neutron skin and giant resonances
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NEUTRON SKIN AND GIANT RESONANCES
Shalom Shlomo
Cyclotron Institute
Texas A&M University
Outline
1. Introduction
Isovector giant dipole resonance,
Giant resonances (GR) and bulk properties of nuclei
2. Experimental and theoretical approaches for GR
Hadron excitation of giant resonances
Hartree-Fock plus Random Phase Approximation (RPA)
3. Density dependence of symmetry energy and neutron skin
A study within the Energy Density Functional Approach (EDF)
4. Giant resonances and symmetry energy density
ISGMR—Incompressibility and Symmetry energy
IVGDR and ISGMR in Ca isotopes
5. Nuclear + Coulomb excitations of GR and neutron skin
6. Conclusions
The total photoabsorption cross-section for 197Au, illustrating the absorption of photons on a giant resonating electric dipole state. The solid curve show a Breit-Wigner shape. (Bohr and Mottelson, Nuclear Structure, vol. 2, 1975).
The isovector giant dipole resonance
Macroscopic picture of giant resonances
L = 0 L = 1 L = 2
Theorists: calculate transition strength S(E) within HF-RPA using a simple scattering operator F ~ rLYLM:
Experimentalists: calculate cross sections within Distorted Wave Born Approximation (DWBA):
or using folding model.
Hadron excitation of giant resonances
Nucleusα
χi
χf
Ψi
Ψf
VαN
DWBA-Folding model description
,4 1,
22
A
ji ji
ijijCoulij
rr
eV
jiij
,))((
)(2
)1(6
1)()1(
])()()[1(2
1)()1(
0
3322
221100
ijjijiij
jiji
ijijjiijij
ijjijiijijjiijNN
krrkiW
rrrr
PxtkrrkPxt
krrrrkPxtrrPxtVij
�
�
�
we adopt the standard Skyrme type interaction NN
ijV
For the nucleon-nucleon interaction
Hartree-Fock with Skyrme interaction
0,,, Wxt ii
are 10 Skyrme parameters.
.),( Coulij
NNijji VVrrV
Carry out the minimization of energy, we obtain the HF equations:
)(
)()(4
3)1()1(
)(2
1)(
)()(2
)()1(
)()(2
*
2
'*
2
2"
*
2
rR
rRrWr
lljj
rmdr
d
rrU
rRrmdr
drR
r
llrR
rm
)()2
1()
2
1(
4
1)()
2
11()
2
11(
4
1
2)(222112211
2
*
2
rxtxtrxtxtmrm
,'
)'(')()(
2
1
)()2
1()
2
1(3
8
1)()
2
11()
2
11(3
8
1
)()2
1(
6
1)()()()
2
1(
12
)()2
11(
12
2)()
2
1()
2
1(
4
1
)()2
11()
2
11(
4
1)()
2
1()()
2
11()(
.20
22211
22211
33221
33
1332211
22110000
2
1,
rr
rrderJrJW
rxtxtrxtxt
rxtrrrxt
rxtrxtxt
rxtxtrxtrxtrU
ch
)(][8
1)()(
8
1)()(
2
1)( 2211210 rJxtxtrJttrrWrW
Hartree-Fock (HF) - Random Phase Approximation (RPA)
''
2 )(
hpphph
EV
1) Assume a form of Skyrme interaction ( - type).
2) Carry out HF calculations for ground states and determine the Skyrme parameters by a fit to binding energies and radii.
3) Determine the particle-hole interaction,
4) Carry out RPA calculations of the strength function, transition density, etc.
In fully self-consistent calculations:
Giant Resonance
In the Green’s Function formulation of RPA, one starts with the RPA-Green’s function which is given by
1)1( opho GVGG
where Vph is the particle-hole interaction and the free particle-hole Green’s function is defined as
)'(11
)(*),',( rrrr iioioi
io EhEhEG
where φi is the single-particle wave function, єi is the single-particle energy, and ho is the single-particle Hamiltonian.
We use the scattering operator F )(1
A
iifF r
obtain the strength function
)](Im[1
)(0)(2
fGfTrEEnFESn
n
and the transition density.
')],',(Im1
[)'()(
),( 3rrrrr dEGfEES
EEt
RPA
is consistent with the strength in RPA 2/EE
ErdrfErES RPA 2
)(),()(
The energy density functional is decomposed as
)(),()(),( 1 pcmsymmmpn Where ρn and ρp are the density distributions of neutrons and protons respectively, and
).()()( ),()()( 1 rrrrrr pnpnm
E.Friedman and S. Shlomo, Z. Phyzik, A283, 67 (1977)
Density dependence of symmetry energy and
Neutron skin within EDF
For the Coulomb energy density, εc, one usually uses the form
3/42 )]([)()(2
1)( rCerVre pcppc
where the first term is the direct Coulomb term with Vc(r) given by
.'')'('')'(1
4)(0
2
r
r
ppv drrrdrrrr
erV
For the symmetry energy density, εsym, we assume the form
.)]()[(4
1),( 2
111 rrVmsym
The interaction V1(r) is taken to be of the form
03/2
042/1
0321
011 /)/()/()/()( mmm aaaarV
where ρm(r) is the nuclear matter density distribution, ρ0=0.165 fm-3. In accordance with the semiemperical mass formula we impose the constraint
.)(
100)]()[(2
3211 A
ZNrdrrV
The terms with a2, a3, and a4 have been used previously in nuclear matter calculations and in applications of the EDF to finite nuclei.
Considering now the constraint
,)( 31 ZNrdr
We introduce a Lagrange multiplier λ and minimize
,)(),( 311 rdpcmsym
using δρm=δρp+δρn=0.
We obtain
,)(
1/)(
)(
1)( 3
111
rd
rVZNVrVe
rVr cc
with
,
)(
1/
)(
)( 3
1
3
1
rdrV
rdrV
rVV c
c
where Vc(r) and V1(r) are given by previous equations.
The EDF is not known for low density. Thus the variational equation for ρ1(r) must be used only in an internal region r<RM where RM is a prescribed matching radius. For r > RM the resulting ρn(r) and ρp(r) should be positive and decay exponentially with r. Taking RM=R, then for the internal region, r < R, we have )(/))()(())()((
)(
1)( 1
11
1 RVRZRNRVrVerV
r cc
where,
R R
pn drrrRZdrrrRN0 0
22 ,)(4)( ,)(4)(
R
rV
drrRV
0 1
21
1 ,)(
4)(
).(/)(
)(4)( 1
1
0 1
2
RVrV
drrrVRV
Rc
c
For the external region, r > R, we choose
rpp
rnn
pn er
Crer
Cr 22
1)( ,
1)(
where the coefficients C and γ are determined by imposing (i) the continuity of the densities and (ii) the total normalizations
0
2
0
2 .)(4 ,)(4 ZdrrrNdrrr pn
A surface enhancement parameter y is defined by
.21
1
aa
ay
Values of rn-rp
Parameterization calculations have been made for 48Ca and 208Pb using a parabolic Fermi for the proton distribution,
,)/)exp((1/1)(2
0 acrc
rr pp
with c=3.74 fm, a=0.53 fm and ω=-0.03, leading to rp = 3.482 fm for 48Ca,
and c=6.66 fm, a=0.50 fm and ω=0 leading to
rp = 5.483 fm for 208Pb
Giant Resonances and Symmetry Energy
ISGMR --Incompressibility and symmetry energy
ISGMR in Ca isotopes
IVGDR in Ca isotopes and symmetry energy
Nucleus ω1-ω2 Expt. NL3 SK255 SGII KDE090Zr 0-60 18.7 18.9 17.9 18.0
10-35 17.81±0.30 18.9 17.9 18.0116Sn 0-60 17.1 17.3 16.4 16.6
10-35 15.85±0.20 17.3 16.4 16.6144Sm 0-60 16.1 16.2 15.3 15.5
10-35 15.40±0.40 16.2 15.2 15.5208Pb 0-60 14.2 14.3 13.6 13.8
10-35 13.96±0.30 14.4 13.6 13.8
K (MeV) 272 255 215 229
J (MeV) 37.4 37.4 26.8 33.0
Fully self-consistent HF-RPA results for ISGMR centroid energy (in MeV) with the Skyrme interaction SK255, SGII and KDE0 and compared with the RRPA results using the NL3 interaction. Note the coressponding values of the nuclear matter incompressibility, K, and the symmetry energy , J, coefficients. ω1-ω2 is the range of excitation energy. The experimental data are from TAMU.
Nuclear and Coulomb Excitations of Giant Resonances
Neutron skin and nuclear excitation of IVGDR
by alpha (T=0) scattering
Interference between Nuclear and Coulomb
excitations of GR and neutron skin
2/)(
2/)()0(/)0(
ZNN
ZNZrr np
2/)( pn RR 03
1R
A
ZN
2/)(0 pn RRR
)0()0( , ,0 nppn ZNRR
)0()0( ,3
2 ,1 0 nppn R
A
ZNRR
Definitions: Assuming uniform density distributions
For:
lMMMi
iiiii Ydr
grgr *2/1
3
4)()(
dr
lMMMi
i Ydr
gr *2/1
tr 3
4)(
(exchange)')',()'(Tr tr rrrr dvU
')'1)(1(
4
1')',()',()',(
2
3310 rrrrrrrr
evvv
For Isovector Dipole (T=1, L=1) oscillations;
CoM: g = -Z/A for a neutron and N/A for a proton.
Transition density and transition potential are:
CNp UUU
lMMMnpnp
N Ydr
U
A
Z
r
U
A
N
r
U
A
Z
r
U
A
NU *
2/1
1100
3
4
For a proton projectile the transition potential is:
')',()'(
')',()'(
11
00
rrrr
rrrr
dvU
dvU
ii
ii
With
Note: Un and Up are of different geometry
rA
ZNR
A
ZNiii
000 62
)1(2
1
)0()0(0 pn
Expanding the ground state densities:
Where,
And
lMMMCC YdrFA
NU *
2/1
)(3
4
3
3
2
2
)()()(C
ccC R
rrRRr
r
zZerF
lMMMCNN YdrFA
NrFrFU *1
2/1
)()()(2
1
3
4
2
02
00
3)(
r
UR
r
U
A
ZNrFN
2
121
01
2
1
21
31)(
r
UR
r
U
A
ZN
r
U
A
ZNrFN
We obtain for a proton projrectile
MM
MM ccdd 1
110
* )1(
2/122/120 )2/()2/( xx NZmEAEd
lMMM
mM
CCCNNNNp
Ycc
rFRrFRrFRU
])1([
)3/1)](()()([
11
1
2/1110
10
For excitaion of IVGDR by a proton:
With
CCNN RN
ARR
210
10
2/1
2
2
2
4
xCC EmRAZ
N
lMM
MmM
CCCNN
Ycc
rFRrFRU
11
1
2/10
alpha
)(
)3/1()()(
2
02
00
3)(
r
UR
r
U
A
ZNrFN
2/)(
2/)()0(/)0(
ZNN
ZNZrr np
For excitaion of IVGDR by an alpha particle (T=0), adding
the contributions of the two neutron and two protons,
we have
Note that;
0000 )()()( YccrFrFU CCNN
r
UrUrFN
003)(
)()(2
3)( 22
3
2
rRrRR
zZerF CC
CC
2/1
20
2 1
23
20
xNN EAmR
For excitaion of ISGMR by an alpha particle;
CONCLUSIONS 1. Fully self-consistent HF-based RPA calculations of the
ISGMR lead to K = 210-250 MeV with uncertainty due to the uncertaint in the symetry energy density.
2. The neutron skin depends strongly on the density dependence of the symmetry energy.
3. The dependence of the centroid energy of the Isovector giant dipole resonance is clouded by the effects of (i) momentum dependence of the interaction (ii) the spin-orbit interaction.
4. Interference between Nuclear and Coulomb excitations of GR can be used to determine the depependence of neutron skin on N-Z.
5. Accurate determination of the magnitude of the neutron skin in neutron rich nuclei is very much need.
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