Shell Structure of Exotic Nuclei (a Paradigm Shift?) Witold Nazarewicz (UTK, ORNL) Rutgers University Seminar, Nov. 5, 2008 Introduction Shell structure revisited Nuclear Density Functional Theory Questions and Challenges, Homework Perspectives Emphasis on: novel aspects recent results problems

Introduction

Shell effects and classical periodic orbits One-body field Not external (self-bound) Hartree-Fock Shells Product (independent-particle) state is often an excellent starting point Localized densities, currents, fields Typical time scale: babyseconds (10-22s) Closed orbits and s.p. quantum numbers But… Nuclear box is not rigid: motion is seldom adiabatic The walls can be transparent

Shell effects and classical periodic orbits Balian & Bloch, Ann. Phys. 69 (1971) 76 Bohr & Mottelson, Nuclear Structure vol 2 (1975) Strutinski & Magner, Sov. J. Part. Nucl. 7 (1976) 138 Trace formula, Gutzwiller, J. Math. Phys. 8 (1967) 1979 The action integral for the periodic orbit  Condition for shell structure Principal shell quantum number Distance between shells (frequency of classical orbit)

gap shell Pronounced shell structure Shell structure absent (quantum numbers) Shell structure absent gap shell closed trajectory (regular motion) trajectory does not close 

Shells Nuclei Sodium Clusters Jahn-Teller Effect (1936) experiment theory discrepancy 20 60 100 -10 10 Nuclei Number of Neutrons Shell Energy (MeV) 28 50 82 126 diff. P. Moller et al. S. Frauendorf et al. -1 1 50 100 150 200 Number of Electrons Shell Energy (eV) 58 92 138 198 experiment theory deformed clusters spherical Sodium Clusters Jahn-Teller Effect (1936) Symmetry breaking and deformed (HF) mean-field

Magicity is a fragile concept Near the drip lines nuclear structure may be dramatically different.

Phys. Rev. Lett. 99, 192501 (2007) Nature 449, 1022 (2007) No shell closure for N=8 and 20 for drip-line nuclei; new shells at 14, 16, 32…

What is the next magic nucleus beyond 208Pb?

Weinberg’s Laws of Progress in Theoretical Physics From: “Asymptotic Realms of Physics” (ed. by Guth, Huang, Jaffe, MIT Press, 1983) First Law: “The conservation of Information” (You will get nowhere by churning equations) Second Law: “Do not trust arguments based on the lowest order of perturbation theory” Third Law: “You may use any degrees of freedom you like to describe a physical system, but if you use the wrong ones, you’ll be sorry!”

Physics of the large neutron excess Interactions Many-body Correlations Open Channels Interactions Isovector (N-Z) effects Poorly-known components come into play Long isotopic chains crucial Configuration interaction Mean-field concept often questionable Asymmetry of proton and neutron Fermi surfaces gives rise to new couplings (Intruders and the islands of inversion) New collective modes; polarization effects Open channels Nuclei are open quantum systems Exotic nuclei have low-energy decay thresholds Coupling to the continuum important Virtual scattering Unbound states Impact on in-medium Interactions

Prog. Part. Nucl. Phys. 59, 432 (2007)

Modern Mean-Field Theory = Energy Density Functional mean-field ⇒ one-body densities zero-range ⇒ local densities finite-range ⇒ gradient terms particle-hole and pairing channels Hohenberg-Kohn Kohn-Sham Negele-Vautherin Landau-Migdal Nilsson-Strutinsky Nuclear DFT two fermi liquids self-bound superfluid

Nuclear Local s.p. Densities and Currents isoscalar (T=0) density isovector (T=1) density isoscalar spin density isovector spin density current density spin-current tensor density kinetic density kinetic spin density + analogous p-p densities and currents

Most general second order expansion in densities and their derivatives Construction of the functional Perlinska et al., Phys. Rev. C 69, 014316 (2004) isoscalar (T=0) density +isoscalar and isovector densities: spin, current, spin-current tensor, kinetic, and kinetic-spin + pairing densities isovector (T=1) density p-h density p-p density (pairing functional) Most general second order expansion in densities and their derivatives Constrained by microscopic theory: ab-initio functionals provide quasi-data! Not all terms are equally important. Usually ~12 terms considered Some terms probe specific experimental data Pairing functional poorly determined. Usually 1-2 terms active. Becomes very simple in limiting cases (e.g., unitary limit)

Example: Spin-Orbit and Tensor Force (among many possibilities) The origin of SO splitting can be attributed to 2-body SO and tensor forces, and 3-body force R.R. Scheerbaum, Phys. Lett. B61, 151 (1976); B63, 381 (1976); Nucl. Phys. A257, 77 (1976); D.W.L. Sprung, Nucl. Phys. A182, 97 (1972); C.W. Wong, Nucl. Phys. A108, 481 (1968); K. Ando and H. Bando, Prog. Theor. Phys. 66, 227 (1981); R. Wiringa and S. Pieper, Phys. Rev. Lett. 89, 182501 (2002) The maximum effect is in spin-unsaturated systems Discussed in the context of mean field models: Fl. Stancu, et al., Phys. Lett. 68B, 108 (1977); M. Ploszajczak and M.E. Faber, Z. Phys. A299, 119 (1981); J. Dudek, WN, and T. Werner, Nucl. Phys. A341, 253 (1980); J. Dobaczewski, nucl-th/0604043; Otsuka et al. Phys. Rev. Lett. 97, 162501 (2006); Lesinski et al., arXiv:0704.0731,… …and the nuclear shell model: T. Otsuka et al., Phys. Rev. Lett. 87, 082502 (2001); Phys. Rev. Lett. 95, 232502 (2005) 28, 50, 82, 126 2, 8, 20 F j< j< F j> j> Spin-saturated systems Spin-unsaturated systems

(strongly depend on shell filling) acts in s and d states of relative motion acts in p states SO densities (strongly depend on shell filling) Additional contributions in deformed nuclei Particle-number dependent contribution to nuclear binding It is not trivial to relate theoretical s.p. energies to experiment.

16O 13C7 13N6 The nucleus is a correlated open quantum many-body system Environment: continuum of decay channels `Alignment’ of w.b. state with the decay channel 16O Thomas-Ehrmann effect 13C7 13N6 1943 2365 3502 1/2 3089 4946 3685 12C+n 12C+p 1/2 3/2 7162 6049 Spectra and matter distribution modified by the proximity of scattering continuum

unification of structure and reactions The importance of the particle continuum was discussed in the early days of the multiconfigurational Shell Model and the mathematical formulation within the Hilbert space of nuclear states embedded in the continuum of decay channels goes back to H. Feshbach (1958-1962), U. Fano (1961), and C. Mahaux and H. Weidenmüller (1969) unification of structure and reactions resonance phenomena generic to many small quantum systems coupled to an environment of scattering wave functions: hadrons, nuclei, atoms, molecules, quantum dots, microwave cavities, … consistent treatment of multiparticle correlations Open quantum system many-body framework Continuum (real-energy) Shell Model (1977 - 1999 - 2005) H.W.Bartz et al, NP A275 (1977) 111 R.J. Philpott, NP A289 (1977) 109 K. Bennaceur et al, NP A651 (1999) 289 J. Rotureau et al, PRL 95 (2005) 042503 Gamow (complex-energy) Shell Model (2002 -) N. Michel et al, PRL 89 (2002) 042502 R. Id Betan et al, PRL 89 (2002) 042501 N. Michel et al, PRC 70 (2004) 064311 G. Hagen et al, PRC 71 (2005) 044314

Rigged Hilbert space Gamow Shell Model (2002) One-body basis J. Rotureau et al., DMRG Phys. Rev. Lett. 97, 110603 (2006) bound, anti-bound, and resonance states non-resonant continuum

N. Michel et al. PRC 75, 0311301(R) (2007) 5He+n 6He 6He+n 7He WS potential depth decreased to bind 7He. Monopole SGI strength varied 5He+n 6He Overlap integral, basis independent! 6He+n 7He Anomalies appear at calculated thresholds (many-body S-matrix unitary) Scattering continuum essential see also Nucl. Phys. A 794, 29 (2007)

Questions and challenges

How to extend DFT to finite, self-bound systems? Intrinsic-Density Functionals J. Engel, Phys. Rev. C75, 014306 (2007) Generalized Kohn-Sham Density-Functional Theory via Effective Action Formalism M. Valiev, G.W. Fernando, cond-mat/9702247 B.G. Giraud, B.K. Jennings, and B.R. Barrett, arXiv:0707.3099 (2007); B.G. Giraud, arXiv:0707.3901 (2007)

Can dynamics be incorporated directly into the functional? Example: Local Density Functional Theory for Superfluid Fermionic Systems: The Unitary Gas, Aurel Bulgac, Phys. Rev. A 76, 040502 (2007) See also: Density-functional theory for fermions in the unitary regime T. Papenbrock Phys. Rev. A72, 041603 (2005) Density functional theory for fermions close to the unitary regime A. Bhattacharyya and T. Papenbrock Phys. Rev. A 74, 041602(R) (2006)

Connections to computational science 1Teraflop=1012 flops 1peta=1015 flops (next 2-3 years) 1exa=1018 flops (next 10 years)

How to root nuclear DFT in a microscopic theory? ab-initio - DFT connection NN+NNN - EDF connection (via EFT+RG)

Broyden’s Mixing Procedure: Phys. Rev. C 78, 014318 (2008) A. Baran, A. Bulgac, M. McNeil Forbes, G. Hagen, W. Nazarewicz, N. Schunck and M.V. Stoitsov 300,000 200 108 3,000,000

Example: Large Scale Mass Table Calculations Science scales with processors M. Stoitsov HFB+LN mass table, HFBTHO INCITE award Dean et al. 17.5M hours Even-Even Nuclei The SkM* mass table contains 2525 even-even nuclei A single processor calculates each nucleus 3 times (prolate, oblate, spherical) and records all nuclear characteristics and candidates for blocked calculations in the neighbors Using 2,525 processors - about 4 CPU hours (1 CPU hour/configuration) Jaguar Cray XT4 at ORNL All Nuclei 9,210 nuclei 599,265 configurations Using 3,000 processors - about 25 CPU hours see MassExplorer.org

A. Staszczak, J. Dobaczewski, W. Nazarewicz, in preparation Bimodal fission in nuclear DFT A. Staszczak, J. Dobaczewski, W. Nazarewicz, in preparation S. Umar and V. Oberacker Phys. Rev. C 76, 014614 (2007) nucl-th/0612017 TDHF description of heavy ion fusion

Conclusions Why is the shell structure changing at extreme N/Z ? Can we talk about shell structure at extreme N/Z ? Interactions Many-body Correlations Open Channels Thank You