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The first systematic study of the ground-state properties of finite nuclei in the relativistic mean field model. Lisheng Geng. Research Center for Nuclear Physics, Osaka University School of Physics, Beijing University. Long-time collaborators. Hiroshi Toki - PowerPoint PPT Presentation
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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!
9
Introduction I: Nuclear masses
Up to 1988!
10
Introduction I: Nuclear masses
Up to 1994!
11
Introduction I: Nuclear masses
Up to 2004!
12
Introduction II: Charge radiusE. G. Nadjakov, At. Data Nucl. Data Tables 56 (1994)133-157
523
13
Introduction II: Charge radiusI. Angeli, At. Data Nucl. Data Tables 87 (2004)185-206
798
14
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
17
Introduction IV: Nuclear deformation
18
Introduction V: The odd-even effect and pairing correlation
19
Introduction V: The odd-even effect and pairing correlation
20
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!
24
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
27
The effect of deformation and pairing
Binding energy per nucleon of Zirconium isotopes
deformation
pairing
28
Extending RMF to incorporate the pairing correlation
From RMF to RMF+BCS
Total energy:
BCS equations:
Or gap equation
Occupation probability
RMF RMF +BCS
29
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!
30
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
31
Resonant states exist due to centrifugal barriers.
1g7/2 (0.55 MeV)
this barrier traps 1g7/2
32
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
33
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.
34
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
35
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
36
Quaqrupole-constrained calculation and the true ground-state
The potential energy surfaces of 14 N=116 isotones
Z=58
Z=64 Z=71
37
Z=64 Z=66
Z=68 Z=70
Model predictions: Binding energy per nucleon
38
Z=64 Z=66
Z=68 Z=70
Model predictions: Two neutron separation energy
39
Model predictions: Deformation
Z=64 Z=66
Z=68 Z=70
40
Model predictions: Charge radius
Z=64 Z=66
Z=68 Z=70
41
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!
42
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
43
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
44
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
45
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
46
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)
49
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)
50
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?
51
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
53
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.
54
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