                                                                                                                                                         

Figure: Neutron star masses for various binary systems, measured with relativistic timing effects. The upper 5 systems consist of a radio pulsar with a neutron star as companion, the lower systems of a radio pulsar with a White Dwarf as companion. All the masses seem to cluster around the value of 1.4 solar masses.

All these results seem to indicate that the presently measured masses are very close to the maximum possible mass. This could indicate that neutron stars are always formed close to the maximum mass.

J.M. Lattimer and M. Prakash, Sience 304 (2004)

0

10

20

30

40

50

0.00 0.10 0.20 0.30

n ( fm-3 )

Fig. 3

13 14 15 10 5 3

9

11

86712

20

2

1

4

AV14+3body

Neutron Matter

-10

0

10

20

30

40

50

0.00 0.10 0.20 0.30

n ( fm-3 )

Fig. 4

13 1414

15

10

359

11

8671220

2

1

4

0.05

0.10

0.15

0.20

0.25

0.30

-5.0 0.0 5.0 10.0 15.0 20.0

P ( n=0.2 fm-3 ) ( MeV fm-3 )

Fig. 5

(b)

4 1

212 8

76 11

93

5

10

1513

20

14

0.05

0.10

0.15

0.20

0.25

0.30

0.0 0.5 1.0 1.5 2.0

P ( n=0.1 fm-3 ) ( MeV fm-3 )

Fig. 5

(a)

41

212

20

8

11

7

96

3 5

10

13

15

14

2 2np n pr r

0.05

0.10

0.15

0.20

0.25

0.30

0.35

25 30 35 40

asym

( MeV )

Fig. 8

41

27

12 8

11 96

35

10

15

13

14

20

Volume symmetry energy J=asym as well as the neutron matter pressure acts to increase linearly the neutron surface thickness in finite nuclei.

J=

0.05

0.10

0.15

0.20

0.25

0.30

0.0 0.5 1.0 1.5 2.0

P ( n=0.1 fm-3 ) ( MeV fm-3 )

41

212

20

8

11

7

9

6

3 5

10

13

15

14

np

= 0.109 P + 0.0776

Fig. 3

Neutron skin of 208Pb and Pressure

GDR

(p,p)

Pigmy GDR

Sum Rule of Charge Exchange Spin Dipole Excitations

Polarized electron beam experiment at Jefferson Lab.

---- scheduled in summer 2008 ---

1ˆ [ ]lO r Y t

2 2

f f

ˆ ˆ = | | | |S S f O i f O i

2 22 1

4 n pN r Z r

Model independent observation of neutron skin

Electron scattering parity violation experiments

Polarized electron scattering (Jafferson laboratory)

* 0 boson interscts with neutronsZ

e e e e

More precise (p,p’) experiments (RCNP)

Future experiments

Multipole decomposition analysis

0-, 1-, 2-: inseparable

JxJphJx EaE ),(),( cm

calc

;cmexp

90Zr(n,p) angular dist.ω= 20 MeV

• NN interaction: t-matrix by Franey & Love• optical model parameters: Global optical potential (Cooper et al.)• one-body transition density: pure 1p-1h configurations

• n-particle 1g7/2, 2d5/2, 2d3/2, 1h11/2, 3s1/2

• p-hole

1g9/2, 2p1/2, 2p3/2, 1f5/2, 1f7/2

radial wave functions … W.S. / RPA

MDA

DWIA

DWIA inputs

3 ,2 ,1 ,0L)1 ,1( 1

2/92/7gg

)1 ,1( 1 2/92/9gg

Results of MDA for 90Zr(p,n) & (n,p) at 300 MeV(K.Yako et al.,PLB 615, 193 (2005))

• Multipole Decomposition (MD) Analyses– (p,n)/(n,p)　data　have　been　

analyzed　with　the　same　MD　technique

– (p,n)　data　have　been　re-analyzed　up　to　70　MeV　

• Results– (p,n)

• Almost L=0 for GTGR region(No Background)

• Fairly large L=1 strength up to 50 MeV excitation at

around (4-5)o

– (n,p)• L=1 strength up to 30MeV at around (4-5)o

L=0 L=1 L=2

Neutron skin thickness

pn

rZrNSS 22

4

9

222 fm 17207 pn

rZrN

fm 19.42 p

re scattering & proton form factor

fm 04.007.0 np

Neutron thickness

Sum rule value ⇒

method nucleus (fm) Ref.

p elastic scatt. 90Zr 0.09±0.07 Ray, PRC18(1978)1756

IVGDR by α scatt. 116,124Sn … ±0.12 Krasznahorkay, PRL66(1991)1287

SDR by (3He,t) 114--124Sn … ±0.07 Krasznahorkay, PRL82(1999)3216

Yako(2006) 90Zr 0.07±0.04

np

Isoscalar and Isovector nuclear matter properties and Giant Resonances

H. Sagawa, University of Aizu

1. Introduction

2. Incompressibility and ISGMR

3. Neutron Matter EOS and Neutron Skin Thickness

4. Isotope Dependence of ISGMR and symmetry term of Incompressibility

5. Summary

---nuclear structure from laboratory to stars----

Istanbul, Turkey, June 30, 2008

Isovector properties of energy density functional by extended Thomas-Fermi approximation

nmJ

1 SI 2 SIII 3 SIV

4 SVI 5 Skya 6 SkM

7 SkM* 8 SLy4 9 MSkA

10 SkI3 11 SkI4 12 SkX

13 SGII

Notation for the Skyrme interactions

14 NL3 15 NLC

16 NLSH 17 TM1

18 TM2 19 DD-ME1

20 DD-ME2

Notation for the RMF parameter sets

Parameter sets of SHF and relativistic mean field (RMF) model

Correlation among nuclear matter properties

2 2

3 3

1 SI 2 SIII 3 SIV

4 SVI 5 Skya 6 SkM

7 SkM* 8 SLy4 9 MSkA

10 SkI3 11 SkI4 12 SkX

13 SGII　　　

14 SGI

Notation for the Skyrme interactions

15 NLSH 16 NL3

17 NLC 18 TM1

19 TM2 20 DD-ME1

21 DD-ME2

Notation for the RMF parameter sets

Parameter sets of SHF and relativistic mean field (RMF) model

Correlation among nuclear matter properties

1. Nuclear incompressibility K is determined empirically to be K~230MeV(Skyrme,Gogny), K~250MeV(RMF).

2. A clear correlation between neutron skin thickness and neutron

matter EOS, and volume symmetry energy.

3. The pressure of RMF is higher than that of SHF in general.

4. Neutron skin thickness can be obtained by the sum rules of charge exchange SD and also spin monopole excitations.

5. The SD strength gives a critical information both on the neutron EOS and mean field models. 90Zr

6. is extracted from isotope dependence of ISGMR

7. J=(32+/-1)MeV, L=(60+/-5)MeV, Ksym= -(100+/-40)MeV

Summary

K (500 50)MeV

fm 04.007.0 np

S. Yoshida and H.S., Phys. Rev. C69, 024318 (2004), C73,024318(2006).

K. Yako, H.S. and H. Sakai, PRC74,051303(R) (2007).

H.S., S. Yoshida, G.M.Zeng, J.Z. Gu, X.Z. Zhang, PRC76,024301(2007).

Collaborators

Theory: Satoshi Yoshida, Guo-Mo Zeng, Jian-Zhong Gu,

Xi-Zhen Zhang

Experiment: Kentaro Yako, Hideyuki Sakai

Publications

-0.1

0.0

0.1

0.2

0.3

-0.5 0.0 0.5 1.0 1.5

32Mg38Ar44Ar100Sn132Sn138Ba182Pb208Pb214Pb

P ( n=0.1 fm-3 ) ( MeV fm-3 )

4

1

2 20

1211

8

9

7 6 3 5

10

Neutron skin and pressure

Fig. 5

4

164

252213

19

0

TZ

208Pb analysis

Rn – Rp = 0.18 ± 0.035 fm

∑Bpdr(E1)=1.98 e2 fm2

from N.Ryezayeva et al., PRL 89(2002)272501

∑Bgdr(E1)=60.8 e2 fm2 from A.Veyssiere et al.,NPA 159(1970)561

RQRPA-

N.Paa

r

RQRPA-

N.Paar

RQRPA-

N.Paa

r

LAND

C.Satlos et al. NPA 719(2003)304

A.Krasznahorkay et al.NPA 567(1994)521

C.J.Batty et al.Adv.Nucl.Phys. (1989)1

B.C. Clark et al. PRC 67(2003)044306