Theoretical Description of the Fission Process Witold Nazarewicz (Tennessee)

1. Theoretical Description of the Fission Process Witold Nazarewicz (Tennessee) Stewardship Science Academic Alliances Program, NNSA January 29, 2008, Washington, DC Introduction and motivation Accomplishments and recent examples Summary Theoretical Description of the Fission Process NNSA Grant DE-FG03-03NA00083

2. 240Pu 1938 - Hahn & Strassmann 1939 - Meitner & Frisch 1939 - Bohr & Wheeler 1940 - Petrzhak & Flerov Fission N,Z elongation necking N=N1+N2 Z=Z1+Z2 split N2,Z2 N1,Z1

3. Spontaneous and Induced Fission Lifetimes n, d, EM… Direct time propagation Impossible ! Fragment distributions

4. Nuclear Structure Weinberg’s Laws of Progress in Theoretical Physics From: “Asymptotic Realms of Physics” (ed. by Guth, Huang, Jaffe, MIT Press, 1983) 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!”

5. http://www.scidacreview.org/0704/html/unedf.html

6. Powerful phenomenology exists… • … but no satisfactory microscopic understanding of: • Barriers • Fission half-lives • Fission dynamics • Cross sections • … • What is needed? • Effective interaction (UNEDF) • Microscopic many-body technique (SSAA)

7. http://www.phys.utk.edu/witek/fission/fission.html

8. Project Participants • Andrzej Baran Maria Sklodowska-Curie University, Lublin, Poland • David Dean Oak Ridge National Laboratory • Jacek Dobaczewski University of Warsaw, Warsaw, Poland • Witold Nazarewicz University of Tennessee, Knoxville • Oak Ridge National Laboratory • Nikolai Nikolov University of Tennessee, Knoxville • Nicolas Schunck University of Tennessee, Knoxville • Javid Sheikh University of Tennessee, Knoxville • University of Kashmir, Srinagar, India • Janusz Skalski Soltan Institute for Nuclear Studies, Warsaw, Poland • Andrzej Staszczak Maria Sklodowska-Curie University, Lublin, Poland • Mario Stoitsov University of Tennessee, Knoxville • Bulgarian Academy of Sciences, Sofia

9. Collaborators & Consultants • Peter Möller LANL • Arnie Sierk LANL • Walid Younes LLNL • Daniel Gogny LLNL • Arthur Kerman MIT/NNSA • Heloise Goutte CEA-Bruyères-le-Chatel • Ludovic Bonneau LANL/Bordeaux

10. V(q) E q Adiabatic Approaches to Fission WKB: collective inertia (mass parameter) multidimensional space of collective parameters Several collective coordinates The action has to be minimized

11. 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

12. 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

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

14. Nuclear DFT Global properties, global calculations S. Goriely et al., ENAM’04 M. Stoitsov et al. • * Global DFT mass calculations: HFB mass formula: m~700keV • Taking advantage of high-performance computers

15. Large-scale Calculations S. Cwiok, P.H. Heenen, W. Nazarewicz Nature, 433, 705 (2005) Stoitsov et al., PRL 98, 132502 (2007) NNSA Grant DE-FG03-03NA00083

16. Collective potential V(q) • Universal nuclear energy density functional is yet to be developed • Choice of collective parameters • How to define a barrier? • How to connect valleys? • Dynamical corrections going beyond mean field important • Center of mass • Rotational and vibrational • (zero-point quantum • correction) • Particle number

17. N. Nikolov, N. Schunck, W. Nazarewicz, M. Bender Leptodermous expansion show stability of deformed neutron-rich nuclei caused by (poorly constrained) surface-symmetry energy Benchmarking of excitation energy of super-deformed states and fission isomers with microscopic interactions show fluctuations larger than theoretical and experimental uncertainties [1] P.-G. Reinhard et al, Phys. Rev. C73, 014309 (2006) [2] M. Bender et al, In preparation

18. Pairing properties A. Staszczak, J. Dobaczewski, and W. Nazarewicz Int. J. Mod. Phys. E 16, 310 (2007)

19. Collective inertia B(q) and ZPE Various prescriptions for collective inertia and ZPE exist: GOA of the GCMRing and P. Schuck, The Nuclear Many-Body Problem, 1980 ATDHF+Cranking Giannoni and Quentin, Phys. Rev. C21, 2060 (1980); Warda et al., Phys. Rev. C66, 014310 (2002) Goutte et al., Phys. Rev. C71, 024316 (2005) A.Baran et al., nucl-th/0610092 HFODD+BCS+Skyrme

20. A. Baran, A. Staszczak, J. Dobaczewski, and W. Nazarewicz, Int. J. Mod. Phys. E 16, 443 (2007).

21. Self-consistent Static Fission Paths Calculations A. Staszczak, J. Dobaczewski W. Nazarewicz, in preparation See also: A. Warda et al., Phys. Rev. C66, 014310 (2002) and IJMP E13, 169 (2004) Gogny L. Bonneau, Phys. Rev. C74, 014301 (2006) Skyrme

22. nucl-th/0612017 A. Staszczak, J. Dobaczewski W. Nazarewicz, Acta. Phys. Pol. B38, 1589 (2007)

23. Triaxial! A. Staszczak, J. Dobaczewski W. Nazarewicz, in preparation

24. Bimodal fission in nuclear DFT Elongation Mass asymmetry nucl-th/0612017 A. Staszczak, J. Dobaczewski W. Nazarewicz, in preparation

25. Elongation Necking In practice, several collective coordinates should be considered

26. FJWKB: Two Peaked Barrier • A.V. Ignatiuk et al. • Phys. Lett.29B, 209 (1969) Very recent result: spontaneous fission lifetimes from static path A.Baran et al., in preparation

27. b a The shortest path determined by dynamic programming method. The grid points represent the mesh in a 2D space {x} in which the functional S[x(s)] is defined. Thick zig-zag line represents the shortest path (S=0) starting from the point (a) and ending at the final point (b). A. Baran et al., Nucl. Phys. A361, 83 (1981)

28. Related problem: heavy-ion fusion • nucleus-nucleus potentials • adiabatic fusion barriers J. Skalski, Phys.Rev. C 74, 051601 (2006) Coordinate-space Skyrme-Hartree-Fock

29. Adiabatic fusion barriers from self-consistent calculations J. Skalski, Phys. Rev. C76, 044603 (2007) Center of mass!!!

30. Interactions 240Pu energy versus elongation & asymmetry(Younes and Gogny, LLNL) Densities along most likely path 240Pu EHFB (MeV) Q3 (b3/2) Q2 (b) Reflection symmetry is spontaneously broken  asymmetric fission (Q3 0)

31. Interactions (cont.) 238U Kinetic Energy and Mass Distributions in HFB+TDGCM(GOA) one-dimensional dynamical Wahl (experiment) • Time-dependent microscopic collective Schroedinger equation • Two collective degrees of freedom • TKE and mass distributions reproduced • Dynamical effects are responsible for the large widths of the mass distributions • No free parameters HFB + Gogny D1S + Time-Dependent GOA H. Goutte, P. Casoli, J.-F. Berger, D. Gogny, Phys. Rev. C71, 024316 (2005)

32. Research goals for the next five years • Applications of modern adiabatic and time-dependent theories • Use modern NEDF optimized for deformation effects • Perform action minimization on 3D mesh • Develop symmetry restoration schemes for DFT • Increase the number of collective coordinates in GCM, including pairing channel • Tests of non-adiabatic approaches • Proper quantum treatment of many-body tunneling • Non adiabatic, properly accounts for level crossings and symmetry breaking effects (collective path strongly influenced by level crossings). HF and ATDHF not adequate • Evolution in an imaginary time • The lifetime is expressed by the sum of bounces • Difficulty in solving the periodic mean-field equations (fission bounce equations) • Important role of pairing correlations (restore adiabaticity) • Unclear how to restore broken symmetries bounce trajectory governing fission J. Skalski: http://arxiv.org/abs/0712.3030 Nuclear fission with mean-field instantons ATDHFB inertia

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

34. “Developing codes and technology that can be freely used by NNSA is also one of our goals.” The program HFBTHO (v1.66p) provides an axially deformed solution of the Skyrme-Hartree-Fock-Bogoliubov equations using a Transformed Harmonic Oscillator Basis. The program can be used in a variety of applications, including systematic studies of wide ranges of nuclei, both spherical and axially deformed, extending all the way out to nucleon drip lines. The program is available from the CPC Program Library. Recently, we developed a particle-number-projected variant of HFBTHO. Another code developed by our group is HFODD (v2.31f) that solves the Skyrme-Hartree-Fock-Bogolyubov equations in the Cartesian deformed harmonic-oscillator basis. This code is currently used in all our advanced fission calculations that require breaking of most self-consistent symmetries. Based on HFODD, we developed codes to calculate collective inertia (tensor of mass parameters; A.3) and the density-dependent pairing used in HFB calculations. Future developments: ATDHFB dynamics based on wavelets

35. Example: Large Scale Mass Table Calculations Science scales with processors Jaguar@ M. Stoitsov, HFB+LN mass table, HFBTHO 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) Odd and odd-odd Nuclei • The even-even calculations define 250,754 configurations in odd-A and odd-odd nuclei assuming 0.5 MeV threshold for the blocking candidates • Using 10,000 processors - about 25 CPU hours

37. What is needed? Computing (today) Efficient symmetry-unrestricted HFB (Kohn-Sham) solver UNEDF fitting: optimization/linear regression, distributed computing Action minimization in many dimensions, fission pathways Performance evaluation for existing codes Computing (tomorrow) Beyond-mean-field dynamics (GCM, projections) Real-time evolution for excited states (at and above the barrier) Imaginary-time evolution at subbarier energies Theory Symmetry restoration in DFT Symmetry-conserving formalism of LACM needs to be developed Can imaginary-time propagation be replaced by variational approach?

39. Nuclear Science Applications LRP’07 report

40. Relevance… Addressing national needs • Advanced Fuel Cycles • neutron-reaction cross sections from eV to 10 MeV • the full range of (n,f), (n,n’), (n,xn), (n,g) reactions • heavy transuranics, rare actinides, and some light elements (iron, sulfur) • Quantified nuclear theory error bars • Cross sections input to core reactor simulations (via data evaluation) • BETTER CROSS SECTIONS AFFECT both SAFETY and COST of AFC reactors. • Science Based Stockpile Stewardship • Radiochemical analysis from days of testing: inference on device performance shows final products but not how they came to be. • Typical example Yttrium charged particle out reaction. LESS THAN 10% of cross sections in region measured. • Theory with quantifiable error bars is needed. Stewardship Center at HRIBF

41. Pre-Conclusions • Quest for the universal interaction/functional • The major challenge for low-energy nuclear theory • Many-dimensional problem • Tunneling of the complex system • Coupling between collective and single-particle • Time dependence on different scales • All intrinsic symmetries broken • Large elongations, necking, mass asymmetry, triaxiality, time reversal (odd, odd-odd systems),… • Correlations important • Pairing makes the LACM more adiabatic. Quantum corrections impact dynamics. UNEDF (SCIDAC-2) DOE, NNSA, ASCR

42. Conclusions • Fission is a fundamental many-body phenomenon that possess the ultimate challenge for theory • Understanding crucial for many areas: • Nuclear structure and reactions (superheavies) • Astrophysics (n-rich fission and fusion, neutrino-induced fission) • Numerous applications (energy, AFC, Stockpile Stewardship…) • … • The light in the end of the tunnel: coupling between modern microscopic many-body theory and high-performance computing

43. Thank You!