Nuclear Energy Density Functional from Chiral Two- and Three-Nucleon - - PowerPoint PPT Presentation

Nuclear Energy Density Functional from Chiral Two- and Three-Nucleon Interactions

• N. Kaiser

EMMI seminar, 3.May 2012

Introduction: nuclear energy density functional Tool: (improved) density-matrix expansion Chiral two- and three-nucleon interactions Diagrammatic calculation of energy density functional Results for isospin-symmetric nuclear systems Isovector part of nuclear energy density functional

Publications:

• J. Holt, N. Kaiser, W. Weise, Eur. Phys. J. A47 (2011) 128; Eur. Phys. J. A48 (2012) 36.
• N. Kaiser

Nuclear Energy Density Functional from 2N +3N Chiral Interactions

Introduction

Nuclear energy density functional: many-body method of choice for systematic calculation of medium-mass and heavy nuclei Non-relativistic (parametrized) Skyrme functionals and relativistic mean-field models (σ + ω-mesons) are widely and successfully used RMF: strong Lorentz scalar and vector mean-fields of opposite sign act coherently to generate nuclear spin-orbit interaction Complementary approach: constrain form of a predictive energy density functional and its couplings by many-body perturbation theory and the underlying two- and three-nucleon interactions Switch from hard-core NN-potentials to low-momentum interactions: with Vlow−k nuclear many-body problem becomes more perturbative Non-local Fock contributions to energy: approximate them by functionals expressed in terms of local densities and currents only Key ingredient: Density-matrix expansion Negele and Vautherin, Phys. Rev. C5 (1972) 1472 Gebremariam, Bogner, Duguet, Nucl. Phys. A851 (2011) 17: used N2LO chiral NN-potential + Skyrme → got small but systematic reduction of χ2 This work: Improved chiral NN-potential N3LO + chiral 3N-interaction

• N. Kaiser

Nuclear Energy Density Functional from 2N +3N Chiral Interactions

Improved density-matrix expansion

Improved density-matrix expansion via phase-space averaging: Gebremariam, Duguet and Bogner, Phys. Rev. C82 (2010) 14305

• α

Ψα

• r −

a 2

• Ψ†

α

• r +

a 2

• = 3ρ

akf j1(akf) − a 2kf j1(akf)

• τ − 3

5ρk 2

f − 1

4

• ∇2ρ
• + 3i

2akf j1(akf) σ · ( a × J ) + . . . ρ = 2k 3

f /3π2 local nucleon density,

J =

αΨ† αi

σ × ∇Ψα spin-orbit density Few % accuracy for Fock contrib. from central and tensor interactions Spin-dependent part ( a × J ) of Negele-Vautherin DME makes 50% error Fourier-transform: ”medium insertion” for inhomogenous nuclear system Γ( p, q ) =

• d3r e−i

q·

• r
• θ(kf − |

p |) + π2 4k 4

f

• kf δ′(kf − |

p |) − 2δ(kf − | p |)

• ×
• τ − 3

5ρk 2

f − 1

4

• ∇2ρ
• − 3π2

4k 4

f

δ(kf − | p |) σ · ( p × J )

• generalizes step-function θ(kf − |

p |) for infinite nuclear matter

• N. Kaiser

Nuclear Energy Density Functional from 2N +3N Chiral Interactions

Improved density-matrix expansion

Comparison of density-matrix expansions: central interaction

• N. Kaiser

Nuclear Energy Density Functional from 2N +3N Chiral Interactions

Improved density-matrix expansion

Comparison of density-matrix expansions: tensor interaction INM: quadrupolar deformation of local Fermi-moment. distribution neglected

• N. Kaiser

Nuclear Energy Density Functional from 2N +3N Chiral Interactions

Nuclear energy density functional

Energy density functional for N = Z even-even nuclei: E[ρ, τ, J ] = ρ ¯ E(ρ) +

• τ − 3

5ρk 2

f

1 2M − k 2

f

4M3 + Fτ(ρ)

• + (

∇ρ)2 F∇(ρ) + ∇ρ · J Fso(ρ) + J 2 FJ(ρ) effective nucleon mass M∗(ρ), surface term, spin-orbit coupling, J2 term Relation to slope of single-particle potential at Fermi surface: Fτ(ρ) = 1 2kf ∂U(p, kf) ∂p

• p=kf

= − kf 3π2 f1(kf) i.e. same effective nucleon mass M∗(ρ) as in Fermi-liquid theory Decomposition: for Fd(ρ), factor ( ∇ρ)2 emerges directly from interaction F∇(ρ) = 1 4 ∂Fτ(ρ) ∂ρ + Fd(ρ) For zero-range Skyrme force: improved density-matrix expansion and Negele-Vautherin DME give identical results (quadratic p-dependence) Differences expected for long-range 1π- and 2π-exchange interaction

• N. Kaiser

Nuclear Energy Density Functional from 2N +3N Chiral Interactions

Chiral NN and 3N interactions

Preferred 2-body interact.: universal low-momentum NN-potential Vlow−k Partial wave matrix elements, explicit spin-isospin operators better suited Easier tractable substitute for Vlow−k: Chiral N3LOW potential,Λ=414MeV Finite-range part of N3LOW: one- and two-pion exchange of the form V (π)

NN

= VC(q) + τ1 · τ2 WC(q) +

• VS(q) +

τ1 · τ2 WS(q)

• σ1 ·

σ2 +

• VT(q) +

τ1 · τ2 WT(q)

• σ1 ·

q σ2 · q +

• VSO(q) +

τ1 · τ2 WSO(q)

• i(

σ1 + σ2) · ( q × p ) , dependence only on momentum transfer q, no quadratic spin-orbit comp.

100 200 300 400 500 q [MeV]

• 100
• 80
• 60
• 40
• 20

20 40 60 [GeV

• 2]

finite-range central potentials VC(q) WC(q) 100 200 300 400 500 q [MeV]

• 40
• 30
• 20
• 10

10 20 30 40 50 60 [GeV

• 2]

finite-range spin-spin potentials VS(q) WS(q) 100 200 300 400 500 q [MeV]

• 2.5
• 2
• 1.5
• 1
• 0.5

[10

3 GeV

• 4]

finite-range tensor potentials VT(q) WT(q) 100 200 300 400 500 q [MeV]

• 0.6
• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1 0.2 0.3 [10

3 GeV

• 4]

finite-range spin-orbit potentials VSO(q) WSO(q)

Short-range part: 24 contact terms up 4th power of momenta, CST, Cj, Dj determined in fits to NN-phase shifts and deuteron (→ Machleidt’s code)

• N. Kaiser

Nuclear Energy Density Functional from 2N +3N Chiral Interactions

Two-body contributions at 1st order

Finite-range pieces: Hartree-Fock, employing Γ p1, q ) Γ( p2, − q )

¯ E(ρ) = ρ 2 VC(0) − 3ρ 2 1 dx x2(1 − x)2(2 + x)[VC(q) + 3VS(q) + q2VT (q) + ...] Fτ (ρ) = kf 2π2 1 dx(x − 2x3)[VC(q) + 3VS(q) + q2VT (q) + 3Wcomb(q)] Fd(ρ) = 1 4V ′′

C (0)

Fso(ρ) = 1 2 VSO(0) + 1 dx x3[VSO(2xkf ) + 3WSO(2xkf )] FJ(ρ) = 3 8k2

f

1 dx

• (2x3 − x)[VC(q) − VS(q)] − x3q2VT (q) + 3Wcomb(q)
• Short-range pieces:

¯ E(ρ) = 3ρ 8 (CS − CT ) + 3ρk2

f

20 (C2 − C1 − 3C3 − C6) + 9ρk4

f

140 (D2 − 4D1 + ...) Fτ (ρ) = ρ 4 (C2 − C1 − 3C3 − C6) + ρk2

f

4 (D2 − 4D1 − 12D5 − 4D11) Fd(ρ) = 1 32 (16C1 − C2 − 3C4 − C7) + k2

f

48 (9D3 + 6D4 − 9D7 − 6D8 + ...) Fso(ρ) = 3 8 C5 + k2

f

6 (2D9 + D10)

• N. Kaiser

Nuclear Energy Density Functional from 2N +3N Chiral Interactions

Three-body contributions at 1st order

Leading order chiral 3N-interaction: contact + 1π-exchange + 2π-exch. LECs cE = −0.625,cD = 2.06 fitted to binding energies of 3H and 4He 3-body correlations in inhomogeneous nuclear many-body systems: factorized density-matrices in p-space Γ( p1, q1) Γ( p2, q2) Γ( p3, − q1 − q2) cE- and cD-terms:

¯ E(ρ) = − cEk6

f

12π4f 4

πΛχ

¯ E(ρ) = gAcDm6

π

(2πfπ)4Λχ u6 3 − 3u4 4 + u2 8 + u3 arctan 2u − 1 + 12u2 32 ln(1 + 4u2)

• Fτ(ρ) = 2gAcDm4

π

(4πfπ)4Λχ

• (1 + 2u2) ln(1 + 4u2) − 4u2

Fd(ρ) = gAcDmπ (4fπ)4π2Λχ 1 2u ln(1 + 4u2) − 2u 1 + 4u2

• FJ(ρ) =

3gAcDmπ (4fπ)4π2Λχ

• 2u − 1

u + 1 4u3 ln(1 + 4u2)

• ,

u = kf mπ

• N. Kaiser

Nuclear Energy Density Functional from 2N +3N Chiral Interactions

Three-body contributions at 1st order

2π-exchange Hartree diagram prop. to c1 = −0.76, c3 = −4.78 (GeV−1)

¯ E(ρ) = g2

Am6 π

(2πfπ)4

• (12c1 − 10c3)u3 arctan 2u − 4

3 c3u6 + 6(c3 − c1)u4 +(3c1 − 2c3)u2 + 1 4 (2c3 − 3c1) + 3u2 2 (3c3 − 4c1)

• ln(1 + 4u2)
• Fso(ρ)

= 3g2

Amπ

(8π)2f 4

π

2 u (4c1 − 3c3) − 4c3u + 4 u (c3 − c1) + 3c3 − 4c1 2u3

• ln(1 + 4u2)
• ,

u = kf mπ

3-body spin-orbit coupling originally suggested by Fujita and Miyazawa Most tedious to evaluate: 2π-exchange Fock diagram, c4 = 3.96 GeV−1

• N. Kaiser

Nuclear Energy Density Functional from 2N +3N Chiral Interactions

Results for isospin-symmetric nuclear systems

Energy per particle: for 2-body part Vlow−k ≃ VN3LOW

0.05 0.1 0.15 0.2 ρ [fm

• 3]
• 40
• 30
• 20
• 10

10 20 E(ρ) [MeV]

2-body 3-body total Epot Epot+Ekin Vlow-k

energy per particle

Improved description: treat 2-body interaction to second order etc. Effective nucleon mass M∗(ρ0): in phenomenological reasonable range

0.05 0.1 0.15 0.2 ρ [fm

• 3]

2 4 6 8 10 12 Fτ(ρ) [MeVfm

2] 2-body 3-body total Vlow-k

0.05 0.1 0.15 0.2 ρ [fm

• 3]

0.6 0.7 0.8 0.9 1 M

*(ρ) / M

effective nucleon mass

• N. Kaiser

Nuclear Energy Density Functional from 2N +3N Chiral Interactions

Results for isospin-symmetric nuclear systems

Strength of surface term ( ∇ρ)2: fair agreement with phen. Skyrme forces

0.05 0.1 0.15 0.2 ρ [fm

• 3]
• 100
• 50

50 100 Fgrad(ρ) [MeVfm

5] 2-body 3-body total

(density-gradient)

2 term

Skyrme forces

Spin-orbit coupling strength

0.05 0.1 0.15 0.2 ρ [fm

• 3]

25 50 75 100 125 150 175 Fso(ρ) [MeVfm

5] 2-body 3-body total

spin-orbit coupling strength

Skyrme forces

0.05 0.1 0.15 0.2 ρ [fm

• 3]

50 100 150 200 FJ(ρ) [MeVfm

5] 2-body 3-body total 1π-exchange

quadratic spin-orbit coupling

2-body contrib. mainly of short-range origin, sizeable 3-body spin-orbit c3 Expect reduction by second order π-exchange tensor interaction

• N. Kaiser

Nuclear Energy Density Functional from 2N +3N Chiral Interactions

Isovector part of nuclear energy density functional

Isovector terms pertaining to different proton and neutron densities: relevant for long chains of stable isotopes and nuclei far from stability Up to second order in proton-neutron differences and spatial gradients

Eiv[ρp, ρn, τp, τn, Jp, Jn] = 1 ρ (ρp − ρn)2 ˜ A(ρ) + 1 ρ (τp − τn)(ρp − ρn) Gτ (ρ) +( ∇ρp − ∇ρn)2 G∇(ρ) + ( ∇ρp − ∇ρn) · ( Jp − Jn) Gso(ρ) + ( Jp − Jn)2 GJ(ρ)

˜ A(ρ) interacting part of nuclear matter asymmetry energy Gτ(ρ) splits effective proton and neutron masses prop. to local ρp − ρn G∇(ρ) isovector surface term, Gso(ρ) isovector spin-orbit coupl. strength Adapt density-matrix expansion to asym. situation: → Fourier-transform

Γiv( p, q ) =

• d3r e−i

q·

• r

1 + τ 3 2 θ(kp − | p |) + 1 − τ 3 2 θ(kn − | p |) + π2 4k4

f

• kf δ′(kf − |

p |) − 2δ(kf − | p |)

• τp − τn −
• k2

f +

• ∇2

4

• ×(ρp − ρn)
• τ 3 − 3π2

4k4

f

δ(kf − | p |) ( σ × p ) · ( Jp − Jn)τ 3

• Fermi momenta: ρp = k 3

p /3π2, ρn = k 3 n /3π2, ρ = ρp + ρn = 2k 3 f /3π2

• N. Kaiser

Nuclear Energy Density Functional from 2N +3N Chiral Interactions

Isovector part of nuclear energy density functional

Asymmetry energy: A(ρ0) = 26.5 MeV, empirical value (34 ± 2)MeV

0.05 0.1 0.15 0.2 ρ [fm

• 3]

5 10 15 20 25 30 A(ρ) [MeV]

2-body 3-body total Apot Apot+Akin

asymmetry energy

Hartree-Fock approx. seems to work better for isovector quantities

0.05 0.1 0.15 0.2 ρ [fm

• 3]
• 3
• 2
• 1

1 Gτ(ρ) [MeVfm

2] 2-body 3-body total 1π-exch.

0.05 0.1 0.15 0.2 ρ [fm

• 3]
• 10
• 5

5 10 Ggrad(ρ) [MeVfm

5] 2-body 3-body total Gd(ρ)

isovector surface term

|Gτ(ρ)| << Fτ(ρ), from modern Sly forces G∇ = −(11 ± 5) MeVfm5

• N. Kaiser

Nuclear Energy Density Functional from 2N +3N Chiral Interactions

Isovector part of nuclear energy density functional

Isovector spin-orbit coupling strength

0.05 0.1 0.15 0.2 ρ [fm

• 3]

5 10 15 20 25 30 Gso(ρ) [MeVfm

5] 2-body 3-body total

isovector spin-orbit coupling 0.05 0.1 0.15 0.2

ρ [fm

• 3]
• 50
• 40
• 30
• 20
• 10

GJ(ρ) [MeVfm

5] 2-body 3-body total 1π-exch.

Small 3-body contrib., result close to Skyrme Gso = 1

3Fso ≃ 30 MeVfm5

Isovector spin-orbit coupling in nuclei presently not well determined GJ(ρ) and FJ(ρ) are dominated by 1π-exchange, strong ρ-dependence

Challenge:

Consistent calc. of EDF to 2nd order in many-body pertubation theory Energy denominators of intermediate states → further non-localities Generalized density-matrix expansion to 2nd order not yet formulated

• N. Kaiser

Nuclear Energy Density Functional from 2N +3N Chiral Interactions