Nicolas Michel CEA / IRFU / SPhN Shell Model approach for two-proton radioactivity Nicolas Michel (CEA / IRFU / SPhN) Marek Ploszajczak (GANIL) Jimmy Rotureau (ORNL – University of Tennessee) Witek Nazarewicz (ORNL – University of Tennessee) October 11-13, 2008

Nicolas Michel CEA / IRFU / SPhN 2 Plan Experimental data Experimental data R-matrix for diproton emission R-matrix for diproton emission Shell Model Embedded in the Continuum (SMEC) Shell Model Embedded in the Continuum (SMEC) SMEC with one and two particles in the continuum SMEC with one and two particles in the continuum Used approximations for diproton emission and results Used approximations for diproton emission and results Gamow Shell Model with valence protons Gamow Shell Model with valence protons Berggren completeness relation and Coulomb interaction Berggren completeness relation and Coulomb interaction Mirror effects in 6 He and 6 Be : spectroscopic factors Mirror effects in 6 He and 6 Be : spectroscopic factors Conclusion et perspectives Conclusion et perspectives

October 11-13, 2008 Nicolas Michel CEA / IRFU / SPhN 3 Experimental data Three diproton emitters discovered: 45 Fe, 54 Zn, ( 48 Ni) Theoretical description: new models to be developped B. Blank et al., Phys. Rev. Lett., 94, (2005) C. Dossat et al., Phys. Rev. C, 72, (2005)

October 11-13, 2008 Nicolas Michel CEA / IRFU / SPhN 4 (A. Brown, F.C. Barker, Phys. Rev. C 67, (R) (2003)) Standard Shell Model: spectroscopic factors only Standard Shell Model: spectroscopic factors only R-matrix reaction formulas: single particle fit, s-wave phase shifts (p+p) R-matrix reaction formulas: single particle fit, s-wave phase shifts (p+p) No mixing between channels: no continuum coupling No mixing between channels: no continuum coupling Extension of R-matrix standard formulas: Extension of R-matrix standard formulas: Q,P,S: available energy, penetration and shift factors ½ (U): density of p+p states from s-wave phase shifts M,a, µ sp 2 : reduced mass, channel radius, single particle reduced width R-matrix formulation

October 11-13, 2008 Nicolas Michel CEA / IRFU / SPhN Feshbach space separation: Feshbach space separation: Q: A bound and quasi-bound (narrow resonant) one-body states Q: A bound and quasi-bound (narrow resonant) one-body states P: A-1 bound and quasi-bound one-body states, 1 scattering state P: A-1 bound and quasi-bound one-body states, 1 scattering state Hamiltonian, wave functions: Hamiltonian, wave functions: 5 (K.Bennaceur, N.Michel, F. Nowacki, J. Okolowicz and M. Ploszajczak, Phys. Lett. B, 488, 75 (2000)) (J. Okolowicz, M. Ploszajczak, and I. Rotter, Phys. Rep., 374, 271 (2003)) SMEC: one particle in the continuum

October 11-13, 2008 Nicolas Michel CEA / IRFU / SPhN Two-body clusters added: Two-body clusters added: Q: A bound and resonant one-body states Q: A bound and resonant one-body states P: A-1 bound and resonant one-body states, 1 scattering state P: A-1 bound and resonant one-body states, 1 scattering state T: A-2 bound and resonant one-body states, 2 scattering states T: A-2 bound and resonant one-body states, 2 scattering states Approximations necessary: Approximations necessary: Full problem currently impossible to treat (zero-range interaction divergence) Full problem currently impossible to treat (zero-range interaction divergence) Cluster emission: diproton considered as a closed system Cluster emission: diproton considered as a closed system Sequential emission: Two independent protons emitted, two two-body decays Sequential emission: Two independent protons emitted, two two-body decays system resonant: standard sequential emission system resonant: standard sequential emission system scattering: virtual sequential emission system scattering: virtual sequential emission 6 (J. Rotureau, J. Okolowicz and M. Ploszajczak, Phys. Rev. Lett., 95, (2005) ; Nucl. Phys. A, 767, 13 (2006)) SMEC: two particles in the continuum

October 11-13, 2008 Nicolas Michel CEA / IRFU / SPhN Effective Hamiltonian: H PT and H TP couplings neglected Effective Hamiltonian: H PT and H TP couplings neglected Two-body cluster treatment: Two-body cluster treatment: Internal diproton degrees of freedom: phenomenological Internal diproton degrees of freedom: phenomenological Integration over energy of cluster U, weighted by p+p s-wave density of states ½ (U) Integration over energy of cluster U, weighted by p+p s-wave density of states ½ (U) Effective two-body reaction Effective two-body reaction 7 SMEC: cluster approximation

October 11-13, 2008 Nicolas Michel CEA / IRFU / SPhN Effective Hamiltonian: H QT and H TQ couplings neglected Effective Hamiltonian: H QT and H TQ couplings neglected Two independent decays: Two independent decays: h: mean field of the first emitted proton on the A-1 daughter nucleus h: mean field of the first emitted proton on the A-1 daughter nucleus Q’,P’ : subspaces associated to system Q’,P’ : subspaces associated to system All interactions between the emitted protons averaged or suppressed All interactions between the emitted protons averaged or suppressed 8 SMEC: sequential decay

October 11-13, 2008 Nicolas Michel CEA / IRFU / SPhN 9 SMEC: diproton decay Q space: 1s 0d 0f 1p Q space: 1s 0d 0f 1p Interaction in Q space: USD, KB3, G-matrix Interaction in Q space: USD, KB3, G-matrix Interaction in P space: Interaction in P space: B. Blank, M. Ploszajczak, to be published

October 11-13, 2008 Nicolas Michel CEA / IRFU / SPhN 10 SMEC: 45 Fe (J. Rotureau, J. Okolowicz and M. Ploszajczak, Nucl. Phys. A,767, 13 (2006))

October 11-13, 2008 Nicolas Michel CEA / IRFU / SPhN 11 SMEC: 48 Ni (J. Rotureau, J. Okolowicz and M. Ploszajczak, Nucl. Phys. A, 767, 13 (2006))

October 11-13, 2008 Nicolas Michel CEA / IRFU / SPhN 12 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 : Straightforward generalization to non-local potentials (HF) Straightforward generalization to non-local potentials (HF)

October 11-13, 2008 Nicolas Michel CEA / IRFU / SPhN 13 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

October 11-13, 2008 Nicolas Michel CEA / IRFU / SPhN 14 Complex energy states bound states broad resonances narrow resonances L + : arbitrary contour antibound states capturing states Im(k) Re(k) Berggren completeness relation

October 11-13, 2008 Nicolas Michel CEA / IRFU / SPhN 15 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, (2008)) Extended to proton case (N. Michel, J. Math. Phys., 49, (2008)) Continuum discretization: Continuum discretization: N-body completeness relation: N-body completeness relation:

October 11-13, 2008 Nicolas Michel CEA / IRFU / SPhN 16 Model for 6 He and 6 Be 6 He, 6 Be: valence particles, 4 He core : H n = T + WS( 5 He) + SGI 6 He, 6 Be: valence particles, 4 He core : H n = T + WS( 5 He) + SGI H p = T + WS( 5 Li) + SGI + V c H p = T + WS( 5 Li) + SGI + V c WS( 5 Li) = WS nucl + U c (Z=2) WS( 5 Li) = WS nucl + U c (Z=2) 0p 3/2, 0p 1/2 (resonant), contours of p 3/2 and p 1/2 scattering states 0p 3/2, 0p 1/2 (resonant), contours of p 3/2 and p 1/2 scattering states SGI : Surface Gaussian Interaction: SGI : 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 0 one-body basis: H 0 one-body basis:

October 11-13, 2008 Nicolas Michel CEA / IRFU / SPhN 17 Nuclear energies WS potentials: V 0 = 47 MeV ( 5 He/ 6 He), V 0 = 47.5 MeV ( 5 Li/ 6 Be) WS potentials: V 0 = 47 MeV ( 5 He/ 6 He), V 0 = 47.5 MeV ( 5 Li/ 6 Be) SGI interaction: V(J=0) = -403 MeV fm 3, V(J=2) = -610 MeV fm 3 SGI interaction: V(J=0) = -403 MeV fm 3, V(J=2) = -610 MeV fm 3

October 11-13, 2008 Nicolas Michel CEA / IRFU / SPhN 18 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 : non resonant components necessary. 5 He / 6 He, 5 Li / 6 Be : non resonant components necessary.

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October 11-13, 2008 Nicolas Michel CEA / IRFU / SPhN 22 Conclusion et perspectives Conclusion Conclusion SMEC and GSM: Two complementary models SMEC and GSM: Two complementary models SMEC: convenient for proton emitters, very small widths can be calculated SMEC: convenient for proton emitters, very small widths can be calculated Approximations needed for the moment, very complicated formulas Approximations needed for the moment, very complicated formulas Emission channels interfere: model-dependent description (cluster, sequential) Emission channels interfere: model-dependent description (cluster, sequential) GSM: First calculations with several valence protons GSM: First calculations with several valence protons Simple and powerful model Simple and powerful model Spectroscopic factors: mirror effects due to continuum Spectroscopic factors: mirror effects due to continuum Perspectives Perspectives More realistic interactions to be used with SMEC and GSM More realistic interactions to be used with SMEC and GSM SMEC: full problem with three-body asymptotics possible SMEC: full problem with three-body asymptotics possible GSM: study of a larger set of light nuclei GSM: study of a larger set of light nuclei