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