microscopic theories of nuclear masses
TRANSCRIPT
Microscopic Theories of Nuclear Masses
Thomas DUGUET
MSU/NSCL
JINA - Pizza Lunch seminar, MSU, 02/26/2007
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Outline
1 Introduction
2 Nuclear Energy Density Functional approach: general characteristics
3 EDF mass tables from the Montreal-Brussels group
4 Towards more microscopic EDF methods and mass tables
5 Conclusions
6 Nuclear Energy Density Functional approach: elements of formalism
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Outline
1 Introduction
2 Nuclear Energy Density Functional approach: general characteristics
3 EDF mass tables from the Montreal-Brussels group
4 Towards more microscopic EDF methods and mass tables
5 Conclusions
6 Nuclear Energy Density Functional approach: elements of formalism
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Some of the big questions
What binds protons and neutrons into stable nuclei and rare isotopes?
What is the origin of simple/complex patterns in nuclei?
When and how were the elements from iron to uranium created?
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Nuclear masses and the r -process: basics
Overall impact
Implicate masses of the most neutron-rich nuclei
Mostly through (n, γ)-(γ, n) competition
Masses also impact beta-decay rates, fission probabilities
Mass differences are in fact important: SN , Qβ
Impact of measured masses on theoretical models
1995 → 2003: only 45 of the 382 new masses are neutron-rich
Almost no mass of nuclei involved in the r -process are currently known
Little help so far in constraining theoretical models
Impressive progress currently made, i.e. NSCL’s penning trap, G. Bollen
Data STRONGLY needed but theory will still fill the gap
Disclaimer: many other nuclear inputs are of crucial importance
Low-energy dipole strength S(ω, E1) in neutron-rich nuclei for (n, γ)
Gamow-Teller strength for beta-decay, e-capture rates, neutrino-induced excitations
. . .
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Nuclear masses and the r -process: basics
Overall impact
Implicate masses of the most neutron-rich nuclei
Mostly through (n, γ)-(γ, n) competition
Masses also impact beta-decay rates, fission probabilities
Mass differences are in fact important: SN , Qβ
Impact of measured masses on theoretical models
1995 → 2003: only 45 of the 382 new masses are neutron-rich
Almost no mass of nuclei involved in the r -process are currently known
Little help so far in constraining theoretical models
Impressive progress currently made, i.e. NSCL’s penning trap, G. Bollen
Data STRONGLY needed but theory will still fill the gap
Disclaimer: many other nuclear inputs are of crucial importance
Low-energy dipole strength S(ω, E1) in neutron-rich nuclei for (n, γ)
Gamow-Teller strength for beta-decay, e-capture rates, neutrino-induced excitations
. . .
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Nuclear masses and the r -process: basics
Overall impact
Implicate masses of the most neutron-rich nuclei
Mostly through (n, γ)-(γ, n) competition
Masses also impact beta-decay rates, fission probabilities
Mass differences are in fact important: SN , Qβ
Impact of measured masses on theoretical models
1995 → 2003: only 45 of the 382 new masses are neutron-rich
Almost no mass of nuclei involved in the r -process are currently known
Little help so far in constraining theoretical models
Impressive progress currently made, i.e. NSCL’s penning trap, G. Bollen
Data STRONGLY needed but theory will still fill the gap
Disclaimer: many other nuclear inputs are of crucial importance
Low-energy dipole strength S(ω, E1) in neutron-rich nuclei for (n, γ)
Gamow-Teller strength for beta-decay, e-capture rates, neutrino-induced excitations
. . .
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Theoretical predictions of nuclear masses
Look for a global theory of
Mass differences and absolute masses (for consistency)
Hopefully other observables
Necessarily of semi-empirical character
”Few” parameters fitted to all known masses
Theory used to extrapolate to unknown nuclei
Bethe-Weizsacker formula (1935)
Negative binding energy of a liquid drop with A = N + Z/I = N − Z
E = avol A + asf A2/3 +3e2
5r0Z2A−1/3 + (asym A + ass A2/3) I 2 + δ(N, Z)
∼ 7 parameters
Fit is surprisingly good: σ(E) = 2.97 MeV
Other qualitative features: location of drip-line, limits of α instability. . .
Fails to incorporate shell effects
Not satisfactory for astrophysical purposes and our fundamental understanding
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Theoretical predictions of nuclear masses
Look for a global theory of
Mass differences and absolute masses (for consistency)
Hopefully other observables
Necessarily of semi-empirical character
”Few” parameters fitted to all known masses
Theory used to extrapolate to unknown nuclei
Bethe-Weizsacker formula (1935)
Negative binding energy of a liquid drop with A = N + Z/I = N − Z
E = avol A + asf A2/3 +3e2
5r0Z2A−1/3 + (asym A + ass A2/3) I 2 + δ(N, Z)
∼ 7 parameters
Fit is surprisingly good: σ(E) = 2.97 MeV
Other qualitative features: location of drip-line, limits of α instability. . .
Fails to incorporate shell effects
Not satisfactory for astrophysical purposes and our fundamental understanding
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Theoretical predictions of nuclear masses
Look for a global theory of
Mass differences and absolute masses (for consistency)
Hopefully other observables
Necessarily of semi-empirical character
”Few” parameters fitted to all known masses
Theory used to extrapolate to unknown nuclei
Bethe-Weizsacker formula (1935)
Negative binding energy of a liquid drop with A = N + Z/I = N − Z
E = avol A + asf A2/3 +3e2
5r0Z2A−1/3 + (asym A + ass A2/3) I 2 + δ(N, Z)
∼ 7 parameters
Fit is surprisingly good: σ(E) = 2.97 MeV
Other qualitative features: location of drip-line, limits of α instability. . .
Fails to incorporate shell effects
Not satisfactory for astrophysical purposes and our fundamental understanding
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Theoretical predictions of nuclear masses
Finite Range Droplet Model ; Moller et al (1995)
Microscopic-Macroscopic approaches (”mic-mac”)
Combine drop-model and shell effects through Strutinsky method
E = Emac +∑
i
ni εi −∑̃
i
ni εi
∼ 30 parameters
Excellent data fit: σ(E) = 0.656MeV (1654 nuclei)
Lack of coherence between ”mic” and ”mac”
Basic treatment of pairing and other correlations (i.e. Wigner energy)
Strong interest for mass models that are as microscopic as possible
For a better fundamental understanding
For reliable extrapolation beyond fitted data, i.e. r-process
Can we use methods treating the N-body problem Quantum Mechanically?
Can this N-body problem treated in terms of ”fundamental” NN-NNN interactions?
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Theoretical predictions of nuclear masses
Finite Range Droplet Model ; Moller et al (1995)
Microscopic-Macroscopic approaches (”mic-mac”)
Combine drop-model and shell effects through Strutinsky method
E = Emac +∑
i
ni εi −∑̃
i
ni εi
∼ 30 parameters
Excellent data fit: σ(E) = 0.656MeV (1654 nuclei)
Lack of coherence between ”mic” and ”mac”
Basic treatment of pairing and other correlations (i.e. Wigner energy)
Strong interest for mass models that are as microscopic as possible
For a better fundamental understanding
For reliable extrapolation beyond fitted data, i.e. r-process
Can we use methods treating the N-body problem Quantum Mechanically?
Can this N-body problem treated in terms of ”fundamental” NN-NNN interactions?
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Theoretical tools
Figure by W. NazarewiczThomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Recent ”microscopic” mass tables
Duflo-Zuker mass table (1999)
Based on the Shell-model formalism
Parameterized monopole and multipole terms of effective shell-model hamiltonian
H = Hm + HM
Constrain Hm through scaling arguments to account for saturation
Constrain HM to account for main features of Kuo-Brown residual interaction
28 parameters / σ(E) = 0.360 MeV (1751 nuclei)
Current connection to underlying NN-NNN interactions is weak
Montreal-Brussels mass tables (2000-now)
Based on the Energy Density Functional (EDF) formalism
E =
∫d~r E[ρT (~r), τT (~r), ~J(~r), . . .]
Reconcile single-particle and collective dynamics
Allows a coherent calculation of many other quantities of interest
∼ 19 parameters / σ(E) ' 0.700MeV (2135 nuclei)
Current connection to underlying NN-NNN interactions is weak
Several types of correlations are still poorly treated
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Recent ”microscopic” mass tables
Duflo-Zuker mass table (1999)
Based on the Shell-model formalism
Parameterized monopole and multipole terms of effective shell-model hamiltonian
H = Hm + HM
Constrain Hm through scaling arguments to account for saturation
Constrain HM to account for main features of Kuo-Brown residual interaction
28 parameters / σ(E) = 0.360 MeV (1751 nuclei)
Current connection to underlying NN-NNN interactions is weak
Montreal-Brussels mass tables (2000-now)
Based on the Energy Density Functional (EDF) formalism
E =
∫d~r E[ρT (~r), τT (~r), ~J(~r), . . .]
Reconcile single-particle and collective dynamics
Allows a coherent calculation of many other quantities of interest
∼ 19 parameters / σ(E) ' 0.700MeV (2135 nuclei)
Current connection to underlying NN-NNN interactions is weak
Several types of correlations are still poorly treated
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Some elements of comparison
Fit to known masses
Goriely and Pearson (2006)
Overall precision is impressive ∼ 0.05% of the mass of a heavy nucleus
FRDM and EDF on the same footing
DZ which includes explicit configuration mixing is significantly better
Extrapolate according to their intrinsic uncertainty (FRDM better in that respect)
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Some elements of comparison
Fit over known masses
Evolution of shell effects through two-neutron shell-gap
δ2N = S2N(N, Z)− S2N(N+2, Z)
Different patterns already for near-stable nuclei (i.e. mutually enhanced magicity)
Different extrapolation towards the neutron drip-line
Shell-quenching predicted by EDF-mass tables not seen in others
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Some elements of comparison
Fit over known masses
Shell effects through two-neutron shell-gap
Network calculations of r -process abundances, Wanajo et al (2004)
Solar abundance vs yields from ”prompt-supernova explosion”
Significant difference just below the A = 130 (A = 195) peak
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Some elements of comparison
Fit over known masses
Shell effects through two-neutron shell-gap
Network calculations of r -process abundances, Wanajo et al (2004)
Chart of ”SN” = S2N/2
Yields below A = 130 (A = 195) reflect the evolution of the N = 82 (N=126) shell
Missing the data to assess the existence of a shell quenching
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Outline
1 Introduction
2 Nuclear Energy Density Functional approach: general characteristics
3 EDF mass tables from the Montreal-Brussels group
4 Towards more microscopic EDF methods and mass tables
5 Conclusions
6 Nuclear Energy Density Functional approach: elements of formalism
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
EDF method: spirit and characteristics
Aim at the whole nuclear chart (A ' 16)
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
EDF method: spirit and characteristics
Aim at the whole nuclear chart (A ' 16)
”Basic” EDF method dedicated to G. S. properties
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
EDF method: spirit and characteristics
Aim at the whole nuclear chart (A ' 16)
”Basic” EDF method dedicated to G. S. properties
Binding energy E
Separation energies SN , S2N , Qα, Qβ
Matter/charge density ρq(~r)
r.m.s. radii Rrms(q)
Deformation properties
Single-particle energies and shell structure
Nuclear matter Equation of State
Pairing properties
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
EDF method: spirit and characteristics
Aim at the whole nuclear chart (A ' 16)
”Basic” EDF method dedicated to G. S. properties
Universal functional but only ”Universal” parametrization
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
EDF method: spirit and characteristics
Aim at the whole nuclear chart (A ' 16)
”Basic” EDF method dedicated to G. S. properties
Universal functional but only ”Universal” parametrization
Looks like Hartree-Fock but includes ALL correlations in principle
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
EDF method: spirit and characteristics
Aim at the whole nuclear chart (A ' 16)
”Basic” EDF method dedicated to G. S. properties
Universal functional but only ”Universal” parametrization
Looks like Hartree-Fock but includes ALL correlations in principle
In practice, certain correlations are difficult to incorporate
Use of symmetry breaking to capture most important correlations
Symmetries must eventually be restored through extensions of the method
Same for correlations associated with shape/pair fluctuations
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
EDF method: spirit and characteristics
Aim at the whole nuclear chart (A ' 16)
”Basic” EDF method dedicated to G. S. properties
Universal functional but only ”Universal” parametrization
Looks like Hartree-Fock but includes ALL correlations in principle
Excited states through extensions of the method (Cranking, QRPA, Proj/GCM)
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
EDF method: spirit and characteristics
Aim at the whole nuclear chart (A ' 16)
”Basic” EDF method dedicated to G. S. properties
Universal functional but only ”Universal” parametrization
Looks like Hartree-Fock but includes ALL correlations in principle
Excited states through extensions of the method (Cranking, QRPA, Proj/GCM)
Rotational, vibrational and s.p. excitations (i.e. high-K isomers)
Fission isomers and fission barriers
Multipole strength (i.e. E1) and reduced transition probability (i.e. B(E2))
Beta-decay
Systematic microscopic calculations limited at this point (odd, deformed. . . )
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
EDF method: spirit and characteristics
Aim at the whole nuclear chart (A ' 16)
”Basic” EDF method dedicated to G. S. properties
Universal functional but only ”Universal” parametrization
Looks like Hartree-Fock but includes ALL correlations in principle
Excited states through extensions of the method (Cranking, QRPA, Proj/GCM)
Phenomenological energy functionals used
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
EDF method: spirit and characteristics
Aim at the whole nuclear chart (A ' 16)
”Basic” EDF method dedicated to G. S. properties
Universal functional but only ”Universal” parametrization
Looks like Hartree-Fock but includes ALL correlations in principle
Excited states through extensions of the method (Cranking, QRPA, Proj/GCM)
Phenomenological energy functionals used
”Mean-field” part = Skyrme (quasi-local) or Gogny (non-local) functional forms
”Pairing” part = local and density-dependent
⇒ necessitates ultra-violet regularization/renormalization
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Outline
1 Introduction
2 Nuclear Energy Density Functional approach: general characteristics
3 EDF mass tables from the Montreal-Brussels group
4 Towards more microscopic EDF methods and mass tables
5 Conclusions
6 Nuclear Energy Density Functional approach: elements of formalism
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
First generation of EDF mass tables
First EDF mass table in (2000) including around 19 fitted parameters (10+5+4)
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
First generation of EDF mass tables
First EDF mass table in (2000) including around 19 fitted parameters (10+5+4)
Fits made to all available masses (∼ 2000) + other (evolving) constraints
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
First generation of EDF mass tables
First EDF mass table in (2000) including around 19 fitted parameters (10+5+4)
Fits made to all available masses (∼ 2000) + other (evolving) constraints
Odd-even and odd-odd nuclei not treated on the same level as even-even ones
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
First generation of EDF mass tables
First EDF mass table in (2000) including around 19 fitted parameters (10+5+4)
Fits made to all available masses (∼ 2000) + other (evolving) constraints
Odd-even and odd-odd nuclei not treated on the same level as even-even ones
Some correlations are included in a phenomenological way or simply omitted
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
First generation of EDF mass tables
First EDF mass table in (2000) including around 19 fitted parameters (10+5+4)
Fits made to all available masses (∼ 2000) + other (evolving) constraints
Odd-even and odd-odd nuclei not treated on the same level as even-even ones
Some correlations are included in a phenomenological way or simply omitted
Wigner energy (binding cusp for N ≈ Z nuclei) included through
EW = VW exp {λ|N − Z |/A}
⇒ recent attempt to explain Wigner energy through neutron-proton T = 0 pairing
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
First generation of EDF mass tables
First EDF mass table in (2000) including around 19 fitted parameters (10+5+4)
Fits made to all available masses (∼ 2000) + other (evolving) constraints
Odd-even and odd-odd nuclei not treated on the same level as even-even ones
Some correlations are included in a phenomenological way or simply omitted
Restoration of intrinsically-broken rotational symmetry in deformed nuclei
Erot = E crankrot tanh(cβ2) =
〈Φ|J2|Φ〉2Icrank
tanh(cβ2)
Bender et al (2006)Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
First generation of EDF mass tables
First EDF mass table in (2000) including around 19 fitted parameters (10+5+4)
Fits made to all available masses (∼ 2000) + other (evolving) constraints
Odd-even and odd-odd nuclei not treated on the same level as even-even ones
Some correlations are included in a phenomenological way or simply omitted
Correlations associated with shape fluctuations are omitted
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Highlights: two examples
HFB-9 mass table (2005)
Fit to ab-initio Neutron matter EOS
Increase asym from 28 to 30 MeV (cf. neutron-skin thickness)
Impact the isotopic composition of neutron star core and inner crust
Goriely et al (2005)
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Highlights: two examples
HFB-9 mass table (2005)
HFB-13 mass table (2006)
Weakening of too strong pairing
Theoretically motivated renormalization scheme
Improved calculations of level densities
Anticipate improved fission barriers
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Highlights: two examples
HFB-9 mass table (2005)
HFB-13 mass table (2006)
Difference between their predictions for the most neutron-rich nuclei
Differences are within intrinsic uncertainties
Shell evolution towards neutron-rich nuclei very similar
Proof of consistency because only minor modifications within this first generation
Still, interesting differences regarding other observables
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Outline
1 Introduction
2 Nuclear Energy Density Functional approach: general characteristics
3 EDF mass tables from the Montreal-Brussels group
4 Towards more microscopic EDF methods and mass tables
5 Conclusions
6 Nuclear Energy Density Functional approach: elements of formalism
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
How to go beyond σ(E ) = 0.6 MeV ?
Nucleus-dependent correlations must be included microscopically
Qualitatively discussed in terms of ”chaotic” layer in the nucleonic dynamics
Evaluation from semi-classical periodic-orbit theory
σ(E) as a function of A
DZ (pink crosses)
FRDM (blue dots)
EDF (red squares)
Typical ”chaotic” contribution to E (solid line)
Bohigas and Leboeuf (2002) and (2006)
Partly included in DZ but not in FRDM/EDF-based mass tables
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
How to go beyond σ(E ) = 0.6 MeV ?
Nucleus-dependent correlations must be included microscopically
Extensions of the standard EDF method (Proj-GCM, QRPA)
Symmetry restorations and shape/pair fluctuations
Very involved numerically
Axial quadrupole correlations for 500 even-even nuclei, Bender et al (2006)
Modify significantly shell gaps around doubly-magic nuclei
More modes needed (triaxiality, octupole, pair vibrations, diabatic effects. . . )
Even more difficult for mass tables because nuclei are calculated many times
Formal problems being addressed, Bender and T. D. (2006), Lacroix et al. (2007)
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
How to go beyond σ(E ) = 0.6 MeV ?
Nucleus-dependent correlations must be included microscopically
Fitting strategies must be improved
Parts of the functional are under constrained, Bertsch et al. (2005)
Better use of (new) data in exotic/odd/rotating/elongated nuclei
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
How to go beyond σ(E ) = 0.6 MeV ?
Nucleus-dependent correlations must be included microscopically
Fitting strategies must be improved
Some physics is missing in current functionals
Parts of the functional are over constrained, Lesinski et al. (2006)
Tensor terms, Otsuka et al. (2006), Brown et al. (2006), Lesinski et al. (2007)
Connection to NN-NNN interactions needed (UNEDF collab.)
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
How to go beyond σ(E ) = 0.6 MeV ?
Nucleus-dependent correlations must be included microscopically
Fitting strategies must be improved
Some physics is missing in current functionals
Microscopic pairing functional from (direct) NN interaction
First step towards microscopic pairing functional, T. D. (2004)
Non-locality can be handled by codes in coordinate space
Eth−Eexp for 134 spherical nuclei
DFTM = phenomenological local functionalσ(E) = 2.964MeV
FR = functional from bare NN interactionσ(E) = 2.144MeV
Refit of the p-h part to be meaningful. . .
Lesinski et al. (2007)
Controlled approximations being worked out for systematic calculations
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
How to go beyond σ(E ) = 0.6 MeV ?
Nucleus-dependent correlations must be included microscopically
Fitting strategies must be improved
Some physics is missing in current functionals
Microscopic pairing functional from (direct) NN interaction
Odd-even and odd-odd nuclei must be treated in a better way
Fully self-consistent treatment is difficult on a large scale
Perturbative treatment, T. D. and Bonneau (2007)
Incorporates time reversal symmetry breaking, blocking of pairing. . .
Systematic calculations (for mass tables) become feasible
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Outline
1 Introduction
2 Nuclear Energy Density Functional approach: general characteristics
3 EDF mass tables from the Montreal-Brussels group
4 Towards more microscopic EDF methods and mass tables
5 Conclusions
6 Nuclear Energy Density Functional approach: elements of formalism
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Conclusions
First generation of (almost) microscopic mass tables exist
Accuracy of the same order as mic-mac models
Interested differences when extrapolated to unknown regions
Are those extrapolations trustable?
Nucleus-dependent correlations must be included to go beyond σ(E) = 0.6 MeV
Better treatment of odd-even and odd-odd nuclei mandatory
EDF methods are being further developed
Extensions allowing for symmetry restorations and configuration mixing
More modes needed (triaxiality, octupole, pair vibrations, diabatic effects. . . )
Formulation of those extensions within a truly EDF framework
First attempts to connect to underlying NN-NNN interactions are being made
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Conclusions
First generation of (almost) microscopic mass tables exist
Accuracy of the same order as mic-mac models
Interested differences when extrapolated to unknown regions
Are those extrapolations trustable?
Nucleus-dependent correlations must be included to go beyond σ(E) = 0.6 MeV
Better treatment of odd-even and odd-odd nuclei mandatory
EDF methods are being further developed
Extensions allowing for symmetry restorations and configuration mixing
More modes needed (triaxiality, octupole, pair vibrations, diabatic effects. . . )
Formulation of those extensions within a truly EDF framework
First attempts to connect to underlying NN-NNN interactions are being made
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Outline
1 Introduction
2 Nuclear Energy Density Functional approach: general characteristics
3 EDF mass tables from the Montreal-Brussels group
4 Towards more microscopic EDF methods and mass tables
5 Conclusions
6 Nuclear Energy Density Functional approach: elements of formalism
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Foundation: Hohenberg-Kohn theorem (1964)
Theorem
H + v
Ground state energy minimizes E [ρ(~r)] = F [ρ(~r)] +
∫d~r v(~r) ρ(~r)
EGS obtained for ρ(~r) = ρGS(~r) such that∫
d~r ρGS(~r) = N
F [ρ(~r)] = universal functional for given H
Reduces the problem from 3(4)N to 3(4) variables
The one-body local field ρ(~r) is the relevant degree of freedom
The difficulty resides in constructing F [ρ(~r)]. . . especially from first principles
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Foundation: Hohenberg-Kohn theorem (1964)
Theorem
H + v
Ground state energy minimizes E [ρ(~r)] = F [ρ(~r)] +
∫d~r v(~r) ρ(~r)
EGS obtained for ρ(~r) = ρGS(~r) such that∫
d~r ρGS(~r) = N
F [ρ(~r)] = universal functional for given H
Reduces the problem from 3(4)N to 3(4) variables
The one-body local field ρ(~r) is the relevant degree of freedom
The difficulty resides in constructing F [ρ(~r)]. . . especially from first principles
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Foundation: Hohenberg-Kohn theorem (1964)
Theorem
H + v
Ground state energy minimizes E [ρ(~r)] = F [ρ(~r)] +
∫d~r v(~r) ρ(~r)
EGS obtained for ρ(~r) = ρGS(~r) such that∫
d~r ρGS(~r) = N
F [ρ(~r)] = universal functional for given H
Reduces the problem from 3(4)N to 3(4) variables
The one-body local field ρ(~r) is the relevant degree of freedom
The difficulty resides in constructing F [ρ(~r)]. . . especially from first principles
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Foundation: Hohenberg-Kohn theorem (1964)
Theorem
H + v
Ground state energy minimizes E [ρ(~r)] = F [ρ(~r)] +
∫d~r v(~r) ρ(~r)
EGS obtained for ρ(~r) = ρGS(~r) such that∫
d~r ρGS(~r) = N
F [ρ(~r)] = universal functional for given H
Reduces the problem from 3(4)N to 3(4) variables
The one-body local field ρ(~r) is the relevant degree of freedom
The difficulty resides in constructing F [ρ(~r)]. . . especially from first principles
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Foundation: Hohenberg-Kohn theorem (1964)
Theorem
H + v
Ground state energy minimizes E [ρ(~r)] = F [ρ(~r)] +
∫d~r v(~r) ρ(~r)
EGS obtained for ρ(~r) = ρGS(~r) such that∫
d~r ρGS(~r) = N
F [ρ(~r)] = universal functional for given H
Reduces the problem from 3(4)N to 3(4) variables
The one-body local field ρ(~r) is the relevant degree of freedom
The difficulty resides in constructing F [ρ(~r)]. . . especially from first principles
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Foundation: Hohenberg-Kohn theorem (1964)
Theorem
H + v
Ground state energy minimizes E [ρ(~r)] = F [ρ(~r)] +
∫d~r v(~r) ρ(~r)
EGS obtained for ρ(~r) = ρGS(~r) such that∫
d~r ρGS(~r) = N
F [ρ(~r)] = universal functional for given H
Reduces the problem from 3(4)N to 3(4) variables
The one-body local field ρ(~r) is the relevant degree of freedom
The difficulty resides in constructing F [ρ(~r)]. . . especially from first principles
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Foundation: Hohenberg-Kohn theorem (1964)
Theorem
H + v
Ground state energy minimizes E [ρ(~r)] = F [ρ(~r)] +
∫d~r v(~r) ρ(~r)
EGS obtained for ρ(~r) = ρGS(~r) such that∫
d~r ρGS(~r) = N
F [ρ(~r)] = universal functional for given H
Reduces the problem from 3(4)N to 3(4) variables
The one-body local field ρ(~r) is the relevant degree of freedom
The difficulty resides in constructing F [ρ(~r)]. . . especially from first principles
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Implementation : Kohn-Sham (1965)
Introduce the non-interacting system |Φ〉
EKS [ρ(~r)] = TKS [ρ(~r)] +
∫d~r vKS(~r) ρ(~r) / TKS [ρ(~r)] ≡ ~2
2m
∫d~r
∑i
|∇ϕi (~r)|2
Choose vKS(~r) / ρKS(~r) =∑
i |ϕi (~r)|2 = ρGS(~r)
Re-write F [ρ(~r)] ≡ TKS [ρ(~r)] + EH [ρ(~r)] + Exc [ρ(~r)]
Minimization /∫
d~r ρ(~r) = N leads to{−~24
2m+ vKS(~r)
}ϕi (~r) = εi ϕi (~r)
Kohn-Sham equations with the local potential
vKS(~r) = vH [ρ(~r)] + vxc [ρ(~r)] / vxc [ρ(~r)] =δExc [ρ(~r)]
δρ(~r)
Koopmans’ Theorem ε0 = EN0 − EN−1
0 ; other εi have no meaning
Looks like solving a Hartree problem BUT it is in principle exact
Difficulty = approximating Exc [ρ(~r)]: LDA, GGA, Meta-GGA, Hybrid, . . .
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Implementation : Kohn-Sham (1965)
Introduce the non-interacting system |Φ〉
EKS [ρ(~r)] = TKS [ρ(~r)] +
∫d~r vKS(~r) ρ(~r) / TKS [ρ(~r)] ≡ ~2
2m
∫d~r
∑i
|∇ϕi (~r)|2
Choose vKS(~r) / ρKS(~r) =∑
i |ϕi (~r)|2 = ρGS(~r)
Re-write F [ρ(~r)] ≡ TKS [ρ(~r)] + EH [ρ(~r)] + Exc [ρ(~r)]
Minimization /∫
d~r ρ(~r) = N leads to{−~24
2m+ vKS(~r)
}ϕi (~r) = εi ϕi (~r)
Kohn-Sham equations with the local potential
vKS(~r) = vH [ρ(~r)] + vxc [ρ(~r)] / vxc [ρ(~r)] =δExc [ρ(~r)]
δρ(~r)
Koopmans’ Theorem ε0 = EN0 − EN−1
0 ; other εi have no meaning
Looks like solving a Hartree problem BUT it is in principle exact
Difficulty = approximating Exc [ρ(~r)]: LDA, GGA, Meta-GGA, Hybrid, . . .
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Implementation : Kohn-Sham (1965)
Introduce the non-interacting system |Φ〉
EKS [ρ(~r)] = TKS [ρ(~r)] +
∫d~r vKS(~r) ρ(~r) / TKS [ρ(~r)] ≡ ~2
2m
∫d~r
∑i
|∇ϕi (~r)|2
Choose vKS(~r) / ρKS(~r) =∑
i |ϕi (~r)|2 = ρGS(~r)
Re-write F [ρ(~r)] ≡ TKS [ρ(~r)] + EH [ρ(~r)] + Exc [ρ(~r)]
Minimization /∫
d~r ρ(~r) = N leads to{−~24
2m+ vKS(~r)
}ϕi (~r) = εi ϕi (~r)
Kohn-Sham equations with the local potential
vKS(~r) = vH [ρ(~r)] + vxc [ρ(~r)] / vxc [ρ(~r)] =δExc [ρ(~r)]
δρ(~r)
Koopmans’ Theorem ε0 = EN0 − EN−1
0 ; other εi have no meaning
Looks like solving a Hartree problem BUT it is in principle exact
Difficulty = approximating Exc [ρ(~r)]: LDA, GGA, Meta-GGA, Hybrid, . . .
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Implementation : Kohn-Sham (1965)
Introduce the non-interacting system |Φ〉
EKS [ρ(~r)] = TKS [ρ(~r)] +
∫d~r vKS(~r) ρ(~r) / TKS [ρ(~r)] ≡ ~2
2m
∫d~r
∑i
|∇ϕi (~r)|2
Choose vKS(~r) / ρKS(~r) =∑
i |ϕi (~r)|2 = ρGS(~r)
Re-write F [ρ(~r)] ≡ TKS [ρ(~r)] + EH [ρ(~r)] + Exc [ρ(~r)]
Minimization /∫
d~r ρ(~r) = N leads to{−~24
2m+ vKS(~r)
}ϕi (~r) = εi ϕi (~r)
Kohn-Sham equations with the local potential
vKS(~r) = vH [ρ(~r)] + vxc [ρ(~r)] / vxc [ρ(~r)] =δExc [ρ(~r)]
δρ(~r)
Koopmans’ Theorem ε0 = EN0 − EN−1
0 ; other εi have no meaning
Looks like solving a Hartree problem BUT it is in principle exact
Difficulty = approximating Exc [ρ(~r)]: LDA, GGA, Meta-GGA, Hybrid, . . .
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Implementation : Kohn-Sham (1965)
Introduce the non-interacting system |Φ〉
EKS [ρ(~r)] = TKS [ρ(~r)] +
∫d~r vKS(~r) ρ(~r) / TKS [ρ(~r)] ≡ ~2
2m
∫d~r
∑i
|∇ϕi (~r)|2
Choose vKS(~r) / ρKS(~r) =∑
i |ϕi (~r)|2 = ρGS(~r)
Re-write F [ρ(~r)] ≡ TKS [ρ(~r)] + EH [ρ(~r)] + Exc [ρ(~r)]
Minimization /∫
d~r ρ(~r) = N leads to{−~24
2m+ vKS(~r)
}ϕi (~r) = εi ϕi (~r)
Kohn-Sham equations with the local potential
vKS(~r) = vH [ρ(~r)] + vxc [ρ(~r)] / vxc [ρ(~r)] =δExc [ρ(~r)]
δρ(~r)
Koopmans’ Theorem ε0 = EN0 − EN−1
0 ; other εi have no meaning
Looks like solving a Hartree problem BUT it is in principle exact
Difficulty = approximating Exc [ρ(~r)]: LDA, GGA, Meta-GGA, Hybrid, . . .
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Implementation : Kohn-Sham (1965)
Introduce the non-interacting system |Φ〉
EKS [ρ(~r)] = TKS [ρ(~r)] +
∫d~r vKS(~r) ρ(~r) / TKS [ρ(~r)] ≡ ~2
2m
∫d~r
∑i
|∇ϕi (~r)|2
Choose vKS(~r) / ρKS(~r) =∑
i |ϕi (~r)|2 = ρGS(~r)
Re-write F [ρ(~r)] ≡ TKS [ρ(~r)] + EH [ρ(~r)] + Exc [ρ(~r)]
Minimization /∫
d~r ρ(~r) = N leads to{−~24
2m+ vKS(~r)
}ϕi (~r) = εi ϕi (~r)
Kohn-Sham equations with the local potential
vKS(~r) = vH [ρ(~r)] + vxc [ρ(~r)] / vxc [ρ(~r)] =δExc [ρ(~r)]
δρ(~r)
Koopmans’ Theorem ε0 = EN0 − EN−1
0 ; other εi have no meaning
Looks like solving a Hartree problem BUT it is in principle exact
Difficulty = approximating Exc [ρ(~r)]: LDA, GGA, Meta-GGA, Hybrid, . . .
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Implementation : Kohn-Sham (1965)
Introduce the non-interacting system |Φ〉
EKS [ρ(~r)] = TKS [ρ(~r)] +
∫d~r vKS(~r) ρ(~r) / TKS [ρ(~r)] ≡ ~2
2m
∫d~r
∑i
|∇ϕi (~r)|2
Choose vKS(~r) / ρKS(~r) =∑
i |ϕi (~r)|2 = ρGS(~r)
Re-write F [ρ(~r)] ≡ TKS [ρ(~r)] + EH [ρ(~r)] + Exc [ρ(~r)]
Minimization /∫
d~r ρ(~r) = N leads to{−~24
2m+ vKS(~r)
}ϕi (~r) = εi ϕi (~r)
Kohn-Sham equations with the local potential
vKS(~r) = vH [ρ(~r)] + vxc [ρ(~r)] / vxc [ρ(~r)] =δExc [ρ(~r)]
δρ(~r)
Koopmans’ Theorem ε0 = EN0 − EN−1
0 ; other εi have no meaning
Looks like solving a Hartree problem BUT it is in principle exact
Difficulty = approximating Exc [ρ(~r)]: LDA, GGA, Meta-GGA, Hybrid, . . .
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Nuclear EDF: implementation
Use more local ”sources” to derive HK theorem E ≡ E [ρT (~r), τT (~r), . . .]
ρT (~r) ≡ Iso-scalar/vector matter density
τT (~r) ≡ Iso-scalar/vector kinetic density
~JT (~r) ≡ Iso-scalar/vector spin-orbit density
Allows the explicit inclusion of
non-locality effects
spin-orbit and tensor correlations
Authorize the breaking of all symmetries: ~P, I , Π, N, T
Fields break spatial symmetries + new local fields
~sT (~r) ≡ Iso-scalar/vector spin density
~jT (~r) ≡ Iso-scalar/vector current density
ρ̃T (~r) ≡ Iso-scalar/vector pair density
Way to easily incorporate static correlations
Static correlations leave their prints on experimental spectra
BUT symmetries must be eventually restored ⇒ requires extensions!
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Nuclear EDF: implementation
Use more local ”sources” to derive HK theorem E ≡ E [ρT (~r), τT (~r), . . .]
ρT (~r) ≡ Iso-scalar/vector matter density
τT (~r) ≡ Iso-scalar/vector kinetic density
~JT (~r) ≡ Iso-scalar/vector spin-orbit density
Allows the explicit inclusion of
non-locality effects
spin-orbit and tensor correlations
Authorize the breaking of all symmetries: ~P, I , Π, N, T
Fields break spatial symmetries + new local fields
~sT (~r) ≡ Iso-scalar/vector spin density
~jT (~r) ≡ Iso-scalar/vector current density
ρ̃T (~r) ≡ Iso-scalar/vector pair density
Way to easily incorporate static correlations
Static correlations leave their prints on experimental spectra
BUT symmetries must be eventually restored ⇒ requires extensions!
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Nuclear EDF: implementation
Use more local ”sources” to derive HK theorem E ≡ E [ρT (~r), τT (~r), . . .]
ρT (~r) ≡ Iso-scalar/vector matter density
τT (~r) ≡ Iso-scalar/vector kinetic density
~JT (~r) ≡ Iso-scalar/vector spin-orbit density
Allows the explicit inclusion of
non-locality effects
spin-orbit and tensor correlations
Authorize the breaking of all symmetries: ~P, I , Π, N, T
Fields break spatial symmetries + new local fields
~sT (~r) ≡ Iso-scalar/vector spin density
~jT (~r) ≡ Iso-scalar/vector current density
ρ̃T (~r) ≡ Iso-scalar/vector pair density
Way to easily incorporate static correlations
Static correlations leave their prints on experimental spectra
BUT symmetries must be eventually restored ⇒ requires extensions!
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Skyrme functional
Local fields up to second order in spatial derivatives + symmetry constraints
No power counting but motivated from the DME (Negele and Vautherin (1972))
E =
∫d~r
∑T=0,1
[Cρρ
T ρ2T + C ss
T ~sT ·~sT + Cρ∆ρT ρT ∆ρT + C s∆s
T ~sT ·∆~sT
+CρτT
(ρT τT −~jT ·~jT
)+ C J2
T
(~sT · ~TT − J 2
T
)+Cρ∇J
T
(ρT
~∇ · ~JT +~sT · ~∇∧~jT
)+ C∇s∇s
T
(~∇ ·~sT
)2]
+C ρ̃ρ̃T |ρ̃T (~r)|2
Density-dependent couplings
Historical guidance from HF + density-dependent Skyrme ”interaction”
Local pairing functional ⇐⇒ density-dependent delta ”interaction”
V ρ̃ρ̃(~r1 −~r2) = C ρ̃ρ̃1 [ρ0(~r)] δ(~r1 −~r2)
which necessitates an ultra-violet regularization/renormalization
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Skyrme functional
Local fields up to second order in spatial derivatives + symmetry constraints
No power counting but motivated from the DME (Negele and Vautherin (1972))
E =
∫d~r
∑T=0,1
[Cρρ
T ρ2T + C ss
T ~sT ·~sT + Cρ∆ρT ρT ∆ρT + C s∆s
T ~sT ·∆~sT
+CρτT
(ρT τT −~jT ·~jT
)+ C J2
T
(~sT · ~TT − J 2
T
)+Cρ∇J
T
(ρT
~∇ · ~JT +~sT · ~∇∧~jT
)+ C∇s∇s
T
(~∇ ·~sT
)2]
+C ρ̃ρ̃T |ρ̃T (~r)|2
Density-dependent couplings
Historical guidance from HF + density-dependent Skyrme ”interaction”
Local pairing functional ⇐⇒ density-dependent delta ”interaction”
V ρ̃ρ̃(~r1 −~r2) = C ρ̃ρ̃1 [ρ0(~r)] δ(~r1 −~r2)
which necessitates an ultra-violet regularization/renormalization
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Skyrme functional
Local fields up to second order in spatial derivatives + symmetry constraints
No power counting but motivated from the DME (Negele and Vautherin (1972))
E =
∫d~r
∑T=0,1
[Cρρ
T ρ2T + C ss
T ~sT ·~sT + Cρ∆ρT ρT ∆ρT + C s∆s
T ~sT ·∆~sT
+CρτT
(ρT τT −~jT ·~jT
)+ C J2
T
(~sT · ~TT − J 2
T
)+Cρ∇J
T
(ρT
~∇ · ~JT +~sT · ~∇∧~jT
)+ C∇s∇s
T
(~∇ ·~sT
)2]
+C ρ̃ρ̃T |ρ̃T (~r)|2
Density-dependent couplings
Historical guidance from HF + density-dependent Skyrme ”interaction”
Local pairing functional ⇐⇒ density-dependent delta ”interaction”
V ρ̃ρ̃(~r1 −~r2) = C ρ̃ρ̃1 [ρ0(~r)] δ(~r1 −~r2)
which necessitates an ultra-violet regularization/renormalization
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Skyrme functional
Local fields up to second order in spatial derivatives + symmetry constraints
No power counting but motivated from the DME (Negele and Vautherin (1972))
E =
∫d~r
∑T=0,1
[Cρρ
T ρ2T + C ss
T ~sT ·~sT + Cρ∆ρT ρT ∆ρT + C s∆s
T ~sT ·∆~sT
+CρτT
(ρT τT −~jT ·~jT
)+ C J2
T
(~sT · ~TT − J 2
T
)+Cρ∇J
T
(ρT
~∇ · ~JT +~sT · ~∇∧~jT
)+ C∇s∇s
T
(~∇ ·~sT
)2]
+C ρ̃ρ̃T |ρ̃T (~r)|2
Density-dependent couplings
Historical guidance from HF + density-dependent Skyrme ”interaction”
Local pairing functional ⇐⇒ density-dependent delta ”interaction”
V ρ̃ρ̃(~r1 −~r2) = C ρ̃ρ̃1 [ρ0(~r)] δ(~r1 −~r2)
which necessitates an ultra-violet regularization/renormalization
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Skyrme functional
Local fields up to second order in spatial derivatives + symmetry constraints
No power counting but motivated from the DME (Negele and Vautherin (1972))
E =
∫d~r
∑T=0,1
[Cρρ
T ρ2T + C ss
T ~sT ·~sT + Cρ∆ρT ρT ∆ρT + C s∆s
T ~sT ·∆~sT
+CρτT
(ρT τT −~jT ·~jT
)+ C J2
T
(~sT · ~TT − J 2
T
)+Cρ∇J
T
(ρT
~∇ · ~JT +~sT · ~∇∧~jT
)+ C∇s∇s
T
(~∇ ·~sT
)2]
+C ρ̃ρ̃T |ρ̃T (~r)|2
Density-dependent couplings
Historical guidance from HF + density-dependent Skyrme ”interaction”
Local pairing functional ⇐⇒ density-dependent delta ”interaction”
V ρ̃ρ̃(~r1 −~r2) = C ρ̃ρ̃1 [ρ0(~r)] δ(~r1 −~r2)
which necessitates an ultra-violet regularization/renormalization
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Equation of motions
Introduce (quasi-)particle state |Φ〉 to construct the local fields, i.e.
ρT (~r) ≡∑
ij
∑σ
ϕ∗j (~rσ)ϕi (~rσ) ρij with ρij = 〈Φ|a†j ai |Φ〉
ρ̃T (~r) ≡∑
ij
∑σ
2σ ϕj(~r −σ)ϕi (~rσ) κij with κij = 〈Φ|ajai |Φ〉
Minimizing E / ρij and κij leads to HFB/Bogoliubov-De Gennes equations(h − λ ∆−∆∗ −(h∗ − λ)
) (UV
)µ
= Eµ
(UV
)µ
(U, V )µ/Eµ = quasi-particle states/energies ⇒ (ϕi (~rσ), ρij , κij)
h = δE/δρ = ”Hartree-Fock” field / ∆ = δE/δκ = pairing field
h and ∆ depend on solutions through densities ⇒ iterative method
All G.S. properties discussed before can be calculated from there
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Equation of motions
Introduce (quasi-)particle state |Φ〉 to construct the local fields, i.e.
ρT (~r) ≡∑
ij
∑σ
ϕ∗j (~rσ)ϕi (~rσ) ρij with ρij = 〈Φ|a†j ai |Φ〉
ρ̃T (~r) ≡∑
ij
∑σ
2σ ϕj(~r −σ)ϕi (~rσ) κij with κij = 〈Φ|ajai |Φ〉
Minimizing E / ρij and κij leads to HFB/Bogoliubov-De Gennes equations(h − λ ∆−∆∗ −(h∗ − λ)
) (UV
)µ
= Eµ
(UV
)µ
(U, V )µ/Eµ = quasi-particle states/energies ⇒ (ϕi (~rσ), ρij , κij)
h = δE/δρ = ”Hartree-Fock” field / ∆ = δE/δκ = pairing field
h and ∆ depend on solutions through densities ⇒ iterative method
All G.S. properties discussed before can be calculated from there
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Equation of motions
Introduce (quasi-)particle state |Φ〉 to construct the local fields, i.e.
ρT (~r) ≡∑
ij
∑σ
ϕ∗j (~rσ)ϕi (~rσ) ρij with ρij = 〈Φ|a†j ai |Φ〉
ρ̃T (~r) ≡∑
ij
∑σ
2σ ϕj(~r −σ)ϕi (~rσ) κij with κij = 〈Φ|ajai |Φ〉
Minimizing E / ρij and κij leads to HFB/Bogoliubov-De Gennes equations(h − λ ∆−∆∗ −(h∗ − λ)
) (UV
)µ
= Eµ
(UV
)µ
(U, V )µ/Eµ = quasi-particle states/energies ⇒ (ϕi (~rσ), ρij , κij)
h = δE/δρ = ”Hartree-Fock” field / ∆ = δE/δκ = pairing field
h and ∆ depend on solutions through densities ⇒ iterative method
All G.S. properties discussed before can be calculated from there
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Equation of motions
Introduce (quasi-)particle state |Φ〉 to construct the local fields, i.e.
ρT (~r) ≡∑
ij
∑σ
ϕ∗j (~rσ)ϕi (~rσ) ρij with ρij = 〈Φ|a†j ai |Φ〉
ρ̃T (~r) ≡∑
ij
∑σ
2σ ϕj(~r −σ)ϕi (~rσ) κij with κij = 〈Φ|ajai |Φ〉
Minimizing E / ρij and κij leads to HFB/Bogoliubov-De Gennes equations(h − λ ∆−∆∗ −(h∗ − λ)
) (UV
)µ
= Eµ
(UV
)µ
(U, V )µ/Eµ = quasi-particle states/energies ⇒ (ϕi (~rσ), ρij , κij)
h = δE/δρ = ”Hartree-Fock” field / ∆ = δE/δκ = pairing field
h and ∆ depend on solutions through densities ⇒ iterative method
All G.S. properties discussed before can be calculated from there
Thomas DUGUET Microscopic Theories of Nuclear Masses
Introduction Nuclear Energy Density Functional approach: general characteristics EDF mass tables from the Montreal-Brussels group Towards more microscopic EDF methods and mass tables Conclusions Nuclear Energy Density Functional approach: elements of formalism
Equation of motions
Introduce (quasi-)particle state |Φ〉 to construct the local fields, i.e.
ρT (~r) ≡∑
ij
∑σ
ϕ∗j (~rσ)ϕi (~rσ) ρij with ρij = 〈Φ|a†j ai |Φ〉
ρ̃T (~r) ≡∑
ij
∑σ
2σ ϕj(~r −σ)ϕi (~rσ) κij with κij = 〈Φ|ajai |Φ〉
Minimizing E / ρij and κij leads to HFB/Bogoliubov-De Gennes equations(h − λ ∆−∆∗ −(h∗ − λ)
) (UV
)µ
= Eµ
(UV
)µ
(U, V )µ/Eµ = quasi-particle states/energies ⇒ (ϕi (~rσ), ρij , κij)
h = δE/δρ = ”Hartree-Fock” field / ∆ = δE/δκ = pairing field
h and ∆ depend on solutions through densities ⇒ iterative method
All G.S. properties discussed before can be calculated from there
Thomas DUGUET Microscopic Theories of Nuclear Masses