1. Shell Structure of Exotic Nuclei ( a Paradigm Shift?) Witold Nazarewicz (University of Tennessee/ORNL) Introduction Shell structure revisited Nuclear Density Functional Theory Questions and Challenges, Homework Perspectives Emphasis on:novel aspects recent results problems JUSTIPEN ジャスティペン

2. 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!”

3. Introduction

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

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

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

7. 0 1 50100150200 0 1 Number of Electrons Shell Energy (eV) 58 92 138 198 experiment theory deformed clusters spherical clusters Sodium Clusters experiment theory discrepancy 2060100 -10 0 10 -10 0 0 Nuclei Number of Neutrons Shell Energy (MeV) 20 28 50 82 126 experiment theory diff. Shells P. Moller et al. S. Frauendorf et al. Jahn-Teller Effect (1936) Symmetry breaking and deformed (HF) mean-field

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

9. No shell closure for N=8 and 20 for drip-line nuclei; new shells at 14, 16, 32… First experimental indications demonstrate significant changes

10. What is the next magic nucleus beyond 208 Pb?

11. Physics of the large neutron excess Interactions Isovector (N-Z) effects Poorly-known components come into play Long isotopic chains crucial Interactions Many-body Correlations Open Channels 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 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

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

13. 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

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

15. Justification of the standard Skyrme functional: DME In practice, the one-body density matrix is strongly peaked around r=r’. Therefore, one can expand it around the mid-point: The Skyrme functional was justified in such a way in, e.g., Negele and Vautherin, Phys. Rev. C5, 1472 (1972); Phys. Rev. C11, 1031 (1975) Campi and Bouyssy, Phys. Lett. 73B, 263 (1978) … but nuclear EDF does not have to be related to any given effective two-body force! Actually, many currently used nuclear energy functionals are not related to a force DME and EFT+RG

16. Not all terms are equally important. Some probe specific observables pairing functional Construction of the functional Perlinska et al., Phys. Rev. C 69, 014316 (2004) Most general second order expansion in densities and their derivatives p-h densityp-p density

17. 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) Example: Spin-Orbit and Tensor Force (among many possibilities) j<j< j>j> FF 2, 8, 20 Spin-saturated systems j<j< j>j> FF 28, 50, 82, 126 Spin-unsaturated systems

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

19. Importance of the tensor interaction far from stability [523]7/2 [411]1/2 Proton emission from 141 Ho

20. 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 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)

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

22. Questions and challenges

23. 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) How to extend DFT to finite, self-bound systems?

24. What are the missing pieces? What is density dependence? (ph and pp channels) Spin-isospin sector (e.g., tensor) Momentum dependence of the effective mass? Induced interaction Isovector and isoscalar

25. Density Matrix Expansion for RG-Evolved Interactions S.K. Bogner, R.J. Furnstahl et al. see also: EFT for DFT R.J. Furnstahl nucl-th/070204

26. J. Dobaczewski and J. Dudek, Phys. Rev. C52, 1827 (1995) M. Bender et al., Phys. Rev. C65, 054322 (2002) H. Zdunczuk, W. Satula and R. Wyss, Phys. Rev. C71, 024305 (2005) very poorly determined Can be adjusted to the Landau parameters Important for all I>0 states (including low-spin states in odd-A and odd-odd nuclei) Impact beta decay Influence mass filters (including odd-even mass difference) Limited experimental data available How to parameterize time-odd pieces?

27. Isospin dynamics important! High-spin terminating states Zdunczuk et al.,Phys.Rev. C71, 024305(2005) Stoitcheva et al., Phys. Rev. C 73, 061304(R) (2006) Excellent examples of single- particle configurations Weak configuration mixing Spin polarization; probing time- odd terms! Experimental data available

28. How to restore broken symmetry in DFT? J. Dobaczewski et al., Phys. Rev. C76, 054315 (2007) The transition density matrices contains complex poles. Some cancellation appears if the ph and pp Hamiltonians are the same The projection operator cannot be defined uniquely Problems with fractional density dependence Projected DFT yields questionable results when the pole appears close to the integration contour. This often happens when dealing with PESs see also:M. Bender, T. Duguet, D. Lacroix, in preparation. S. Krewald et al.,Phys. Rev. C 74, 064310 (2006).

29. 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)

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

31. Ab-initio - DFT Connection One-body density matrix is the key quantity to study “local DFT densities” can be expressed through  (x,x’) Testing the Density Matrix Expansion and beyond UNEDF Homework Introduce external potential HO for spherical nuclei (amplitude of zero-point motion=1 fm) 2D HO for deformed nuclei Density expressed in COM coordinates Calculate  x,x’) for 12 C, 16 O and 40,48,60 Ca (CC) Perform Wigner transform to relative and c-o-m coordinates q and s Extract , J,  Analyze data by comparing with results of DFT calculations and low- momentum expansion studies. Go beyond I=0 to study remaining densities (for overachievers) Negele and Vautherin: PRC 5, 1472 (1972) isospin UNEDF Pack Forest meeting

32. Jaguar Cray XT4 at ORNL No. 2 on Top500 11,706 processor nodes Each compute/service node contains 2.6 GHz dual-core AMD Opteron processor and 4 GB/8 GB of memory Peak performance of over 119 Teraflops 250 Teraflops after Dec.'07 upgrade 600 TB of scratch disk space 1Teraflop=10 12 flops 1peta=1000 tera

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