1. Nicolas Michel CEA / IRFU / SPhN / ESNT April 26-29, 2011 Isospin mixing and the continuum coupling in weakly bound nuclei Nicolas Michel (University of Jyväskylä) Marek Ploszajczak (GANIL) Witek Nazarewicz (ORNL – University of Tennessee)

2. April 26-29, 2011 Nicolas Michel CEA / IRFU / SPhN / ESNT 2 Plan Experimental motivation Experimental motivation Berggren completeness relation and Gamow Shell Model Berggren completeness relation and Gamow Shell Model Cluster orbital shell model and Hamiltonian definition Cluster orbital shell model and Hamiltonian definition Spectroscopic factor definition Spectroscopic factor definition Treatment of Coulomb interaction and recoil term Treatment of Coulomb interaction and recoil term Isospin symmetry breaking in 6 He, 6 Be and 6 Li Isospin symmetry breaking in 6 He, 6 Be and 6 Li Spectroscopic factors, energies, T +/- and T 2 expectation values Spectroscopic factors, energies, T +/- and T 2 expectation values Conclusion Conclusion

3. April 26-29, 2011 Nicolas Michel CEA / IRFU / SPhN / ESNT 3 Halos, resonant states

4. April 26-29, 2011 Nicolas Michel CEA / IRFU / SPhN / ESNT 4

5. April 26-29, 2011 Nicolas Michel CEA / IRFU / SPhN / ESNT 5 Gamow states Georg Gamow : simple model for  decay Georg Gamow : simple model for  decay G.A. Gamow, Zs f. Phys. 51 (1928) 204; 52 (1928) 510 G.A. Gamow, Zs f. Phys. 51 (1928) 204; 52 (1928) 510 Definition : Definition :

6. April 26-29, 2011 Nicolas Michel CEA / IRFU / SPhN / ESNT 6 Complex scaling Calculation of radial integrals: exterior complex scaling Calculation of radial integrals: exterior complex scaling Analytic continuation : integral independent of R and θ Analytic continuation : integral independent of R and θ

7. April 26-29, 2011 Nicolas Michel CEA / IRFU / SPhN / ESNT 7 Complex energy states bound states broad resonances narrow resonances L + : arbitrary contour antibound states capturing states Im(k) Re(k) Berggren completeness relation

8. April 26-29, 2011 Nicolas Michel CEA / IRFU / SPhN / ESNT 8 Completeness relation with Gamow states Berggren completeness relation (l,j) : Berggren completeness relation (l,j) : T. Berggren, Nucl. Phys. A 109, (1967) 205 (neutrons only) T. Berggren, Nucl. Phys. A 109, (1967) 205 (neutrons only) Extended to proton case (N. Michel, J. Math. Phys., 49, 022109 (2008)) Extended to proton case (N. Michel, J. Math. Phys., 49, 022109 (2008)) Continuum discretization: Continuum discretization: N-body completeness relation: N-body completeness relation:

9. April 26-29, 2011 Nicolas Michel CEA / IRFU / SPhN / ESNT 9 Cluster orbital shell model Shell model : 3A degrees of freedom (particles coordinates) Shell model : 3A degrees of freedom (particles coordinates) 3(A-1) physically (translational invariance) → spurious states 3(A-1) physically (translational invariance) → spurious states Lawson method (standard shell model) : Lawson method (standard shell model) : Nħω spaces only : unavailable for Berggren bases Nħω spaces only : unavailable for Berggren bases Solution : cluster orbital shell model, core coordinates. Solution : cluster orbital shell model, core coordinates. Relative coordinates: no center of mass excitation Relative coordinates: no center of mass excitation

10. April 26-29, 2011 Nicolas Michel CEA / IRFU / SPhN / ESNT 6 He, 6 Be, 6 Li: valence particles, 4 He core : 6 He, 6 Be, 6 Li: valence particles, 4 He core : H = T 1b + WS( 5 Li/ 5 He) + MSGI + V c + T rec H = T 1b + WS( 5 Li/ 5 He) + MSGI + V c + T rec 0p 3/2 (resonant), contours of s 1/2, p 3/2, p 1/2, d 5/2, d 3/2 scattering states, recoil included 0p 3/2 (resonant), contours of s 1/2, p 3/2, p 1/2, d 5/2, d 3/2 scattering states, recoil included MSGI : Modified Surface Gaussian Interaction: MSGI : Modified Surface Gaussian Interaction: 6 Be: Coulomb interaction necessary 6 Be: Coulomb interaction necessary Problem: long-range, lengthy 2D complex scaling, divergences Problem: long-range, lengthy 2D complex scaling, divergences Solution: one-body long-range / two-body short-range separation Solution: one-body long-range / two-body short-range separation H 1b one-body basis: H 1b one-body basis: 10 Hamiltonian definition

11. April 26-29, 2011 Nicolas Michel CEA / IRFU / SPhN / ESNT 11 Spectroscopic factors in GSM One particle emission channel: (l,j,  ) One particle emission channel: (l,j,  ) Basis-independent definition: Basis-independent definition: Experimental : all energies taken into account Experimental : all energies taken into account Standard : representation dependence (n,l,j,  ) Standard : representation dependence (n,l,j,  ) 5 He / 6 He, 5 Li / 6 Be, 5 He / 6 Li, 5 Li / 6 Li 5 He / 6 He, 5 Li / 6 Be, 5 He / 6 Li, 5 Li / 6 Li non resonant components necessary. non resonant components necessary.

12. April 26-29, 2011 Nicolas Michel CEA / IRFU / SPhN / ESNT 12 Coulomb interaction and recoil term Harmonic oscillator expansion Physical precision of the order of 1 keV Sufficient for practical applications N. Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315

13. April 26-29, 2011 Nicolas Michel CEA / IRFU / SPhN / ESNT 13 6 Be/ 5 Li – 6 He/ 5 He 6 Li/ 5 He – 6 Li/ 5 Li Cusps (ν) π asymptotic ≠ ν asymptotic π asymptotic ≠ ν asymptotic N. Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315 Cusps (π)

14. April 26-29, 2011 Nicolas Michel CEA / IRFU / SPhN / ESNT 14 Spectroscopic factors distribution Re[S 2 ] > 1, Im[S 2 ] ≠ 0 Large occupation of non-resonant continuum Large occupation of non-resonant continuum N. Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315

15. April 26-29, 2011 Nicolas Michel CEA / IRFU / SPhN / ESNT 15 0+ state 6 He 6 Be (V1) 6 Be (V2) 6 Li (V1) 6 Li (V2) E calc (MeV)-0.9741.6531.3710.0866-0.0706 E exp (MeV)-0.9731.371 0.136 Γ calc (keV)041148.85 ·10 -3 9.13 ·10 -3 Γ exp (keV)092 8.2 ·10 -3 S 2 (π)————1.015-i0.1471.015-i0.1771.061-i0.2801.028-i0.300 S 2 (ν)0.87-i0.383————— 0.911-i0.3610.898-i0.369 2+ state 6 He 6 Be (V1) 6 Be (V2) 6 Li (V1) 6 Li (V2) E calc (MeV)0.8232.8872.6791.6671.569 E exp (MeV)0.8243.041 1.667 Γ calc (keV)89986804404329 Γ exp (keV)1131160 541 S 2 (π)—————0.973-i0.0140.978-i0.0160.987-i0.0030.993-i0.003 S 2 (ν)1.061+i0.001————— 1.034-i0.0241.043-i0.022 Observables of 0 +, 2 + (T=1) states V1 : WS nucl (π) = WS nucl (ν) V2 : WS(π) fitted to 6 Be binding energy N. Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315

16. April 26-29, 2011 Nicolas Michel CEA / IRFU / SPhN / ESNT 16 (C k ) 2 6 He 6 He (rig. core) 6 Be (V1) 6 Be (V2) 6 Li (V1) 6 Li (V2) (0p 3/2 ) 2 0.750-i 0.6920.798-i 0.7321.090-i 0.2431.107-i 0.2880.994-i0.5870.949-i0.614 S1(πp 3/2) ————— -0.115+i0.218-0.143+i0.255-0.084+i0.226-0.050+i0.244 S1(νp 3/2) 0.243+i0.6190.244+i0.668————— 0.066+i0.3080.0797+i0.314 S2(s 1/2 ) 0.009+i0.00.0+i0.00.022+i0.00.023+i0.0040.011+i0.00.010+i0.0 S2(p 1/2 ) 0.012+i0.00.013+i0.00.008+i0.0010.009+i0.00.011+i0.00.012+i0.0 S2(p 3/2 ) -0.049-i0.074-0.063+i0.065-0.030+i0.029-0.028+i0.034-0.033+i0.054-0.033+i0.055 S2(d 3/2 ) 0.002+i0.00.001+i0.00.002+i0.00.002-i0.00.002+i0.0 S2(d 5/2 ) 0.032+i0.00.006+i0.00.025-i0.00.031-i0.040.031+i0.0 Configuration mixing of 0 + (T=1) states V1 and V2 fits, recoil : slight change of basis states occupation Redistribution of basis states occupation from Coulomb Hamiltonian N. Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315

17. April 26-29, 2011 Nicolas Michel CEA / IRFU / SPhN / ESNT Isospin operators: Isospin operators: Same basis demanded for protons and neutrons Same basis demanded for protons and neutrons Coulomb infinite-range part in 1/r to diagonalize Coulomb infinite-range part in 1/r to diagonalize 1/r matrix representation with Berggren basis 1/r matrix representation with Berggren basis Infinities appear on the diagonal with scattering states : Infinities appear on the diagonal with scattering states : Possible treatments: Possible treatments: Cut after r >R : no infinities but very crude Cut after r >R : no infinities but very crude Analytical subtraction of integrable singularities : Analytical subtraction of integrable singularities : Off-diagonal method : replacement of diverging by Off-diagonal method : replacement of diverging by 17 Isospin operators expectation values N. Michel, Phys. Rev. C, 83 (2011) 034325

18. April 26-29, 2011 Nicolas Michel CEA / IRFU / SPhN / ESNT 18 1/r treatment precision Cut method Off-diagonal method method Subtractionmethod Numerical precision obtained with off-diagonal method N. Michel, Phys. Rev. C, 83 (2011) 034325

19. April 26-29, 2011 Nicolas Michel CEA / IRFU / SPhN / ESNT 19 6 Li(V1) 6 Be(V1) ‹ 0+ | 0+ IAS › 2 0.9950.951-i0.050 T av 0.99941 Application to 0 + (T=1) states 0 + of 6 Li almost isospin invariant 0 + of 6 Be shows large isospin asymmetry 6 Be : two valence protons → T=1 exactly Partial dynamical symmetry N.Michel, W. Nazarewicz, M. Ploszajczak, Phys. Rev. C, 82 (2010) 044315 IAS:Isobaricanalogstate

20. April 26-29, 2011 Nicolas Michel CEA / IRFU / SPhN / ESNT 20 Conclusion GSM: Exact calculations with valence protons and neutrons GSM: Exact calculations with valence protons and neutrons Recoil exactly taken into account with COSM formalism Recoil exactly taken into account with COSM formalism Coulomb interaction: exact asymptotic via Z = Z val potential introduction Coulomb interaction: exact asymptotic via Z = Z val potential introduction Theoretical and numerical errors of the model controlled Theoretical and numerical errors of the model controlled Isospin asymmetry: Proton and neutron spectroscopic factors Isospin asymmetry: Proton and neutron spectroscopic factors 0 + and 2 + T=1 triplets of 6 He, 6 Li and 6 Be 0 + and 2 + T=1 triplets of 6 He, 6 Li and 6 Be Same separation energies for all A=6 systems Same separation energies for all A=6 systems Differences from Coulomb Hamiltonian only: continuum coupling Differences from Coulomb Hamiltonian only: continuum coupling Spectroscopic factors : neutron with cusps, proton without cusps Spectroscopic factors : neutron with cusps, proton without cusps Different configuration mixings for isobaric analog states Different configuration mixings for isobaric analog states T 2 and T - expectation values : partial dynamical symmetry T 2 and T - expectation values : partial dynamical symmetry Origin : Coulomb+continuum, no charge-dependent effective forces Origin : Coulomb+continuum, no charge-dependent effective forces