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Astrophysics and Nuclear Structure / International Workshop XXXIV on Gross Properties of Nuclei and Nuclear Excitations / Hirschegg, January 15 - 21, 2006. Nuclear Ground State and Collective Excitations based on Correlated Realistic NN Interactions. N. Paar , P. Papakonstantinou, A. Zapp,

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slide1

Astrophysics and Nuclear Structure / International Workshop XXXIV on Gross Properties of Nuclei and Nuclear Excitations / Hirschegg, January 15 - 21, 2006

Nuclear Ground State and Collective Excitations

based on Correlated Realistic NN Interactions

N. Paar, P. Papakonstantinou, A. Zapp,

H. Hergert, and R. Roth

slide2

REALISTIC AND EFFECTIVE NUCLEON-NUCLEON INTERACTIONS

Realistic nucleon-nucleon interactions are determined from the phase-shift analysis of nucleon-nucleon scattering

Wiringa et al., PRC 51, 38 (1995)

ARGONNE V18

One-pion exchange

intermediate and short-range

phenomenological terms

NN interactions in the nuclear structure models  truncation of the full many-body Hilbert space to a subspace of tractable size (e.g. Slater det.) is problematic

Effective interactions!

AV18, central part (S,T)=(0,1)

Very large matrix elements of interaction (relative wave functions penetrate the core)

4He

slide3

SHORT-RANGE NN INTERACTION-INDUCED CORRELATIONS

Short-range centraland tensor correlations are included in

the simple many body states via unitary transformation

CORRELATED MANY-BODY STATE

UNCORRELATED MANY-BODY STATE

Both the central and tensor correlations are necessary to

obtain a bound nuclear system:

AV18

Expectation value of the

Hamiltonian for Slater

determinant of harmonic

oscillator states

slide4

THE UNITARY CORRELATION OPERATOR METHOD (UCOM)

FELDMEIER, NEFF, ROTH (GSI&TUD)

40

Argonne V18 Potential

Correlation functions are constrained by the energy minimization in the two-body system

12

Central Correlator Cr

TWO-NUCLEON SYSTEM

6

Additional constraints necessary to restrict the ranges of the correlation functions

Tensor Correlator CΩ

Two-Body Approximation

VUCOM

Hartree-Fock

FINITE NUCLEI

?

Fermionic Molecular Dynamics

3-Nucleon Interaction

Random-Phase Approximation

No-core Shell Model

T. Neff

slide5

CORRELATED WAVE FUNCTIONS - DEUTERON

The short-range correlations are encapsulated in correlation functions:

The range of tensor correlation function becomes a parameter

slide6

CORRELATED OPERATORS IN THE UCOM SCHEME

Instead of correlated states with uncorrelated operators,

the correlated operators are employed

All observables need to be transformedconsistently !

Correlated Hamiltonian in two-body approximation

Correlated realistic NN interaction is given in an explicit operator form

Momentum-space matrix elements of the correlated interaction VUCOM and low-momentum interaction Vlow-k are similar

slide7

TEST OF CONVERGENCE FOR THE VUCOM POTENTIAL: 4He

V

V

bare

UCOM

NCSM code by P. Navrátil, Phys. Rev. C 61, 044001 (2000)

slide8

TJON-LINE AND THE RANGE OF THE TENSOR CORRELATOR

NO-CORE SHELL MODEL CALCULATIONS USING VUCOM

Increasing range of

the tensor correlator

SELECT A CORRELATOR

WITH ENERGY CLOSE TO

EXPERIMENTAL VALUE

CANCELLATION OF OMMITED 3-BODY CONTRIBUTION OF CLUSTER EXPANSION AND GENUINE 3-BODY FORCE

VNN+V3N

R. Roth et al., Phys. Rev. C 72, 034002 (2005)

slide10

HARTREE-FOCK BASED ON CORRELATED NN-POTENTIAL

The nuclear ground state is described by a Slater determinant

(short-range correlations are included in the correlated Hamiltonian)

Expansion of the single-particle states in harmonic-oscillator basis

NMAX=12

Hartree-Fock matrix equations as a generalized eigenvalue problem

Single-particle energies and wave functions are determined from

iterative solution of the Hartree-Fock equations

slide12

UCOM HARTREE-FOCK BINDING ENERGIES & CHARGE RADII

?

?

WHAT IS MISSING?

  • Long-range correlations
  • Genuine three-body forces
  • Three-body cluster contributions
slide13

BEYOND THE HARTREE-FOCK

Long-range correlations can be

recovered by the Many-Body Perturbation Theory (MBPT) or by evaluating RPA Correlations

MBPT

MBPT

I

RPA (preliminary)

slide14

UCOM RANDOM-PHASE APPROXIMATION

Low-amplitude collective oscillations

Vibration creation operator (1p-1h):

Equations of motion  RPA

&

Fully-self consistent RPA model: there is no mixing between the spurious 1- state and excitation spectra

VUCOM

EXCITATION

ENERGIES

N. Paar et al., nucl-th/0511041 (2005)

nucl-th/0601026 (2006)

Sum rules: Є=±3%

slide15

UCOM-RPA ISOSCALAR GIANT MONOPOLE RESONANCE

Various ranges of the UCOM tensor correlation functions

Relativistic RPA (DD-ME1 interaction)

slide16

UCOM-RPA ISOVECTOR GIANT DIPOLE RESONANCE

Various ranges of the UCOM tensor correlation functions

Improved description of the single-particle spectra (smaller range of the tensor correlator) pushes the IVGDR to lower energies.

slide17

UCOM-RPA GIANT QUADRUPOLE RESONANCE

EXP.

EXP.

EXP.

THE CORRELATED REALISTIC NN INTERACTION GENERATES THE LOW-LYING 2+ STATE AND GIANT QUADRUPOLE RESONANCE (THE EXCITATION ENERGY TOO HIGH)

slide18

SUMMARY

The correlated realistic nucleon-nucleon interaction (AV18) is employed in

different nuclear structure methods: NCSM, HF, RPA

UCOM Hartree-Fock results in underbinding and small radii

long-range correlations are recovered by many-body

perturbation theory and RPA

Fully self-consistent UCOM Random-Phase Approximation (RPA) is constructed in the Hartree-Fock single-nucleon basis

Correlated realistic NN interaction generates collective excitation

modes, however it overestimates the energies of giant resonances

Optimization of the ranges of correlators

Three-body interaction, complex configurations in RPA

WHAT ARE PERSPECTIVES FOR NUCLEAR ASTROPHYSICS?