17 November 2005 SPhN Saclay 1

The spin-orbit interaction and effective masses in mean field

models

17 November 2005 SPhN Saclay 2

Dirac equation and Schroedinger equation (1)

17 November 2005 SPhN Saclay 3

Dirac equation and Schroedinger equation (2)

17 November 2005 SPhN Saclay 4

p+40Ca at 497MeV (left) and 800 MeV (right)

17 November 2005 SPhN Saclay 5

DBHF (J. Rong, Z. Ma, NVG (2005))

17 November 2005 SPhN Saclay 6

Differential cross sections and analyzing powers

17 November 2005 SPhN Saclay 7

Spin rotation functions

17 November 2005 SPhN Saclay 8

17 November 2005 SPhN Saclay 9

Skyrme vs. RMF

17 November 2005 SPhN Saclay 10

S.O. potentials: Skyrme-HF vs. RMF

17 November 2005 SPhN Saclay 11

S.O. potentials in RMF (Na & Mg)

17 November 2005 SPhN Saclay 12

S.O. potentials in RMF (Ni)

17 November 2005 SPhN Saclay 13

S.O. potentials in RMF (Sn)

17 November 2005 SPhN Saclay 14

S.O. splittings in Ne and Mg

17 November 2005 SPhN Saclay 15

17 November 2005 SPhN Saclay 16

Density-dependent Relativistic Hartree-Fock(Wenhui Long, thesis, Orsay 2005)

• Lagrangian density

where

35

2 2 2 2

2

1

2

1 1 1 1 1 1

2 2 4 2 4 21 1 1

2 2 4

fi M g g g e

m

m m R R m

m F

A

F

L

, ,R F A A

17 November 2005 SPhN Saclay 17

Density-dependence of Coupling constantsFig. 4 Isovector channelsFig. 3 Isoscalar channels

Contributions from the exchange terms

Isospin properties and exchange terms

Density-dependences of -N and -N

17 November 2005 SPhN Saclay 18

Root mean square deviations from the data　 　 RHF1 RHF2 RHF3 PK1 PKDD NL3 DD-ME1

Eb

S.N. 1.8745 1.6177 2.0489 1.8825 2.3620 2.2506 2.7561

Pb 1.5797 1.8995 1.5627 2.0336 2.7007 2.0021 2.1491

Sn 2.3136 1.2665 1.5260 1.9552 2.4567 1.6551 0.9168

S2n

Pb 0.7264 0.6831 0.7262 0.9192 1.3139 0.9359 1.2191

Sn 1.0203 0.6813 0.5867 0.7762 1.0629 0.8463 0.7646

rc

S.N. 0.0299 0.0269 0.0225 0.0204 0.0188 0.0177 0.0163

Pb 0.0071 0.0056 0.0061 0.0061 0.0060 0.0143 0.0150

Iso. Pb 0.1122 0.0760 0.0790 0.0784 0.0784 0.0679 0.0567

S.O.

　 0.5434 0.3992 0.3859 0.5727 0.7802 0.6588 0.5635

S. N.: Selected Nuclei in Tab. 6

Iso. : Isotope shift of Charge radii in Pb isotopes

S. O.: Spin-Orbit splittings in 16O, 40Ca, 48Ca, 56Ni, 90Zr, 132Sn, 208Pb

17 November 2005 SPhN Saclay 19

The isotope shift of charge radii in Pb isotopes

17 November 2005 SPhN Saclay 20

Effective Mass(M. Jaminon and C. Mahaux, Phys. Rev. C 40:354, 1989)

• Scalar Mass:

• Effective Mass

S SM M

, ,E k T k E V k E

2

eff.

kdE dk

M

eff.

kdkM

dE

eff.

0ˆ ˆ1 V S

pM

d dp P d dp M d dp

RMF *eff. 0M E E

RMFeff. 01M M M

2 2. 0 0 0

1

2eq S S

EV

M M

17 November 2005 SPhN Saclay 21

What is the sign of M*(n)-M*(p) in Skyrme-HF?

17 November 2005 SPhN Saclay 22

Extended Brueckner-HF (Cao, Lombardo, Shen, NVG, 2005)

17 November 2005 SPhN Saclay 23

Isospin dependence of masses

DBHF: Zhong-Yu Ma, et al, Phys. Lett. B 604(2004)170.

17 November 2005 SPhN Saclay 24

Momentum dependence of masses

E. N. E. van Dalen, et al, arXiv: nucl-th/0502064

Filled symbols: Effective Mass

Open symbols: Scalar Mass (Dirac Mass)

17 November 2005 SPhN Saclay 25

Density-dependence of masses

Fig. 11 Effective mass as function of b

Fig. 12 Scalar mass as function of b

17 November 2005 SPhN Saclay 26

Effective Mass in 208Pb

Eff.2 2Eff.

,n pb b

a aa

MMG F dr

M M

02 2Eff. 1a aa

rMG F dr

M M

17 November 2005 SPhN Saclay 27

Summary

• The s.o. component of Skyrme forces is too restrictive. • It can be extended (like SkI4) if one work with energy functionals,

not forces.• The s.o. of RMF predicts a definite decrease of splittings for

increasing (N-Z).• The s.o. of DDRHF looks promising, more study needed.• Effective masses: EBHF and DBHF agree that M*(n)>M*(p) when N>Z- This prediction disagrees with RMF, agrees with DDRHF.

- As for Skyrme-HF it depends on the parametrization!