Finite Nuclei and Nuclear Matter in Relativistic Hartree-Fock Approach

Long Wenhui 1,2, Nguyen Van Giai 2, Meng Jie 1

1School of Physics, Peking University, China2Institut de Physique Nucleaire, Universite Paris-Sud, France

Contents

Introduction and motivations Theoretical Framework Numerical Calculations Results and Discussions Summary

Introduction Relativistic Hartree-Fock (RHF)

• Without self-interactionsA. Bouyssy, J.-F. Mathiot, N. V. Giai, S. Marcos, Phys. Rev. C36-380(1987).

• With -meson self-interactions P. Bernardos, V. N. Fomenko, N. V. Giai et. al., Phys. Rev. C48-2665(1993).

• With zero-range self-interactionsS. Marcos, L. N. Savushkin, V. N. Fomenko et. al., arXiv: nucl-th/0307063.

Advantage of RHF approach• More fundamental theory• Nuclear structure: spin-orbit interaction

Motivations

RMF theory and RHF approach• Hartree Hartree-Fock• Contributions of -meson• Pairing force in RHB theory

Proposal:• The effective interactions in RHF approach

PK1: PHYSICAL REVIEW C 69, 034319 (2004)

• The contributions of -meson• Different nonlinear mechanism

Lagrangian and Hamiltonian Lagrangian Density

Hamiltonian Density

35

2 2 2 2

3 422 3

1

2

1 1 1 1 1 1

2 2 4 2 4 21 1 1 1 1

2 2 4 3 4

fi M g g g e

m

m m R R m

m F F

A

g g

L

3

2 2 2 2

3 422 3

1

2

1 1 1

2 2 21 1 1 1

2 2 3 4

i ii i

ii

fi g g g e A

m

m m R m

F A m g g

H

Hartree-Fock Approach 0

Hartree-Fock Trial State

Expectations (see -meson as representative)

Fierz transformation (n=2, 3, 4)

†0

† † † †0

iE t iE t

iE t iE t

x f e c g e d

x f e c g e d

x x

x x

†0 0c

0 0E H

, ,n

s b Tf Fierz Transformation

Radial Dirac Equation Dirac Equation

G and F separations

0

0

0

0

T S

T S

dG G E M F X

dr r

dF F E M G Y

dr r

2 2 2 2G FG F

W W W WG F G F

0

0

0

0

G FT S

F GT S

dX G E M X F

dr r

dY F E M Y G

dr r

PKA 938.5 586.707 11.576 13.537 3.184 0.0 -38.962 41.967

HF(e) 938.9 440.0 7.2302 11.2100 2.6290 1.0027 0.0 0.0

HFSI 939.0 412.0 7.0942 11.4320 2.6290 1.0027 -67.18 -14.61

ZRL1 939.0 497.8 8.3695 12.1420 2.6290 1.0027 -29.646 51.00

M mg g g f 410b 310c

6 8

2 3783.0, 770.0, 138.0, ,g g

m m m g bM g cm m

Tab. I Effective Interactions

HF(e): (, , and )HF

A. Bouyssy, J.-F. Mathiot, N. V. Giai, S. Marcos, Phys. Rev. C36-380(1987).

HFSI: (, , and )HF + self-interactions

P. Bernardos, V. N. Fomenko, N. V. Giai et. al., Phys. Rev. C48-2665(1993).

ZRL1: (, , and )HF + zero-range self-interactionsS. Marcos, L. N. Savushkin, V. N. Fomenko et. al., arXiv: nucl-th/0307063.

Observables Nuclear Matter: 0, K, EB, asym

Binding energies of the following nuclei:

16O, 40Ca, 48Ca, 56Ni, 68Ni, 90Zr, 116Sn, 132Sn, 182Pb, 194Pb, 208Pb

0 EB K asym M*

Empirical data 0.166±0.018 -16.±1. 240±50 32.±8.

PKA 0.150 -15.99 276.15 29.51 0.54

HF(e) 0.149 -16.40 465 28 0.56

HFSI 0.14 -15.75 250 35.0 0.61

ZRL1 0.155 -16.39 250 35.0 0.58

Tab. II Bulk Properties of Nuclear Matter

Tab. III Binding energies and charge radii

16O 40Ca 48Ca 56Ni 68Ni 90Zr 116Sn 132Sn 182Pb 194Pb 208Pb

127.6 342.1 416.0 484.0 590.4 783.8 988.7 1102.9 1411.7 1525.9 1636.4

PKA 127.7 342.2 416.2 473.6 586.3 784.6 984.4 1103.5 1412.2 1526.2 1635.9

HF(e) 89.8 272.8 340.8 666.0 1401.9

HFSI 118.9 333.2 405.6 772.2 1618.2

ZRL1 117.9 333.2 408.5 780.3 1632.8

2.730 3.478 3.479 4.270 4.625 5.442 5.504

PKA 2.732 3.444 3.550 3.847 3.930 4.304 4.650 4.750 5.392 5.461 5.502

HF(e) 2.73 3.47 3.47 4.26 5.50

HFSI 2.73 3.48 3.48 4.26 5.52

ZRL1 2.71 3.44 3.49 4.25 5.49

Fig.1 The binding Energies of Pb isotopes

184 188 192 196 200 204 208 212

-6

-4

-2

0

2

4

6

A

Eex

p. -

Eca

l. (M

eV)

PKA PK1 NL3

Fig. 2 Single Particle Energies of 132Sn

Fig. 3 Single Particle Energies of 208Pb

Fig.4 Charge density distributions

0 1 2 3 4 5 6 7 8 9 100.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 1 2 3 4 5 6 7 80.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 1 2 3 4 5 60.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0 1 2 3 4 5 60.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

0 1 2 3 4 50.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Exp. PKA PK1

208Pb

90Zr48Ca

40Ca

16O

Summary Programs for RHF approach are constructed New effective interaction PKA with -, -, -mesons

and nonlinear high order terms is obtained Better descriptions for nuclear matter and finite

nuclei are obtained. Perspective

• Isotopic shifts

• hard equation of state

• Contributions of -meson

• Density-dependent RHF

Thank you!

Sigma Field -meson field

Hamiltonian for -meson

4 ,x g d yD x y y y

2 3 4

0

1,

2 tH g x y D x y y x d xd y

2' '

; ' '

1,

2g f f D x y f f

H x y y x† †

' 'c c c c

0 0

', '

', '

D

E

EE H

E

2' '

; ' '

1,

2g f f D x y f f

H x y y x † †' 'c c c c

Multipole Expansion of the propagator

Potential Energy and Self-Energy

2 2 200

22 2 22

, 1 1 '0; 2 2

2 200

2 22

1 12 2

14 ' ' ' ; ',

2ˆˆ ˆ1

' ; ',02 4

' ' ' ; , '

ˆˆ

4

Ds s

Eq q Lr r

L

S s

q q

E g r r drdr r V m r r r

j j Lj j LE g drdr G G F F V m r r G G F F

g r dr r V m r r

j jj LX g

2

'0;

22 22

1 1 '0; 2 2

' ; ',0

ˆˆ' ; ',

04

LrL

q q LrL

Ldr G G F F V m r r F r

j j Lj LY g dr G G F F V m r r G r

1 12 2

0

1/ 2

1; , ' ; , ' '

4

1; , '

'

L L LL

L L L

eD r r V r r

V r r I r K rrr

Y Y r-r'

r - r'

-meson Pseudo-vector coupling

Exchange potential

5NN

f

m

L

2

2 21 2 1 2 1 22 2

1 13

3T c

fV

m m

V V

q q q q qq

q q

2 2

1 2 1 2 2 2

11

3c f m

Vm m

q

q

23

1 2 1 2 3

4

12

m rc m f e

Vm m m r

r r