1. 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

2. 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

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

4. 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

5. 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

6. 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.

7. 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

8. 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

9. Application to the Sn-isotopes

10. 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

11. 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

12. N3LO Gmat
Gmat 10ħω

13. Vlow
Gmat and Vlow
N3LO 10ħω

14. 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

15. N3LO Gmat 10ħω

16. Binding energies
rel 100Sn
One-part sep energies
ε + nα + β
ε + nα
α(exp) = MeV
β(exp) = MeV
α(calc) = MeV
β(calc) = MeV

17. Sqp states calc
Sqp states exp

18. BE2(0→2)

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

20. 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

21. Back-up slides

22. 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)

23. 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

24. 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

25. N3LO Gmat
Gmat 10ħω

26. Vlow
N3LO 10ħω

27. N3LO Gmat 10ħω

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

29. Sqp states calc
Sqp states exp

30. BE2(0→2)