Density functional theory and energy density functionals in …dobaczew/Tokyo2010/CNSEFES10... ·...

Jacek Dobaczewski

Density functional theory and energydensity functionals in nuclear physics

Jacek DobaczewskiUniversity of Warsaw & University of Jyväskylä

The 9th CNS-EFES International Summer School18-24 August 2010

Center for Nuclear StudyUniversity of Tokyo in RIKEN

Jacek Dobaczewski

Reading material:

Jacek Dobaczewski:2004 RIA Summer Schoolhttp://www.fuw.edu.pl/~dobaczew/RIA.Summer.Lectures/slajd01.htmlJacek Dobaczewski:2005 Ecole Doctorale de Physique, Strasbourghttp://www.fuw.edu.pl/~dobaczew/Strasbourg/slajd01.htmlWitek Nazarewicz:2007 Lectures at the University of Knoxvillehttp://www.phys.utk.edu/witek/NP622/NuclPhys622.htmlJacek Dobaczewski: 2008 the 18th Jyväskylä Summer Schoolhttp://www.fuw.edu.pl/~dobaczew/JSS18/JSS18.htmlJacek Dobaczewski: 2008 Euroschool on Exotic Beamshttp://www.fuw.edu.pl/~dobaczew/Euroschool/Euroschool.htmlJacek Dobaczewski: 2008 Lectures at University of Jyväskylähttp://www.fuw.edu.pl/~dobaczew/FYSN305/FYSN305.htmlJacek Dobaczewski: 2009 Lectures at University of Stellenboschhttp://www.fuw.edu.pl/~dobaczew/Stellenbosch/dobaczewski_lecture.pdf

Home page: http://www.fuw.edu.pl/~dobaczew/

Jacek Dobaczewski

Nuclear Structure

Energy scales in nuclear physics

Jacek Dobaczewski

ttp://

Universal Nuclear Energy Density FunctionalUniversal Nuclear Energy Density Functional

Jacek Dobaczewski

Mean-Field Theory ⇒ Density Functional Theory

• mean-field ⇒ one-body densities

• zero-range ⇒ local densities

• finite-range ⇒ gradient terms

• particle-hole and pairing channels

• Has been extremely successful. A broken-symmetry generalized product state does surprisingly good job for nuclei.

Nuclear DFT

• two fermi liquids• self-bound• superfluid

Jacek Dobaczewski

Price of land in Poland per voivodship

Energy density

functional

647 Price voivodshipfunctional

654 763

295 580 446

631 356

549 548

623 362

Jacek Dobaczewski

Price of land in Poland per

district

Energy density

functional

Price district functional

Jacek Dobaczewski

Price of land in Eurpe per

country

Energy density

functional

Price country functional

Jacek Dobaczewski

What is DFT?

Density Functional Theory:

A variational method that uses observables as variational

parameters.

Jacek Dobaczewski

Which DFT?

Jacek Dobaczewski

What is the DFT good for?

1) Exact: Minimization of E(Q) gives the exact E and exact Q

2) Impractical: Derivation of E(Q) requires the full variation (bigger effort than to find the exact ground state)

3) Inspirational: Can we build useful models E’(Q) of the exact E(Q)?

4) Experiment-driven: E’(Q) works better or worse depending on the physical input used to build it.

Energy E is afunction(al) of Q

Jacek Dobaczewski

Nuclear Energy Density

Functionals

Jacek Dobaczewski

How the nuclear EDF is built?Local energy density is afunction of

local densityLDA

Non-local energy density is afunction of

non-local densityGogny, M3Y,…

Jacek Dobaczewski

How the nuclear EDF is built?

Quasi-local energy density is a function of

local densities and gradientsSkyrme, BCP,

point-coupling,…

Non-local energy density is a function of

local densitiesRMF (Hartree)

Jacek Dobaczewski

Neutrons in external Woods-Saxon well

Jacek Dobaczewski

Hohenberg-Kohn theorem

Jacek Dobaczewski

Hohenberg-Kohn theorem (trivial version)

Jacek Dobaczewski

Nuclear Energy Density Functional (physical insight)

Jacek Dobaczewski

Effective Theories

Jacek Dobaczewski

Hydrogen atom perturbed near the center

Relative errors in the S-wave binding energies are plotted versus:(i) the binding energy for the Coulomb theory(ii) the Coulomb theory augmented with a delta function in first-order perturbation theory(iii) the non-perturbativeeffective theory through a2, and(iv) the effective theory through a4.

Jacek Dobaczewski

Dimensional analysis - regularization

Jacek Dobaczewski

Dimensional analysis – the hydrogen-like atom

Jacek Dobaczewski

Emission of long electromagnetic waves

Jacek Dobaczewski

Emission of long electromagnetic waves

Jacek Dobaczewski

Blue-sky problem - Compton scattering

Jacek Dobaczewski

N3LO in the chiral perturbation effective field theory

Jacek Dobaczewski

EFT phase-shift analysis

np phase parameters below 300 MeV lab. energy for partial waves with J=0,1,2. The solid line is the result at N3LO. The dotted and dashed lines are the phase shifts at NLO and NNLO, respectively, as obtained by Epelbaum et al. The solid dots show the Nijmegen multi-energy np phase shift analysis and the open circles are the VPI single-energy np analysis SM99. D

Jacek Dobaczewski

Many-fermion Hilbert space

Jacek Dobaczewski

Indistinguishability principle

Jacek Dobaczewski

Fock space

Jacek Dobaczewski

Creation and annihilation operators

Jacek Dobaczewski

Operators in the Fock space

Jacek Dobaczewski

0 1 2 3

0 0.4 0.8 1.2

n-n distance (fm)

O -O distance (nm)22

O -O system22

n-n system (1S0)

n-n versusO2 -O2 interaction

Jacek Dobaczewski

Lessons learned

1) Energy density functional exists due to the two-step variational method and gives exact ground state energy and its exact particle density.

2) Whenever the energy scales (or range scales) between the interactions and observations are different, the observations can be described by a series of pseudopotentials with coupling constants adjusted to data (an effective theory).

Jacek Dobaczewski

Density-matrix expansion

Jacek Dobaczewski

Density matrices and Wick theorem

Jacek Dobaczewski

Coulomb force – the direct self-consistent potential

Jacek Dobaczewski

Coulomb force – the exchange self-consistent potential

Jacek Dobaczewski

s(R,r)

Particle and spin densities in 208Pb

Jacek Dobaczewski

Density-matrix expansion (Negele-Vautherin)(or do we need the non-local density)

Nonlocal energy density

Jacek Dobaczewski

0 2 4 6 8

(n)100Zn

2 4 6 8 10R (fm)

mean field one-body densitieszero range local densities

finite range non-local densities

Hohenberg-KohnKohn-ShamNegele-VautherinLandau-MigdalNilsson-Strutinsky

Modern Mean-Field Theory Energy Density Functionalj,, , J,

Nuclear densities as composite fields

Jacek Dobaczewski

Density-matrix expansion (2)

Localenergy density

Jacek Dobaczewski

Localenergy density

Jacek Dobaczewski

0 5 10 15 20

0 O(4)

0 Gauss

2 O(2)

2 Gauss

a = kFr J.

, J. P

Jacek Dobaczewski

Exchange interaction energy in infinite matter

Jacek Dobaczewski

Lessons learned

1) Energy density functional exists due to the two-step variational method and gives exact ground state energy and its exact particle density.

3) In nuclei, the non-local energy density functionals can be replaced by the local ones. This is because the range of the interaction is shorter than the range of variations in the local and non-local density matrix.

Jacek Dobaczewski

t al.,

010)Negele-Vautherin density-matrix expansion

Jacek Dobaczewski

t al.,

Jacek Dobaczewski

t al.,

Jacek Dobaczewski

Density-matrix expansion and the Skyrme force

Jacek Dobaczewski

Negele-Vautherin density-matrix expansion

Based on the Negele-Vautherin density-matrix expansion, we have derived the NLO Skyrme-functional parameters corresponding to the finite-range Gogny interaction. The method has been extended to derive the coupling constants of local N3LO functionals

t al.,

Jacek Dobaczewski

t al.,

Jacek Dobaczewski

0 50 100 150 200

Number of nucleons A

0 50 100 150 200

Number of nucleons A

Quasi-local vs. non-local functionals

40Ca48Ca

56Ni78Ni

100Sn132Sn

0 100 200

48Ca56Ni

208Pbquasi-local (Gogny+DME NLO)non-local (Gogny)

0 100 200

The Negele-Vautherin density-matrix expansion (DME) up to NLO (2nd order) applied to the Gogny non-local functional gives a

Skyrme-like quasi-local functional. The results for self-consistent observables, obtained for both functionals are very similar.

t al.,

Jacek Dobaczewski

Open symbols show the results obtained directly by using the Negele-Vautherin DME, whereas the full symbols show the results inferred from the time-even sector by using the Gogny-equivalent Skyrme force. Solid and dashed lines (left and right panels) show the values of the isoscalar and isovector coupling constants, respectively.

t al.,

The quasi-local EDF derived from the finite-range force does not correspond to a zero-range force

Jacek Dobaczewski

Nuclear densities as composite fields

Jacek Dobaczewski

Local energydensity:(no isospin,no pairing)

Jacek Dobaczewski

Nuclear Energy Density Functional

Jacek Dobaczewski

Complete local energy density

Mean field Pairing

E. Perlińska, et al.,Phys. Rev. C69 (2004) 014316

Jacek Dobaczewski

Mean-field equations

Jacek Dobaczewski

Lessons learned1) Energy density functional exists due to the two-step

variational method and gives exact ground state energy and its exact particle density.

4) Systematic energy density functionals with derivative corrections can be constructed and the resulting self-consistent equations solved.

Jacek Dobaczewski

Applications

Jacek Dobaczewski

Phenomenological effective interactions

Jacek Dobaczewski

0 20 40 60 80 100 120 140 160 180Neutron Number

deformation Prot

Theory

< -0.2-0.2 ÷ -0.1

-0.1 ÷ +0.1+0.1 ÷ +0.2

+0.2 ÷ +0.3+0.3 ÷ +0.4

> +0.4

SLy4volume pairing

, et a

Jacek Dobaczewski

Gogny D1S

Jacek Dobaczewski

t al.,

Nuclear binding energies (masses)

The first Gogny HFB mass model. An explicit and self-consistent account of all the quadrupole correlation energies are included within the 5D collective Hamiltonian approach.

2149 masses

RMS = 798 keV

The new Skyrme HFB nuclear-mass model, in which the contact-pairing force is constructed from microscopic pairing gaps of symmetric nuclear matter and neutron matter.

2149 masses

RMS = 581 keV

Jacek Dobaczewski

t al.,

0)537 2+ energies 359 2+ energies

First 2+ excitations of even-even nuclei

Skyrme HF+BCS calculations plus the particle-number and angular-momentum projection and shape mixing.

Gogny HFB calculations plus the 5D collective Hamiltonian approach.

Jacek Dobaczewski

UNEDF collaboration: SciDAC Review 6, 42 (2007)

Spontaneous fission

Symmetry-unrestricted Skyrme HF+BCS calculations.

Jacek Dobaczewski

Collective states in even-even nuclei

J.M. Y

t al.,

Jacek Dobaczewski

Giant resonances in deformed nuclei

t al.,

Jacek Dobaczewski

Fast RPA and QRPA + Arnoldi method

J. Toivanen et al., Phys. Rev. C 81, 034312 (2010)

Jacek Dobaczewski

Fast RPA and QRPA + Arnoldi method

Jacek Dobaczewski

Isoscalar 1- strength functions

0 10 20 30 40 50

standard RPAArnoldi (spurious)Arnoldi

E [MeV]

Removal of spurious modes

Jacek Dobaczewski

QRPA timing

Spherical QRPA+Arnoldiscales linearly with the size of the single-particle space

Deformed QRPA+Arnoldiexpected to scale quadratically, that is, as

Standard QRPA scales quartically, that is, as

Scaling properties

Future plans: Full implementation and testing of the spherical QRPA + Arnoldi methodin the code HOSPHE with new-generation separable pairing interactions. Systematic calculations of multipole giant-resonance modes to be used in the EDF adjustments.Deformed QRPA + Arnoldi method implemented in the code HFODD. Systematic calculations of -decay strengths functions and -delayed neutron emission probabilities to be used in the EDF adjustments.

Jacek Dobaczewski

T. Inakura et al., Phys. Rev. C80, 044301 (2009)

Iterative methods to solve (Q)RPA

Photoabsorption cross sections

Theory

Finite-amplitude method to solve the QRPA equations.

Jacek Dobaczewski

J. Toivanen et al., Phys. Rev. C81, 034312 (2010)

Iterative methods to solve (Q)RPA

Monopole resonance

Arnoldi method to solve the RPA equations.

Jacek Dobaczewski

Dipole oscillationsin 14Be

t al.,

UNEDF collaboration: Quantum Dynamics with TDSLDA,

A. Bulgac, et al.,http://www.phys.washington.edu/gr

oups/qmbnt/index.html

Time-dependent solutions

– 8Be collision(tip configuration)

neutrons

protons

Jacek Dobaczewski

Fits of s.p. energies

Singular value decomposition

EXP: M.N. Schwierz, I. Wiedenhover,and A. Volya, arXiv:0709.3525

(b)1.0

0 2 4 6 8 10 12

SLy5SkPSkO'SIII

Number of singular value

Jacek Dobaczewski

Fits of single-particle energies

101520

-4 -2 0 2 4

SLy5 (fit)

i - iex

-4 -2 0 2 4

SIII(fit)

-4 -2 0 2 4

SkP(fit)

-4 -2 0 2 4

SkO'(fit)

101520

i - iex

(a) SkP (b) SkO' (c) SIII (d)

M. Kortelainen et al., Phys. Rev. C77, 064307 (2008)

Jacek Dobaczewski

Fit residuals for centroids of SO partners (SkP)

Before

Jacek Dobaczewski

Fit residuals for splittings of SO partners (SkP)

Before

Jacek Dobaczewski

Global (masses)

Bertsch, Sabbey, and Uusnakki

Phys. Rev. C71, 054311 (2005)

How many parameters are really needed?

Jacek Dobaczewski

SciDAC 2 UNEDF Project (USA)

Building a Universal Nuclear Energy Density Functional

• Understand nuclear properties “for element formation, for properties of stars, and for present and future energy and defense applications”

• Scope is all nuclei, with particular interest in reliable calculations of unstable nuclei and in reactions

• Order of magnitude improvement over present capabilities Precision calculations

• Connected to the best microscopic physics• Maximum predictive power with well-quantified uncertainties

FIDIPRO Project (Finland)

http://www.unedf.org/

http://www.jyu.fi/accelerator/fidipro/

Jacek Dobaczewski

UNEDF Skyrme Functionals

t al.,

Jacek Dobaczewski

UNEDF Skyrme Functionals

t al.,

Jacek Dobaczewski

New functionals

Jacek Dobaczewski

Derivatives of higher order: Negele & Vautherindensity matrix expansion

Jacek Dobaczewski

Energy density functional up to N3LO

Jacek Dobaczewski

Numbers of terms in the density functional up to N3LO

Eq. (30) ≡ density independent CCEq. (28) ≡ density dependent CC

Jacek Dobaczewski

Energy density functional for spherical nuclei (I)

Jacek Dobaczewski

Energy density functional for spherical nuclei (II)

B.G. Carlsson et al., Phys. Rev. C 78, 044326 (2008)Phys. Rev. C 81, 029904(E) (2010)

Jacek Dobaczewski

Convergence of density-matrix expansions for nuclear interactions (the direct term)

, J. D

Jacek Dobaczewski

Convergence of density-matrix expansions for nuclear interactions (the exchange term)

, J. D

Jacek Dobaczewski

0 2 4 6 8 10 12

EXP-1EXO-2EXP-1 (NM)EXP-2 (NM)

2nd orderEXP-1:M.N. Schwierz,I. Wiedenhover, andA. Volya, arXiv:0709.3525

EXP-2:M.G. Porquet et al.,to be published

NM:Nuclear-matterconstraints on: saturation density energy per particle incompressibility effective mass

Fits of s.p. energies – regression analysis

Jacek Dobaczewski

0.0001

0 4 8 12 16 20 24 28

EXP-1EXP-2EXP-1 (NM)EXP-2 (NM)

4th orderEXP-1:M.N. Schwierz,I. Wiedenhover, andA. Volya, arXiv:0709.3525

(Galilean invariance)

Jacek Dobaczewski

EXP-1:M.N. Schwierz,I. Wiedenhover, andA. Volya, arXiv:0709.3525

0.0001

0 10 20 30 40 50 60

EXP-1EXP-2EXP-1 (NM)EXP-2 (NM)

6th orderN3LO

(Galilean invariance)

Problem of insta

bilities u

nsolved yet

M. Korte

lainen and T. Lesin

J. Phys. G

: Nucl.

Part. Phys. 3

7 064039

Jacek Dobaczewski

Program HOSPHE

Solution of self-consistent

equations for the N3LO nuclear

energy density functional in

spherical symmetry

Jacek Dobaczewski

Program HOSPHE

Solution of self-consistent

equations for the N3LO nuclear

energy density functional in

spherical symmetry

HOSPHE

Jacek Dobaczewski

Spontaneous symmetry breaking

Jacek Dobaczewski

Nitrogen atom

Hydrogen atom

Ammonia molecule NH3

left state right state

Jacek Dobaczewski

Ammonia molecule NH3

Distance of N from the H3 plane (a.u.)

.)Symmetry-conserving

configuration

Symmetry-breakingconfigurations

Jacek Dobaczewski

ExperimentR.G. Helmer et al., Nucl. Phys. A474 (1987) 77

Skyrme-Hartree-FockJ. Dobaczewski, J. Engel,

Phys. Rev. Lett. 94, 232502 (2005)

1/2- 55

Jz=1/2

Jacek Dobaczewski

7.6×10-63.2 ×10125.5 ×1081.8 ×10-9

~0.1 e× fm0.76 e×nmD~5 ns16 ksT1/2 (E.M.)

0.012 as6.6 psT1/2 (Q.M.)55 keV0.1 meV

225RaNH3

Jacek Dobaczewski

Lessons learned1) Energy density functional exists due to the two-step variational

method and gives exact ground state energy and its exactparticle density.

3) In nuclei, the non-local energy density functionals can be replaced by the local ones. This is because the range of the interaction is shorter than the range of variations in the localand non-local density matrix.

5) In finite systems, the phenomenon of spontaneous symmetry breaking is best captured by the mean-field or energy-density-functional methods.

Jacek Dobaczewski

Nuclear deformation

Elongation (a.u.)

.)Symmetry-conserving

configuration

Symmetry-breakingconfigurations

Jacek Dobaczewski

Origins of nuclear deformation

Elongation (a.u.)

.u.) Open-shell system:

8 particles on 8 doublydegenrate levels

Jacek Dobaczewski

0 20 40 60 80 100 120 140 160 180Neutron Number

deformation Prot

Theory

< -0.2-0.2 ÷ -0.1

-0.1 ÷ +0.1+0.1 ÷ +0.2

+0.2 ÷ +0.3+0.3 ÷ +0.4

> +0.4

SLy4volume pairing

, et a

Jacek Dobaczewski

Tensor effects

Jacek Dobaczewski

M.Honma et al., Eur. Phys. J. 25,s01 (2005) 499B. Fornal, XXIX Mazurian Lakes Conference on Physics (2005)

Jacek Dobaczewski

24 26 28 30 32 34 36

Neutron Number N

2+ ) (

e2 b2 )

26 28 30 32 34

E(2+ ) (

Neutron Number N

Jacek Dobaczewski

Evolution of the single particle orbitalswith Z going from 28 to 20

V couples j> and j< orbitals and favors charge exchange processes

f7/2 f5/2T. Otsuka et al. Phys. Rev. Lett 87, 082502 (2001)

Tensor InteractionVT = (2) Y(2 Z(r)

Jacek Dobaczewski

Tensor-even, tensor-odd, and spin-orbit interactions

Jacek Dobaczewski

Tensor energy densities

For conserved spherical and time-reversal symmetries,averaged tensor and SO interactions give the followingenergy densities:

where the particle and SO densities read

Jacek Dobaczewski

Single-particle spin-orbit potentials

Variation of the energy densities with respect tothe single-particle wave functions givesform factors of the single-particle spin-orbit potentials:

Jacek Dobaczewski

-0.005

0 2 4 6 8R (fm)

te=200

Neutron S-O Density

Jacek Dobaczewski

-0.005

0 2 4 6 8R (fm)

te=200

Neutron S-O Density

Jacek Dobaczewski

-0.005

0 2 4 6 8R (fm)

N=38 N=40

te=200

Neutron S-O Density

Jacek Dobaczewski

02s_1/2

1d_3/2

1d_5/2

2p_1/2

2p_3/2

1f_5/2

1f_7/2

102030

Proton Number Z

HFB+SLy4

te=0N=32

Jacek Dobaczewski

102030

Proton Number Z

1d2p1f

HFB+SLy4

te=0 N=32

Jacek Dobaczewski

1d2p1f

102030

Proton Number Z

HFB+SLy4

te=-100 N=32

Jacek Dobaczewski

1d2p1f

102030

Proton Number Z

HFB+SLy4

te=200 N=32

Jacek Dobaczewski

Polarizationeffects

for neutronspin-orbitsplitting

M. Zalewski et al.,Phys. Rev. C77,024316 (2008)

Jacek Dobaczewski

Fits of C0J,

C0J , and C1

M. Zalewski et al.,Phys. Rev. C77,024316 (2008)

Jacek Dobaczewski

Fits of spin-orbit and tensor coupling constants

t al.,

Jacek Dobaczewski

littin

1234567

16 O 40 Ca48 Ca

56 Ni90 Zr

132 Sn208 Pb

16 O 40 Ca48 Ca

56 Ni90 Zr

132 Sn208 Pb

f7/2-f5/2

1g 1h1i

d5/2-d3/2p3/2-p1/2

SkPSkP T

SkP originalSkP T

ensor + SO*0.8

t al.,

Jacek Dobaczewski

littin

1234567

f7/2-f5/2

SkOSkO T

SkOoriginal

ensor + SO*0.81g 1h

d5/2-d3/2p3/2-p1/2

16 O 40 Ca48 Ca

56 Ni90 Zr

132 Sn208 Pb

16 O 40 Ca48 Ca

56 Ni90 Zr

132 Sn208 Pb

f7/2-f5/2

t al.,

Jacek Dobaczewski

Challenges

Jacek Dobaczewski

Collectivity

beyond mean field, ground-state correlations, shape coexistence, symmetry restoration, projection on good quantum numbers, configuration interaction, generator coordinate method, multi-reference DFT, etc….

True formean field

Jacek Dobaczewski

I. Range separation and exact long-range effects

II. Derivatives of higher order:

III. Products of more than two densities:

Extensions

Jacek Dobaczewski

Lessons learned1) Energy density functional exists due to the two-step variational

method and gives exact ground state energy and its exact particle density.

5) In finite systems, the phenomenon of spontaneous symmetry breaking is best captured by the mean-field or energy-density-functional methods.

6) Energy density functionals up to the second order in derivatives(Skyrme functionals) provide for a fair but not very precise description of global properties of nuclear ground states.

Jacek Dobaczewski

Thank you

Density functional theory and energy density functionals in …dobaczew/Tokyo2010/CNSEFES10... ·...

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