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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-Fock-Bogoliubov Approaches Part 2: Nuclear Excitations: The Random Phase Approximation. Nguyen Van Giai.

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Mean field methods for nuclear structure

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-Fock-Bogoliubov Approaches

Part 2: Nuclear Excitations: The Random Phase Approximation

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The random phase approximation in nuclear physics
The Random Phase Approximation in Nuclear Physics

  • Linear response theory: a brief reminder

  • Non-relativistic RPA (Skyrme)

  • Relativistic RPA (RMF)

  • Extension to QRPA

  • Beyond RPA .

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Linear response theory
Linear Response Theory

  • In the presence of a time-dependent external field, the response of the system reveals the characteristics of the eigenmodes.

  • In the limit of a weak perturbing field, the linear response is simply related to the exact two-body Green’s function.

  • The RPA provides an approximation scheme to calculate the two-body Green’s function. .

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.

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First order response as a function of time
First order response as a function of time

.

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Two body green s function and density density correlation function
Two-body Green’s Function and density-density correlation function

.

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Linear response function and strength distribution
Linear response function and Strength distribution function

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Main results
Main results: function

  • The knowledge of the retarded Green’s function gives access to:

  • Excitation energies of eigenmodes (the poles)

  • Transition probabilities (residues of the response function)

  • Transition densities (or form factors), transition currents, etc… of each excited state .

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Tdhf and rpa 1
TDHF and RPA (1) function

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Tdhf and rpa 2
TDHF and RPA (2) function

And by comparing with p.6

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Residual p h interaction
Residual p-h interaction function

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Analytic summation of single particle continuum
Analytic summation of single-particle continuum function

1) u, w are regular and irregular solutions satisfying appropriate asymptotic conditions

2) This analytic summation is not possible if potential U is non-local .

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Transition densities and divergence of transition currents
Transition densities and divergence of transition currents function

Solid: GQR

Dotted: empirical

Dashed: low-lying 2+

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Convection current distributions
Convection current distributions function

GQR in 208Pb

Low-lying 2+ in 208Pb

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Finite temperature
Finite temperature function

Applications: evolution of escape widths and Landau

damping of IVGDR with temperature .

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Rpa on a p h basis
RPA on a p-h basis function

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A and b matrices
A and B matrices function

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Restoration of symmetries
Restoration of symmetries function

  • Many symmetries are broken by the HF mean-field approximation: translational invariance, isospin symmetry, particle number in the case of HFB, etc…

  • If RPA is performed consistently, each broken symmetry gives an RPA (or QRPA) state at zero energy (the spurious state)

  • The spurious state is thus automatically decoupled from the physical RPA excitations

  • This is not the case in phenomenological RPA .

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Sum rules
Sum rules function

  • For odd k, RPA sum rules can be calculated from HF, without performing a detailed RPA calculation.

  • k=1: Thouless theorem

  • k=-1: Constrained HF

  • k=3: Scaling of HF .

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Qrpa 1
QRPA (1) function

  • The scheme which relates RPA to linearized TDHF can be repeated to derive QRPA from linearized Time-Dependent Hartree-Fock-Bogoliubov (cf. E. Khan et al., Phys. Rev. C 66, 024309 (2002))

  • Fully consistent QRPA calculations, except for 2-body spin-orbit, can be performed (M. Yamagami, NVG, Phys. Rev. C 69, 034301 (2004)) .

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Qrpa 2
QRPA (2) function

  • If Vpp is zero-range, one needs a cut-off in qp space, or a renormalisation procedure a la Bulgac. Then, one cannot sum up analytically the qp continuum up to infinity

  • If Vpp is finite range (like Gogny force) one cannot solve the Bethe-Salpeter equation in coordinate space

  • It is possible to sum over an energy grid along the positive axis ( Khan - Sandulescu et al., 2002) .

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The qrpa green s function
The QRPA Green’s Function function

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External field and strength distribution
External field and Strength distribution function

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2 states in 120sn
2+ states in 120Sn function

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2 states in 120sn with smearing
2+ states in 120Sn, with smearing function

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3 states in 120sn with smearing
3- states in 120Sn, with smearing function

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Relativistic rpa on top of relativistic mean field
Relativistic RPA on top of Relativistic Mean Field function

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Fermi states and dirac states
Fermi states and Dirac states function

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Single particle spectrum
Single-particle spectrum function

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The hartree polarization operator
The Hartree polarization operator function

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Fermi and dirac contributions
Fermi and Dirac contributions function

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The rrpa polarization operator
The RRPA polarization operator function

  • Generalized meson propagator for density-dependent case (Z.Y. Ma et al., 1997) .

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Diagrammatic representation
Diagrammatic representation function

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Rrpa and tdrmf
RRPA and TDRMF function

  • One can derive RRPA from the linearized version of the time-dependent RMF

  • At each time, one assumes the no-sea approximation, i.e., ones keeps only the positive energy states

  • These states are expanded on the complete set (at positive and negative energies) of states calculated at time t=0

  • This is how the Dirac states appear in RRPA. How important are they?

  • From the linearized TDRMF one obtains the matrix form of RRPA, but the p-h configuration space is much larger than in RPA! .

  • P.Ring, Z. Ma, NVG, et al. Nucl. Phys. A 694, 249 (2001)

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Including continuum in rrpa
Including continuum in RRPA function

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Effect of dirac states on isgmr
Effect of Dirac states on ISGMR function

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Effect of dirac states on isgqr
Effect of Dirac states on ISGQR function

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Effect of dirac states on ivgdr
Effect of Dirac states on IVGDR function

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Concluding remarks
Concluding Remarks function

  • More studies are needed in the following topics:

  • 1.In non-relativistic approach:

  • - RPA, QRPA for deformed systems.

  • - second RPA.

  • 2.In relativistic approach:

  • - RPA, QRPA on top of RHF.

  • - deformed systems.

  • - particle-vibration coupling.

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Lectures on mean field methods for nuclear structure list of references for further reading
Lectures on: functionMean Field Methods for Nuclear StructureList of references for further reading

  • 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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Skyrme Hartree-Fock Method: Computer Programs function

- P.-G. Reinhard, in Computational Nuclear Physics 1

(eds. K. Langanke, J.A. Maruhn, S.E. Koonin), Springer ‘93

- Spherical, SHF+BCS (monopole pairing)

- ev8: Bonche, Flocard, and Heenen, Comp. Phys. Comm. 171(’05)49

- 3D mesh, SHF+BCS (density dependent pairing)

- K. Bennaceur and J. Dobaczewski, Comp. Phys. Comm. 168(’05)96

- Spherical SHFB with density dependent pairing

  • - M.V. Stoitsov, J. Dobaczewski, W. Nazarewicz, P. Ring,

  • Comp. Phys. Comm. 167(’05)43

    • Axially deformed SHFB with density dependent pairing

  • transformed HO basis

  • - J. Dobaczewski and P. Olbratowski, Comp. Phys. Comm. 158(’04)158

    • Axially deformed SHFB with density dependent pairing

    • deformed HO basis

    Special thanks to

    Kouichi Hagino

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    Collaborators
    Collaborators function

    • Nicu Sandulescu (Bucharest)

    • Marcella Grasso (Catania, Orsay)

    • Elias Khan (Orsay)

    • Gianluca Colò (Milano)

    • Hiro Sagawa (Aizu)

    • Zhongyu Ma (Beijing)

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    Beyond rpa 1
    Beyond RPA (1) function

    • Large amplitude collective motion: Generator Coordinate Method

    • RPA can describe escape widths if continuum is treated, and it contains Landau damping, but spreading effects are not in the picture

    • Spreading effects are contained in Second RPA

    • Some applications called Second RPA are actually Second TDA: consistent SRPA calculations of nuclei are still waited for.

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    Beyond rpa 2
    Beyond RPA (2) function

    • There exist models to approximate SRPA:

    • The quasiparticle-phonon model (QPM) of Soloviev et al. Recently, attempts to calculate with Skyrme forces (A. Severyukhin et al.)

    • The ph-phonon model: see G. Colo. Importance of correcting for Pauli principle violation

    • Not much done so far in relativistic approaches .

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    Beyond rpa 3
    Beyond RPA (3) function

    • Particle-vibration coupling

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    Effect of particle vibration coupling
    Effect of particle-vibration coupling function

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