Mean Field Methods for Nuclear Structure

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


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


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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• Adding a time-dependent external field:

.

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First order response as a function of time

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

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Linear response function and Strength distribution

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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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TDHF and RPA (1)

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TDHF and RPA (2)

And by comparing with p.6

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Residual p-h interaction

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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

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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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RPA on a p-h basis

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A and B matrices

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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 QRPA Green’s Function

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External field and Strength distribution

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2+ states in 120Sn

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2+ states in 120Sn, with smearing

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3- states in 120Sn, with smearing

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Relativistic RPA on top of Relativistic Mean Field

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Fermi states and Dirac states

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Single-particle spectrum

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The Hartree polarization operator

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Fermi and Dirac contributions

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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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Diagrammatic representation

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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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Including continuum in RRPA

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Effect of Dirac states on ISGMR

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Effect of Dirac states on ISGQR

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Effect of Dirac states on IVGDR

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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


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


Collaborators

• Nicu Sandulescu (Bucharest)

• Marcella Grasso (Catania, Orsay)

• Elias Khan (Orsay)

• Gianluca Colò (Milano)

• Hiro Sagawa (Aizu)

• Zhongyu Ma (Beijing)


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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Beyond RPA (3)

• Particle-vibration coupling

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Effect of particle-vibration coupling

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Mean Field Methods for Nuclear Structure