Large-Scale Shell-Model Study of the Sn-isotopes Talk at the FAIR Workshop, ICNFP, 28 July – 6 August 2014 T. Engeland, M. Hjorth-Jensen, E. Osnes Department of Physics, University of Oslo

Objective Explore the nuclear landscape using the ‘workhorse’ of nuclear structure - - the shell model Despite recent developments, the shell model remains the most ‘universal’ microscopic approach to nuclear structure See how far it can take us

Outline Objective Reminder Effective interactions 4. Large-scale shell-model calculations (~ 10 ± valence nucleons) 5. Summary and outlooks

Reminder: ’Ancient’ history Success of the independent-particle shell model (~1950) Explicit inclusion of interactions among the valence nucleons - phenomenological forces of simple mathematical form - fit effective force to experimental level energies From 1960’s on, introduction of ‘realistic’ effective interactions pioneered by Kuo and Brown - nucleon-nucleon forces fitting two-nucleon data - inclusion of higher-order effects by many-body perturbation theory

Reminder: The basic steps Get rid of the repulsive hard core in the nucleon-nucleon potential by some renormalization procedure - Bethe-Brueckner-Goldstone reaction matrix G - T-matrix type V(lowk) Include higher order corrections in G or V(lowk) Apply the emerging effective interaction to some shell-model calculation

Our ”philosophy” Aim: Obtain a realistic effective interaction with predictive power for nuclear systems with many valence nucleons Effective interaction: - Use Q-box formulation of Kuo et al. - To ensure interaction contain relevant properties, consider general trends over wide range of nuclei rather than detailed spectra of selected nuclei. With general properties in place, proceed to look at details. Many-particle problem: - Early calculations dealt with nuclei with only a few valence nucleons and considered detailed spectroscopy - Now we are able to study systematic features of series of nuclei from a few to many valence nucleons - Present case: Sn-isotopes with up to 32 neutrons beyond a 100Sn closed-shell core - Evaluate ground-state binding energies, yrast state energies and transition rates, shell closure, etc.

Model-space eigenvalue problem Instead of solving the full eigenvalue we solve a model/valence-space eigenvalue problem with an effective Hamiltonian operator P projecting onto model space, and

Q-box formulation of effective interaction Construct effective interaction from Q-box formulation of Kuo and collaborators Include folded diagrams by differentiation wrt starting energy

Application to the Sn-isotopes

Input to the Sn-calculation (1) Up to 32 valence neutrons in N = 4 oscillator shell 1d5/2 [0] 2s1/2 [2.45 MeV] 1d3/2 [2.55 MeV] + intruders from below and above 0g7/2 [0.20 MeV] 0h11/2 [3.00 MeV] High dimensionality e.g.16 million basis states for n = 16 valence nucleons Use Lanzcos iteration obtain tri-diagonal energy matrix

Input to the Sn-calculation (2) Explorative calculations Q-box to 3rd order evaluated to 2, 4, 6, 8, 10 ħω intermediate excitation Start from G-matrix and V(lowk) Three different nucleon-nucleon potentials V18, CD Bonn, N3LO

N3LO Gmat Gmat 10ħω

Vlow Gmat and Vlow N3LO 10ħω

Input to the Sn-calculation (3) Final calculations Q-box to 3rd order evaluated to 10 ħω intermediate excitation Start from G-matrix Use the chiral nucleon-nucleon potential N3LO

N3LO Gmat 10ħω

Binding energies rel 100Sn One-part sep energies ε + nα + β ε + nα α(exp) = -0.21 MeV β(exp) = 2.92 MeV α(calc) = -0.08 MeV β(calc) = 2.40 MeV

Sqp states calc Sqp states exp

BE2(0→2)

Morales, Van Isacker, Talmi Subshell effects? Morales, Van Isacker, Talmi Phys Lett B 703 (2011) 606 N3LO 10ħω Present calculation

Summary and outlooks Capable of handling large shell-model systems Present realistic two-body effective interactions show predictive power when it comes reproducing general systematics of energy spectra for nuclei with many valence nucleons, however they still fail on binding energies (monopole component) Effective three-body forces due to truncation of the valence space could produce the desired repulsion, but elimination of disconnected diagrams is tricky in practice With the current and prospective interest in the properties of rare isotopes, efforts should be continued with the aim of deriving effective forces with truly reliable predictive power

Back-up slides

Effective interactions De-Shalit & Talmi: αj and βj well-defined linear combinations of two- particle interaction matrix elements Empirically in Ca-isotopes: α small and repulsive (0.23 MeV) β large and attractive (-3.33 MeV)

Starting from NN-interaction (Brown, Kuo et al.) Bare G-matrix α small, attractive (-0.21) β attractive, too small (-0.66) Adding 2nd order-core polarization α repulsive, (too) small (0.15) β attractive, too small (-1.96) Adding 3-body little effect

Further developments Separate summation of classes of diagrams to arbitrary order (Ellis, Kirson, Zamick, ...) - e.g. TDA, RPA served to enhance core polarization - screening of particle- hole interaction reduced to 2nd order result Barrett and Kirson (1970) Including 3rd order diagrams - 3rd order typically of similar magnitude but oppsite sign to 2nd order - no obvious order-by- order convergence

N3LO Gmat Gmat 10ħω

Vlow N3LO 10ħω

N3LO Gmat 10ħω

ε + nα + β ε + nα Binding energies rel 100Sn One-part sep energies α(exp) = 0.21 MeV β(exp) = -2.92 MeV α(calc) = 0.08 MeV β(calc) = -2.40 MeV

Sqp states calc Sqp states exp

BE2(0→2)