15 june, 2006istanbul, part 21 mean field methods for nuclear structure part 1: ground state...
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15 June, 2006 Istanbul, part 2 1
Mean Field Methods for Nuclear Structure
Part 1: Ground State Properties: Hartree-Fock and Hartree-Fock-
Bogoliubov Approaches
Part 2: Nuclear Excitations: The Random Phase Approximation
Nguyen Van Giai
Nguyen Van GiaiInstitut de Physique Nucléaire Université Paris-Sud, Orsay
15 June, 2006 Istanbul, part 2 2
The Random Phase Approximation in Nuclear Physics
1. Linear response theory: a brief reminder
2. Non-relativistic RPA (Skyrme)
3. Relativistic RPA (RMF)
4. Extension to QRPA
5. Beyond RPA .
Nguyen Van Giai
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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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Two-body Green’s Function and density-density correlation 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
Dashed: low-lying 2+Dotted: empirical
Nguyen Van Giai
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Convection current distributionsGQR in 208Pb Low-lying 2+ in 208Pb
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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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Relativistic RPA on top of Relativistic Mean Field
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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 Structure
List 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 .
Nguyen Van Giai