Microscopic Theories of Nuclear Masses

MSU/NSCL JINA - Pizza Lunch seminar, MSU, 02/26/2007

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

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: S N, Q β 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...

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 = a vol A + a sf A 2/3 + 3e2 5r 0 Z 2 A 1/3 + (a sym A + a ss A 2/3 ) 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

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 = E mac + n i ɛ i n i ɛ i i i 30 parameters Excellent data fit: σ(e) = 0.656 MeV (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. Nazarewicz

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 = H m + H M Constrain H m through scaling arguments to account for saturation Constrain H M 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[ρ T ( r), τ T ( r), J( r),...] Reconcile single-particle and collective dynamics Allows a coherent calculation of many other quantities of interest 19 parameters / σ(e) 0.700 MeV (2135 nuclei) Current connection to underlying NN-NNN interactions is weak 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)

Some elements of comparison Fit over known masses Evolution of shell effects through two-neutron shell-gap δ 2N = S 2N (N, Z) S 2N (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

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

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 S N = S 2N /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

EDF method: spirit and characteristics Aim at the whole nuclear chart (A 16)

EDF method: spirit and characteristics Aim at the whole nuclear chart (A 16) Basic EDF method dedicated to G. S. properties

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 S N, S 2N, Q α, Q β Matter/charge density ρ q( r) r.m.s. radii R rms(q) Deformation properties Single-particle energies and shell structure Nuclear matter Equation of State Pairing properties

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

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

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... )

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

First generation of EDF mass tables First EDF mass table in (2000) including around 19 fitted parameters (10+5+4)

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

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 E W = V W exp {λ N Z /A} recent attempt to explain Wigner energy through neutron-proton T = 0 pairing

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 E rot = E crank rot tanh(cβ 2 ) = Φ J2 Φ 2I crank tanh(cβ 2 )

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

Highlights: two examples HFB-9 mass table (2005) Fit to ab-initio Neutron matter EOS Increase a sym from 28 to 30 MeV (cf. neutron-skin thickness) Impact the isotopic composition of neutron star core and inner crust Goriely et al (2005)

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

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

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

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)

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

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.)

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 E th E exp for 134 spherical nuclei DFTM = phenomenological local functional σ(e) = 2.964 MeV FR = functional from bare NN interaction σ(e) = 2.144 MeV Refit of the p-h part to be meaningful... Lesinski et al. (2007) Controlled approximations being worked out for systematic calculations

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

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

Foundation: Hohenberg-Kohn theorem (1964) Theorem H + v Ground state energy minimizes E[ρ( r)] = F [ρ( r)] + d r v( r) ρ( r) E GS obtained for ρ( r) = ρ GS ( r) such that d r ρ GS ( 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

Implementation : Kohn-Sham (1965) Introduce the non-interacting system Φ E KS [ρ( r)] = T KS [ρ( r)] + d r v KS ( r) ρ( r) / T KS [ρ( r)] 2 2m Choose v KS ( r) / ρ KS ( r) = i ϕ i( r) 2 = ρ GS ( r) Re-write F [ρ( r)] T KS [ρ( r)] + E H [ρ( r)] + E xc[ρ( r)] d r i ϕ i ( r) 2 Minimization / d r ρ( r) = N leads to { } 2 2m + v KS( r) ϕ i ( r) = ɛ i ϕ i ( r) Kohn-Sham equations with the local potential v KS ( r) = v H [ρ( r)] + v xc[ρ( r)] / v xc[ρ( r)] = δexc[ρ( r)] δρ( r) Koopmans Theorem ɛ 0 = E N 0 E N 1 0 ; other ɛ i have no meaning Looks like solving a Hartree problem BUT it is in principle exact Difficulty = approximating E xc[ρ( r)]: LDA, GGA, Meta-GGA, Hybrid,...

Nuclear EDF: implementation Use more local sources to derive HK theorem E E[ρ T ( r), τ T ( r),...] ρ T ( r) Iso-scalar/vector matter density τ T ( 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 s T ( r) Iso-scalar/vector spin density jt ( r) Iso-scalar/vector current density ρ T ( 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)) E = d r [ C ρρ T ρ2 T + C T ss s T s T + C ρ ρ T ρ T ρ T + CT s s s T s T T =0,1 +C ρτ T (ρ T τ T ) ( j T j T + CT J2 s T T ) T JT 2 Density-dependent couplings +C ρ J T (ρ T JT + s T ) j T ρ ρ +CT ρ T ( r) 2 ] ( ) 2 + CT s s st Historical guidance from HF + density-dependent Skyrme interaction Local pairing functional density-dependent delta interaction V ρ ρ ( r 1 r 2) = C ρ ρ 1 [ρ0( r)] δ( r1 r2) which necessitates an ultra-violet regularization/renormalization

Equation of motions Introduce (quasi-)particle state Φ to construct the local fields, i.e. ρ T ( r) ij ρ T ( r) ij ϕ j ( rσ)ϕ i ( rσ) ρ ij σ 2σ ϕ j ( r σ)ϕ i ( rσ) κ ij σ with ρ ij = Φ a j a i Φ with κ ij = Φ a j a i Φ Minimizing E / ρ ij and κ ij leads to HFB/Bogoliubov-De Gennes equations ( h λ (h λ) ) ( U V ) µ = E µ ( U V ) µ (U, V ) µ/e µ = quasi-particle states/energies (ϕ i ( rσ), ρ ij, κ ij ) 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