Density functional theory and energy density functionals in nuclear physics

Transkrypt

1 Density functional theory and energy density functionals in nuclear physics University of Warsaw & University of Jyväskylä The 9th CNS-EFES International Summer School August 2010 Center for Nuclear Study University of Tokyo in RIKEN

2 Reading material: :2004 RIA Summer School :2005 Ecole Doctorale de Physique, Strasbourg Witek Nazarewicz:2007 Lectures at the University of Knoxville : 2008 the 18th Jyväskylä Summer School : 2008 Euroschool on Exotic Beams : 2008 Lectures at University of Jyväskylä : 2009 Lectures at University of Stellenbosch Home page:

3 Energy scales in nuclear physics Nuclear Structure G. Bertsch et al., SciDAC Review 6, 42 (2007)

4

5 Universal Nuclear Energy Density Functional UNEDF Collaboration,

6 Mean-Field Theory Density Functional Theory Nuclear DFT two fermi liquids self-bound superfluid 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.

7 Price of land in Poland per voivodship Price voivodship functional Energy density functional

8 Price of land in Poland per district Price district functional Energy density functional

9 Price of land in Eurpe per country Price country functional Energy density functional

10 What is DFT? Density Functional Theory: A variational method that uses observables as variational parameters.

11 Which DFT?

12 What is the DFT good for? Energy E is a function(al) of Q 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.

13 Nuclear Energy Density Functionals

14 How the nuclear EDF is built? LDA Gogny, M3Y, Local energy density is a function of local density Non-local energy density is a function of non-local density

15 How the nuclear EDF is built? Skyrme, BCP, point-coupling, RMF (Hartree) Quasi-local energy density is a function of local densities and gradients Non-local energy density is a function of local densities

16 R.B. Wiringa & UNEDF Collaboration Neutrons in external Woods-Saxon well

17 Hohenberg-Kohn theorem

18 Hohenberg-Kohn theorem (trivial version)

19 Nuclear Energy Density Functional (physical insight)

20 Effective Theories

21 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-perturbative effective theory through a 2, and (iv) the effective theory through a 4.

22 Dimensional analysis - regularization

23 Dimensional analysis the hydrogen-like atom

24 Emission of long electromagnetic waves

25 Emission of long electromagnetic waves

26 Blue-sky problem - Compton scattering

27 N 3 LO in the chiral perturbation effective field theory W.C. Haxton, Phys. Rev. C77, (2008)

28 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 N 3 LO. 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.R. Entem and R. MachleidtPhys.Rev. C68 (2003)

29 Many-fermion Hilbert space

30 Indistinguishability principle

31 Fock space

32 Creation and annihilation operators

33 Operators in the Fock space

34 n-n versus O 2 -O 2 interaction O 2 O 2 d n d n O 2 -O 2 potential (mev) n-n potential (MeV) n-n system (1S 0 ) O -O system n-n distance (fm) O 2-O 2 distance (nm)

35

36 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).

37 Density-matrix expansion

38 Density matrices and Wick theorem

39 Coulomb force the direct self-consistent potential

40 Coulomb force the exchange self-consistent potential

41 B. Gebremariam et al., Phys. Rev. C82, (2010) Particle and spin densities in 208 Pb (R,r) s(r,r)

42 J. D., B.G Carlsson, M. Kortelainen, J. Phys. G 37, (2010) Density-matrix expansion (Negele-Vautherin) (or do we need the non-local density) Nonlocal energy density

43 Nuclear densities as composite fields Particle density (fm -3 ) 100 Sn 0.10 (n) (p) R (fm) (n) 100 Zn (p) Modern Mean-Field Theory Energy Density Functional,, J, j, T, s, F, Hohenberg-Kohn Kohn-Sham Negele-Vautherin Landau-Migdal Nilsson-Strutinsky mean field one-body densities zero range local densities finite range non-local densities

44 Density-matrix expansion (2) Local energy density

45 Density-matrix expansion (3)

46 Density-matrix expansion (4) Local energy density

47 Density-matrix expansion (6) O(4) 0 0 Gauss 2 O(2) 2 2 Gauss Infinite-matter 0 (a) & 2 (a) J. D., B.G Carlsson, M. Kortelainen, J. Phys. G 37, (2010) a = k F r

48 Exchange interaction energy in infinite matter

49 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). 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.

50 Density-matrix expansion (5)

51 J. Dobaczewski et al., J. Phys. G: 37, (2010) Negele-Vautherin density-matrix expansion

52 J. Dobaczewski et al., J. Phys. G: 37, (2010) Negele-Vautherin density-matrix expansion

53 J. Dobaczewski et al., J. Phys. G: 37, (2010) Negele-Vautherin density-matrix expansion

54 Density-matrix expansion and the Skyrme force

55 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 N 3 LO functionals J. Dobaczewski et al., J. Phys. G: 37, (2010)

56 J. Dobaczewski et al., J. Phys. G: 37, (2010) Negele-Vautherin density-matrix expansion

57 Proton RMS radius R p (fm) Quasi-local vs. non-local functionals The Negele-Vautherin density-matrix expansion (DME) up to NLO (2 nd 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. 56 Ni 48 Ca 40 Ca R p / A 1/3 (fm) 132 Sn 100 Sn 78 Ni Pb Number of nucleons A Binding Energy (MeV) quasi-local (Gogny+DME NLO) 1600 non-local (Gogny) Ni 48 Ca 40 Ca 100 Sn 78 Ni 132 Sn B / A (MeV) Pb Number of nucleons A J. Dobaczewski et al., J. Phys. G: 37, (2010)

58 Negele-Vautherin density-matrix expansion The quasi-local EDF derived from the finite-range force does not correspond to a zero-range force 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. J. Dobaczewski et al., J. Phys. G: 37, (2010)

59 Nuclear densities as composite fields

60 Local energy density: (no isospin, no pairing)

61 Nuclear Energy Density Functional

62 Complete local energy density E. Perlińska, et al., Phys. Rev. C69 (2004) Mean field Pairing

63 Mean-field equations

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

65 Applications

66 Phenomenological effective interactions

67 Proton Number < SLy4 volume pairing N=Z Neutron Number N=2Z deformation Theory > +0.4 M.V. Stoitsov, et al., Phys. Rev. C68, (2003)

68 J.-P. Delaroche et al., Phys. Rev. C81, (2010) Gogny D1S

69 Nuclear binding energies (masses) 2149 masses RMS = 798 kev The first Gogny HFB mass model. An explicit and selfconsistent account of all the quadrupole correlation energies are included within the 5D collective Hamiltonian approach. S. Goriely et al., Phys. Rev. Lett. 102, (2009) 2149 masses RMS = 581 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. S. Goriely et al., Phys. Rev. Lett. 102, (2009)

70 First 2 + excitations of even-even nuclei energies energies Gogny HFB calculations plus the 5D collective Hamiltonian approach. J.-P. Delaroche et al., Phys. Rev. C81, (2010) Skyrme HF+BCS calculations plus the particle-number and angular-momentum projection and shape mixing. B. Sabbey et al., Phys. Rev. C75, (2007)

71 Spontaneous fission UNEDF collaboration: SciDAC Review 6, 42 (2007) Symmetry-unrestricted Skyrme HF+BCS calculations. A. Staszczak et al., Phys. Rev. C80, (2009)

72 M. Bender et al., Phys. Rev. C78, (2008) Collective states in even-even nuclei J.M. Yao et al., Phys. Rev. C81, (2010)

73 S. Péru et al., Phys. Rev. C77, (2008) J. Terasaki et al., arxiv: Giant resonances in deformed nuclei K. Yoshida et al., Phys. Rev. C78, (2008) 24 Mg 24 Mg 174 Yb

74 Fast RPA and QRPA + Arnoldi method J. Toivanen et al., Phys. Rev. C 81, (2010)

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79 J. Toivanen et al., Phys. Rev. C 81, (2010) Fast RPA and QRPA + Arnoldi method Sn 2 +

80 Removal of spurious modes e 2 fm 6 / MeV Isoscalar 1 - strength functions standard RPA Arnoldi (spurious) Arnoldi 132 Sn E [MeV] J. Toivanen et al., Phys. Rev. C 81, (2010)

81 QRPA timing Scaling properties Spherical QRPA+Arnoldi scales linearly with the size of the single-particle space Deformed QRPA+Arnoldi expected to scale quadratically, that is, as Standard QRPA scales quartically, that is, as Future plans: Full implementation and testing of the spherical QRPA + Arnoldi method in 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.

82 Iterative methods to solve (Q)RPA Photoabsorption cross sections Theory Exp T. Inakura et al., Phys. Rev. C80, (2009) Finite-amplitude method to solve the QRPA equations.

83 Iterative methods to solve (Q)RPA Monopole resonance 132 Sn J. Toivanen et al., Phys. Rev. C81, (2010) Arnoldi method to solve the RPA equations.

84 Time-dependent solutions 8 Be collision (tip configuration) A.S. Umar et al., Phys. Rev. Lett. 104, (2010) Dipole oscillations in 14 Be protons neutrons UNEDF collaboration: Quantum Dynamics with TDSLDA, A. Bulgac, et al., oups/qmbnt/index.html T. Nakatsukasa et al., Nucl. Phys. A788, 349 (2007)

85 Nuclear Energy Density Functional

86 RMS deviation of s.p. energies (MeV) Singular value SLy5 SkP SkO' SIII (a) (b) Number of singular value Fits of s.p. energies EXP: M.N. Schwierz, I. Wiedenhover, and A. Volya, arxiv: Singular value decomposition = M. Kortelainen et al., Phys. Rev. C77, (2008)

87 Fits of single-particle energies i - i exp i - i exp SLy5 SLy5 (fit) (a) (e) SkP SkP (fit) (b) (f) SkO' SkO' (fit) (MeV) (MeV) (MeV) (MeV) (c) (g) SIII SIII (fit) M. Kortelainen et al., Phys. Rev. C77, (2008) (d) (h) Before After

88 Fit residuals for centroids of SO partners (SkP) Before After n n Before After L L M. Kortelainen et al., to be published

89 Fit residuals for splittings of SO partners (SkP) Before After n n Before After L L M. Kortelainen et al., to be published

90 How many parameters are really needed? Bertsch, Sabbey, and Uusnakki Phys. Rev. C71, (2005) Global (masses)

91 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)

92 UNEDF Skyrme Functionals M. Kortelainen, et al., arxiv:

93 UNEDF Skyrme Functionals M. Kortelainen, et al., arxiv:

94 New functionals

95 Nuclear Energy Density Functional

96 B.G. Carlsson et al., Phys. Rev. C 78, (2008) Derivatives of higher order: Negele & Vautherin density matrix expansion

97 Energy density functional up to N 3 LO B.G. Carlsson et al., Phys. Rev. C 78, (2008)

98 Numbers of terms in the density functional up to N 3 LO Eq. (28) density dependent CC Eq. (30) density independent CC B.G. Carlsson et al., Phys. Rev. C 78, (2008)

99 B.G. Carlsson et al., Phys. Rev. C 78, (2008) Phys. Rev. C 81, (E) (2010) Energy density functional for spherical nuclei (I)

100 Energy density functional for spherical nuclei (II) B.G. Carlsson et al., Phys. Rev. C 78, (2008) Phys. Rev. C 81, (E) (2010)

101 Convergence of density-matrix expansions for nuclear interactions (the direct term) B.G. Calsson, J. Dobaczewski, arxiv:

102 Convergence of density-matrix expansions for nuclear interactions (the exchange term) B.G. Calsson, J. Dobaczewski, arxiv:

103 RMS deviation of s.p. energies (MeV) Singular value SLy4 Fits of s.p. energies regression analysis EXP-1 EXO-2 EXP-1 (NM) EXP-2 (NM) Number of singular value 2nd order NLO B.G. Carlsson et al., to be published EXP-1: M.N. Schwierz, I. Wiedenhover, and A. Volya, arxiv: EXP-2: M.G. Porquet et al., to be published NM: Nuclear-matter constraints on: saturation density energy per particle incompressibility effective mass

104 RMS deviation of s.p. energies (MeV) Singular value SLy4 Fits of s.p. energies regression analysis EXP-1 EXP-2 EXP-1 (NM) EXP-2 (NM) (Galilean invariance) Number of singular value 4th order N 2 LO B.G. Carlsson et al., to be published EXP-1: M.N. Schwierz, I. Wiedenhover, and A. Volya, arxiv: EXP-2: M.G. Porquet et al., to be published NM: Nuclear-matter constraints on: saturation density energy per particle incompressibility effective mass

105 RMS deviation of s.p. energies (MeV) Singular value Fits of s.p. energies regression analysis SLy4 EXP-1 EXP-2 EXP-1 (NM) EXP-2 (NM) (Galilean invariance) 6th order N 3 LO Number of singular value B.G. Carlsson et al., to be published EXP-1: M.N. Schwierz, I. Wiedenhover, and A. Volya, arxiv: EXP-2: M.G. Porquet et al., to be published NM: Nuclear-matter constraints on: saturation density energy per particle incompressibility effective mass Problem of instabilities unsolved yet M. Kortelainenand T. Lesinski J. Phys. G: Nucl. Part. Phys

106 Program HOSPHE Solution of selfconsistent equations for the N 3 LO nuclear energy density functional in spherical symmetry B.G. Carlsson et al., arxiv:

107 HFODD HOSPHE Program HOSPHE Solution of selfconsistent equations for the N 3 LO nuclear energy density functional in spherical symmetry B.G. Carlsson et al., arxiv:

108 Spontaneous symmetry breaking

109 Ammonia molecule NH 3 Nitrogen atom Hydrogen atom left state right state

110 Ammonia molecule NH 3 Symmetry-conserving configuration Total energy (a.u.) Symmetry-breaking configurations Distance of N from the H 3 plane (a.u.)

111

112 Skyrme-Hartree-Fock J. Dobaczewski, J. Engel, Phys. Rev. Lett. 94, (2005) Experiment R.G. Helmer et al., Nucl. Phys. A474 (1987) 77 J z =1/2 1/ Ra 1/2+ 0

113 NH Ra ratio T 1/2 (Q.M.) T 1/2 (E.M.) D 0.1 mev 6.6 ps 16 ks 0.76 e nm 55 kev as ~5 ns ~0.1 e fm

114 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). 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. 4) Systematic energy density functionals with derivative corrections can be constructed and the resulting self-consistent equations solved. 5) In finite systems, the phenomenon of spontaneous symmetry breaking is best captured by the mean-field or energy-densityfunctional methods.

115 Nuclear deformation Symmetry-conserving configuration Total energy (a.u.) Symmetry-breaking configurations Elongation (a.u.)

116 Origins of nuclear deformation Single-particle energy (a.u.) Open-shell system: 8 particles on 8 doubly degenrate levels Elongation (a.u.)

117 Proton Number < SLy4 volume pairing N=Z Neutron Number N=2Z deformation Theory > +0.4 M.V. Stoitsov, et al., Phys. Rev. C68, (2003)

118 Tensor effects

119 B. Fornal, XXIX Mazurian Lakes Conference on Physics (2005) M.Honma et al., Eur. Phys. J. 25,s01 (2005) 499 B. Fornal et al., Phys. Rev. C 70, (2004) D.-C. Dinca et al., Phys. Rev. C 71, (2005)

120 E(2 + ) (MeV) Neutron Number N A 22 Ti BE2( ) (e 2 b 2 )

121 Evolution of the single particle orbitals with Z going from 28 to 20 Tensor Interaction V T = ( 2) Y (2 Z(r) B. Fornal, XXIX Mazurian Lakes Conference on Physics (2005) V couples j > and j < orbitals and favors charge exchange processes f 7/2 f 5/2 T. Otsuka et al. Phys. Rev. Lett 87, (2001)

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

123 Tensor energy densities For conserved spherical and time-reversal symmetries, averaged tensor and SO interactions give the following energy densities: where the particle and SO densities read

124 Single-particle spin-orbit potentials Variation of the energy densities with respect to the single-particle wave functions gives form factors of the single-particle spin-orbit potentials:

125 Neutron S-O Density J (R) [fm -4 ] n N=28 N= R (fm) t e =200 Z=28 J. Dobaczewski, nucl-th/

126 Neutron S-O Density J (R) [fm -4 ] n R (fm) N=50 N=40 t e =200 Z=28 J. Dobaczewski, nucl-th/

127 Neutron S-O Density J (R) [fm -4 ] n N=28 N=50 N=38 N= R (fm) t e =200 Z=28 J. Dobaczewski, nucl-th/

128 N (MeV) t e =0 N=32 N=28 N=20 HFB+SLy4 2s_1/2 1d_3/2 1d_5/2 2p_1/2 2p_3/2 1f_5/2 1f_7/2 J. Dobaczewski, nucl-th/ Proton Number Z 10

129 d 2p 1f 0 2s t e =0 N=32 1d -10 1d 2p HFB+SLy4 20 Proton Number Z 10 2p 1f 1f cent (MeV) SO (MeV) J. Dobaczewski, nucl-th/

130 d 2p 1f 0 2s t e =-100 N=32 1d -10 1d 2p HFB+SLy4 20 Proton Number Z 10 2p 1f 1f cent (MeV) SO (MeV) J. Dobaczewski, nucl-th/

131 d 2p 1f 0 2s t e =200 N=32 1d -10 1d 2p HFB+SLy4 20 Proton Number Z 10 2p 1f 1f cent (MeV) SO (MeV) J. Dobaczewski, nucl-th/

132 exp Polarization effects for neutron spin-orbit splitting M. Zalewski et al., Phys. Rev. C77, (2008)

133 Fits of C 0 J, C 0 J, and C 1 J SkP L M. Zalewski et al., Phys. Rev. C77, (2008)

134 M. Zalewski et al., Phys. Rev. C77, (2008) Fits of spin-orbit and tensor coupling constants

135 M. Zalewski et al., Phys. Rev. C77, (2008) SkP original SkP T Tensor + SO* Sn208Pb 56 Ni90Zr 16 O40Ca48Ca 56 Ni90Zr 132 Sn208Pb 16 O40Ca48Ca Spin-orbit splittings [MeV] SkP SkP T n n d 5/2 -d 3/2 p 3/2 -p 1/2 p d 5/2 -d 3/2 p 3/2 -p 1/2 p f 7/2 -f 5/2 f 7/2 -f 5/2 1g 1h 1i 2g 1g 1h

136 M. Zalewski et al., Phys. Rev. C77, (2008) SkO original SkO T Tensor + SO* Sn208Pb Spin-orbit splittings [MeV] n n SkO SkO T d 5/2 -d 3/2 p 3/2 -p 1/2 p d 5/2 -d 3/2 p 3/2 -p 1/2 p f 7/2 -f 5/2 1g 1h 1i 2g 1g 1h 56 Ni90Zr f 7/2 -f 5/2 56 Ni90Zr 132 Sn208Pb 16 O40Ca48Ca 16 O40Ca48Ca

137 Challenges

138 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 for mean field

139 Extensions I. Range separation and exact long-range effects II. Derivatives of higher order: III. Products of more than two densities:

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

141 Thank you