Description of weakly bound nuclei with PTG/HFB and Gamow/HFB approaches

Nicolas MichelNicolas MichelCross-fertilization between shell model and Cross-fertilization between shell model and energy density functional methodsenergy density functional methods

Description of weakly Description of weakly bound nuclei with bound nuclei with

PTG/HFB and Gamow/HFB PTG/HFB and Gamow/HFB approachesapproaches

Nicolas MichelNicolas Michel (ESNT/SPhN/CEA)(ESNT/SPhN/CEA)

Kenichi MatsuyanagiKenichi Matsuyanagi (Kyoto University) (Kyoto University)

Mario StoitsovMario Stoitsov (ORNL – University of Tennessee) (ORNL – University of Tennessee)

April 7-10, 2008April 7-10, 2008ESNT/SPhN/CEAESNT/SPhN/CEA


Cross-fertilization between shell model Cross-fertilization between shell model and energy density functional methodsand energy density functional methods

Nicolas MichelNicolas Michel 22

Plan Plan

• Scientific motivation: drip-line nucleiScientific motivation: drip-line nuclei

• Gamow states, Berggren completeness relation, Gamow Shell ModelGamow states, Berggren completeness relation, Gamow Shell Model

• Gamow quasi-particle states and HFB densitiesGamow quasi-particle states and HFB densities

• Applications: Nickel chain (spherical)Applications: Nickel chain (spherical)

• PPööschl-Teller-Ginocchio (PTG) basis for loosely bound systemsschl-Teller-Ginocchio (PTG) basis for loosely bound systems

• Resonant structure with PTG basisResonant structure with PTG basis

• Applications: Nickel chain (spherical)Applications: Nickel chain (spherical)

Zirconium and Magnesium (deformed)Zirconium and Magnesium (deformed)

• Conclusion and perspectivesConclusion and perspectives




Scientific motivation Scientific motivation




Gamow statesGamow states

• Georg GamowGeorg Gamow : : decay decay

G.A. Gamow, Zs f. Phys. G.A. Gamow, Zs f. Phys. 5151 (1928) 204; (1928) 204; 5252 (1928) 510 (1928) 510

• DefinitionDefinition : :




Complex scaling methodComplex scaling method

• Radial integral calculationRadial integral calculation : : complex scalingcomplex scaling

• Analytic continuationAnalytic continuation : integral : integral independentindependent of R and of R and θθ




Gamow states locationGamow states location

Choice of contourChoice of contour

arbitraryarbitrary

narrownarrow

broadbroad




Completeness relations with Completeness relations with Gamow states Gamow states

• Berggren completeness relationBerggren completeness relation (l,j) : (l,j) :

T. Berggren, Nucl. Phys. A T. Berggren, Nucl. Phys. A 109109, (1967) 205, (1967) 205

• Continuum discretizationContinuum discretization : :

• N-body discretized completeness relationN-body discretized completeness relation (all l,j) : (all l,j) :




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

• He, Li chainsHe, Li chains : valence particles above : valence particles above 44HeHe core : core : H = WS (H = WS (55He) + SGIHe) + SGI

0p0p3/23/2, 0p, 0p1/21/2 (resonant), p (resonant), p3/23/2 and p and p1/21/2 scattering continuums scattering continuums

SGI : Surface Gaussian Interaction :SGI : Surface Gaussian Interaction :

• Dependence on number of nucleons for T=0Dependence on number of nucleons for T=0

• Spherical Gamow Hartree-Fock basisSpherical Gamow Hartree-Fock basis from H = WS + SGI from H = WS + SGI




N. Michel et al.,N. Michel et al., Phys. Rev C, Phys. Rev C, 6767 054311 (2003) 054311 (2003)Rev. Mex. Fis., Rev. Mex. Fis., 50 S250 S2 74 (2004) 74 (2004)

Bound from Bound from unbound basis unbound basis

Helium anomaly Helium anomaly

reproducedreproduced




Halo densityHalo density




E (

MeV

)E

(M

eV)

S-components S-components

missingmissing

SatisfactorySatisfactory

resultsresults

(schematic model) (schematic model)

N. Michel et al., Phys. Rev. C N. Michel et al., Phys. Rev. C 7070, 064313 (2004), 064313 (2004)




Gamow HFB spaceGamow HFB space




Densities with Gamow HFBDensities with Gamow HFB• HFB equations:HFB equations:

• Complex particle and pairing densities:Complex particle and pairing densities:

• HF associated bound and narrow resonant states in HF associated bound and narrow resonant states in discrete sumdiscrete sum




Quasi-particle pole statesQuasi-particle pole states• Bound, resonant states:Bound, resonant states:

S matrix poles => outgoing wave function behavior S matrix poles => outgoing wave function behavior




Quasi-particle scattering statesQuasi-particle scattering states• Scattering states:Scattering states:

u(r):u(r): incoming and outgoing componentsincoming and outgoing components v(r):v(r): outgoingoutgoing wave function behavior wave function behavior




Gamow Gamow quasi-particle states normquasi-particle states norm

• Normalization:Normalization:

S-matrix polesS-matrix poles:: complex scaling complex scaling

Scattering statesScattering states:: Dirac delta normalization Dirac delta normalization

Continuum discretizationContinuum discretization::




Gamow Hartree-Fock Gamow Hartree-Fock diagonalization methoddiagonalization method

• Two-basis methodTwo-basis method

BasisBasis generated bygenerated by ph part ph part of HFB hamiltonian: of HFB hamiltonian:

B. Gall et al., Z. Phys. B. Gall et al., Z. Phys. A348A348 183 (1994) 183 (1994)

• HFB matrix structure:HFB matrix structure:

• Diagonalization of HFB matrix in Gamow HF basisDiagonalization of HFB matrix in Gamow HF basis




Description of Nickel calculationsDescription of Nickel calculations

• Considered nuclei:Considered nuclei: 8484Ni, Ni, 8686Ni, Ni, 8888Ni, Ni, 9090NiNi

• Interaction and space:Interaction and space:

Skyrme interaction:Skyrme interaction: Sly4, E Sly4, Ecutcut = 60 MeV, l: 0 = 60 MeV, l: 0 →→ 10 10

RRcutcut = 20 fm, k = 20 fm, kmaxmax = 4 fm = 4 fm-1-1, N, Nscatscat (l,j) = 100 for GHF basis. (l,j) = 100 for GHF basis.

• Interest:Interest: Resonant structure Resonant structure directlydirectly put in HFB basis put in HFB basis




PTG basis for HFB calculationsPTG basis for HFB calculations

• Gamow HF(B) basis:Gamow HF(B) basis: Advantages:Advantages: good asymptoticsgood asymptotics, , smoothly varying continuumssmoothly varying continuums Inconvenients:Inconvenients: complex arithmetic, complex arithmetic, long calculationslong calculations

• Weakly bound systems:Weakly bound systems: realreal continuouscontinuous bases bases sufficientsufficient• Real Gamow HF basis:Real Gamow HF basis: problematicproblematic due to due to resonant structureresonant structure in in continuumcontinuum

• PTG basis:PTG basis: resonancesresonances replaced by replaced by bound statesbound states No resonant stateNo resonant state in (l,j) partial wave: in (l,j) partial wave: Hankel/Coulomb functionsHankel/Coulomb functions Smooth continuumsSmooth continuums for all partial waves for all partial waves Weakly bound systems asymptotics Weakly bound systems asymptotics well describedwell described




Accepted in Phys. Rev. CAccepted in Phys. Rev. C

HF/PTG potentials HF/PTG potentials




HF/PTG wave functionsHF/PTG wave functions

r (fm)r (fm) Accepted in Phys. Rev. CAccepted in Phys. Rev. C

---- : PTG---- : PTG : HF: HF




r (fm)r (fm) r (fm)r (fm) Accepted in Phys. Rev. CAccepted in Phys. Rev. C

---- : PTG p---- : PTG p---- : PTG n---- : PTG n

: Box p: Box p : Box n: Box n

.… .… : HO: HO





----- : prot.----- : prot. : neut.: neut.

… … : THO: THO





----- : prot.----- : prot. : neut.: neut.




Conclusion and perspectivesConclusion and perspectives• Gamow Shell Model:Gamow Shell Model: Hybrid methodHybrid method: Gamow Hartree-Fock basis necessary: Gamow Hartree-Fock basis necessary He, Li chains with He, Li chains with schematic Hamiltoniansschematic Hamiltonians Next stepNext step : : realisticrealistic interactions, interactions, effectiveeffective interactions with continuum interactions with continuum

• HFB expansions with Gamow and PTG basesHFB expansions with Gamow and PTG bases Precise tool to study dripline heavy nucleiPrecise tool to study dripline heavy nuclei PTG basis near-optimal for weakly bound systemsPTG basis near-optimal for weakly bound systems Nickel chain, Nickel chain, 4040Mg and Mg and 110110Zn : spherical and deformed ground statesZn : spherical and deformed ground states • Conclusions and perspectivesConclusions and perspectives Weakly bound nuclei : Weakly bound nuclei : fast and stable method with PTG basisfast and stable method with PTG basis Future Future QRPAQRPA calculationscalculations with with quasi-particle basis quasi-particle basis fromfrom HFB/PTG HFB/PTG Gamow-HFBGamow-HFB :: problems remain for unbound nuclei problems remain for unbound nuclei

Description of weakly bound nuclei with PTG/HFB and Gamow/HFB approaches

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