The first systematic study of the ground-state properties of finite nuclei

in the relativistic mean field model

Lisheng GengResearch Center for Nuclear Physics, Osaka

University

School of Physics, Beijing University

2

Long-time collaborators

Jie Meng

School of physics, Beijing University

China

Hiroshi Toki

Research Center for Nuclear Physics

Osaka University, Japan

3

Outline

① A brief review of relevant experimental quantities: nuclear masses, charge radii, 2+ energies, deformations, odd-even effects

② Theoretical framework a. The relativistic mean field (RMF) modelb. The BCS methodc. Model parameters

③ The first systematic study of over 7000 nucleia. Comparison with experimental data and other theoretical predictionsb. The causes of some discrepancies: the not-well-constrained isovector channel

④ Summary and perspective

4Up to 1940!

Introduction I: Nuclear masses

5

Introduction I: Nuclear masses

Up to 1948!

6

Introduction I: Nuclear masses

Up to 1958!

7

Introduction I: Nuclear masses

Up to 1968!

8

Introduction I: Nuclear masses

Up to 1978!

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Introduction I: Nuclear masses

Up to 1988!

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Introduction I: Nuclear masses

Up to 1994!

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Introduction I: Nuclear masses

Up to 2004!

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Introduction II: Charge radiusE. G. Nadjakov, At. Data Nucl. Data Tables 56 (1994)133-157

523

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Introduction II: Charge radiusI. Angeli, At. Data Nucl. Data Tables 87 (2004)185-206

798

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Introduction II: Charge radius

15

Introduction III: The energy of the first excited 2+ stateS. Raman, At. Data Nucl. Data Tables 78 (2001)1-128

16

Introduction III: The energy of the first excited 2+ stateS. Raman, At. Data Nucl. Data Tables 78 (2001)1-128

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Introduction IV: Nuclear deformation

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Introduction V: The odd-even effect and pairing correlation

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Introduction V: The odd-even effect and pairing correlation

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1. The spin-oribit interaction: i.e. the magic number effect

2. The deformation effect: most nuclei are deformed except a few magic nuclei

3. The pairing correlation: important to describe open-shell nuclei and responsible for the very existences of drip line nuclei

The essential ingredients to build a nuclear structure model

21

It was found that without the spin-orbit interaction, only the first three magic numbers can be reproduced: 2, 8, 20

However, if one introduces by hand the so-called spin-orbit potential of the following form:

All the magic numbers come out correctly

Z=2, 8, 20, 28, 50,82 & N=2,8,20,28,50,82,126

Elementary theory of nuclear shell model, M. G. Mayer and J. Hans D. Jensen, 1956

no spin-orbit spin-orbit

Spin-orbit interaction in non-relativistic models

Therefore, in all non-relativistic nuclear structure

models, a similar form of spin-orbit potential has to be introduced by hand and adjusted to reproduce the experimentally observed magic number effects

22

The relativistic mean field (RMF) modelThe RMF model starts from the following Lagrangian density:

Dirac equation Klein-Gordon equation

Scalar and Vector potentials

23

Spin-orbit interaction in the RMF model For a spherical nucleus, the Dirac spinor has the following form:

Substitute it into the Dirac equation

one obtains the coupled one-order differential equations for the large and small components:

By eliminating the small component, one obtains a second-order differential equation for the large component, namely

spin-orbit interaction The scalar and vector potentials are of the order of several hundred MeV!

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1. Spin-orbit interaction

2. Nuclear matter saturation

3. Polarization (spin) observables in nuclear reaction

4. Study of high density and high temperature nuclear matter

5. Connection to QCD

6. Pseudospin symmetry

The necessity of a relativistic model

“The atomic nucleus as a Relativistic system”, L. N. Savushkin and H. Toki, springer, 2005

25

Non-relativistic calculations: not successful.

Coester band

Empirical saturation

non-relativisticrelativistic Bruckner Hartree-Fock

Relativistic Bruckner Hartree-Fock calculations: encouraging!

relativistic mean field theory

The RMF model: parameterized to describe the nuclear matter saturation.

Nuclear matter saturation

26

The basis expansion method: Treating the deformationThe Dirac wave functions can be expanded by the eigen-functions

of an axially-symmetric harmonic oscillator potential

more specifically

Therefore, solving the Dirac equation is transformed to diagonalizing the following matrix

The meson fields can be treated similarly

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The effect of deformation and pairing

Binding energy per nucleon of Zirconium isotopes

deformation

pairing

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Extending RMF to incorporate the pairing correlation

From RMF to RMF+BCS

Total energy:

BCS equations:

Or gap equation

Occupation probability

RMF RMF +BCS

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But for weakly bound nuclei, which are the subjects of present research, it fails.

The pairing correlation in weakly bound nuclei The constant-gap BCS method: very successful for

stable nuclei

A zero-range delta force in the particle-particle channel is found to be useful!

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The pairing correlation in weakly bound nuclei

Yadav and Toki, Mod. Phys. Lett A 17 (2002) 2523

2d3/2

0.59

0.57

0.55

-0.56

1g7/2

4s1/23p3/2

S.P.E [MeV]

The state dependent BCS method can describe weakly bound nuclei properly

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Resonant states exist due to centrifugal barriers.

1g7/2 (0.55 MeV)

this barrier traps 1g7/2

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The state-dependent BCS method: extremely important!!!!

Self-consistent description of spin-orbit interaction: RMF (1980-)

Deformation effect: basis expansion method (1990-)

Proper pairing correlation: state-dependent BCS method (2002-)

Spherical case: Yadav and Toki, MPLA (2002)

Sandulescu, Geng and Toki, PRC (2003)

Deformed case: Geng and Toki, PTP (2003), NPA (2004)

The advantage of the state-dependent BCS method:

1. Effective: valid for all nuclei

2. Numerically simple: systematic study possible

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Stars: twelve nuclei used in DD-ME2

Nuclear matter: saturation density,

binding energy per nucleon

symmetry energy

compression modulus

Finite nuclei: Binding energy

Charge radius

Model parameters of the mean-field channel Free parameters in the RMF model: the sigma meson mass, the sigma-nucle

on, omega-nucleon, rho-nucleon couplings, the sigma non-linear self couplings (2) and the omega non-linear self coupling. In total, there are 7 parameters.

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The effective force TMA:

Parameter values of TMA

Saturation properties of SNM

To describe simultaneously both light and heavy nuclei

To simulate the nuclear surface effect

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The pairing strength and the cutoff energy are determined by fitting experimental

one- and two-nucleon separation energies of a large number of nuclei!

Model parameters of the pairing channel

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Quaqrupole-constrained calculation and the true ground-state

The potential energy surfaces of 14 N=116 isotones

Z=58

Z=64 Z=71

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Z=64 Z=66

Z=68 Z=70

Model predictions: Binding energy per nucleon

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Z=64 Z=66

Z=68 Z=70

Model predictions: Two neutron separation energy

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Model predictions: Deformation

Z=64 Z=66

Z=68 Z=70

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Model predictions: Charge radius

Z=64 Z=66

Z=68 Z=70

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First, we want to construct a mass table for all the nuclei throughout the periodic table, which could be used in astrophysical studies and could be compared with other non-relativistic predictions.

Second, for those nuclei that we have experimental data, we want to know, to what extent, the RMF+BCS model can describe them.

Finally, through such a study, we hope to know the limitations of the current RMF model and how to further improve it.

The first systematic study: Motivation

current RMF?

current RMF?

Nature!

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The first systematic study: Statistics

The pairing correlation properly treated: the state-dependent BCS method

Axial degree of freedom included: Quadrupole constrained calculation performed for each nucleus, i.e. the potential energy surface of each of the 6969 nuclei is obtained, to ensure that the absolute energy minimum is reached.

The blocking of nuclei with odd numbers of nucleons properly treated

6969 nuclei, even and odd, compared to two previous works

Hirata@1997, about 2000 even-even, no pairing;

Lalazissis@1999, about 1000 even-even, the constant gap BCS method

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The sigma is about 2.1 MeV--a small deviation compared to the nuclear mass of the order of several hundred or thousand MeV.

Somewhat inferior to FRDM and HFB-2.

about 10 free parameters (FRDM 30, HFB-2 20)

only 10 nuclei to fit our parameters (FRDM 1000, HFB-2 2000).

In this sense, the predictions of FRDM and HFB-2 are not really predictions.

Nuclear mass: theory vs. experiment Experimental data divided into three groups:

Group I: experimental error not limited, 2882 nuclei

Group II: experimental error less than 0.2 MeV, 2157 nuclei

Group III: experimental error less than 0.1 MeV, 1960 nuclei

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Experimental data divided into four groups:

Group I: experimental error not limited, 2790 nuclei

Group II: experimental error less than 0.2 MeV, 1994 nuclei

Group III: experimental error less than 0.1 MeV, 1767 nuclei

Group IV: experimental error less than 0.02 MeV, 1767 nuclei

One-neutron separation energy: theory vs. experiment

Our results become comparable to those of FRDM and HFB-2 for one-neutron separation energies, which are more important in studies of nuclear structure

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Most deviations are in the range of minus 2.5 MeV and plus 2.5 MeV

The largest overbinding is seen around (82,58) and (126,92)

Underbindings are observed in several regions, which might indicate possible shape coexistence, i.e. occurrence of triaxial degree of freedom.

Nuclear mass: theory vs. experiment

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Nuclear mass: How about other effective forces? NL3

G. A. Lalazissis, S. Raman, and P. Ring, At. Data Nucl. Data Tables 71 (1999)1-40

TMA

NL3

47

Z=58

Z=92

Strongly deformed prolate and oblate shapes coexist in 8<Z<20 and 28<Z<50 regions.

Anomalies seen at Z=92 and Z=58: many nuclei with these proton numbers are spherical

Nuclear deformation: Theoretical predictions

48 523 0.037 0.045 0.028

Z RMF FRDM HFB

The rms deviation for 523 nuclei over 42 isotopic chains is only 0.037 fm!

Nuclear charge radii: theory vs. experiment (I)

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The agreement is particularly good for Z between 40 and 70

Nuclei with less protons are generally underestimated.

Nuclei with more protons are generally overestimated.

Nuclear charge radii: theory vs. experiment (II)

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The shell closure at Z=58 is comparable to that at Z=50, and the shell closure at Z=92 is even larger that that Z=82.

Therefore, we conclude that the spurious shell closures at Z=58 and Z=92 are the reasons behind the observed anomalies

Discrepancies at (82,58) and (92,126): Spurious shell closures?

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The shell closure at Z=92

The shell closure at Z=58

The not well constrained isovector channel in the RMF model

Z=92

Z=58

Overestimated neutron shell closure, underestimated proton shell closure

Overestimated neutron shell closure, underestimated proton shell closure

The present RMF model does not constrain the isovector channel very well!

52

Underbindings: Missing triaxial degree of freedom?

H. Toki and A. Faessler, Nucl. Phys. A 253(1975)231-152

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Summary

1.We have built a theoretical model which is valid for all nuclei.

2.Using this model, we have conducted the first systematic study of over 7000 nuclei from the proton drip line to the neutron drip line.

3.Extensive applications of our model to various regions demonstrate that our model is very good in all respects.

4.Further improvement of the current formulation is expected.

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Future works:Further improvements

New effective forces

Spurious shell closures at Z=58 and Z=92

Triaxial degree of freedom

Higher order correlations

Shell-model like approach treatment of the pairing correlation

Angular momentum projection

The residual proton and neutron pairing

Numerical methodsThe basis expansion method with the woods-saxon basis

New mechanism

The effect of the Dirac sea

The contribution of pions