Description of medium and heavy dripline nuclei within the PTGHFB and GamowHFB frameworks

1
Description of medium and heavy drip-line nuclei
within the PTG/HFB and Gamow/HFB frameworks
Nicolas Michel (CEA/IRFU/SPhN) Kenichi
Matsuyanagi (Kyoto University) Mario Stoitsov
(ORNL University of Tennessee)
2
Plan

• Scientific motivation drip-line nuclei
• Gamow states definition and normalization,
  Berggren basis
• Gamow HFB framework HF basis diagonalization,
  direct integration
• Applications Nickel chain
• Pöschl-Teller-Ginocchio (PTG) basis for loosely
  bound systems
• Applications Nickel chain (spherical)
• Zirconium and
  Magnesium (deformed)
• Optimized HFB matrix diagonalization Takagi
  factorization
• Conclusion and perspectives

3
Scientific motivation
4
Gamow states

• Georg Gamow simple model of a decay
• G.A. Gamow, Zs f. Phys. 51 (1928) 204 52
  (1928) 510
• Definition
• Straightforward generalization to non-local
  potentials (HF)

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Complex scaling

• Normalization complex scaling
• Analytic continuation integral independent of R
  and ?
• Normalization of bound and resonant states
• Scattering states normalization impossible with
  complex scaling
• Normalization with Dirac delta

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Complex energy states
Berggren completeness relation
Im(k)
bound
narrow resonances
Re(k)
antibound states
L arbitrary contour
broad resonances
capturing states
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Completeness relations with Gamow states

• Berggren completeness relation (l,j)
• T. Berggren, Nucl. Phys. A 109, (1967) 205
• Continuum discretization

8
HFB framework

• HFB ground state product of independent
  quasi-particles
• HFB equations
• Standard methods of resolution
• HO basis diagonalization well bound states
  only
• THO basis diagonalization basis dependence
  from scaling function ?
• Direct integration very precise, but long
  (box boundary conditions)

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PTG/HFB and Gamow/HFB models

• Continuous basis methods
• Complex-energy formalism
• Bound, resonant and scattering states
  (Berggren basis)
• Berggren basis of HF particle states
  (two-basis method)
• Berggren quasi-particles states calculated
  in coordinate space
• Real-energy formalism
• Basis generated by a Pöschl-Teller-Ginocchi
  o (PTG) potential
• Examples Ni chain, 110Zr and 40Mg, HFB density
  functional Sly4 surface pairing
• orbital momentum l0 to
  l10, Ecut 60 MeV

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Gamow HFB framework HF basis diagonalization

• Two-basis method
• Basis generated by ph part of HFB
  hamiltonian
• B. Gall et al., Z. Phys. A348 183 (1994)
• HFB matrix structure
• Diagonalization of HFB matrix in PTG or Gamow HF
  basis

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Gamow HFB frameworkdirect integration

N. Michel, K.Matsuyanagi, M. Stoitsov Phys. Rev.
C, 78 044319 (2008)
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HF and PTG potentials
M. Stoitsov, N. Michel, K.Matsuyanagi, Phys.
Rev. C, 77, 054301 (2008)
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HF/PTG wave functions



---- PTG HF
M. Stoitsov, N. Michel, K.Matsuyanagi, Phys.
Rev. C, 77, 054301 (2008)
r (fm)
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Long axis
PTG basis



Short axis
densities ----- prot. neut. THO
M. Stoitsov, N. Michel, K.Matsuyanagi, Phys.
Rev. C, 77, 054301 (2008)
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PTG basis



Pairing densities ----- prot.
neut. THO
M. Stoitsov, N. Michel, K.Matsuyanagi, Phys.
Rev. C, 77, 054301 (2008)
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Gamow HFB Nickel densities
Black solid line HFB box Dashed green line
GHFB coord. Dotted blue line GHF basis
N. Michel, K.Matsuyanagi, M. Stoitsov Phys. Rev.
C, 78 044319 (2008)
17
Gamow HFB Nickel pairing densities
Black solid line HFB box Dashed green line
GHFB coord. Dotted blue line GHF basis
N. Michel, K.Matsuyanagi, M. Stoitsov Phys. Rev.
C, 78 044319 (2008)
18
HFB equations

• Supermatrix standard representation
• Wasteful 2Nx2N matrix diagonalization for
  N vectors
• Equivalent formulation for real case
• Takagi factorization
• _Special case of SVD
• _ positive eigenvalues only
  quasi-particle energies
• _ up to
  complex phases only
• _Phase provided by phase-dependent equation

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Numerical results

• Comparison of different methods (Householder
  QL)
• Conclusion
• Takagi factorization much faster than full
  diagonalization (gain 3.5)
• Speed comparable to NxN complex hermitian
  diagonalization
• Divide-and-conquer or twisted factorization
  to be considered in the future

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Conclusion and perspectives

• HFB expansions with Gamow and PTG bases
• Precise tool to study dripline medium and
  heavy nuclei
• Continuum fully taken into account
• PTG basis very good for weakly bound systems
• First applications
• Nickel chain close to neutron dripline
• Deformed nuclei with Mg and Zn, prolate and
  oblate deformation
• Perspectives
• PTG/HFB, Gamow/HFB mean field and beyond
  (QRPA)
• Unbound many-body states at HFB and QRPA
  level
• Takagi factorization to calculate HFB wave
  functions