Isospin dependence and effective forces of the Relativistic Mean Field

Model

Isospin dependence and effective forces of the Relativistic Mean Field

ModelGeorgios A. Lalazissis

Aristotle University of Thessaloniki, Greece

Georgios A. LalazissisAristotle University of Thessaloniki, Greece

Collaborators: Collaborators: T. Niksic (Zagreb), N. Paar (Darmstadt), T. Niksic (Zagreb), N. Paar (Darmstadt), P. Ring (Munich), D. Vretenar (Zagreb)P. Ring (Munich), D. Vretenar (Zagreb)

Need for improved isovector channel ofthe effective nuclear interaction.

EOS of asymmetric nuclear matter and neutron matter

Structure and stability of exotic nuclei with extremeproton/neutron asymmetries

Formation of neutron skin and halo structures

Isoscalar and isovectordeformations Mapping the drip-lines

Evolution of shell structure Structure of superheavy elements

Covariant density functional theory:Covariant density functional theory:Covariant density functional theory:Covariant density functional theory:

system of Dirac nucleons coupled by the exchange mesons and the photon field through an effective Lagrangian.

(J,T)=(0+,0) (J,T)=(1-,0) (J,T)=(1-,1)

Sigma-meson: attractive scalar field:

Omega-meson: short-range repulsive

Rho-meson:isovector field

)()( rr gS )()()()( rrrr eAggV

Covariant density functional theory Covariant density functional theory

A

iii

1

)()(),(ˆ r'rr'r

Dirac operator:

)(

)(

r

r

i

ii g

f

No sea approximation: i runs over all states in the Fermi sea

..)()()()(ˆ ),(ˆˆˆ rαVrrrrpα VSmh

)(

)(ˆSVm

SVmh

αp

αp

EFFECTIVE INTERACTIONSEFFECTIVE INTERACTIONS (NL1, NL2, NL3, NL-Z2, …) (NL1, NL2, NL3, NL-Z2, …)

model parameters: meson masses m, m, m, meson-nucleon coupling constants g, g, g, nonlinear self-interactions coupling constants g2, g3, ...

The parameters are determined from properties of nuclear matter (symmetric and asymmetric) and bulk properties of finite nuclei (binding energies, charge radii, neutron radii, surface thickeness ...)

Effective density dependence Effective density dependence

through a non-linear potential: Boguta and Bodmer, NPA. 431, 3408 (1977)NL1,NL3,TM1..

43

32

2222

4

1

3

1

2

1)(

2

1 ggmUm 4

33

22222

4

1

3

1

2

1)(

2

1 ggmUm

through density dependent coupling constants:

T.W.,DD-ME..Here, the meson-nucleon couplings

)(),(),(,, gggggg )(),(),(,, gggggg

are replaced by functions depending on the density r)

number of param.

How many parameters ?How many parameters ?

symmetric nuclear matter: E/A, ρ0

finite nuclei (N=Z):

E/A, radii

spinorbit for free

m

g

m

g

m

Coulomb (N≠Z): a4

m

g

density dependence: T=0 K∞

7 parameters

rn - rpT=1

g2 g3

aρ

One- and two-neutron separation energies

surface thicknesssurface diffuseness

Neutron densities

groundstates of Ni-SnGround states of Ni and Sn isotopesGround states of Ni and Sn isotopes

combination of the NL3 effective interaction for the RMF Lagrangian, and the Gogny interaction with the parameter set D1S in the pairing channel.

G.L., Vretenar, Ring, Phys. Rev. C57, 2294 (1998)

RHB description of neutron rich N=28 nuclei. NL3+D1S effective interaction.

Strong suppression of the spherical N=28 shell gap.

1f7/2 -> fp core breaking Shape coexistence

G.L., Vretenar, Ring, Stoitsov, Robledo, Phys. Rev. C60, 014310 (1999)

Ground-state quadrupole deformation

Average neutronpairing gaps

Shape coexistence in the N=28 regionShape coexistence in the N=28 region

Neutron single-particle levels for 42Si, 44S, and 46Ar against of the deformation. The energies in the canonical basis correspond to qround-state RHB solutions with constrained quadrupole deformation.

Total binding energy curves

SHAPE COEXISTENCE

Evolution of the shell structure, shell gaps and magicity with neutron number!Evolution of the shell structure, shell gaps and magicity with neutron number!

Ground-state proton emitters

Self-consistent RHB calculations -> separation energies, quadrupole deformations, odd-proton orbitals, spectroscopic factors

G.L., Vretenar, RingPhys.Rev. C60, 051302 (1999)

Proton emitters I

characterized by exotic ground-state decay modes such as the direct emission of charged particles and -decays with large Q-values.

Vretenar, G.L., Ring, Phys.Rev.Lett. 82, 4595 (1999)

Nuclei at the proton drip line:Nuclei at the proton drip line:

How far is the proton-drip line from the experimentallyknown superheavy nuclei?

G.L. Vretenar, Ring, PRC 59 (2004) 017301

Proton drip-line in the sub-Uranium region Proton drip-line in the sub-Uranium region

Possible ground-state protonemitters in this mass region?

Proton drip-line for super-heavy elements: Proton drip-line for super-heavy elements:

Pygmy: 208-PbPaar et al, Phys. Rev. C63, 047301 (2001)

Exp GDR at 13.3 MeV

Exp PYGMY centroid at 7.37 MeV

In heavier nuclei low-lying dipole states appear that are characterized by a more distributed structure of the RQRPA amplitude.Among several single-particle transitions, a single collective dipole state is

found below 10 MeV and its amplitude represents a coherent superposition of many neutron particle-hole configurations.

208Pb

208Pb 208Pb

Neutron radiiNeutron radii

RHB/NL3RHB/NL3

NaNa SnSn

ME2ME2

2. 2. MODELS WITH DENSITY-DEPENDENTMODELS WITH DENSITY-DEPENDENTMESON-NUCLEON COUPLINGSMESON-NUCLEON COUPLINGS

2. 2. MODELS WITH DENSITY-DEPENDENTMODELS WITH DENSITY-DEPENDENTMESON-NUCLEON COUPLINGSMESON-NUCLEON COUPLINGS

A. THE LAGRANGIANA. THE LAGRANGIAN

B. B. DENSITY DEPENDENCE OF THEDENSITY DEPENDENCE OF THE COUPLINGSCOUPLINGS

the meson-nucleon couplings g, g, g -> functions of Lorentz-scalar bilinear forms of the nucleon operators. The simplest choice:

a) functions of the vector density

b) functions of the scalar density

PARAMETRIZATION OF THE DENSITY DEPENDENCEPARAMETRIZATION OF THE DENSITY DEPENDENCE

MICROSCOPIC: Dirac-Brueckner calculations of nucleon self-energies in symmetric and asymmetric nuclear matter g

PHENOMENOLOGICAL:

S.Typel and H.H.Wolter, NPA 656, 331 (1999) Niksic, Vretenar, Finelli, Ring, PRC 66, 024306 (2002)

g()

g()

g()

saturation density

Fit: DD-ME2Nuclei used in the fit for DD-ME2Nuclei used in the fit for DD-ME2

(%)

(%)

Nuclear matter: E/A=-16 MeV (5%), o=1,53 fm-1 (10%)

K = 250 MeV (10%), a4 = 33 MeV (10%)

Neutron Matter

DD-ME2 DD-ME1 TW-99 NL3 NL3*

ρο (fm-3) 0.152 0.152 0.153 0.149 0.150

Ε/Α (MeV) -16.14 -16.20 -16.25 -16.25 -16.31

K (MeV) 250.89 244.5 240.0 271.8 258.5

J (MeV) 32.3 33.1 32.5 37.9 38.3

m*/m 0.572 0.578 0.556 0.60 0.595

Nuclear Matter PropertiesNuclear Matter Properties

Masses: 900 keV

rms-deviations: masses: m = 900 keV radii: r = 0.015 fmrms-deviations: masses: m = 900 keV radii: r = 0.015 fmG.L., Niksic, Vretenar, Ring, PRC 71, 024312 (2005)

DD-ME2

SH-Elements

DD-ME2 Exp: Yu.Ts.Oganessian et al, PRC 69, 021601(R) (2004)

Superheavy Elements: Q-valuesSuperheavy Elements: Q-values

IS-GMRIsoscalar Giant Monopole: IS-GMRIsoscalar Giant Monopole: IS-GMR

The ISGMR represents the essential source of

experimental information on the nuclear

incompressibility

constraining the nuclear matter compressibility

RMF models reproduce the experimental data only if

250 MeV K0 270 MeV

Blaizot-concept:

T. Niksic et al., PRC 66 (2002) 024306

IV-GDRIsovector Giant Dipole: IV-GDRIsovector Giant Dipole: IV-GDR

the IV-GDR represents one of the sources of experimental informations on the nuclear matter symmetry energy

constraining the nuclear matter symmetry energy

32 MeV a4 36 MeV

the position of IV-GDR isreproduced if

T. Niksic et al., PRC 66 (2002) 024306

saturation density

LombardoLombardo

Relativistic (Q)RPA calculations of giant resonances

Isoscalar monopole response

Sn isotopes: DD-ME2 effectiveinteraction + Gogny pairing

Conclusions:Conclusions:

- Covariant Density Functional Theory provides Covariant Density Functional Theory provides a unified description of properties for ground a unified description of properties for ground states and excited states all over the periodic states and excited states all over the periodic tabletable

- The present functionals have 7-8 parameters.The present functionals have 7-8 parameters.- The density dependence (DD) is crucial:The density dependence (DD) is crucial: NL3 is has only DD in the T=0 channelNL3 is has only DD in the T=0 channel DD-ME1,… have also DD in the T=1 channelDD-ME1,… have also DD in the T=1 channel better neutron radiibetter neutron radii better neutron EOSbetter neutron EOS better symmetry energybetter symmetry energy consistent description of GDR and GMRconsistent description of GDR and GMR