Nuclear Structure from Scratch

1. Nuclear Structure from Scratch Erich Ormand Petr Navratil Christian Forssen Vesselin Gueorguiev Lawrence Livermore National Laboratory Collaborators: Bruce Barrett, U of Arizona Ionel Stecu, U of Arizona Andreas Nogga, U of Washington James Vary, Iowa State University E. Caurier, CNRS, Strasbourg Calvin Johnson, San Diego State University This work was carried out under the auspices of the U.S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. This work was supported in part from LDRD contract 00-ERD-028 Feb 6, 2004 UCRL-PRES-205602

2. Nuclear structure at the dawn of the 21st century • Nuclear structure has been studied for a long time • But there are still many mysteries • Especially for light p-shell nuclei • But they are a rich laboratory to test a basic understand of nuclear physics • Vast array of physics: deformation, clustering, super-allowed beta decay, three-body forces, parity inversion, halos, etc. • Newer methods and faster computers are making a comprehensive study from first principles possible • GFMC • No-core shell model • Coupled cluster methods Exactly how nuclei are put together is one of the most fundamental questions in nuclear physics

3. Goal: Exact calculation of Nuclear Structure for light nuclei • Two nucleons  A-nucleons • Properties: binding energies, decay mechanisms, etc. • Do we need three-body forces? (Yes) • How about four-body? • Hasn’t the Shell Model already solved this problem? • Yes andNO! • The Shell Model as practiced is an empirical tool that often requires arbitrary and unsatisfying approximations to make it work The goal of the No-core Shell Model is to implement effective interaction theory to apply the shell model correctly We want to take the model out of the shell model

4. Add the center-of-mass oscillator potential • The Bad: • Radial behavior is not right for large r • Provides a confining potential, so all states are effectively bound • Some dependence on oscillator parameter (goes away with larger model space) The many-body Hamiltonian • The Good: • Provides a convenient basis to build the many-body Slater determinants • Does not affect the intrinsic motion • Exact separation between intrinsic and center-of-mass motion

5. Construct many-body states |i so that 0f1p N=4 • Calculate Hamiltonian matrix Hij=j|H|i • Diagonalize to obtain eigenvalues Strong repulsion at ~ 0.5 fm j|H|i large 0d1s N=2 1p N=1 0s N=0 Problem: Short-range repulsion requires an infinite space Many-body solutions • Typical eigenvalue problem

6. Lee-Suzuki: Heff=PXHX-1P Can we get around this problem?Effective interactions • Choose subspace of for a calculation (P-space) • Include most of the relevant physics • Q -space (excluded - infinite) • Effective interaction: Q P QXHX-1P=0

7. e Q P-space defined by Heffhas one-, two, three-, … A-body terms eN+2 eN+1 eN eN Heff QXHX-1P=0 Exact reproduction of N eigenvalues P e3 e2 e3 e1 e2 e1 Q P The general idea behind effective interactions H

8. Choose P-space for A-body calculation, with dimension dp • P-space basis states: and Q-space basis states: • Need dP solutions, |k, in the “infinite” space, i.e., • Write X=e- Hermitian effective interaction The matrix bP|Heff|aPexactly reproduces dp solutions of the full problem Effective interactions with the Lee-Suzuki method

9. Argonne potentials • Potential model, v18, v8’ Local Fit to Nijmegan scattering data • Bonn potentials • Based on meson exchange Non-local, off shell behavior Charge-dependence, CD-Bonn • Effective field theory EFT • Guided by QCD Idaho-A, N3LO, etc Softer,  faster convergence • INOY • Two-body Inside non-local, outside Yukawa Fit to NN data and 3H and 3He • Three-body • n=3 clusters in Heff Tucson-Melbourne Urbana and EFT-based The NCSM uses a variety of realistic interactions

10. Application to A=3 and convergence • 3H with CD-Bonn and N3LO

11. Results with two-body - frequency dependence • 9Be with CD-Bonn • NCSM result taken in region with least dependence on frequency • Note NCSM is not variational

12. 400 keV 1.8 MeV, Clustering? Oscillator parameter Model space Results with three-body effective interactions Binding energies with Av8’ Three-body effective interactions represent a significant improvement Results are within 400 keV of the GFMC

13. Excitation spectra with NN-interactions So far, things are looking pretty good!

14. The NN-interaction clearly has problems Note inversion of spins! • The NN-interaction by itself does not describe nuclear structure • Also true for A=11

15. Three-body interaction in a nucleus The three-nucleon interaction plays a critical role in determining the structure of nuclei

16. Future directions for the No-core Shell Model • We want to get better convergence • Extend Nmax • For three-body interactions up to Nmax=6 • Include higher clusters in the effective interaction • Four-body • Shell-model program REDSTICK • Two-body • Competitive with ANTOINE • Three-body • Now finished and production runs expected this Fall!

17. Two-body interactions • Model-space limitation ~ 108 states A=3, Nmax = 36, Jacobi coordinates A=4, Nmax = 20, Jacobi coordinates A=6, Nmax = 14, M-scheme 7  A  11, Nmax ~ 10, M-scheme A  12, Nmax = 6, M-scheme Some computational aspects of the NCSM • Three-body interactions • Limited by number of three-body matrix elements For Nmax=4, 39,523,066 For Nmax=6 over 600,000,000 Practical limit is Nmax = 6 for all the p-shell Four-body - Nmax = 4

18. REDSTICK3B – Timing for three-body 12C: N ~ 32.5x106 G.S. 10 iterations ~ 1305 hr Four-body interaction G.S. 10 iterations ~454 hr

19. +  7Be 3He 4He A new formalism for nuclear reactions • Nuclear reactions from scratch - Petr Navratil • Ingredients: • Exact intrinsic wave functions for entrance and exit channels - ab initio shell model • Wave function for composite • Radial-cluster form factor • Solve couple integro-differential equation for relative wave function • Extract cross section from asymptotic normalization  Entrance-channel wave function Relative wave function Orbital angular momentum L Radial-cluster overlap enters in these matrix elements of H Harmonic oscillator functions Sum over all channels

20. Example: 4He + t 7Li and 6Li + n 7Li  4He + t

21. Example: 7Be(p,)8B • NCSM cluster overlap • Fit tail to binding energy • Potential model for capture Preliminary P. Navratil and C. Bertulani

22. Extensions and improvements: • Determine the form of the NNN-interaction • Implementation of effective operators for transitions • Four-body effective interactions • Effective field-theory potentials; are they any good for structure? • Integrate the structure into some reaction models (underway) • Questions and open problems to be addressed: • Is it possible to improve the mean field? • Can we improve the convergence of the higher states? • Unbound states. Can we use a continuum shell model? • How high in A can we go? • Can we use this method to derive effective interactions for conventional nuclear structure studies? Summary • Significant progress towards an exact understanding of nuclear structure is being made! • These are exciting times!!