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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

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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

- 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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Main results:

- 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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Analytic summation of single-particle continuum

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

Solid: GQR

Dotted: empirical

Dashed: low-lying 2+

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Finite temperature

Applications: evolution of escape widths and Landau

damping of IVGDR with temperature .

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Restoration of symmetries

- 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

- 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)

- 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)

- 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 RRPA polarization operator

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

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RRPA and TDRMF

- 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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Concluding Remarks

- 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 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

- 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

- 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)

- 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)

- 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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