description of drip-line nuclei with gsm and gamow/hfb frameworks
DESCRIPTION
Description of drip-line nuclei with GSM and Gamow/HFB frameworks. Nicolas Michel (CEA/IRFU/SPhN) Marek Ploszajszak (GANIL) Witek Nazarewicz (UT/ORNL) Kenichi Matsuyanagi (Kyoto University) Mario Stoitsov (ORNL – University of Tennessee). Plan. Scientific motivation: drip-line nuclei - PowerPoint PPT PresentationTRANSCRIPT
Nicolas MichelNicolas MichelImportance of continuum for nuclei close to Importance of continuum for nuclei close to drip-linedrip-line
May 20th, 2009May 20th, 2009
Description of drip-line Description of drip-line nuclei with GSM and nuclei with GSM and
Gamow/HFB frameworks Gamow/HFB frameworks
Nicolas MichelNicolas Michel (CEA/IRFU/SPhN)(CEA/IRFU/SPhN)Marek PloszajszakMarek Ploszajszak (GANIL) (GANIL)
Witek NazarewiczWitek Nazarewicz (UT/ORNL) (UT/ORNL)Kenichi MatsuyanagiKenichi Matsuyanagi (Kyoto University) (Kyoto University)
Mario StoitsovMario Stoitsov (ORNL – University of Tennessee) (ORNL – University of Tennessee)
May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines
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Plan Plan
• Scientific motivation: drip-line nucleiScientific motivation: drip-line nuclei• Gamow states definition and normalization, Berggren basisGamow states definition and normalization, Berggren basis• Spectroscopic factors, overlap functionsSpectroscopic factors, overlap functions• 66He/He/55He, He, 77He/He/66He, He, 1818O*/O*/1717O*, O*, 66Be/Be/55LiLi
• Gamow HFB framework: HF basis diagonalization, direct integrationGamow HFB framework: HF basis diagonalization, direct integration• Applications: Nickel chainApplications: Nickel chain• PPööschl-Teller-Ginocchio (PTG) basis for loosely bound systemsschl-Teller-Ginocchio (PTG) basis for loosely bound systems• Applications: Nickel chain (spherical)Applications: Nickel chain (spherical) Zirconium and Magnesium (deformed)Zirconium and Magnesium (deformed)• Conclusion and perspectivesConclusion and perspectives
May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines
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Scientific motivation Scientific motivation
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Gamow statesGamow states
• Georg GamowGeorg Gamow : simple model of : simple model of 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 : :
• Straightforward generalization to non-local potentials (HF)Straightforward generalization to non-local potentials (HF)
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• NormalizationNormalization: : complex scalingcomplex scaling
• Analytic continuationAnalytic continuation: integral : integral independentindependent of R and of R and θθ
• Normalization of bound and resonant states: Normalization of bound and resonant states:
• Scattering states:Scattering states: normalization normalization impossibleimpossible with complex scaling with complex scaling Normalization with Dirac delta: Normalization with Dirac delta:
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Complex scalingComplex scaling
May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines
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Complex energy statesComplex energy states
boundbound
broad resonancesbroad resonances
narrow resonancesnarrow resonances
LL++ : arbitrary contour : arbitrary contour
antibound statesantibound states
capturing statescapturing states
Im(k)Im(k)
Re(k)Re(k)
Berggren completeness relationBerggren completeness relation
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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 : :
May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines
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• O isotopesO isotopes : valence particles above : valence particles above 1616OO core : core : H = T + WS (H = T + WS (1717O) + SDIO) + SDI
0d0d5/25/2, 1s, 1s1/21/2 (bound), 0d (bound), 0d3/23/2 (resonant), d(resonant), d3/23/2 scattering continuum scattering continuum
• He and Be isotopesHe and Be isotopes : valence particles above : valence particles above 44HeHe core core
H = T + WS (H = T + WS (55He/He/55Li) + SGILi) + 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 :
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Application : O, He and Be isotopesApplication : O, He and Be isotopes
May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines
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Spectroscopic factors in GSMSpectroscopic factors in GSM
• One particle emission channelOne particle emission channel: (l,j,: (l,j,))
• Basis-independent definitionBasis-independent definition::
• ExperimentalExperimental : : cuspscusps appear in appear in cross-sectionscross-sections
• StandardStandard : representation : representation dependencedependence ( (nn,l,j,,l,j,))
• 55He / He / 66He, He, 55Li / Li / 66BeBe : non resonant components : non resonant components necessarynecessary..
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Spectroscopic factorSpectroscopic factor::
Nucleon-nucleon Nucleon-nucleon correlationscorrelations measure measure
Universal behaviorUniversal behaviorclose to emission thresholdclose to emission threshold
N. Michel et al.,N. Michel et al.,Phys. Rev. C, (Rap. Comm.), Phys. Rev. C, (Rap. Comm.), 7575 031301(R) (2007) 031301(R) (2007)
VVSGISGI
variesvaries
VVSGISGI
variesvaries
WSWSvariesvaries
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Real WS fitReal WS fit
Complex WS fitComplex WS fit
O(r)O(r)
N.Michel, W.Nazarewicz, M.Ploszajczak,N.Michel, W.Nazarewicz, M.Ploszajczak,
Nucl. Phys. A Nucl. Phys. A 794794 29 (2007) 29 (2007)
May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines
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N.Michel, W.Nazarewicz, M.Ploszajczak,N.Michel, W.Nazarewicz, M.Ploszajczak,
Nucl. Phys. A Nucl. Phys. A 794794 29 (2007) 29 (2007)
WSWSvariesvaries
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N.Michel, W.Nazarewicz, M.Ploszajczak,N.Michel, W.Nazarewicz, M.Ploszajczak,
Nucl. Phys. A Nucl. Phys. A 794794 29 (2007) 29 (2007)
VVSGISGI variesvaries
May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines
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VVSGISGI variesvaries VVSGISGI variesvaries
Mirror reactions : Mirror reactions : 66He/He/55HeHe and and 66Be/Be/55LiLi
66He/He/55HeHe
66Be/Be/55LiLi
May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines
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Mirror reactions : Mirror reactions : 66He/He/55HeHe and and 66Be/Be/55LiLi
VVSGISGI variesvaries
66Be/Be/55LiLi
66He/He/55HeHe
May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines
Nicolas MichelNicolas Michel 1616
VVSGISGI variesvaries
Mirror reactions : Mirror reactions : 66He/He/55HeHe and and 66Be/Be/55LiLi
66Be/Be/55LiLi
66He/He/55HeHe
May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines
Nicolas MichelNicolas Michel
HFB frameworkHFB framework• HFB ground state HFB ground state : product of independent quasi-particles: product of independent quasi-particles
• HFB equations: HFB equations:
• Standard methods of resolutionStandard methods of resolution
HO basis diagonalizationHO basis diagonalization: : well bound states onlywell bound states only
THO basis diagonalizationTHO basis diagonalization: : basis dependence from scaling function ?basis dependence from scaling function ?
Direct integrationDirect integration: : very precisevery precise, , but long but long (box boundary conditions)(box boundary conditions)
1717
May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines
Nicolas MichelNicolas Michel 1818
PTG/HFB and Gamow/HFB modelsPTG/HFB and Gamow/HFB models
• Continuous basis methodsContinuous basis methods
Complex-energy formalismComplex-energy formalism
Bound, resonant Bound, resonant andand scattering scattering states (Berggren basis)states (Berggren basis)
Berggren basis of Berggren basis of HF particle states HF particle states ((two-basis methodtwo-basis method) )
BerggrenBerggren quasi-particles states quasi-particles states calculated in coordinate spacecalculated in coordinate space
Real-energy formalismReal-energy formalism
Basis generated by a Basis generated by a Pöschl-Teller-Ginocchio (PTG) potentialPöschl-Teller-Ginocchio (PTG) potential
• Examples:Examples: Ni chain, Ni chain, 110110Zr and Zr and 4040MgMg, HFB density functional: , HFB density functional: Sly4 + surface pairing Sly4 + surface pairing
orbital momentum: l=0 to l=10, Eorbital momentum: l=0 to l=10, Ecutcut = 60 MeV = 60 MeV
May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines
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Gamow HFB framework: Gamow HFB framework: HF basis diagonalizationHF basis diagonalization
• 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 PTG or Gamow HF basisDiagonalization of HFB matrix in PTG or Gamow HF basis
May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines
Nicolas MichelNicolas Michel 2020
Gamow HFB framework:Gamow HFB framework:direct integrationdirect integration
N. Michel, K.Matsuyanagi, M. StoitsovN. Michel, K.Matsuyanagi, M. StoitsovPhys. Rev. C, Phys. Rev. C, 7878 044319 (2008) 044319 (2008)
May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines
Nicolas MichelNicolas Michel 2121
HF and PTG potentialsHF and PTG potentials
M. Stoitsov, N. Michel, K.Matsuyanagi,M. Stoitsov, N. Michel, K.Matsuyanagi,Phys. Rev. C, Phys. Rev. C, 7777, 054301 (2008), 054301 (2008)
May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines
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HF/PTG wave functionsHF/PTG wave functions
r (fm)r (fm)
---- : PTG---- : PTG : HF: HF
M. Stoitsov, N. Michel, K.Matsuyanagi,M. Stoitsov, N. Michel, K.Matsuyanagi,Phys. Rev. C, Phys. Rev. C, 7777, 054301 (2008), 054301 (2008)
May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines
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densitiesdensities----- : prot.----- : prot. : neut.: neut.
… … : THO: THO
M. Stoitsov, N. Michel, K.Matsuyanagi,M. Stoitsov, N. Michel, K.Matsuyanagi,Phys. Rev. C, Phys. Rev. C, 7777, 054301 (2008), 054301 (2008)
PTG PTG basisbasis
Long axisLong axis
Short axisShort axis
May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines
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M. Stoitsov, N. Michel, K.Matsuyanagi,M. Stoitsov, N. Michel, K.Matsuyanagi,Phys. Rev. C, Phys. Rev. C, 7777, 054301 (2008), 054301 (2008)
PairingPairingdensitiesdensities
----- : prot.----- : prot. : neut.: neut.
… … : THO: THO
PTG PTG basisbasis
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Black solid line : Black solid line : HFB boxHFB box
Dashed green line :Dashed green line : GHFB coord. GHFB coord.
Dotted blue line :Dotted blue line : GHF basisGHF basis
Gamow HFBGamow HFBNickel Nickel
densitiesdensities
N. Michel, K.Matsuyanagi, M. StoitsovN. Michel, K.Matsuyanagi, M. StoitsovPhys. Rev. C, Phys. Rev. C, 7878 044319 (2008) 044319 (2008)
May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines
Nicolas MichelNicolas Michel 2626
Black solid line : Black solid line : HFB boxHFB box
Dashed green line :Dashed green line : GHFB coord. GHFB coord.
Dotted blue line :Dotted blue line : GHF basisGHF basis
Gamow HFBGamow HFBNickel Nickel pairingpairing
densitiesdensities
N. Michel, K.Matsuyanagi, M. StoitsovN. Michel, K.Matsuyanagi, M. StoitsovPhys. Rev. C, Phys. Rev. C, 7878 044319 (2008) 044319 (2008)
May 20th, 2009May 20th, 2009 Importance of continuum for nuclei close to Importance of continuum for nuclei close to drip-lines drip-lines
Nicolas MichelNicolas Michel 2727
Conclusion and perspectivesConclusion and perspectives• GSMGSM : Standard shell model : Standard shell model simplicitysimplicity and and powerpower beyond Hilbert space. beyond Hilbert space. CompleteComplete description of description of looselyloosely boundbound and and resonantresonant states.. states.. Spectroscopic factors:Spectroscopic factors: continum coupling necessary continum coupling necessary Cusps:Cusps: clear l-dependence clear l-dependence Important differences Important differences between between protonproton and and neutronneutron case case
• HFB expansions with Gamow and PTG bases:HFB expansions with Gamow and PTG bases: Precise tool Precise tool to study dripline to study dripline medium and heavy nucleimedium and heavy nuclei ContinuumContinuum fully taken into accountfully taken into account PTG basis PTG basis very good for very good for weakly bound systemsweakly bound systems Nickel chain Nickel chain close to neutron driplineclose to neutron dripline Deformed nuclei Deformed nuclei with Mg and Zn, with Mg and Zn, prolate and oblate deformation prolate and oblate deformation • PerspectivesPerspectives GSM:GSM: Coherent framework Coherent framework unifying unifying structurestructure and and reactionreaction,, many body many body open channels.open channels. Realistic interactions Realistic interactions and and DMRG method DMRG method to be applied in calculations.to be applied in calculations. PTG/HFB, Gamow/HFB: PTG/HFB, Gamow/HFB: mean field mean field and and beyond (QRPA)beyond (QRPA)