Effects of self-consistence violations in HF based RPA

calculations for giant resonances

Shalom Shlomo

Texas A&M University

Outline

1. Introduction

Definitions: nuclear matter incompressibility coefficient K

Background: isoscalar giant monopole resonance,

isoscalar giant dipole resonance

2. Theoretical approaches for giant resonances

Hartree-Fock plus Random Phase Approximation (RPA)

Comments: self-consistency ?

Relativistic mean field (RMF) plus RPA

ρ [fm-3]

ρ = 0.16 fm-3

E/A [MeV]

E/A = -16 MeV

The EOS is an important ingredient in the study of properties of nuclei, heavy-ion collisions, and in astrophysics (neutron stars, supernova).

The nuclear matter equation of state (EOS)

Hartree-Fock (HF)

Within the HF approximation: the ground state wave function

),,(...),,(),,(

),,(...),,(),,(

),,(...),,(),,(

!

1

21

22222222221

11111121111

AAAAAAAAAA

A

A

rrr

rrr

rrr

A

In spherical case

)(),()(

),,(

mjlmi rY

r

rRr i

HF equations: minimize totalHE ˆ

,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 NNijV

For the nucleon-nucleon interaction

Skyrme interaction

0,,, Wxt ii

are 10 Skyrme parameters.

.),( Coulij

NNijji VVrrV

The total energy

rdrHrHrH

VVTHE

SkyrmeCoulKinetic

NNij

Coulijtotal

)()()(

ˆ

),(2

)(2

)(22

rm

rm

rH nn

pp

Kinetic

'

'

)',('

'

)'()(

2)(

22

rdrr

rrrd

rr

rr

erH chch

chCoul

2

1)()()(

2

11)(

12

1

))()()()()()(2

1)(

16

1)()(

16

1

)()()()(2

11

2

113

16

1

)()(2

11

2

113

16

1)()()()(

2

11

2

11

4

1

)()(2

11

2

11

4

1)()(

2

11

2

1)(

2

11

2

1)(

322

32

3

02

221122

21

222211

222112211

221122

002

00

xrrrxrt

rJrrJrrJrWrJxtxtrJrJtt

rrrrxtxt

rrxtxtrrrrxtxt

rrxtxtrrxtrxtrH

np

nnppnp

nnpp

nnpp

npSkyrme

Where

)()( rr ),,(),,()(

1

rrr i

A

ii

A

ii rr

1

2),,()(

)()( rr

A

iii rrirJ

1 ,

''*

'

),,(),,()(

)()( rJrJ

',,

''* )2

1,,()

2

1,,()',(

iiich rrrr

.

Now we apply the variation principle to derive the Hartree-Fock equations. We minimize

totalHE ˆ

(*)0,

,

,,

,

i

i

ii

rdE

rdE

,

*

2

)()()()()()(2

rdrJrWrrUrrm

E

where

',

* ),',(),',(

i

ii rr

',

* ),',(),',()(

i

ii rrr

'',',

* "'),'',(),',()(

i

ii rrirJ

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)

In fully self-consistent calculations:

1. Assume a form for the Skyrme parametrization (δ-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 residual p-h interaction

4. Carry out RPA calculations of strength function, transition density etc.

''

2

''hpph

hphp

EV

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

)(),()(

Are mean-field RPA calculations fully self-consistent ?

NO ! In practice, one makes approximations.

A. Mean field and Vph determined independently → no information on K∞.

B. In HF-RPA one

1. neglects the Coulomb part in Vph;

2. neglects the two-body spin-orbit;

3. uses limited upper energy for s.p. states (e.g.: Eph(max) = 60 MeV);

4. introduces smearing parameters.

Main effects:

change in the moments of S(E), of the order of 0.5-1 MeV; note:

spurious state mixing in the ISGDR;

inaccuracy of transition densities.

KE

Commonly used scattering operators:

• for ISGMR

• for ISGDR

In fully self-consistent HF-RPA calculations the (T=0, L=1) spurious state (associated with the center-of-mass motion) appears at E=0 and no mixing (SSM) in the ISGDR occurs.

In practice SSM takes place and we have to correct for it.

Replace the ISGDR operator with

(prescriptions for η: discussion in the literature)

NUMERICS:

Rmax = 90 fm Δr = 0.1 fm (continuum RPA)

Ephmax ~ 500 MeV

ω1 – ω2 ≡ Experimental range

Self-consistent calculation within constrained HF

The steps involved in the relativistic mean field based RPA calculations are analogous to those for the non-relativistic HF-RPA approach. The nucleon-nucleon interaction is generated through the exchange of various effective mesons. An effective Lagrangian which represents a system of interacting nucleons looks like

It contains nucleons (ψ) with mass M; σ, ω, ρ mesons; the electromagnetic field; non linear self-interactions for the σ (and possibly ω) field.

Values of the parameters for the most widely used NL3 interaction are mσ=508.194 MeV, mω=782.501 MeV, mρ=763.000 MeV, gσ=10.217, gω=12.868, gρ=4.474, g2=-10.431 fm-1 and g3=-28.885 (in this case there is no self-interaction for the ω meson).

NL3: K∞=271.76 MeV, G.A.Lalazissis et al., PRC 55 (1997) 540.

RMF-RPA: J. Piekarewicz PRC 62 (2000) 051304; Z.Y. Ma et al., NPA 686 (2001) 173.

Relativistic Mean Field + Random Phase Approximation

Ephmax Ess E0 E1 E2

50 4.7 23.92 35.34 16.11

75 3.3 23.51 35.76 15.51

100 2.9 23.25 35.66 15.14

200 1.5 23.09 35.55 14.82

400 1.0 23.02 35.51 14.73

600 0.9 23.02 35.51 14.72

∞ 0.7 23.01 35.46 14.70

Dependence of the energy Ess of the spurious state (T=0, L=1) and the centroid energies EL of the isoscalar multipole giant resonances (L=0, 1, and 2), in MeV, on the value of Eph

max (in MeV) adopted in HF-discretized RPA calculation for 80Zr using a Skyrme interaction. The corresponding HF-Continuum RPA results are placed in the last row

Strength function for the spurious state and ISGDR calculated using a smearing parameter Г/2 = 1 MeV in CRPA. The transition strength S1, S3 and Sη correspond to the scattering operators f1, f3 and f η, respectively. The SSM caused due to long tail of spurious state is projected out using the operator f η

Isoscalar strength functions of 208Pb for L=0-3 multi-polarities are displayed. SC (full line) corresponds to the fully self-consistent calculation where LS (dashed line) and CO (dotted line) represent the calculations without the ph spin-orbit and the Coulomb interactions in the RPA, respectively. The Skyrme interaction SGII was used.

Isovector strength functions of 208Pb for L=0-3 multi-polarities are displayed. SC (full line) corresponds to the fully self-consistent calculation where LS (dashed line) and CO (dotted line) represent the calculations without the ph spin-orbit and the Coulomb interactions in the RPA, respectively. The Skyrme interaction SGII was used.

S. Shlomo and A.I. Sanzhur, Phys. Rev. C 65, 044310 (2002) ISGDR

MYrrrf 123

3

5

SL1 interaction, K∞=230 MeV

Eα = 240 MeV

)(3

5310 0

022 rdr

drrrcoll

Fully self-consistent HF-RPA results for ISGDR centroid energy (in MeV) with the Skyrme interaction 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, RCNP Osaka.

Nucleus ω1-ω2 Expt. NL3 SGII KDE0

90Zr 18-50 25.7±0.7 32. 28.8 29.1

26.7±0.5

26.9±0.7116Sn 18-45 23.0±0.6 29. 27.4 28.0

25.5±0.6

25.4±0.5144Sm 18-45 26.5 26.4 27.3

24.5±0.4

25.0±0.3208Pb 16-40 19.9±0.7 26. 24.1 24.7

22.2±0.5

22.7±0.2

K (MeV) 272 215 229

J (MeV) 37.4 26.8 33.0

Nucleus ω1-ω2 Expt. NL3 SK255 SGII KDE0

90Zr 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 16.6

144Sm 0-60 16.1 16.2 15.3 15.5

10-35 15.40±0.40 16.2 15.2 15.5

208Pb 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.

CONCLUSION

Fully self-consistent calculations of the ISGMR using Skyrme forces lead to K∞~ 230-240 MeV.

ISGDR: At high excitation energy, the maximum cross section for the ISGDR drops below the experimental sensitivity. There remain some problems in the experimental analysis.

It is possible to build bona fide Skyrme forces so that the incompressibility is close to the relativistic value.

Recent relativistic mean field (RMF) plus RPA: lower limit for K∞ equal to 250 MeV.

→ K∞ = 240 ± 20 MeV.

sensitivity to symmetry energy.