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.