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Isospin mixing and the continuum coupling in weakly bound nuclei

Isospin mixing and the continuum coupling in weakly bound nuclei. Nicolas Michel ( University of Jyväskylä ) Marek Ploszajczak (GANIL) Witek Nazarewicz (ORNL – University of Tennessee). Plan . Experimental motivation Berggren completeness relation and Gamow Shell Model

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Isospin mixing and the continuum coupling in weakly bound nuclei

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  1. Isospin mixing and the continuum coupling in weakly bound nuclei Nicolas Michel(University of Jyväskylä) MarekPloszajczak(GANIL) WitekNazarewicz(ORNL – University of Tennessee)

  2. Plan • Experimental motivation • Berggren completeness relation and Gamow Shell Model • Cluster orbital shell model and Hamiltonian definition • Spectroscopic factor definition • Treatment of Coulomb interaction and recoil term • Isospin symmetry breaking in 6He, 6Be and 6Li • Spectroscopic factors, energies, T+/- and T2 expectation values • Conclusion

  3. Halos, resonant states

  4. Gamow states • Georg Gamow : simple model for a decay G.A. Gamow, Zs f. Phys. 51 (1928) 204; 52 (1928) 510 • Definition :

  5. Complex scaling • Calculation of radial integrals:exteriorcomplex scaling • Analytic continuation : integral independent of R and θ

  6. Complex energy states Berggren completeness relation Im(k) bound states narrowresonances Re(k) antibound states L+ : arbitrary contour broadresonances capturing states

  7. Completeness relation with Gamow states • Berggren completeness relation (l,j) : T. Berggren, Nucl. Phys. A 109, (1967) 205 (neutrons only) Extended to proton case (N. Michel, J. Math. Phys., 49, 022109 (2008)) • Continuum discretization: • N-body completeness relation:

  8. Cluster orbital shell model • Shell model: 3A degrees of freedom (particles coordinates) 3(A-1) physically (translational invariance) → spurious states • Lawson method (standard shell model) : Nħωspacesonly: unavailable for Berggren bases • Solution : cluster orbital shell model, core coordinates. Relative coordinates: no center of mass excitation

  9. Hamiltonian definition • 6He, 6Be, 6Li: valence particles, 4He core: H = T1b + WS(5Li/5He) + MSGI + Vc + Trec 0p3/2 (resonant), contours of s1/2, p3/2, p1/2, d5/2, d3/2 scattering states, recoil included MSGI : Modified Surface Gaussian Interaction: • 6Be:Coulomb interaction necessary Problem:long-range, lengthy 2D complex scaling, divergences Solution:one-body long-range / two-body short-range separation H1b one-body basis:

  10. Spectroscopic factors in GSM • One particle emission channel: (l,j,p/n) • Basis-independent definition: • Experimental: all energies taken into account • Standard : representation dependence (n,l,j,p/n) • 5He / 6He, 5Li / 6Be, 5He / 6Li, 5Li / 6Li non resonant components necessary.

  11. Coulomb interaction and recoil term Harmonicoscillator expansion Physicalprecision of the order of 1 keV Sufficient for practical applications N. Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315

  12. 6Be/5Li – 6He/5He 6Li/5He – 6Li/5Li Cusps (π) Cusps(ν) πasymptotic≠νasymptotic N. Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315

  13. Spectroscopic factors distribution Re[S2] > 1, Im[S2] ≠ 0 Large occupation ofnon-resonant continuum N. Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315

  14. Observables of 0+, 2+ (T=1) states V1 : WSnucl(π) = WSnucl(ν) V2 : WS(π) fitted to 6Be bindingenergy N. Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315

  15. Configuration mixing of 0+ (T=1) states V1 and V2 fits, recoil: slight change of basis states occupation Redistribution of basis states occupation fromCoulomb Hamiltonian N. Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315

  16. Isospin operators expectation values • Isospin operators: Same basis demanded for protons and neutrons Coulomb infinite-range part in 1/r to diagonalize • 1/r matrix representation with Berggren basis Infinities appear on the diagonal withscattering states : • Possible treatments: Cut after r >R : no infinities but very crude Analytical subtraction of integrable singularities : Off-diagonal method : replacement of diverging by N. Michel, Phys. Rev. C, 83 (2011) 034325

  17. 1/r treatment precision Cutmethod Subtraction method Off-diagonal method Numericalprecisionobtained with off-diagonal method N. Michel, Phys. Rev. C, 83 (2011) 034325

  18. Application to 0+ (T=1) states IAS: Isobaric analog state 0+ of 6Li almost isospin invariant 0+ of 6Be shows large isospin asymmetry 6Be : two valence protons → T=1 exactly Partial dynamicalsymmetry N.Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315

  19. Conclusion • GSM:Exact calculations with valence protons and neutrons Recoil exactly taken into account with COSM formalism Coulomb interaction: exact asymptotic via Z = Zval potential introduction Theoretical and numerical errors of the model controlled • Isospin asymmetry:Proton and neutron spectroscopic factors 0+and 2+ T=1 triplets of 6He, 6Li and6Be Same separation energies for all A=6 systems Differences from Coulomb Hamiltonian only: continuum coupling Spectroscopic factors : neutron with cusps, proton without cusps Different configuration mixings for isobaric analog states T2and T- expectation values : partial dynamical symmetry Origin : Coulomb+continuum, no charge-dependent effective forces

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