1. Shell Structure of Exotic Nuclei
(a Paradigm Shift?)
Witold Nazarewicz (UTK, ORNL)
Rutgers University Seminar, Nov. 5, 2008
Introduction
Shell structure revisited
Nuclear Density Functional Theory
Questions and Challenges, Homework
Perspectives
Emphasis on: novel aspects
recent results
problems

2. Introduction

3. Shell effects and classical periodic orbits
One-body field
Not external (self-bound)
Hartree-Fock
Shells
Product (independent-particle) state is often an excellent starting point
Localized densities, currents, fields
Typical time scale: babyseconds (10-22s)
Closed orbits and s.p. quantum numbers
But…
Nuclear box is not rigid: motion is seldom adiabatic
The walls can be transparent

4. Shell effects and classical periodic orbits
Balian & Bloch, Ann. Phys. 69 (1971) 76
Bohr & Mottelson, Nuclear Structure vol 2 (1975)
Strutinski & Magner, Sov. J. Part. Nucl. 7 (1976) 138
Trace formula, Gutzwiller, J. Math. Phys. 8 (1967) 1979
The action integral for the periodic orbit 
Condition for
shell structure
Principal shell
quantum number
Distance between shells
(frequency of classical orbit)

5. gap shell Pronounced shell structure Shell structure absent
(quantum numbers)
Shell structure
absent
gap
shell
closed trajectory
(regular motion)
trajectory does not close 

6. Shells Nuclei Sodium Clusters Jahn-Teller Effect (1936)
experiment
theory
discrepancy 20 60
100
-10 10
Nuclei
Number of Neutrons
Shell Energy (MeV) 28 50 82
126
diff.
P. Moller et al.
S. Frauendorf et al. -1 1 50
100
150
200
Number of Electrons
Shell Energy (eV) 58 92
138
198
experiment
theory
deformed
clusters
spherical
Sodium Clusters
Jahn-Teller Effect (1936)
Symmetry breaking and deformed (HF) mean-field

7. Magicity is a fragile concept
Near the drip lines nuclear structure may be dramatically different.

8. Phys. Rev. Lett. 99, (2007)
Nature 449, 1022 (2007)
No shell closure for N=8 and 20 for drip-line nuclei; new shells at 14, 16, 32…

9. What is the next magic nucleus beyond 208Pb?

10. Weinberg’s Laws of Progress in Theoretical Physics
From: “Asymptotic Realms of Physics” (ed. by Guth, Huang, Jaffe, MIT Press, 1983)
First Law: “The conservation of Information” (You will get nowhere by churning equations)
Second Law: “Do not trust arguments based on the lowest order of perturbation theory”
Third Law: “You may use any degrees of freedom you like to describe a physical system, but if you use the wrong ones, you’ll be sorry!”

11. Physics of the large neutron excess
Interactions
Many-body
Correlations
Open
Channels
Interactions
Isovector (N-Z) effects
Poorly-known components come into play
Long isotopic chains crucial
Configuration interaction
Mean-field concept often questionable
Asymmetry of proton and neutron Fermi surfaces gives rise to new couplings (Intruders and the islands of inversion)
New collective modes; polarization effects
Open channels
Nuclei are open quantum systems
Exotic nuclei have low-energy decay thresholds
Coupling to the continuum important
Virtual scattering
Unbound states
Impact on in-medium Interactions

12. Prog. Part. Nucl. Phys. 59, 432 (2007)

13. Modern Mean-Field Theory = Energy Density Functional
mean-field ⇒ one-body densities
zero-range ⇒ local densities
finite-range ⇒ gradient terms
particle-hole and pairing channels
Hohenberg-Kohn
Kohn-Sham
Negele-Vautherin
Landau-Migdal
Nilsson-Strutinsky
Nuclear DFT
two fermi liquids
self-bound
superfluid

14. Nuclear Local s.p. Densities and Currents
isoscalar (T=0) density
isovector (T=1) density
isoscalar spin density
isovector spin density
current density
spin-current tensor density
kinetic density
kinetic spin density
+ analogous p-p densities and currents

15. Most general second order expansion in densities and their derivatives
Construction of the functional
Perlinska et al., Phys. Rev. C 69, (2004)
isoscalar (T=0) density
+isoscalar and isovector densities:
spin, current, spin-current tensor, kinetic, and kinetic-spin
+ pairing densities
isovector (T=1) density
p-h density
p-p density (pairing functional)
Most general second order expansion in densities and their derivatives
Constrained by microscopic theory: ab-initio functionals provide quasi-data!
Not all terms are equally important. Usually ~12 terms considered
Some terms probe specific experimental data
Pairing functional poorly determined. Usually 1-2 terms active.
Becomes very simple in limiting cases (e.g., unitary limit)

16. Example: Spin-Orbit and Tensor Force (among many possibilities)
The origin of SO splitting can be attributed to 2-body SO and tensor forces, and 3-body force
R.R. Scheerbaum, Phys. Lett. B61, 151 (1976); B63, 381 (1976); Nucl. Phys. A257, 77 (1976); D.W.L. Sprung, Nucl. Phys. A182, 97 (1972); C.W. Wong, Nucl. Phys. A108, 481 (1968); K. Ando and H. Bando, Prog. Theor. Phys. 66, 227 (1981); R. Wiringa and S. Pieper, Phys. Rev. Lett. 89, (2002)
The maximum effect is in spin-unsaturated systems
Discussed in the context of mean field models:
Fl. Stancu, et al., Phys. Lett. 68B, 108 (1977); M. Ploszajczak and M.E. Faber, Z. Phys. A299, 119 (1981); J. Dudek, WN, and T. Werner, Nucl. Phys. A341, 253 (1980); J. Dobaczewski, nucl-th/ ; Otsuka et al. Phys. Rev. Lett. 97, (2006); Lesinski et al., arXiv: ,…
…and the nuclear shell model:
T. Otsuka et al., Phys. Rev. Lett. 87, (2001); Phys. Rev. Lett. 95, (2005)
28, 50, 82, 126
2, 8, 20 F
j<
j< F
j>
j>
Spin-saturated systems
Spin-unsaturated systems

17. (strongly depend on shell filling)
acts in s and d states of
relative motion
acts in p states
SO densities
(strongly depend on shell filling)
Additional contributions in deformed nuclei
Particle-number dependent contribution to nuclear binding
It is not trivial to relate theoretical s.p. energies to experiment.

18. 16O 13C7 13N6 The nucleus is a correlated open
quantum many-body system
Environment: continuum of decay channels
`Alignment’ of w.b. state
with the decay channel
16O
Thomas-Ehrmann effect
13C7
13N6
1943
2365
3502
1/2
3089
4946
3685
12C+n
12C+p
1/2
3/2
7162
6049
Spectra and matter distribution modified by the proximity of scattering continuum

19. unification of structure and reactions
The importance of the particle continuum was discussed in the early days of the multiconfigurational Shell Model and the mathematical formulation within the Hilbert space of nuclear states embedded in the continuum of decay channels goes back to H. Feshbach ( ), U. Fano (1961), and C. Mahaux and H. Weidenmüller (1969)
unification of structure and reactions
resonance phenomena generic to many small quantum systems coupled to an environment of scattering wave functions: hadrons, nuclei, atoms, molecules, quantum dots, microwave cavities, …
consistent treatment of multiparticle correlations
Open quantum system many-body framework
Continuum (real-energy) Shell Model
( )
H.W.Bartz et al, NP A275 (1977) 111
R.J. Philpott, NP A289 (1977) 109
K. Bennaceur et al, NP A651 (1999) 289
J. Rotureau et al, PRL 95 (2005)
Gamow (complex-energy) Shell Model
(2002 -)
N. Michel et al, PRL 89 (2002)
R. Id Betan et al, PRL 89 (2002)
N. Michel et al, PRC 70 (2004)
G. Hagen et al, PRC 71 (2005)

20. Rigged Hilbert space Gamow Shell Model (2002) One-body basis
J. Rotureau et al., DMRG
Phys. Rev. Lett. 97, (2006)
bound, anti-bound, and resonance states
non-resonant continuum

21. N. Michel et al. PRC 75, 0311301(R) (2007) 5He+n 6He 6He+n 7He
WS potential depth decreased to bind 7He. Monopole SGI strength varied
5He+n He
Overlap integral, basis independent!
6He+n He
Anomalies appear at calculated thresholds (many-body S-matrix unitary)
Scattering continuum essential
see also Nucl. Phys. A 794, 29 (2007)

22. Questions and challenges

23. How to extend DFT to finite, self-bound systems?
Intrinsic-Density Functionals
J. Engel, Phys. Rev. C75, (2007)
Generalized Kohn-Sham Density-Functional Theory via Effective
Action Formalism
M. Valiev, G.W. Fernando, cond-mat/
B.G. Giraud, B.K. Jennings, and B.R. Barrett, arXiv: (2007);
B.G. Giraud, arXiv: (2007)

24. Can dynamics be incorporated directly into the functional?
Example: Local Density Functional Theory for Superfluid Fermionic Systems: The Unitary Gas, Aurel Bulgac, Phys. Rev. A 76, (2007)
See also:
Density-functional theory for fermions
in the unitary regime
T. Papenbrock
Phys. Rev. A72, (2005)
Density functional theory for fermions
close to the unitary regime
A. Bhattacharyya and T. Papenbrock
Phys. Rev. A 74, (R) (2006)

25. Connections to computational science
1Teraflop=1012 flops
1peta=1015 flops (next 2-3 years)
1exa=1018 flops (next 10 years)

26. How to root nuclear DFT in a microscopic theory?
ab-initio - DFT connection
NN+NNN - EDF connection (via EFT+RG)

27. Broyden’s Mixing Procedure: Phys. Rev. C 78, 014318 (2008)
A. Baran, A. Bulgac, M. McNeil Forbes, G. Hagen, W. Nazarewicz, N. Schunck and M.V. Stoitsov
300,000
200
108
3,000,000

28. Example: Large Scale Mass Table Calculations
Science scales with processors
M. Stoitsov
HFB+LN mass table, HFBTHO
INCITE award
Dean et al. 17.5M hours
Even-Even Nuclei
The SkM* mass table contains 2525 even-even nuclei
A single processor calculates each nucleus 3 times (prolate, oblate, spherical) and records all nuclear characteristics and candidates for blocked calculations in the neighbors
Using 2,525 processors - about 4 CPU hours (1 CPU hour/configuration)
Jaguar Cray XT4 at ORNL
All Nuclei
9,210 nuclei
599,265 configurations
Using 3,000 processors - about 25 CPU hours
see MassExplorer.org

29. A. Staszczak, J. Dobaczewski, W. Nazarewicz, in preparation
Bimodal fission in nuclear DFT
A. Staszczak, J. Dobaczewski, W. Nazarewicz, in preparation
S. Umar and V. Oberacker
Phys. Rev. C 76, (2007)
nucl-th/
TDHF description
of heavy ion fusion

30. Conclusions
Why is the shell structure changing at extreme N/Z ? Can we talk about shell structure at extreme N/Z ?
Interactions
Many-body
Correlations
Open
Channels
Thank You