The No-Core Shell Model

The No-Core Shell Model New Perspectives on P-shell Nuclei - The Shell Model and Beyond 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 UCRL-PRES-205602 July 22, 2003 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

Nuclear structure at the dawn of the 21 st century P-shell nuclei have been studied for a long time But there are still many mysteries 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

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 and NO! The Shell Model as practiced is an empirical tool that often requires arbitrary and unsatisfying approximations to make it work If we work hard enough, we can use effective interaction theory to apply the shell model correctly

We start with the many-body Hamiltonian The Good: The Bad: r p H = 2 i 2m + V r NN i r i i< j ( r j ) Add the center-of-mass oscillator potential H CM = 1 2 AmΩ2 r R 2 = i 1 2 mωr r Provides a convenient basis to build the many-body Slater determinants Does not affect the intrinsic motion i 2 i< j mω 2 Exact separation between intrinsic and center-of-mass motion 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) 2A r i r ( r j ) 2 hω

Many-body solutions Typical eigenvalue problem H H M H 11 21 N1 HΨ i = E i Ψ i Construct many-body states φ i so that Ψ i = H H 12 22 n L O K C in φ n Calculate Hamiltonian matrix H ij = φ j H φ i Diagonalize to obtain eigenvalues H H 1N NN φ = 1 A! 0f1p N=4 0d1s N=2 1p N=1 0s N=0 ( ) φ i ( r 2 ) K φ i ( r A ) ( ) φ j ( r 2 ) φ j ( r A ) φ i r 1 φ j r 1 M O M φ l ( r 1 ) φ l ( r 2 ) K φ l ( r A ) = a l + Ka j + a i + 0 Problem: Short-range repulsion requires an infinite space

Can we get around this problem? Effective interactions φ n Choose subspace of for a calculation (P-space) Include most of the relevant physics Q -space (excluded - infinite) Effective interaction: H eff ˆ P Ψ i = E i ˆ P Ψ i P Q Lee-Suzuki: H eff =PXHX -1 P QXHX -1 P=0

The general idea behind effective interactions ε Q H eff has one-, two, three-, A-body terms H P-space defined by N max hω ε Ν ε Ν ε Ν+1 ε Ν+2 P ε 3 ε 2 ε 1 Exact reproduction of N eigenvalues H eff QXHX -1 P=0 P Q

Effective interactions with the Lee-Suzuki method Choose P-space for A-body calculation, with dimension d p P-space basis states: α P and Q-space basis states: αq Need d P solutions, k, in the infinite space, i.e., H k = Ek k Write X=e -ω Qe ω He ω P = 0; ω = QωP d P α Q ω α P = α Q k k α P ; k α P α P k = δ k k k d P β P H eff α P = β P 1+ ω + ω k α P α P α P ( ) 1 2 α P α P k E k k α P α P ( 1+ ω + ω) 1 2 α P β P d P ( 1+ ω + ω) α P = β P k k α P k Hermitian effective interaction β P H eff α P exactly reproduces d p solutions of the full problem Essentially the same procedure is used for n-body clusters

The NCSM uses a variety of realistic interactions 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, N 3 LO, etc Softer, faster convergence Three-body n=3 clusters in H eff Tucson-Melbourne INOY Two-body Urbana and EFT-based Inside non-local, outside Yukawa Fit to NN data and 3 H and 3 He

Computational aspects of the NCSM Two-body interactions Three-body interactions Model-space limitation ~ 10 8 states A=3, N max = 36, Jacobi coordinates A=4, N max = 20, Jacobi coordinates A=6, N max = 14, M-scheme 7 A 11, N max ~ 10, M-scheme A 12, N max = 6, M-scheme Limited by number of three-body matrix elements For N max =4, 39,523,066 For N max =6 over 600,000,000 Practical limit is N max = 6 for all the p-shell Four-body - N max = 4

Application to A=3 and convergence 3 H with CD-Bonn and N 3 LO Potential AV18 N 3 LO CD-Bonn INOY Exp E gs (MeV) -7.62-7.86-8.00-8.48-8.48

Application to A=3 and convergence 3 H with CD-Bonn and N 3 LO Potential AV18 N 3 LO CD-Bonn INOY Exp E gs (MeV) -7.62-7.86-8.00-8.48-8.48

Effective interactions really work 4 He with the effective-field theory Idaho-A potential Effective interactions improve convergence! Are EFT potentials useful for nuclear-structure studies?

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

Results with three-body effective interactions Binding energies with Av8 400 kev 1.8 MeV, Clustering? Three-body effective interactions represent a significant improvement and give results within 400 kev of the GFMC Oscillator parameter Model space

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

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

How about the three-body interaction? Tucson-Melbourne

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

Can we explain the parity inversion in p-shell nuclei? 9 Be

Can we explain the parity inversion in p-shell nuclei? 9 Be - Extrapolation

Can we explain the parity inversion in p-shell nuclei? 11 Be

Can we explain the parity inversion in p-shell nuclei? 11 Be - Extrapolation

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 h2 d 2 L(L +1) + 2 2M dr r 2 + V Coul + E 7 Be + E 8 Be E u Γ r Entrance-channel wave function [ 3 He 4 He ] s Y L ( r ˆ )g J ( r) ( ) + drr nl ( r) H Γn, Sum over all channels 3 He + 4 He 7 Be JM Relative wave function Orbital angular momentum L Radial-cluster overlap enters in these matrix elements of H Γ n R n L ( r )u Γ ( r ) = 0 Γ n n 0 Harmonic oscillator functions

Example: 4 He + t 7 Li and 6 Li + n 7 Li 4 He + t 7 Li [ 3 H 4 He ] Formalism is in place for computing radial-cluster overlaps Resonances for 4 He + t and 6 Li + n are in accordance with experiment Model-space renormalization in FY05 7 Li [ 6 Li n ]

Summary Significant progress towards an exact understanding of nuclear structure is being made! These are exciting times!! 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? hω Can we use this method to derive effective interactions for conventional nuclear structure studies?