September 26 to September 26 to December 2, 2005December 2, 2005

Nicolas MichelNicolas MichelNuclear Structure Near the Limits of Nuclear Structure Near the Limits of Stability (INTStability (INT--0505--3)3)

Description of nearDescription of near--threshold threshold states with states with

the the GamowGamow Shell Model Shell Model

Nicolas MichelNicolas Michel (ORNL (ORNL -- University of Tennessee)University of Tennessee)

WitekWitek NazarewiczNazarewicz (ORNL (ORNL -- University of Tennessee)University of Tennessee)

MarekMarek PloszajczakPloszajczak (GANIL)(GANIL)

Jimmy Jimmy RotureauRotureau (ORNL (ORNL –– University of Tennessee)University of Tennessee)

22September 26 toSeptember 26 toDecember 2, 2005December 2, 2005

Nuclear Structure Near the Limits of Nuclear Structure Near the Limits of Stability (INTStability (INT--0505--3)3)

Nicolas MichelNicolas Michel

Plan Plan

•• Standard Shell Model : usefulness and limitationsStandard Shell Model : usefulness and limitations•• GamowGamow states : definition and intereststates : definition and interest•• Completeness relations with Completeness relations with GamowGamow statesstates•• GamowGamow Shell Model (GSM) matrix Shell Model (GSM) matrix diagonalizationdiagonalization•• Applications : O, He and Li chainsApplications : O, He and Li chains•• Spectroscopic factors in GSMSpectroscopic factors in GSM•• Application : Application : 55He/He/66HeHe•• SpinSpin--orbit splitting : 1/2orbit splitting : 1/2-- and 3/2and 3/2-- states of states of 55He and He and 77HeHe

33September 26 toSeptember 26 toDecember 2, 2005December 2, 2005

Nuclear Structure Near the Limits of Nuclear Structure Near the Limits of Stability (INTStability (INT--0505--3)3)

Nicolas MichelNicolas Michel

Standard Shell Model : Standard Shell Model : usefulness and limitationsusefulness and limitations

•• Slater Determinants basis (MSlater Determinants basis (M--scheme) :scheme) :•• Hamiltonian matrix :Hamiltonian matrix :

•• Matrix Matrix diagonalizationdiagonalization ((LanczosLanczos method) :method) :•• Harmonic oscillators statesHarmonic oscillators states : :

MoshinskyMoshinsky transformation, exact center of mass treatment.transformation, exact center of mass treatment.ButBut : well: well--bound states only.bound states only.

44September 26 toSeptember 26 toDecember 2, 2005December 2, 2005

Nuclear Structure Near the Limits of Nuclear Structure Near the Limits of Stability (INTStability (INT--0505--3)3)

Nicolas MichelNicolas Michel

GamowGamow states : states : definition and interestdefinition and interest

•• Spherical SchrSpherical Schröödinger equation :dinger equation :

•• Boundary conditions :Boundary conditions :

•• Expansion of loosely bound and resonant states possible.Expansion of loosely bound and resonant states possible.

55September 26 toSeptember 26 toDecember 2, 2005December 2, 2005

Nuclear Structure Near the Limits of Nuclear Structure Near the Limits of Stability (INTStability (INT--0505--3)3)

Nicolas MichelNicolas Michel

Berggren completeness relationBerggren completeness relation

66September 26 toSeptember 26 toDecember 2, 2005December 2, 2005

Nuclear Structure Near the Limits of Nuclear Structure Near the Limits of Stability (INTStability (INT--0505--3)3)

Nicolas MichelNicolas Michel

Completeness relations with Completeness relations with GamowGamow states states

•• Berggren completeness relation :Berggren completeness relation :

•• Continuum Continuum discretizationdiscretization ::

•• NN--body body discretizeddiscretized completeness relation :completeness relation :

77September 26 toSeptember 26 toDecember 2, 2005December 2, 2005

Nuclear Structure Near the Limits of Nuclear Structure Near the Limits of Stability (INTStability (INT--0505--3)3)

Nicolas MichelNicolas Michel

GSM matrix GSM matrix diagonalizationdiagonalization•• GSM matrix : large complex symmetric matrices.GSM matrix : large complex symmetric matrices.

•• LanczosLanczos method insufficient : method insufficient : resonances hidden among scattering states.resonances hidden among scattering states.

•• SolutionSolution : the overlap method: the overlap method1. Diagonalization without scattering basis states : pole a1. Diagonalization without scattering basis states : pole approximation.pproximation.

Approximate vectors obtained with the Approximate vectors obtained with the LanczosLanczos method.method.

2. Refinement of the approximate vector in the total space.2. Refinement of the approximate vector in the total space.Exact vector has the largest overlap with the approximatExact vector has the largest overlap with the approximate vector.e vector.Complex version of the Davidson method.Complex version of the Davidson method.

88September 26 toSeptember 26 toDecember 2, 2005December 2, 2005

Nuclear Structure Near the Limits of Nuclear Structure Near the Limits of Stability (INTStability (INT--0505--3)3)

Nicolas MichelNicolas Michel

N. Michel et al., Phys. Rev. C, 054311 (2003)N. Michel et al., Phys. Rev. C, 054311 (2003)

99September 26 toSeptember 26 toDecember 2, 2005December 2, 2005

Nuclear Structure Near the Limits of Nuclear Structure Near the Limits of Stability (INTStability (INT--0505--3)3)

Nicolas MichelNicolas Michel

N. Michel et al., Phys. Rev. C, 054311 (2003)N. Michel et al., Phys. Rev. C, 054311 (2003)

1010September 26 toSeptember 26 toDecember 2, 2005December 2, 2005

Nuclear Structure Near the Limits of Nuclear Structure Near the Limits of Stability (INTStability (INT--0505--3)3)

Nicolas MichelNicolas Michel

Application : O, He and Li chainsApplication : O, He and Li chains

•• O : Valence particles above O : Valence particles above 1616O core : H = WS + SDIO core : H = WS + SDI•• He, Li :Valence particles above He, Li :Valence particles above 44He core : H = WS + SGI.He core : H = WS + SGI.

WS : fitted with single particle states of WS : fitted with single particle states of 1717O / O / 55He.He.SGI : Surface Gaussian Interaction :SGI : Surface Gaussian Interaction :

•• Dependence on number of nucleons for T=0.Dependence on number of nucleons for T=0.•• pp3/23/2 and pand p1/21/2 valence shells : poles and scattering states.valence shells : poles and scattering states.•• Spherical Spherical HartreeHartree--FockFock basis from H = WS + SGI.basis from H = WS + SGI.•• No center of mass correction. No center of mass correction.

1111September 26 toSeptember 26 toDecember 2, 2005December 2, 2005

Nuclear Structure Near the Limits of Nuclear Structure Near the Limits of Stability (INTStability (INT--0505--3)3)

Nicolas MichelNicolas Michel

N. Michel et al., Phys. Rev. C, 054311 (2003)N. Michel et al., Phys. Rev. C, 054311 (2003)

1212September 26 toSeptember 26 toDecember 2, 2005December 2, 2005

Nuclear Structure Near the Limits of Nuclear Structure Near the Limits of Stability (INTStability (INT--0505--3)3)

Nicolas MichelNicolas Michel

N. Michel et al., Phys. Rev. C, 054311 (2003)N. Michel et al., Phys. Rev. C, 054311 (2003)

1313September 26 toSeptember 26 toDecember 2, 2005December 2, 2005

Nuclear Structure Near the Limits of Nuclear Structure Near the Limits of Stability (INTStability (INT--0505--3)3)

Nicolas MichelNicolas Michel

1414September 26 toSeptember 26 toDecember 2, 2005December 2, 2005

Nuclear Structure Near the Limits of Nuclear Structure Near the Limits of Stability (INTStability (INT--0505--3)3)

Nicolas MichelNicolas Michel

1515September 26 toSeptember 26 toDecember 2, 2005December 2, 2005

Nuclear Structure Near the Limits of Nuclear Structure Near the Limits of Stability (INTStability (INT--0505--3)3)

Nicolas MichelNicolas Michel

N. Michel et al., Phys. Rev. C, 064313 (2004)N. Michel et al., Phys. Rev. C, 064313 (2004)

1616September 26 toSeptember 26 toDecember 2, 2005December 2, 2005

Nuclear Structure Near the Limits of Nuclear Structure Near the Limits of Stability (INTStability (INT--0505--3)3)

Nicolas MichelNicolas Michel

Spectroscopic factors in GSMSpectroscopic factors in GSM

•• ModelModel--independentindependent definition of spectroscopic factors :definition of spectroscopic factors :

•• 55He / He / 66He : scattering components are He : scattering components are crucial.crucial.

1717September 26 toSeptember 26 toDecember 2, 2005December 2, 2005

Nuclear Structure Near the Limits of Nuclear Structure Near the Limits of Stability (INTStability (INT--0505--3)3)

Nicolas MichelNicolas Michel

1818September 26 toSeptember 26 toDecember 2, 2005December 2, 2005

Nuclear Structure Near the Limits of Nuclear Structure Near the Limits of Stability (INTStability (INT--0505--3)3)

Nicolas MichelNicolas Michel

SpinSpin--orbit splitting : orbit splitting : 1/21/2-- and 3/2and 3/2-- states of states of 55He and He and 77HeHe

1919September 26 toSeptember 26 toDecember 2, 2005December 2, 2005

Nuclear Structure Near the Limits of Nuclear Structure Near the Limits of Stability (INTStability (INT--0505--3)3)

Nicolas MichelNicolas Michel

Conclusion and perspectivesConclusion and perspectivesConclusionConclusion

•• GSM : Shell model simplicity and power beyond GSM : Shell model simplicity and power beyond LL22..•• Complete description of loosely bound and resonant states.Complete description of loosely bound and resonant states.•• No limitation of particles in the continuum : No limitation of particles in the continuum : BorromeanBorromean systems.systems.•• Scattering components necessary for nuclei states close to thresScattering components necessary for nuclei states close to thresholds.holds.

PerspectivesPerspectives•• NoNo--core shell model with a realistic interaction.core shell model with a realistic interaction.•• Density dependent interaction : HF basis and shell model matrix.Density dependent interaction : HF basis and shell model matrix.•• Density Matrix Renormalization Group (DMRG) approach.Density Matrix Renormalization Group (DMRG) approach.

Possibility to handle the numerous scattering configurationPossibility to handle the numerous scattering configurations.s.•• GamowGamow HFBHFB--QRPA : medium and heavy nuclei dripQRPA : medium and heavy nuclei drip--lineslines