1. Towards the Nuclear Universal Energy Density Functional
Witold Nazarewicz (Tennessee/Warsaw)
Milan, May 2007
Introduction
Nuclear DFT
Successes and challenges
Large-amplitude collective motion
Unifying structure and reaction aspects (nuclei are open systems)
Rigged Hilbert space formulation
Summary
Emphasis on: novel aspects
recent results
problems

2. Theory of Nuclei Overarching goal:
Self-bound, two-component quantum many-fermion system
Complicated interaction based on QCD with at least two- and three-nucleon components
We seek to describe the properties of finite and bulk nucleonic matter ranging from the deuteron to neutron stars and nuclear matter; including strange matter
We want to be able to extrapolate to unknown regions
To arrive at a comprehensive and unified microscopic description of all nuclei and low-energy reactions from the the basic interactions between the constituent protons and neutrons
There is no “one size fits all” theory for nuclei, but all our theoretical approaches need to be linked. We are making great progress in this direction.

3. Bottom-up approaches to nuclear structure
Roadmap
Ab initio
Configuration interaction
Density Functional Theory
Theoretical approaches overlap and need to be bridged

4. Old paradigms, universal ideas, are not correct
Near the drip lines nuclear structure may be dramatically different.
No shell closure for N=8 and 20 for drip-line nuclei; new shells at 14, 16, 32…
First experimental indications demonstrate significant changes

5. Issues 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

7. 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!”
D. Furnstahl, INT Fall’05

8. 2002-2006: very successful period for theory of nuclei
many new ideas leading to new understanding
new theoretical frameworks
exciting developments
high-quality calculations
The nucleon-based description works to <0.5 fm
Effective Field Theory/Renormalization Group provides missing links
Accurate ab-initio methods allow for interaction tests
Quantitative microscopic nuclear structure
Integrating nuclear structure and reactions
High-performance computing continues to revolutionize microscopic nuclear many-body problem: impossible becomes possible

9. 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

10. Towards the Universal Nuclear Energy Density Functional
Walter Kohn: Nobel Prize in Chemistry in 1998
isoscalar (T=0) density
isovector (T=1) density
isoscalar spin density
Local densities
and currents
+ pairing…
isovector spin density
current density
spin-current tensor density
kinetic density
kinetic spin density
Example: Skyrme
Functional
Total ground-state HF energy

11. Justification of the standard Skyrme functional: DME
Another motivation… Let us consider a local two-body interaction
The HF interaction energy is:
In practice, the density matrix is strongly peaked around r=r’ (cf. TF expression!). Therefore, one can expand it around the mid-point:
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)
However, the parameters derived in such a way do not reproduce the nuclear bulk properties precisely enough. Hence, the density matrix expansion should be used as a guiding principle, but the actual parameters should be adjusted phenomenologically.
The functional 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.
Exchange-correlation term incorporated

13. From Qualitative to Quantitative!
Nuclear DFT
From Qualitative to Quantitative!
S. Cwiok, P.H. Heenen, W. Nazarewicz
Nature, 433, 705 (2005)
Deformed Mass Table in one day!
HFB mass formula: m~700keV
Good agreement for mass differences

14. Average value: symmetry energy
Empirical Proton-Neutron Interactions and Nuclear Density Functional
Stoitsov, Cakirli, Casten, Nazarewicz, Satula, PRL 98, (2007)
Average value:
symmetry energy

15. Er Ra Pb U spherical Shell systems closure deformed octupole
collectivity
Shell
closure
spherical
systems
deformed Er Ra Pb U
theory (DFT)
experiment

17. Bimodal fission 132Sn + 132Sn nucl-th/0612017
A. Staszczak, J. Dobaczewski
W. Nazarewicz, in preparation
132Sn + 132Sn

18. Fission pathways in the nuclear density functional theory
A. Staszczak, J. Dobaczewski
W. Nazarewicz, in preparation
All intrinsic symmetries
can be broken
Motion non-adiabatic

19. What are the missing pieces?
Density Functional Theory

20. 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); G. Colo et al., Phys. Lett. B646, 227 (2007),…
…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

21. (strongly depend on shell filling)
J. Dobaczewski, nucl-th/
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.

22. Energy differences extracted from calculated masses
J.P. Schiffer et al., PRL 92 (2004)
Z=50, 51
Prog. Part. Nucl. Phys. 59, 432 (2007)
nucl-th/
Energy differences extracted from calculated masses
1g7/2 almost filled
1h11/2 almost empty

23. G. Stoitcheva et al., Phys. Rev. C73, 061304(R) (2006)
J. Dobaczewski and J. Dudek, Phys. Rev. C52, 1827 (1995)
M. Bender et al., Phys. Rev. C65, (2002)
S. Fracasso and G. Colo, preprint, 2007
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)
Important for terminating (maximally aligned) states
Impact beta decay
Influence mass filters (including odd-even mass difference)
Limited experimental data available
G. Stoitcheva et al., Phys. Rev. C73, (R) (2006)

24. nuclear collective dynamics
Beyond Mean Field
nuclear collective dynamics
Variety of phenomena:
symmetry breaking and quantum corrections
LACM: fission, fusion, coexistence
phase transitional behavior
new kinds of deformations
Significant computational resources
required:
Generator Coordinate Method
Projection techniques
Imaginary time method (instanton techniques)
QRPA and related methods
TDHFB, ATDHF, and related methods
M. Bender et al., PRC 69, (2004)
GCM
Shape coexistence
Challenges:
selection of appropriate degrees of freedom
simultaneous treatment of symmetry
coupling to continuum in weakly bound systems
dynamical corrections; fundamental theoretical problems.
rotational, vibrational, translational
particle number
isospin

25. Correlation energy, low-energy collectivity
Gogny
+CSE
518 first
2+ states
PN Projected HFB equations: PRC 66, (2002); nucl-th/
Large-scale GCM+proj correlation energy and 2+ state calculations: PRC 73, (2006); PRL 94, (2005); nucl-th/ ; Gogny+CSE: nucl-th/

26. Microscopic LDM and Droplet Model Coefficients: PRC 73, 014309 (2006)

27. Collective potential V(q)
Universal nuclear energy density functional is yet to be developed
Surface symmetry energy crucial
Different deformabilities!

28. P.H. Heenen et al., Phys. Rev. C57, 1719 (1998)
Shell effects in metastable minima seem to be under control.
Important data needed to fix
the deformability of the NEDF:
absolute energies of SD states
absolute energies of HD states
Advantages:
large elongations
weak mixing with ND structures

29. 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

30. Unique geometries of light nuclei due to the threshold effects
Exotic decay channels
Miernik et al. (Warsaw-MSU-Tennessee)

31. Rigged Hilbert space formulation
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)
One-body basis
J. Rotureau et al., DMRG
Phys. Rev. Lett. 97, (2006)
bound, anti-bound, and resonance states
non-resonant continuum
Michel et al.: Virtual states not
included explicitly in the GSM basis
Phys. Rev. C 74, (2006)

32. 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

33. Homework
Density dependence (insights from EFT)
Momentum dependence of effective mass
Pairing term (induced and direct)
How to incorporate dynamics directly into the functional? GCM, QRPA, VAP, CSE first…

34. Superheavy Elements in Nuclear DFT
long-lived SHE

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

36. Crazy topologies of superheavy nuclei due to the Coulomb frustration

37. rods
Self-consistent calculations confirm the fact that the “pasta phase” might have a rather complex structure, various shapes can coexist, at the same time significant lattice distortions are likely and the neutron star crust could be on the verge of a disordered phase.
Liquid crystal structure?
deformed nuclei
Skyrme HF with SLy4, Magierski and Heenen, Phys. Rev. C 65, (2002)

38. Early 2003 Idea behind RIATG conceived
November 2003 First RIATG meeting, Tucson
November 2003 NSAC Theory Report
March 2004 RIATG Charter approved
October 2004 Second RIATG meeting, Chicago
September 2005 Blue Book finalized
September 2005 Third RIATG meeting, Detroit
July 2006 JUSTIPEN launched
Nov 2006 UNDEF SciDAC-2 funded (DOE and NNSA)
Dec 2006 INCITE award (DOE/SC; 5M processor hours)
Jan 2007 RISAC NRC/NAS report endorses FRIB
May 2007 NSAC LRP endorses FRIB and theory effort

39. Thank You Conclusions CSM and DFT DFT
A comprehensive description of nuclei and their reactions is coming
Exotic nuclei are essential in this quest: they provide missing links
Bridging theoretical approaches
Bridging ab-initio and SM (effective interactions)
Ab initio and DFT (nuclear matter, density dependence)
EFT, RGT and DFT (effective operators)
Fermionic and Bosonic (algebraic)
Bridging structure with reactions (in both directions)
Bridging finite with bulk
CSM and DFT
DFT
Thank You

40. BACKUP

41. Example: Threshold anomaly
E.P. Wigner, Phys. Rev. 73, 1002 (1948), the Wigner cusp
G. Breit, Phys. Rev. 107, 923 (1957)
A.I. Baz’, JETP 33, 923 (1957)
R.G. Newton, Phys. Rev. 114, 1611 (1959).
A.I. Baz', Ya.B. Zel'dovich, and A.M. Perelomov, Scattering Reactions and Decay
in Nonrelativistic Quantum Mechanics, Nauka 1966
A.M. Lane, Phys. Lett. 32B, 159 (1970)
S.N. Abramovich, B.Ya. Guzhovskii, and L.M. Lazarev, Part. and Nucl. 23, 305 (1992).
The threshold is a branching point.
The threshold effects originate in conservation of the flux.
If a new channel opens, a redistribution of the flux in other open channels appears, i.e. a modification of their reaction cross-sections.
The shape of the cusp depends strongly on the orbital angular momentum.
a+X
a1+X1 at Q1
a2+X2 at Q2
an+Xn at Qn
Y(b,a)X
X(a,b)Y
a+X

42. Threshold anomaly (cont.)
Studied experimentally and theoretically in various areas of physics:
pion-nucleus scattering
R.K. Adair, Phys. Rev. 111, 632 (1958)
A. Starostin et al., Phys. Rev. C 72, (2005)
electron-molecule scattering
W. Domcke, J. Phys. B 14, 4889 (1981)
electron-atom scattering
K.F. Scheibner et al., Phys. Rev. A 35, 4869 (1987)
ultracold atom-diatom scattering
R.C. Forrey et al., Phys. Rev. A 58, R2645 (1998)
Low-energy nuclear physics
charge-exchange reactions
neutron elastic scattering
deuteron stripping
The presence of cusp anomaly could provide structural information about reaction products. This is of particular interest for neutron-rich nuclei

43. Coupling between analog states
in (d,p) and (d,n)
C.F. Moore et al.
Phys. Rev. Lett. 17, 926 (1966)

44. C.F. Moore et al., Phys. Rev. Lett. 17, 926 (1966)

45. Density dependence