Ab Initio Shell Model

Goal:
Solving the nuclear structure problem of light nuclei described as systems of nucleons interacting by realistic inter-nucleon forces

Ab Initio : realistic inter-nucleon interactions
Shell model : finite harmonic oscillator (HO) basis
All nucleons active : no-core shell model (NCSM)
Large-scale calculations due to huge basis dimensions: LLNL ASCI computers employed

Nucleon-nucleon interaction has a strong repulsive core whose effects cannot be correctly described using a finite HO basis: Need to derive effective interaction from the underlying realistic inter-nucleon interaction.

Effective Hamiltonian obtained by unitary transformation designed to decouple model space (our finite HO basis space) and the complementary space :

Hamiltonian

This concept is applied on a subcluster level. When a truncated basis is used to solve the nuclear many-body problem, an effective interaction should be used instead of the original nucleon-nucleon interaction to account for the neglected part of the basis. An important property of the effective interaction is the fact that it contains higher-body cluster contributions, in general up to A-body for an A-nucleon system, even if the original interaction was just a two-body interaction. The simplest approximation to the effective interaction is obtained by retaining just the two-body clusters. These can be calculated in a straightforward way by deriving the unitary transformation X from the two-nucleon system solutions. The next improvement is to compute the three-body cluster contribution. For that one first needs to solve exactly the three-nucleon problem and then construct the corresponding unitary transformation X from those three-nucleon solutions.

Power of effective interaction theory to calculate properties of ligh nuclei from the first principles is demonstrated on this figure:
Convergence in 4He

Calculated ground-state energy of ⁴He using the QCD-based Idaho-A nucleon-nucleon potential. Results obtained with the bare (dotted line), two-body effective (dashed line), three-body effective (full line) and four-body effective (dashed-dotted line) interactions as a function of basis size characterized by N_max, the total number of harmonic oscillator excitations, are presented. The calculation was done using a particular HO frequency, but similar results are obtained for a wide range of frequencies.

Note the contrast among the results obtained with the bare (unmodified) and two-body, three-body and the four-body effective interactions. The four-body effective interaction calculation is equivalent to solving the ⁴He system exactly.

We note how the increased clustering improves the convergence. For example, using the three-body correlations in the effective interaction we obtain quite reasonable results already for N_max= 6. We apply the three-body effective interactions in our ab initio calculations for the p-shell nuclei (4<A<17).

We reach a near-convergence in our ⁶Li calculations with the non-local CD-Bonn nucleon-nucleon interaction. Our excitation spectrum exhibits a good stabilty with respect to the basis size change: 6Li

An analogous calculation for a more complex¹⁰B also shows a reasonable stability of the excitation spectra with the basis size change. From our results we conclude that the realistic nucleon-nucleon potentials like the CD-Bonn do not reproduce the experimental ground state spin of this nucleus:

10B

Apparently, realistic nucleon-nucleon potentials are not sufficient for the description of light nuclei. Multi-nucleon forces play a role, most notably the three-nucleon interaction. Ab initio no-core shell model calculations that include the "real" three-nucleon interactions are under way.

July 30, 2002. For information about this page, contact Petr Navratil
UCRL-WEB-149701