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Structure of Resonance and Continuum States

Structure of Resonance and Continuum States. Unbound Nuclei Workshop Pisa, Nov. 3-5, 2008. Hokkaido University. 1. Resolution of Identity in Complex Scaling Method. Bound st. Spectrum of Hamiltonian. Resonant st. Continuum st. Non-Resonant st.

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Structure of Resonance and Continuum States

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  1. Structure of Resonance and Continuum States Unbound Nuclei Workshop Pisa, Nov. 3-5, 2008 Hokkaido University

  2. 1. Resolution of Identity in Complex Scaling Method Bound st. Spectrum of Hamiltonian Resonant st. Continuum st. Non-Resonant st. Completeness Relation (Resolution of Identity) R R.G. Newton, J. Math. Phys. 1 (1960), 319

  3. Among the continuum states, resonant states are considered as an extension of bound states because they result from correlations and interactions. From this point of view, Berggren said “In the present paper,*) we investigate the properties**) of resonant statesand find them in many ways quite analogous to those of the ordinary bound states.” *) NPA 109 (1968), 265.　 **) orthogonality and completeness

  4. Separation of resonant states from continuum states Deformed continuum states Resonant states T. Berggren, Nucl. Phys. A 109, 265 (1968) Deformation of the contour Matrix elements of resonant states Convergence Factor Method Ya.B. Zel’dovich, Sov. Phys. JETP 12, 542 (1961). N. Hokkyo, Prog. Theor. Phys. 33, 1116 (1965).

  5. Complex scaling method reiθ coordinate: r B. Gyarmati and T. Vertse, Nucl. Phys. A160, 523 (1971). momentum: inclination of the semi-circle T. Myo, A. Ohnishi and K. Kato. Prog. Theor. Phys. 99(1998)801]

  6. Resolution of Identity in Complex Scaling Method E k E k Single Channel system B.Giraud and K.Kato, Ann.of Phys. 308 (2003), 115. E| E| b3 b2 b1 r2 r3 r1 B.Giraud, K.Kato and A. Ohnishi, J. of Phys. A37 (2004),11575 Coupled Channel system Three-body system

  7. Structures of three-body continuum states (Complex scaled)

  8. Physical Importance of Resonant States red: 0+ blue: 1- 0+ 1－ M. Homma, T. Myo and K. Kato, Prog. Theor. Phys. 97 (1997), 561.

  9. Kiyoshi Kato B.S. R.S. Sexc=1.5 e2fm2MeV Contributions from B.S. and R.S. to the Sum rule value

  10. ２．Complex Scaled COSM (A) Cluster Orbital Shell Model (COSM) • Y. Suzuki and K. Ikeda, Phys. Rev. C38 (1988), 410 Core+Xn system The total Hamiltonian: where HC : the Hamiltonian of the core cluster AC Ui : the interaction between the core and the valence neutron (Folding pot.) vij : the interaction between the valence neutrons(Minnesota force, Av8, …) X

  11. (B) Extended Cluster Model ー T-type coordinate system ー Y. Tosaka, Y Suzuki and K. Ikeda; Prog. Theor. Phys. 83 (1990), 1140. K. Ikeda; Nucl. Phys. A538 (1992), 355c. The di-neutron like correlation between valence neutrons moving in the spatially wide region θ which has a peak in a region : The two-neutron distance : When R~5-7fm, to describe the short range correlation accurately up to 0.5 fm, the maximum -value is 10~14.

  12. (C) Hybrid-TV Model S. Aoyama, S. Mukai, K. Kato and K. Ikeda, Prog. Theor. Phys. 94, 343-352 (1995) + (p3/2)2 (p1/2)2 Rapid convergence!! (p,sd)+T-base

  13. Two-neutrondensity distribution of 6He (0p3/2) ２ Hybrid-TV S=0 S=1 Total Harmonic oscillator (0p3/2 only) Hybrid-TV model (COSM 9ch + ECM 1ch)

  14. 18O 6He H.Masui, K. Kato and K.Ikeda, PRC75 (2007), 034316.

  15. Excitation of two-neutron halo nuclei (Borromean nuclei) Structure of three-body continuum Three-body resonant states Complex scaling method Resonant state Bound state (divergent) (no-divergent) Soft-dipole mode S. Aoyama, T. Myo, K, Kato and K. Ikeda; Prog. Theor. Phys. 116, (2006) 1.

  16. 1- ( Soft Dipole Resonance) pole in 4He+n+n (CSM+ACCC) Er~3 MeV Γ~32 MeV 1- resonant state?? It is difficult to observe as an isolated resonant state!! Y. Aoyama；Phys. Rev. C68 (2003) 034313.

  17. 7He: 4He+n+n+n COSM T. Myo, K. Kato and K. Ikeda, PRC76 (2007), 054309

  18. 3. Coulomb breakup reactions of Borromean systems Structures of three-body continuum state Coulomb breakup reaction

  19. Strength Functions of Coulomb Breakup Reaction

  20. T. Myo, A. Ohnishi and K. Kato, Prog. Theor. Phys. 99 (1998), 801. in CSM 10Li(1+)+n 10Li(2+)+n Resonances 9Li+n+n

  21. T. Myo, K. Kato, S. Aoyama and K. Ikeda, PRC63(2001),054313

  22. PRL 96, 252502 (2006) coupled channel [9Li+n+n] + [9Li*+n+n] T. Myo

  23. 4. Unified Description of Bound and Unbound States Continuum Level Density Definition of LD: A.T.Kruppa, Phys. Lett. B 431 (1998), 237-241 A.T. Kruppa and K. Arai, Phys. Rev. A59 (1999), 2556 K. Arai and A.T. Kruppa, Phys. Rev. C 60 (1999) 064315

  24. 1 RI in complexscaling Resonance: Rotated Continuum: Descretization

  25. εI εI E E 2θ 2θ

  26. Continuum Level Density: Basis function method:

  27. Phase shift calculationin the complex scaled basis function method S.Shlomo, Nucl. Phys. A539 (1992), 17. In a single channel case,

  28. Phase shift of 8Be=+a calculated with discretized app. Base+CSM: 30 Gaussian basis and =20 deg.

  29. Description of unbound states in the Complex Scaling Method H0=T+VC V；　Short Range Interaction （Ψ0; regular at origin） Solutions of Lippmann-Schwinger Equation Complex Scaling Outgoing waves A. Kruppa, R. Suzuki and K. Kato, phys. Rev.C75 (2007), 044602

  30. T-matrix Tl(k) Tl(k)＝ Second term is approximated aswhere

  31. Lines : Runge-Kutta method • Circles : CSM+Base

  32. Direct breakup Final state interaction (FSI) Complex-scaled Lippmann-Schwinger Eq. • CSLM solution • B(E1) Strength

  33. Dalitz distribution of 6He • Decay process • Di-neutron-like decay is not seen clearly.

  34. ６. Summary and conclusion • It is shown that resonant states play an important role in the continuum phenomena. • The resolution of identity in the complex scaling method is presented to treat the three-body resonant states in the same way as bound states. • The complex scaling method is shown to describe not only resonant states but also non-resonant continuum states on the rotated branch cuts. • We presented several applications of the extended resolution of identity in the complex scaling method; sum rule, break-up strength function and continuum level density.

  35. Collaboration: S. Aoyama(Niigata Univ.),H. Masui(Kitami I. T.), T. Myo (Osaka Tech. Univ.), R. Suzuki(Hokkaido Univ.), C. Kurokawa(Juntendo Univ.), K. Ikeda(RIKEN) Y. Kikuchi(Hokkaido Univ.)

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