Nucleus as an Open System: Continuum Shell Model and

1. Nucleus as an Open System: Continuum Shell Model and New Challenges Vladimir Zelevinsky NSCL/ Michigan State University Supported by NSF Caen, GANIL May 30, 2014

2. OUTLINE • From closed to open many-body systems • Effective non – Hermitian Hamiltonian • Doorways and phenomenon of super-radiance • Continuum shell model • Statistics of complex energies • Cross sections, resonances, correlations and fluctuations • Quantum signal transmission

3. THANKS • NaftaliAuerbach (Tel Aviv University) • Luca Celardo (University of Breschia) • Felix Izrailev (University of Puebla) • Lev Kaplan (Tulane University) • GavriilShchedrin (MSU, TAMU) • Valentin Sokolov (BudkerInstutute) • SurenSorathia (University of Puebla) • Alexander Volya(Florida State University)

4. NSCL and FRIB Laboratory 543 employees, incl. 38 faculty, 59 graduate and 82 undergraduate studentsas of April 21, 2014 • NSCL is funded by the U.S. National Science Foundation to operate a flagship user facility for rare isotope research and education in nuclear science, nuclear astrophysics, accelerator physics, and societal applications • FRIB will be a national user facility for the U.S. Department of Energy Office of Science – when FRIB becomes operational, NSCL will transition into FRIB 2011 2009 2003 User group of over 1300 members with approx. 20 working groups www.fribusers.org

5. The Evolution of Nuclear Science at MSU

6. NSCL Science Is Aligned with National Priorities Articulated by National Research Council RISAC Report (2006), NSAC LRP (2007),NRC Decadal Survey of Nuclear Physics (2012), “Tribble Report” (2013) Properties of nuclei – UNEDF SciDAC, FRIB Theory Center (?) • Develop a predictive model of nuclei and their interactions • Many-body quantum problem: intellectual overlap to mesoscopic science, quantum dots, atomic clusters, etc. – Mesoscopic Theory Astrophysical processes – JINA • Origin of the elements in the cosmos • Explosive environments: novae, supernovae, X-ray bursts … • Properties of neutron stars Tests of fundamental symmetries • Effects of symmetry violations are amplified in certain nuclei Societal applications and benefits • Bio-medicine, energy, material sciences – Varian, isotope harvesting, … • National security – NNSA Reaping benefits from recent investments while creating future opportunities

7. FRIB science answers big questions FRIB Science is Transformational • FRIB physics is at the core of nuclear science:“To understand, predict, and use” (David Dean) • FRIB provides access to a vast unexplored terrain in the chart of nuclides

8. Examples for Cross-Disciplinary and Applied Research Topics Medical research Examples: 47Sc, 62Zn, 64Cu, 67Cu, 68Ge, 149Tb, 153Gd, 168Ho, 177Lu, 188Re, 211At, 212Bi, 213Bi, 223Ra (DOE expert panel) MSU Radiology Dept. interested in 60,61Cu -emitters 149Tb, 211At: potential treatment of metastatic cancer Plant biology: role of metals in plant metabolism Environmental and geosciences: ground water, role of metals as catalysts Engineering: advanced materials, radiation damage, diffusion studies Toxicology: toxicology of metals Biochemistry: role of metals in biological process and correlations to disease Fisheries and Wildlife Sciences: movement of pollutants through environmental and biological systems Reaction rates important for stockpile stewardship – non-classified research Determination of extremely high neutron fluxes by activation analysis Rare-isotope samples for (n,g), (n,n’), (n,2n), (n,f) e.g. 88,89Zr Same technique important for astrophysics Far from stability: surrogate reactions (d,p), (3He,a xn) … Vision: Up to 10 Faculty Positions for Cross-Disciplinary and Applied Research

9. From closed to open (or marginally stable) many-body system CLOSED SYSTEMS: Bound states Mean field, quasiparticles Symmetries Residual interactions Pairing, superfluidity Collective modes Quantum many-body chaos (GOE type) Open systems: Continuum energy spectrum Unstable states, lifetimes Decay channels (E,c) Energy thresholds Cross sections Resonances, isolated or overlapping Statistics of resonances and cross sections Unified approach? (Many…)

10. DooRWAY STATES From giant resonances to superradiance The doorway state is connected directly to external world, other states (next level) only through the doorway. Examples: IAS, single-particle resonance, giant resonances at high excitation energy, intermediate structures. Feshbach resonance in traps, superradiance

12. Single-particle decay in many-body system Evolution of complex energies • 8 s.p. levels, 3 particles • One s.p. level in continuum Total states 8!/(3! 5!)=56; states that decay fast 7!/(2! 5!)=21 – superradiant doorways

13. Examples of superradiance Mechanism of superradiance Interaction via continuum Trapped states - self-organization Narrow resonances and broad superradiant state in 12C in the region of Delta • Optics • Molecules • Microwave cavities • Nuclei • Hadrons • Quantum computing • Measurement theory Bartsch et.al. Eur. Phys. J. A 4, 209 (1999) N. Auerbach, V.Z.. Phys. Lett. B590, 45 (2004)

15. Physics and mathematics of coupling to continuum • New partof Hamiltonian: coupling through continuum [1] C. Mahaux and H. Weidenmüller, Shell-model approach to nuclear reactions, North-Holland Publishing, Amsterdam 1969

16. State embedded in the continuum • Integration region involves no poles Two parts of coupling to continuum • Form of the wave function and probability

17. (+) means + i0 (Eigenchannels in P-space) (off-shell) (on-shell) Factorization (unitarity), energy dependence (kinematic thresholds) , coupling through continuum

18. Self energy, interaction with continuum • Gamow shell model 17O Correction to Harmonic Oscillator Wave Function s,p, and d waves (red, blue, black) N Michel, J. Phys. G: Nucl. Part. Phys. 36 (2009) 013101 • Notes: • Wave functions are not HO • Phenomenological SM is adjusted to observation • No corrections for properly solved mean field • momentum A. Volya, EPJ Web of Conf. 38, 03003 (2012).

19. Traditional shell-model • Continuum physics • The nuclear many-body problem • Effective non-Hermitianand • energy-dependent Hamiltonian • Channels (parent-daughter structure) • Bound states and resonances • Matrix inversion at all energies (time dependent approach) • Single-particles state (particle in the well) • Many-body states (slater determinants) • Hamiltonian and Hamiltonian matrix • Matrix diagonalization Formally exact approach Limit of the traditional shell model Unitarity of the scattering matrix

20. Ingredients • Intrinsic states: Q-space • States of fixed symmetry • Unperturbed energies e1; some e1>0 • Hermitian interaction V • Continuum states: P-space • Channels and their thresholds Ecth • Scattering matrix Sab(E) • Coupling with continuum • Decay amplitudes Ac1(E) - thresholds • Typical partial width =|A|2 • Resonance overlaps: level spacing vs. width “kappa” parameter No approximations until now

21. EFFECTIVE HAMILTONIAN One open channel

24. Interaction between resonances • Real V • Energy repulsion • Width attraction • Imaginary W • Energy attraction • Width repulsion

25. 11Li model Dynamics of states coupled to a common decay channel • Model • Mechanism of binding

26. 11LimodelDynamics of two states coupled to a common decay channel • A1 and A2 • opposite signs • Model H

27. A. Volya and V. Zelevinsky, Phys. Rev. Lett. 94, 052501 (2005); Phys. Rev. C 67, 054322 (2003); Phys. Rev. C 74, 064314 (2006). • Oxygen Isotopes • Continuum Shell Model Calculation • sd space, HBUSD interaction • single-nucleon reactions

28. Measured 2009-2013 • Predictive power of theory • Continuum Shell Model prediction 2003-2006 [1] C. R. Hoffman et al., Phys. Lett. B 672, 17 (2009); Phys.Rev.Lett.102,152501(2009); Phys.Rev.C83,031303(R)(2011); E. Lunderberg et al., Phys. Rev. Lett. 108, 142503 (2012). [2] A. Volya and V. Zelevinsky, Phys. Rev. Lett. 94, 052501 (2005); Phys. Rev. C 67, 054322 (2003); 74, 064314 (2006). [3] G. Hagen et.al Phys. Rev. Lett. 108, 242501 (2012) • http://www.nscl.msu.edu/general-public/news/2012/O26

29. [2] A. Volya and V.Z. Phys. Rev. C 74 (2006) 064314, [3] G. Hagen et al. Phys. Rev. Lett. 108 (2012) 242501

30. Continuum shell model: Detailed predictions For Oxygen isotopes; Color code - for widths [A. Volya]

31. VirVirtualexcitations into continuum 1+ experiment 2+ Figure: 23O(n,n)23O Effect of self-energy term (red curve). Shaded areas show experimental values with uncertainties. Experimental data from: C. Hoffman, et.al. Phys. Lett. B672, 17 (2009)

32. Two-neutron sequential decay of 26O A. Volya and V. Zelevinsky, Continuum shell model, Phys. Rev. C 74, 064314 (2006). • Predicted Q-value: 21 keV • Z. Kohley, et.al PRL 110, 152501 (2013) (experiment)

33. CSM calculation of 18O States marked with longer lines correspond to sd-shell model; only l=0,2 partial waves included in theoretical results.

34. Continuum Shell Model He isotopes • Cross section and structure within the same formalism • Reaction l=1 polarized elastic channel References [1] A. Volya and V. Zelevinsky Phys. Rev. C 74 (2006) 064314 [2] A. Volya and V. Zelevinsky Phys. Rev. Lett. 94 (2005) 052501 [3] A. Volya and V. Zelevinsky Phys. Rev. C 67 (2003) 054322

35. Specific features of thecontinuum shell model • Remnants of traditional shell model • Non-Hermitian Hamiltonian • Energy-dependent Hamiltonian • Decay chains • New effective interaction – unknown… (self – made recipes) …

36. Energy-dependent Hamiltonian • Form of energy-dependence • Consistency with thresholds • Appropriate near-threshold behavior • How to solve energy-dependent H • Consistency in solution • Determination of energies • Determination of open channels

37. Interpretation of complex energies • For isolated narrow resonances all definitions agree • Real Situation • Many-body complexity • High density of states • Large decay widths • Result: • Overlapping, interference, width redistribution • Resonance and width are definition dependent • Non-exponential decay • Solution: Cross section is a true observable (S-matrix )

38. Calculation Details, Time – Dependent • Scale Hamiltonian so that eigenvalues are in [-1 1] • Expand evolution operator in Chebyshev polynomials • Use iterative relation and matrix-vector multiplication to generate • Use FFT to find return to energy representation *W.Press, S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes in C++ the art of scientific computing, Cambrige University Press, 2002 T. Ikegami and S. Iwata, J. of Comp. Chem. 23 (2002) 310-318

39. Green’s function calculation • Advantages of the method • -No need for full diagonalization or inversion at different E • -Only matrix-vector multiplications • -Numerical stability

40. Interplay of collectivities Definitions n - labels particle-hole state n – excitation energy of state n dn - dipole operator An – decay amplitude of n • Two doorway states of different nature • Real energy: multipole resonance • Imaginary energy: super-radiant state Model Hamiltonian Driving GDR externally (doing scattering) Everything depends on angle between multi-dimensional vectors A and d

41. Interplay of collectivities Definitions n - labels particle-hole state n – excitation energy of state n dn - dipole operator An – decay amplitude of n Model Hamiltonian Driving GDR externally (doing scattering) Everything depends on angle between multi dimensional vectors A and d

44. Pygmy resonance Orthogonal: GDR not seen Parallel: Most effective excitation of GDR from continuum At angle: excitation of GDR and pigmy Parallel case: Delta-resonance and particle-hole states with pion quantum numbers A model of 20 equally distant levels is used

45. Loosely stated, the PTD is based on the assumptions that s-wave neutron scattering is a single-channel process, the widths are statistical, and time-reversal invariance holds; hence, an observed departure from the PTD implies that one or more of these assumptions is violated P.E. Koehler et al. PRL 105, 072502 (2010) - • Time-reversal invariance holds • Single-channel process • Widths are statistical (whatever it means…) • Intrinsic “chaotic” states are uncorrelated • Energy dependence of widths is uniform • No doorway states • No structure pecularities (b) and (d) are wrong; (c), (e), (f), (g) depend on the nucleus

47. Resonance width distribution (chaotic closed system, single open channel) G. Shchedrin, V.Z., PRC (2012)

48. Adding many “gamma” - channels

49. Level spacing distribution in an open system with a single decay channel: No level repulsion in the intermediate region 0.5 0.1 1.0 5.0 No level repulsion at short distances! (Energy of an unstable state is not well defined)

50. Super-radiant transition in Random Matrix Ensemble N= 1000, m=M/N=0.25