Transcript Collective Excitations in Unstable Nuclei and in Inner

Mean Field Methods for Nuclear
Structure
Nguyen Van Giai
Institut de Physique Nucléaire
Université Paris-Sud, Orsay
Part 1: Ground State Properties:
Hartree-Fock and Hartree-FockBogoliubov Approaches
Part 2: Nuclear Excitations: The
Random Phase Approximation
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Outline of part 1
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•
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- Introduction
- Non-relativistic energy density functional
- Densities and Potentials
- HF and HFB in spherical symmetry
- Illustrative examples
- Summary
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Microscopic approaches to many-body,
finite nuclear systems
• Theoretical models based on effective interactions
between nucleons:
- Nuclear shell model
- Mean field approaches (and beyond):
-Non-Relativistic (Skyrme forces, Gogny force)
-Relativistic (RMF,RHF)
- Molecular dynamics
• going away from stability regions, we need a theoretical
framework which can be predictive and able to handle
new situations (continuum, pairing correlations in
continuum).
• the Hartree-Fock + Random Phase Approximation (and
their extensions to include pairing effects) can be used
from unstable nuclei to neutron star crust.
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Hartree-Fock, and HF-Bogoliubov for
systems with pairing correlations
 
i1 i 2
HF

iA
a a .....a 0
HFB   a
†
i
i
(u
p
k
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†
i
ik
i
†
p'
p
   (u a  v a )
†
 vp a a ) 0
†
p
ik
i
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k HFB  0
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Energy Density Functional in
Hartree-Fock
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Effective Interaction: Skyrme force
Pairing channel:
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particle-hole
channel:
particleparticle
channel:
Skyrme
interaction
zero-range
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Densities
• Normal density, or
density matrix

ij

  a ai 
†
j
†



†


• Abnormal density, or
pairing tensor

  


a
a

j
i
ij
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T
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One-body densities in Hartree-Fock
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The Energy Density Functional
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The Skyrme-HF equations
Variations with respect to single-particle wave functions:
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The Skyrme-HF effective masses
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The Skyrme-HF central potentials
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The spin-orbit and Coulomb
potentials
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The center-of-mass correction
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Spherical case: radial equations in r-space
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Potentials
Densities
Effective masses
Spin-orbit
potentials
From: Bender et al.,
Revs.Mod.Phys.,
75, 121(2003)
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N-Z
Binding
Energy
Errors
A=N+Z
From: Bender et al.,
Revs.Mod.Phys.,
75, 121(2003)
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2-neutron separation energies
From: Bender et al.,
Revs.Mod.Phys.,
75, 121(2003)
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Single-particle
energies
From: Bender et al.,
Revs.Mod.Phys.,
75, 121(2003)
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r.m.s. radii
From: Bender et al.,
Revs.Mod.Phys.,
75, 121(2003)
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Generalization to Hartree-FockBogoliubov
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HFB densities in spherical case
• Nuclear density
• Abnormal (or pairing)
density
• Kinetic energy density
• Spin density
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The Hartree-Fock-Bogoliubov
Equations
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Hartree-Fock field and pairing field
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Finite-Temperature HFB
1
T 
4
DT(r) = Vpair T(r)
*
(
2
j

1
)
U
 i
i ( r )Vi ( r )(1  2 f i )
where :
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fi=(1+eEi/kT)-1
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Quasiparticle continuum
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Treatment of quasiparticle
continuum (1)
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Treatment of quasiparticle
continuum (2)
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Treatment of quasiparticle
continuum (3)
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Discretization by box boundary
condition
• Alternatively, one can enclose the system
in a box of radius R.
• The quasiparticle spectrum is calculated
with the boundary condition that the wave
function vanishes at r=R.
• One thus obtains a discrete set of states
forming a complete basis in the box.
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illustration: Ni isotopes
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E. Khan,
N. Sandulescu
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Inner Crust Matter
~ 0
~ 0.5
0
~ 0.0010
Crystal lattice structures
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Elementary cells
Wigner-Seitz cell
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Elementary cell
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Lattice
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Density in the Wigner-Seitz Cells
N.Sandulescu, Nguyen Van Giai,R.J.Liotta,
Phys.Rev.C69(2004)045802
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Pairing Field in the Wigner-Seitz Cells
N.Sandulescu, Phys.Rev.C70 (2004) 025801
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SUMMARY
• A self-consistent theory of nuclear ground
states.
• Pairing and continuum effects are treated.
• Applications to the description of unstable
nuclei.
• Applications to the physics of the inner
crust of neutron stars.
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Lectures on:
Mean Field Methods for Nuclear Structure
List of references for further reading
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1. P. Ring, P. Schuck, “The Nuclear Many-Body Problem”, Springer-Verlag (New York, 1980)
2. Hartree-Fock calculations with Skyrme’s interaction. I: spherical nuclei, D. Vautherin, D.M.
Brink, Phys. Rev. C 5, 626 (1972)
3. Hartree-Fock calculations with Skyrme’s interaction. II: axially deformed nuclei, D. Vautherin,
Phys. Rev. C 7, 296 (1973)
4. A Skyrme parametrization from subnuclear to neutron star densities, E. Chabanat, P. Bonche, P.
Haensel, J. Meyer, R. Schaeffer: Part I, Nucl. Phys. A 627, 710 (1997); Part II, Nucl. Phys. A 635,
231 (1998); Erratum to Part II, Nucl. Phys. A 643, 441 (1998)
5. Self-consistent mean-field models for nuclear structure, M. Bender, P.-H. Heenen, P.-G.
Reinhard, Revs. Mod. Phys. 75, 121 (2003)
6. Hartree-Fock-Bogoliubov description of nuclei near the neutron drip line, J. Dobaczewski, H.
Flocard, J. Treiner, Nucl.Phys. A 422, 103 (1984)
7. Mean-field description of ground state properties of drip line nuclei: pairing and continuum
effects, J. Dobaczewski, W. Nazarewicz, T.R. Werner, J.-F. Berger, C.R. Chinn, J. Dechargé,
Phys. Rev. C 53, 2809 (1996)
8. Pairing and continuum effects in nuclei close to the drip line, M. Grasso, N. Sandulescu, N. Van
Giai, R. Liotta, Phys. Rev. C 64, 064321 (2001)
9. Nuclear response functions, G.F. Bertsch, S.F. Tsai, Phys. Rep. 12 C (1975)
10. A self-consistent description of the giant resonances including the particle continuum, K.F. Liu,
N. Van Giai, Phys. Lett. B 65, 23 (1976)
11. Continuum quasiparticle random phase approximation and the time-dependent HFB approach,
E. Khan, N. Sandulescu, M. Grasso, N. Van Giai, Phys. Rev. C 66, 024309 (2002)
12. Self-Consistent Description of Multipole Strength in Exotic Nuclei I: Method, J. Terasaki, J.
Engel, M. Bender, J. Dobaczewski, W. Nazarewicz, M. Stoitsov, Phys. Rev. C 71, 034310 (2005)
13. Self-consistent description of multipole strength: systematic calculations, J. Terasaki, J. Engel,
ArXiv nucl-th/0603062
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