microscopic theories of nuclear masses

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

Outline

1 Introduction

5 Conclusions

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?

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

Overall impact

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

∼ 7 parameters

Finite Range Droplet Model ; Moller et al (1995)

Microscopic-Macroscopic approaches (”mic-mac”)

Combine drop-model and shell effects through Strutinsky method

E = Emac +∑

ni εi −∑̃

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?

Finite Range Droplet Model ; Moller et al (1995)

Microscopic-Macroscopic approaches (”mic-mac”)

Combine drop-model and shell effects through Strutinsky method

E = Emac +∑

ni εi −∑̃

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?

Theoretical tools

Figure by W. NazarewiczThomas DUGUET Microscopic Theories of Nuclear Masses

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

∫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)

Several types of correlations are still poorly treated

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)

Montreal-Brussels mass tables (2000-now)

Based on the Energy Density Functional (EDF) formalism

∫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)

Several types of correlations are still poorly treated

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)

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

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

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

Outline

1 Introduction

5 Conclusions

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

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

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. . . )

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

Outline

1 Introduction

5 Conclusions

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

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

Correlations associated with shape fluctuations are omitted

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)

Weakening of too strong pairing

Theoretically motivated renormalization scheme

Improved calculations of level densities

Anticipate improved fission barriers

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

Outline

1 Introduction

5 Conclusions

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

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)

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

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.)

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

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

Outline

1 Introduction

5 Conclusions

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

Formulation of those extensions within a truly EDF framework

First attempts to connect to underlying NN-NNN interactions are being made

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

Formulation of those extensions within a truly EDF framework

First attempts to connect to underlying NN-NNN interactions are being made

Outline

1 Introduction

5 Conclusions

Foundation: Hohenberg-Kohn theorem (1964)

Theorem

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

Theorem

∫d~r v(~r) ρ(~r)

d~r ρGS(~r) = N

Theorem

∫d~r v(~r) ρ(~r)

d~r ρGS(~r) = N

Theorem

∫d~r v(~r) ρ(~r)

d~r ρGS(~r) = N

Theorem

∫d~r v(~r) ρ(~r)

d~r ρGS(~r) = N

Theorem

∫d~r v(~r) ρ(~r)

d~r ρGS(~r) = N

Theorem

∫d~r v(~r) ρ(~r)

d~r ρGS(~r) = N

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

∫d~r

|∇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, . . .

EKS [ρ(~r)] = TKS [ρ(~r)] +

∫d~r vKS(~r) ρ(~r) / TKS [ρ(~r)] ≡ ~2

∫d~r

|∇ϕi (~r)|2

i |ϕi (~r)|2 = ρGS(~r)

Minimization /∫

2m+ vKS(~r)

}ϕi (~r) = εi ϕi (~r)

δρ(~r)

EKS [ρ(~r)] = TKS [ρ(~r)] +

∫d~r vKS(~r) ρ(~r) / TKS [ρ(~r)] ≡ ~2

∫d~r

|∇ϕi (~r)|2

i |ϕi (~r)|2 = ρGS(~r)

Minimization /∫

2m+ vKS(~r)

}ϕi (~r) = εi ϕi (~r)

δρ(~r)

EKS [ρ(~r)] = TKS [ρ(~r)] +

∫d~r vKS(~r) ρ(~r) / TKS [ρ(~r)] ≡ ~2

∫d~r

|∇ϕi (~r)|2

i |ϕi (~r)|2 = ρGS(~r)

Minimization /∫

2m+ vKS(~r)

}ϕi (~r) = εi ϕi (~r)

δρ(~r)

EKS [ρ(~r)] = TKS [ρ(~r)] +

∫d~r vKS(~r) ρ(~r) / TKS [ρ(~r)] ≡ ~2

∫d~r

|∇ϕi (~r)|2

i |ϕi (~r)|2 = ρGS(~r)

Minimization /∫

2m+ vKS(~r)

}ϕi (~r) = εi ϕi (~r)

δρ(~r)

EKS [ρ(~r)] = TKS [ρ(~r)] +

∫d~r vKS(~r) ρ(~r) / TKS [ρ(~r)] ≡ ~2

∫d~r

|∇ϕi (~r)|2

i |ϕi (~r)|2 = ρGS(~r)

Minimization /∫

2m+ vKS(~r)

}ϕi (~r) = εi ϕi (~r)

δρ(~r)

EKS [ρ(~r)] = TKS [ρ(~r)] +

∫d~r vKS(~r) ρ(~r) / TKS [ρ(~r)] ≡ ~2

∫d~r

|∇ϕi (~r)|2

i |ϕi (~r)|2 = ρGS(~r)

Minimization /∫

2m+ vKS(~r)

}ϕi (~r) = εi ϕi (~r)

δρ(~r)

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!

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))

∫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

(~sT · ~TT − J 2

)+Cρ∇J

~∇ · ~JT +~sT · ~∇∧~jT

)+ C∇s∇s

(~∇ ·~sT

+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

Skyrme functional

∫d~r

∑T=0,1

[Cρρ

T ρ2T + C ss

T ~sT ·∆~sT

+CρτT

)+ C J2

(~sT · ~TT − J 2

)+Cρ∇J

~∇ · ~JT +~sT · ~∇∧~jT

)+ C∇s∇s

(~∇ ·~sT

+C ρ̃ρ̃T |ρ̃T (~r)|2

V ρ̃ρ̃(~r1 −~r2) = C ρ̃ρ̃1 [ρ0(~r)] δ(~r1 −~r2)

Skyrme functional

∫d~r

∑T=0,1

[Cρρ

T ρ2T + C ss

T ~sT ·∆~sT

+CρτT

)+ C J2

(~sT · ~TT − J 2

)+Cρ∇J

~∇ · ~JT +~sT · ~∇∧~jT

)+ C∇s∇s

(~∇ ·~sT

+C ρ̃ρ̃T |ρ̃T (~r)|2

V ρ̃ρ̃(~r1 −~r2) = C ρ̃ρ̃1 [ρ0(~r)] δ(~r1 −~r2)

Skyrme functional

∫d~r

∑T=0,1

[Cρρ

T ρ2T + C ss

T ~sT ·∆~sT

+CρτT

)+ C J2

(~sT · ~TT − J 2

)+Cρ∇J

~∇ · ~JT +~sT · ~∇∧~jT

)+ C∇s∇s

(~∇ ·~sT

+C ρ̃ρ̃T |ρ̃T (~r)|2

V ρ̃ρ̃(~r1 −~r2) = C ρ̃ρ̃1 [ρ0(~r)] δ(~r1 −~r2)

Skyrme functional

∫d~r

∑T=0,1

[Cρρ

T ρ2T + C ss

T ~sT ·∆~sT

+CρτT

)+ C J2

(~sT · ~TT − J 2

)+Cρ∇J

~∇ · ~JT +~sT · ~∇∧~jT

)+ C∇s∇s

(~∇ ·~sT

+C ρ̃ρ̃T |ρ̃T (~r)|2

V ρ̃ρ̃(~r1 −~r2) = C ρ̃ρ̃1 [ρ0(~r)] δ(~r1 −~r2)

Equation of motions

Introduce (quasi-)particle state |Φ〉 to construct the local fields, i.e.

ρT (~r) ≡∑

ϕ∗j (~rσ)ϕi (~rσ) ρij with ρij = 〈Φ|a†j ai |Φ〉

ρ̃T (~r) ≡∑

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)

(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

Equation of motions

ρT (~r) ≡∑

ρ̃T (~r) ≡∑

Equation of motions

ρT (~r) ≡∑

ρ̃T (~r) ≡∑

Equation of motions

ρT (~r) ≡∑

ρ̃T (~r) ≡∑

Equation of motions

ρT (~r) ≡∑

ρ̃T (~r) ≡∑

microscopic theories of nuclear masses

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