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## Nucleus as an Open System: Continuum Shell Model and

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**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**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**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)**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**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**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**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**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…)**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**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**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)**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**State embedded in the continuum**• Integration region involves no poles Two parts of coupling to continuum • Form of the wave function and probability**(+) means + i0**(Eigenchannels in P-space) (off-shell) (on-shell) Factorization (unitarity), energy dependence (kinematic thresholds) , coupling through continuum**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).**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**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**EFFECTIVE HAMILTONIAN**One open channel**Interaction between resonances**• Real V • Energy repulsion • Width attraction • Imaginary W • Energy attraction • Width repulsion**11Li model**Dynamics of states coupled to a common decay channel • Model • Mechanism of binding**11LimodelDynamics of two states coupled to a common decay**channel • A1 and A2 • opposite signs • Model H**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**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**[2] A. Volya and V.Z. Phys. Rev. C 74 (2006) 064314, [3]**G. Hagen et al. Phys. Rev. Lett. 108 (2012) 242501**Continuum shell model:**Detailed predictions For Oxygen isotopes; Color code - for widths [A. Volya]**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)**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)**CSM calculation of 18O**States marked with longer lines correspond to sd-shell model; only l=0,2 partial waves included in theoretical results.**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**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) …**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**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 )**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**Green’s function calculation**• Advantages of the method • -No need for full diagonalization or inversion at different E • -Only matrix-vector multiplications • -Numerical stability**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**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**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**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**Resonance width distribution**(chaotic closed system, single open channel) G. Shchedrin, V.Z., PRC (2012)**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)**Super-radiant transition**in Random Matrix Ensemble N= 1000, m=M/N=0.25

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