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The Random Phase Approximation in The Random Phase Approximation in Nuclear PhysicsNuclear Physics

Lay out of the presentation:Lay out of the presentation:1.1. Linear response theory: a brief reminderLinear response theory: a brief reminder

2.2. Non-relativistic RPA (Skyrme)Non-relativistic RPA (Skyrme)

3.3. Relativistic RPA (RMF)Relativistic RPA (RMF)

4.4. Extension to QRPAExtension to QRPA

5.5. Beyond RPA .Beyond RPA .

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Linear Response TheoryLinear Response Theory

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

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

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

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Adding a time-dependent external field:Adding a time-dependent external field:

.

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

.

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

.

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Linear response function and Linear response function and Strength distributionStrength distribution

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Main results:Main results:

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

Excitation energies of eigenmodes (the poles)Excitation energies of eigenmodes (the poles) Transition probabilities (residues of the response Transition probabilities (residues of the response

function)function) Transition densities (or form factors), transition Transition densities (or form factors), transition

currents, etc… of each excited state .currents, etc… of each excited state .

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TDHF and RPA (1)TDHF and RPA (1)

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TDHF and RPA (2)TDHF and RPA (2)

And by comparing with p.5

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Residual p-h interactionResidual p-h interaction

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Analytic summation of single-Analytic summation of single-particle continuumparticle 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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Approximate treatments of Approximate treatments of continuum (1)continuum (1)

T. Vertse, P. Curutchet, R.J. Liotta, Phys. Rev. C 42, 2605 (1990) .

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Approximate treatments of Approximate treatments of continuum (2)continuum (2)

Calculate positive-energy s.p. states with Calculate positive-energy s.p. states with scattering asymptotic conditions, and sum scattering asymptotic conditions, and sum over an energy grid along the positive over an energy grid along the positive axis, up to some cut-offaxis, up to some cut-off

Sum over discrete states of positive Sum over discrete states of positive energy calculated with a box boundary energy calculated with a box boundary condition .condition .

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Transition densities and divergence Transition densities and divergence of transition currentsof transition currents

Solid: GQR

Dashed: low-lying 2+Dotted: empirical

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Convection current distributionsConvection current distributionsGQR in 208Pb Low-lying 2+ in 208Pb

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Finite temperatureFinite temperature

Applications: evolution of escape widths and Landau damping of IVGDR with temperature .

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RPA on a p-h basisRPA on a p-h basis

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A and B matricesA and B matrices

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Restoration of symmetriesRestoration of symmetries

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

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

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

This is not the case in phenomenological RPA .This is not the case in phenomenological RPA .

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Sum rulesSum rules

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

k=1: Thouless theoremk=1: Thouless theorem k=-1: Constrained HFk=-1: Constrained HF k=3: Scaling of HF .k=3: Scaling of HF .

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Phenomenological RPAPhenomenological RPA

The HF mean field is replaced by a parametrized The HF mean field is replaced by a parametrized mean field (harmonic oscillator, Woods-Saxon mean field (harmonic oscillator, Woods-Saxon potential, …)potential, …)

The residual p-h interaction is adjusted (Landau-The residual p-h interaction is adjusted (Landau-Migdal form, meson exchange, …)Migdal form, meson exchange, …)

Useful in many situations (e.g., double-beta Useful in many situations (e.g., double-beta decay)decay)

Difficulty to relate properties of excitations to Difficulty to relate properties of excitations to bulk properties (K, symmetry energy, effective bulk properties (K, symmetry energy, effective mass, …) .mass, …) .

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Relativistic RPA on top of RMFRelativistic RPA on top of RMF

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Fermi states and Dirac statesFermi states and Dirac states

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Single-particle spectrumSingle-particle spectrum

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The Hartree polarization operatorThe Hartree polarization operator

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Fermi and Dirac contributionsFermi and Dirac contributions

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The RRPA polarization operatorThe RRPA polarization operator

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

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Diagrammatic representationDiagrammatic representation

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RRPA and TDRMFRRPA and TDRMF

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

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

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

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

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

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Effect of Dirac states on ISGMREffect of Dirac states on ISGMR

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Effect of Dirac states on ISGQREffect of Dirac states on ISGQR

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Effect of Dirac states on IVGDREffect of Dirac states on IVGDR

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Including continuum in RRPAIncluding continuum in RRPA

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QRPA (1)QRPA (1)

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

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

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QRPA (2)QRPA (2)

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

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

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

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Pairing window methodPairing window method

K. Hagino, H. Sagawa, Nucl. Phys. A 695, 82 (2001) .

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2+ states in 120Sn2+ states in 120Sn

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2+ states in 120Sn, with smearing2+ states in 120Sn, with smearing

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3- states in 120Sn, with smearing3- states in 120Sn, with smearing

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Beyond RPA (1)Beyond RPA (1)

Large amplitude collective motion: Generator Large amplitude collective motion: Generator Coordinate MethodCoordinate Method

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

Spreading effects are contained in Second RPASpreading effects are contained in Second RPA Some applications called Second RPA are Some applications called Second RPA are

actually Second TDA: consistent SRPA actually Second TDA: consistent SRPA calculations of nuclei are still waited for.calculations of nuclei are still waited for.

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Beyond RPA (2)Beyond RPA (2)

There exist models to approximate SRPA:There exist models to approximate SRPA: The quasiparticle-phonon model (QPM) of The quasiparticle-phonon model (QPM) of

Soloviev et al. Recently, attempts to calculate Soloviev et al. Recently, attempts to calculate with Skyrme forces (A. Severyukhin et al.)with Skyrme forces (A. Severyukhin et al.)

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

Not much done so far in relativistic approaches .Not much done so far in relativistic approaches .

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Beyond RPA (3)Beyond RPA (3)

Particle-vibration Particle-vibration couplingcoupling

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Effect of particle-vibration couplingEffect of particle-vibration coupling

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AcknowledgmentsAcknowledgments

Thanks to Wenhui LONG for Powerpoint Thanks to Wenhui LONG for Powerpoint tutoring . tutoring .