Shell Structure of Exotic Nuclei

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Shell Structure of Exotic Nuclei (a Paradigm
Shift?) Witold Nazarewicz (UTK, ORNL,
UWS) University of Liverpool, UK, Feb. 29, 2008

• Introduction
• Shell structure revisited
• Nuclear Density Functional Theory
• Questions and Challenges, Homework
• Perspectives

Emphasis on novel aspects recent
results problems
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Introduction
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(No Transcript)
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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

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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)
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Pronounced shell structure (quantum numbers)
Shell structure absent
closed trajectory (regular motion)
trajectory does not close
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Shells
P. Moller et al.
S. Frauendorf et al.

• Jahn-Teller Effect (1936)
• Symmetry breaking and deformed (HF) mean-field

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Magicity is a fragile concept
Near the drip lines nuclear structure may be
dramatically different.
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Nature 449, 1022 (2007)
Phys. Rev. Lett. 99, 192501 (2007)
No shell closure for N8 and 20 for drip-line
nuclei new shells at 14, 16, 32
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What is the next magic nucleus beyond 208Pb?
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Prog. Part. Nucl. Phys. 59, 432 (2007)
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Physics of the large neutron excess

• 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

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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

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Nuclear Local s.p. Densities and Currents
isoscalar (T0) density
isovector (T1) density
isoscalar spin density
isovector spin density
current density
spin-current tensor density
kinetic density
kinetic spin density
analogous p-p densities and currents
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Construction of the functional Perlinska et al.,
Phys. Rev. C 69, 014316 (2004)
p-h density
p-p density
Most general second order expansion in densities
and their derivatives
pairing functional
Not all terms are equally important. Some probe
specific observables
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Nuclear DFT Large-scale Calculations
S. Cwiok, P.H. Heenen, W. Nazarewicz Nature, 433,
705 (2005)
Stoitsov et al., PRL 98, 132502 (2007)

• Global DFT mass calculations HFB mass formula
  ?m700keV
• Taking advantage of high-performance computers

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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, 182501 (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/0604043 Otsuka et al. Phys.
Rev. Lett. 97, 162501 (2006) Lesinski et al.,
arXiv0704.0731, and the nuclear shell
model T. Otsuka et al., Phys. Rev. Lett. 87,
082502 (2001) Phys. Rev. Lett. 95, 232502 (2005)
28, 50, 82, 126
2, 8, 20
?F
jlt
jgt
jgt
Spin-saturated systems
Spin-unsaturated systems
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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.

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Importance of the tensor interaction far from
stability
5237/2
Proton emission from 141Ho
4111/2
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The nucleus is a correlated open quantum
many-body system Environment continuum of decay
channels
7162
6049
Spectra and matter distribution modified by the
proximity of scattering continuum
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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
(1958-1962), 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
Gamow (complex-energy) Shell Model
(2002 -) N. Michel et al, PRL 89
(2002) 042502 R. Id Betan et al, PRL 89 (2002)
042501 N. Michel et al, PRC 70 (2004) 064311 G.
Hagen et al, PRC 71 (2005) 044314
Continuum (real-energy) Shell Model
(1977 - 1999 - 2005) 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) 042503
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Rigged Hilbert space Gamow Shell Model (2002)
One-body basis
J. Rotureau et al., DMRG Phys. Rev. Lett. 97,
110603 (2006)
non-resonant continuum
bound, anti-bound, and resonance states
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N. Michel et al. PRC 75, 0311301(R) (2007)
WS potential depth decreased to bind 7He.
Monopole SGI strength varied
Overlap integral, basis independent!
Anomalies appear at calculated thresholds
(many-body S-matrix unitary) Scattering
continuum essential
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Questions and challenges
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How to extend DFT to finite, self-bound systems?
Intrinsic-Density Functionals J. Engel, Phys.
Rev. C75, 014306 (2007) Generalized Kohn-Sham
Density-Functional Theory via Effective Action
Formalism M. Valiev, G.W. Fernando,
cond-mat/9702247 B.G. Giraud, B.K. Jennings, and
B.R. Barrett, arXiv0707.3099 (2007) B.G.
Giraud, arXiv0707.3901 (2007)
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How to root nuclear DFT in a microscopic theory?
ab-initio - DFT connection NNNNN - EDF
connection (via EFTRG)
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Connections to computational science
1Teraflop1012 flops 1peta1015 flops (next 2-3
years) 1exa1018 flops (next 10 years)
Jaguar Cray XT4 at ORNL No. 2 on Top500

• 11,706 processor nodes
• Each compute/service node contains 2.6 GHz
  dual-core AMD Opteron processor and 4 GB/8 GB of
  memory
• Peak performance of over 119 Teraflops
• 250 Teraflops after Dec.'07 upgrade
• 600 TB of scratch disk space

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Example Large Scale Mass Table
Calculations Science scales with processors
Jaguar_at_
M. Stoitsov, HFBLN mass table, HFBTHO
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)

Odd and odd-odd Nuclei

• The even-even calculations define 250,754
  configurations in odd-A and odd-odd nuclei
  assuming 0.5 MeV threshold for the blocking
  candidates
• Using 10,000 processors - about 25 CPU hours

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Bimodal fission in nuclear DFT
A. Staszczak, J. Dobaczewski, W. Nazarewicz, in
preparation
S. Umar and V. Oberacker Phys. Rev. C 76, 014614
(2007)
nucl-th/0612017
TDHF description of heavy ion fusion
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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