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Shell Model approach for two-proton radioactivity

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). Plan. Experimental data R-matrix for diproton emission

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Shell Model approach for two-proton radioactivity

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  1. Shell Model approach for two-proton radioactivity Nicolas Michel(CEA / IRFU / SPhN) MarekPloszajczak(GANIL) Jimmy Rotureau(ORNL – University of Tennessee) WitekNazarewicz(ORNL – University of Tennessee)

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

  3. Experimental data B. Blank et al., Phys. Rev. Lett., 94, 232501 (2005) C. Dossat et al., Phys. Rev. C, 72, 054315 (2005) Threediproton emitters discovered:45Fe,54Zn, (48Ni) Theoretical description: new modelsto bedevelopped

  4. R-matrix formulation • 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,µsp2: reduced mass, channel radius, single particle reduced width • Standard Shell Model: spectroscopic factors only • R-matrix reaction formulas:single particle fit, s-wave phase shifts (p+p) • No mixing between channels:no continuum coupling (A. Brown, F.C. Barker, Phys. Rev. C 67, 041304(R) (2003))

  5. SMEC: one particle in the continuum • Feshbach space separation: • Q:A bound and quasi-bound (narrow resonant) one-body states • P:A-1bound and quasi-bound one-body states, 1 scattering state • Hamiltonian, wave functions: (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))

  6. SMEC: two particles in the continuum • Two-body clusters added: • Q:A bound and resonant one-body states • P:A-1bound and resonant one-body states, 1 scattering state • T:A-2bound and resonant one-body states, 2 scattering states • Approximations necessary: • Full problem currently impossible to treat (zero-range interaction divergence) • Cluster emission:diproton considered as a closed system • Sequential emission:Two independent protons emitted, two two-body decays • system resonant: standardsequential emission • system scattering: virtual sequential emission (J. Rotureau, J. Okolowicz and M. Ploszajczak, Phys. Rev. Lett., 95, 042503 (2005) ; Nucl. Phys. A, 767,13 (2006))

  7. SMEC: cluster approximation • Effective Hamiltonian:HPT and HTP couplings neglected • Two-body cluster treatment: • Internal diproton degrees of freedom: phenomenological • Integration over energy of cluster U, weighted by p+ps-wave density of states ½(U) • Effective two-body reaction

  8. SMEC: sequential decay • Effective Hamiltonian:HQT and HTQ couplings neglected • Two independent decays: • h: mean field of the first emitted proton on the A-1 daughter nucleus • Q’,P’ : subspaces associated to system • All interactions between the emitted protons averaged or suppressed

  9. SMEC: diproton decay • Q space:1s 0d 0f 1p • Interaction in Q space:USD, KB3, G-matrix • Interaction in P space: B. Blank, M. Ploszajczak, to bepublished

  10. SMEC: 45Fe (J. Rotureau, J. Okolowicz and M. Ploszajczak,Nucl. Phys. A ,767,13 (2006))

  11. SMEC: 48Ni (J. Rotureau, J. Okolowicz and M. Ploszajczak,Nucl. Phys. A, 767,13 (2006))

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

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

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

  15. 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:

  16. Model for 6He and 6Be • 6He, 6Be: valence particles, 4He core: Hn = T + WS(5He) + SGI Hp = T + WS(5Li) + SGI + Vc WS(5Li) = WSnucl + Uc(Z=2) 0p3/2, 0p1/2 (resonant), contours of p3/2 and p1/2 scattering states SGI : 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 H0 one-body basis:

  17. Nuclear energies • WS potentials:V0 = 47 MeV (5He/6He), V0 = 47.5 MeV (5Li/6Be) • SGI interaction:V(J=0) = -403 MeV fm3, V(J=2) = -610 MeV fm3

  18. 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: non resonant components necessary.

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

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