Shell Structure of Exotic Nuclei
(a Paradigm Shift?)
Witold Nazarewicz (University of Tennessee/ORNL)
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Introduction
Shell structure revisited
Nuclear Density Functional Theory
Questions and Challenges, Homework
Perspectives
JUSTIPEN
ジャスティペン
Emphasis on: novel aspects
recent results
problems
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!”
Introduction
Shell effects and classical periodic orbits
Shells
One-body field
• Not external (selfbound)
• Hartree-Fock
• 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
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

g   g˜    A  cos S  /   


S     p dq

  
 n1 ,n2 ,n3    n10 ,n20 ,n30   n1  n10   
n1 0
The action
integral for the
periodic orbit 
  
  
 n2  n20    n3  n30   
n2 0
n3 0
       
  :   :    k1 : k2 : k3
n1 0 n2 0 n3 0
Nshell  k1 n1  k2 n2  k3 n3 ,
Principal shell
quantum number
 shell
Condition for
shell structure
1  
  
ki ni 0
Distance between shells
(frequency of classical orbit)
Pronounced
shell structure
(quantum numbers)
Shell structure
absent
shell
gap
shell
gap
shell
closed trajectory
(regular motion)
trajectory does not close
Shell Energy (MeV)
10
experiment
Shells
P. Moller et al.
experiment
0
-10
Nuclei
theory
theory
0
-10
20 28
discrepancy
50
82
126
S. Frauendorf et al.
0
diff.
20
1 experiment
60
100
Number of Neutrons
Sodium Clusters
• Jahn-Teller Effect (1936)
• Symmetry breaking and
deformed (HF) mean-field
0
Shell Energy (eV)
-10
-1
58
92
198
138
spherical
clusters
1 theory
0
deformed
clusters
-1
50
100
150
200
Number of Electrons
Magicity is
a fragile
concept
Near the drip lines
nuclear structure
may be dramatically
different.
First experimental indications
demonstrate significant changes
No shell closure for N=8 and 20 for
drip-line nuclei; new shells at 14, 16,
32…
What is the next magic nucleus beyond 208Pb?
Physics of the large neutron excess
Interactions
Interactions
Many-body
Correlations
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
• Isovector (N-Z) effects
• Poorly-known components come
into play
• Long isotopic chains crucial
Open
Channels
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
Prog. Part. Nucl. Phys. 59, 432 (2007)
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
Nuclear Local s.p. Densities and Currents
0 r   0 r ,r    r  ;r  
isoscalar (T=0) density
1r   1 r ,r    r  ;r  
isovector (T=1) density
s0 r    r  ;r  '   '
isoscalar spin density

s1r    r  ;r  '   ' 

isovector spin density


 ' 
 '


current density


spin-current tensor density
i
'  T r ,r ' r ' r
2
i
JT r   '   sT r ,r ' r ' r
2
 T r    ' T r,r ' r ' r
kinetic density
TT r    ' sT r,r ' r ' r
kinetic spin density
jT r  


0
 n   p 
1
 n   p 
+ analogous p-p densities and currents
Justification of the standard Skyrme functional: DME
In practice, the one-body density matrix is strongly peaked around
r=r’. Therefore, one can expand it around the mid-point:

1 2 
1
r  r 
r, r   qis  j q  s  q   q q, q 
, s  r  r 

2 
4
2


1
2
r , r    q  s q q  j q  q q q


4
2
2
2
The Skyrme functional was justified in such a way in, e.g.,
•Negele and Vautherin, Phys. Rev. C5, 1472 (1972); Phys. Rev. C11, 1031 (1975)
•Campi and Bouyssy, Phys. Lett. 73B, 263 (1978)
… but nuclear EDF does not have to be related to any given
effective two-body force!
Actually, many currently used nuclear energy functionals are not
related to a force
DME and EFT+RG
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
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., arXiv:0704.0731,…
…and the nuclear shell model:
T. Otsuka et al., Phys. Rev. Lett. 87, 082502 (2001); Phys. Rev. Lett. 95, 232502
(2005)
2, 8, 20
j<
F
j>
Spin-saturated systems
28, 50, 82, 126
j<
F
j>
Spin-unsaturated systems
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.
[411]1/2
[523]7/2
Importance of the tensor interaction far from stability
Proton emission
from 141Ho
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
Continuum (real-energy) Shell Model
(1977 - 1999 - 2005)
Gamow (complex-energy) Shell Model
(2002 -)
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
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
One-body basis
bound, anti-bound, and
resonance states
non-resonant
continuum
Rigged Hilbert space
Gamow Shell Model (2002)
J. Rotureau et al., DMRG
Phys. Rev. Lett. 97, 110603 (2006)
Questions and challenges
How to extend DFT to finite, self-bound systems?
E r   F r 
 r V r d r
3
ion
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, arXiv:0707.3099 (2007);
B.G. Giraud, arXiv:0707.3901 (2007)
What are the missing pieces?
What is density dependence?
(ph and pp channels)
Spin-isospin sector (e.g., tensor)
Momentum dependence of the effective mass?
• Induced interaction
• Isovector and isoscalar
Density Matrix Expansion for RG-Evolved Interactions
S.K. Bogner, R.J. Furnstahl et al.
see also:
EFT for DFT
R.J. Furnstahl
nucl-th/070204
How to parameterize time-odd pieces?
J. Dobaczewski and J. Dudek, Phys. Rev. C52, 1827 (1995)
M. Bender et al., Phys. Rev. C65, 054322 (2002)
H. Zdunczuk, W. Satula and R. Wyss, Phys. Rev. C71, 024305 (2005)
very poorly determined
Can be adjusted to the Landau parameters
•Important for all I>0 states (including low-spin states in odd-A
and odd-odd nuclei)
•Impact beta decay
•Influence mass filters (including odd-even mass difference)
•Limited experimental data available
High-spin terminating states
Zdunczuk et al.,Phys.Rev. C71, 024305(2005)
Stoitcheva et al., Phys. Rev. C 73, 061304(R) (2006)
Isospin dynamics
important!
• Excellent examples of singleparticle configurations
• Weak configuration mixing
• Spin polarization; probing timeodd terms!
• Experimental data available
How to restore broken symmetry in DFT?
• The transition density matrices contains complex poles. Some
cancellation appears if the ph and pp Hamiltonians are the same
• The projection operator cannot be defined uniquely
• Problems with fractional density dependence
• Projected DFT yields questionable results when the pole appears close to
the integration contour. This often happens when dealing with PESs
J. Dobaczewski et al., Phys. Rev. C76, 054315 (2007)
see also: M. Bender, T. Duguet, D. Lacroix, in preparation.
S. Krewald et al.,Phys. Rev. C 74, 064310 (2006).
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, 040502 (2007)
See also:
Density-functional theory for fermions
in the unitary regime
T. Papenbrock
Phys. Rev. A72, 041603 (2005)
Density functional theory for fermions
close to the unitary regime
A. Bhattacharyya and T. Papenbrock
Phys. Rev. A 74, 041602(R) (2006)
How to root nuclear DFT in a microscopic theory?
ab-initio - DFT connection
NN+NNN - EDF connection (via EFT+RG)
Ab-initio - DFT Connection
• One-body density matrix is the key quantity to study
• “local DFT densities” can be expressed through (x,x’)
• Testing the Density Matrix Expansion and beyond
UNEDF
Pack Forest meeting
UNEDF Homework
• Introduce external potential
• HO for spherical nuclei (amplitude of
zero-point motion=1 fm)
• 2D HO for deformed nuclei
• Density expressed in COM
coordinates
• Calculate x,x’) for 12C, 16O and
40,48,60Ca
(CC)
isospin
• Perform Wigner transform to relative
and c-o-m coordinates q and s
• Extract , J, 
• Analyze data by comparing with
results of DFT calculations and lowmomentum expansion studies.
• Go beyond I=0 to study remaining
densities (for overachievers)
1 
1

r, r   qis  j q  s2 q   q q, q 

2 
4
r  r 
, s  r  r 
2
Negele and Vautherin:
PRC 5, 1472 (1972)
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
1Teraflop=1012 flops
1peta=1000 tera
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