Description of dripline nuclei with GSM and GamowHFB

Description of drip-line nuclei with GSM and Gamow/HFB frameworks Nicolas Michel (CEA/IRFU/SPh. N) Marek Ploszajszak (GANIL) Witek Nazarewicz (UT/ORNL) Kenichi Matsuyanagi (Kyoto University) Mario Stoitsov (ORNL – University of Tennessee) May 20 th, 2009 Importance of continuum for nuclei close to drip-line Nicolas Michel

Plan • Scientific motivation: drip-line nuclei • Gamow states definition and normalization, Berggren basis • Spectroscopic factors, overlap functions • • • 6 He/5 He, 7 He/6 He, 18 O*/17 O*, 6 Be/5 Li 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) • Conclusion and perspectives May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 2

Scientific motivation May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 3

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) May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 4

Complex scaling • Normalization: complex scaling • Analytic continuation: integral independent of R and θ • Normalization of bound and resonant states: • Scattering states: normalization impossible with complex scaling Normalization with Dirac delta: May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 5

Complex energy states Im(k) bound Berggren completeness relation narrow resonances Re(k) antibound states L+ : arbitrary contour capturing states May 20 th, 2009 broad resonances Importance of continuum for nuclei close to drip-lines Nicolas Michel 6

Completeness relations with Gamow states • Berggren completeness relation (l, j) : T. Berggren, Nucl. Phys. A 109, (1967) 205 • Continuum discretization : May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 7

Application : O, He and Be isotopes • O isotopes : valence particles above 16 O core : H = T + WS (17 O) + SDI 0 d 5/2, 1 s 1/2 (bound), 0 d 3/2 (resonant), d 3/2 scattering continuum • He and Be isotopes : valence particles above 4 He core H = T + WS (5 He/5 Li) + SGI 0 p 3/2, 0 p 1/2 (resonant), p 3/2 and p 1/2 scattering continuums SGI : Surface Gaussian Interaction : May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 8

Spectroscopic factors in GSM • One particle emission channel: (l, j, p/n) • Basis-independent definition: • Experimental : cusps appear in cross-sections • Standard : representation dependence (n, l, j, p/n) • 5 He / 6 He, 5 Li / 6 Be : non resonant components necessary. May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 9

Spectroscopic factor: VSGI varies Nucleon-nucleon correlations measure WS varies Universal behavior close to emission threshold VSGI varies N. Michel et al. , Phys. Rev. C, (Rap. Comm. ), 75 031301(R) (2007) May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 10

Real WS fit Complex WS fit O(r) N. Michel, W. Nazarewicz, M. Ploszajczak, Nucl. Phys. A 794 29 (2007) May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 11

WS varies N. Michel, W. Nazarewicz, M. Ploszajczak, Nucl. Phys. A 794 29 (2007) May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 12

VSGI varies N. Michel, W. Nazarewicz, M. Ploszajczak, Nucl. Phys. A 794 29 (2007) May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 13

6 Be/5 Li 5 He V 6 SGI He/ varies May 20 th, 2009 Mirror reactions : 6 He/5 He and 6 Be/5 Li VSGI varies Importance of continuum for nuclei close to drip-lines Nicolas Michel 14

Mirror reactions : 6 He/5 He and 6 Be/5 Li 6 He/5 He. V SGI varies May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 15

Mirror reactions : 6 He/5 He and 6 Be/5 Li 6 He/5 He VSGI varies May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 16

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) May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 17

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-Ginocchio (PTG) potential • Examples: Ni chain, 110 Zr and 40 Mg, HFB density functional: Sly 4 + surface pairing orbital momentum: l=0 to l=10, Ecut = 60 Me. V May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 18

Gamow HFB framework: HF basis diagonalization • Two-basis method Basis generated by ph part of HFB hamiltonian: B. Gall et al. , Z. Phys. A 348 183 (1994) • HFB matrix structure: • Diagonalization of HFB matrix in PTG or Gamow HF basis May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 19

Gamow HFB framework: direct integration N. Michel, K. Matsuyanagi , M. Stoitsov N. Michel, K. Matsuyanagi, M. Phys. Rev. C, 78 78 044319 (2008) Phys. Rev. C, May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 20

HF and PTG potentials M. Stoitsov , N. Michel, K. Matsuyanagi, , M. Stoitsov, N. Michel, Phys. Rev. C, 77 77, 054301 (2008) Phys. May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 21

HF/PTG wave functions ---- : PTG : HF M. Stoitsov , N. Michel, K. Matsuyanagi, , M. Stoitsov, N. Michel, Phys. Rev. C, 77 77, 054301 (2008) Phys. Rev. C, r (fm) May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 22

Long axis PTG basis Short axis densities ----- : prot. : neut. … : THO M. Stoitsov , N. Michel, K. Matsuyanagi, , M. Stoitsov, N. Michel, Phys. Rev. C, 77 77, 054301 (2008) Phys. May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 23

PTG basis Pairing densities ----- : prot. : neut. … : THO M. Stoitsov , N. Michel, K. Matsuyanagi, , M. Stoitsov, N. Michel, Phys. Rev. C, 77 77, 054301 (2008) Phys. Rev. C, May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 24

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 N. Michel, K. Matsuyanagi, M. Phys. Rev. C, 78 78 044319 (2008) Phys. May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 25

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 N. Michel, K. Matsuyanagi, M. Phys. Rev. C, 78 78 044319 (2008) Phys. May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 26

Conclusion and perspectives • GSM : Standard shell model simplicity and power beyond Hilbert space. Complete description of loosely bound and resonant states. . Spectroscopic factors: continum coupling necessary Cusps: clear l-dependence Important differences between proton and neutron case • 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 Nickel chain close to neutron dripline Deformed nuclei with Mg and Zn, prolate and oblate deformation • Perspectives GSM: Coherent framework unifying structure and reaction, many body open channels. Realistic interactions and DMRG method to be applied in calculations. PTG/HFB, Gamow/HFB: mean field and beyond (QRPA) May 20 th, 2009 Importance of continuum for nuclei close to drip-lines Nicolas Michel 27