New Frontiers in Nuclear Structure Theory

Transcription

1 New Frontiers in Nuclear Structure Theory From Realistic Interactions to the Nuclear Chart Robert Roth Institut für Kernphysik Technical University Darmstadt

2 Overview Motivation Nucleon-Nucleon Interactions Solving the Many-Body Problem Correlations & Unitary Correlation Operator Method Applications

3 Nuclear Structure in the 21 st Century new frontiers in nuclear structure physics Experiment fundamental astrophysical questions need nuclear input possibilities to investigate nuclei far off stability new nuclear structure facilities: RIA, Theory improved understanding of fundamental degrees of freedom / QCD high-precision realistic nucleonnucleon potentials ab initio treatment of the manybody problem

4 Astrophysical Challenges

5 Theoretical Context finite nuclei few-nucleon systems nucleon-nucleon interaction hadron structure quarks & gluons deconfinement better resolution / more fundamental Quantum Chromo Dynamics Nuclear Structure

6 Theoretical Context solve the quantum many-body problem with this interaction derive a realistic nucleon-nucleon interaction from QCD better resolution / more fundamental Quantum Chromo Dynamics Nuclear Structure

7 Realistic Nucleon-Nucleon Potentials

8 Nature of the NN-Interaction NN-interaction is not fundamental induced via mutual polarization of quark & gluon distributions analogous to van der Waals interaction between neutral atoms short-ranged: acts only if the nucleons overlap 1.6fm ρ 1/3 0 = 1.8fm genuine NNN-interaction is important

9 How to Construct the NN-Potential? QCD input symmetries meson-exchange picture chiral perturbation theory short-range phenomenology ansatz for short-range behavior experimental two-body data scattering phase-shifts & deuteron properties Argonne V18 Nijmegen I/II CD Bonn Chiral... reproduced with χ 2 /datum 1

10 Argonne V18 Potential [MeV] v(r) v(r) L 2 (S, T ) (1, 0) (1, 1) (0, 0) (0, 1) 100 v(r) S 12 v(r) ( L S) v(r) ( L S) 2 [MeV] r [fm] r [fm] r [fm]

11 Nuclear Many-Body Problem

12 Ab initio Calculations solve the quantum many-body problem for A nucleons interacting via a realistic NN-potential exact numerical solution possible for small systems at an enormous computational cost Green s Function Monte Carlo: Monte Carlo sampling of the A- body wave function in coordinate space; imaginary time cooling No-Core Shell Model: large-scale diagonalization of the Hamiltonian in a harmonic oscillator basis

13 Green s Function Monte Carlo Energy (MeV) He 6 He Li Li 5/2 5/2 7/2 1/2 3/2 5/2 5/2 7/2 1/2 3/2 8 He Argonne v 18 With Illinois-2 GFMC Calculations 22 June AV18 8 Be Be 9/2 (5/2 ) 7/2 (3/2 ) 1/2 5/2 3/2 4 + (4 + ) Be 10 B CPU-hours -100 [S. Pieper, private comm.] IL2 Exp 12 C C results are preliminary.

14 Our Goal nuclear structure calculations across the whole nuclear chart based on realistic NN-potentials bound to simple Hilbert spaces for large particle numbers need to deal with strong interactioninduced correlations

15 Correlations in Nuclei

16 What are Correlations? correlations = everything beyond the independent particle picture the quantum state of A independent (non-interacting) fermions is a Slater determinant ψ = A( φ1 φ2 φa ) Slater determinants cannot describe correlations by definition

17 Deuteron: Manifestation of Correlations M S = ( + ) M S = ±1, spin-projected two-body density ρ (2) 1,M S ( r) uncorrelated two-body state

18 Deuteron: Manifestation of Correlations M S = ( + ) M S = ±1, z spin-projected two-body density ρ (2) 1,M S ( r) exact deuteron solution for Argonne V18 potential two-body density fully suppressed at small particle distances r central correlations angular distribution depends strongly on relative spin orientation tensor correlations

19 Central Correlations V01(r) [MeV]. ρ (2) 01 (r) [fm 3 ] ρ (2) SD (r) AV18 central part (S, T ) = (0, 1) ρ (2) exact(r) 4 He strong repulsive core in central part of realistic interactions suppression of the probability density for finding two nucleons within the core region central correlations can be described by shifting the nucleons out of the core region r [fm]

20 Tensor Correlations analogy with dipole-dipole interaction anti-parallel spins V tensor (3 ( σ 1 r)( σ 2 r) ) σ r 2 1 σ 2 couples the relative spatial orientation of two nucleons with their spin orientation tensor correlations V tensor parallel spins can be described by rotating nucleons towards pole or equator depending on spin

21 Unitary Correlation Operator Method (UCOM)

22 Unitary Correlation Operator Method Correlation Operator introduce correlations by means of an unitary transformation with respect to the relative coordinates of all pairs [ C = exp[ i G] = exp i ] g ij i<j g = g( r, q; σ 1, σ 2, τ 1, τ 2 ) G = G C C = 1 Correlated States ψ = C ψ Correlated Operators Ô = C O C ψ O ψ = ψ C O C ψ = ψ Ô ψ

23 Central and Tensor Correlators C = C Ω C r Central Correlator C r radial distance-dependent shift in the relative coordinate of a nucleon pair g r = 1 2 [ s(r) qr + q r s(r) ] q r = 1 2[ r r q + q r r ] Tensor Correlator C Ω angular shift depending on the orientation of spin and relative coordinate of a nucleon pair g Ω = 3 2 ϑ(r)[ ( σ 1 q Ω )( σ 2 r) + ( r q Ω ) ] q Ω = q r r q r s(r) and ϑ(r) encapsulate the physics of short-range correlations

24 Correlated States 0.15 L = 0 ρ (2) 1,S M ( r) r φ. r Cr φ. r CΩCr φ r [fm] L = 0 r [fm] L = 2 central correlations tensor correlations [fm] s(r) r [fm] ϑ(r) r [fm] r [fm]

25 Simplistic Shell-Model Calculation He 16 O 40 Ca T expectation values for harmonic osc. Slater determinant [AMeV] C r C Ω H V AV18 nuclei unbound without inclusion of correlations central and tensor correlations essential to obtain bound system

26 Application I: Hartree-Fock Calculations

27 UCOM-Hartree-Fock Approach Standard Hartree-Fock + Matrix Elements of Correlated Realistic NN-Interaction V UCOM many-body state is a Slater determinant of single-particle states obtained by energy minimization correlations cannot be described by Hartree-Fock states bare realistic NN-potential leads to unbound nuclei

28 Correlated Argonne V18 0 E/A [MeV] Rch [fm] He 16 O 24 O 34 Si 40 Ca 48 Ni 68 Ni 88 Sr 100 Sn 132 Sn 208 Pb 48 Ca 56 Ni 78 Ni 90 Zr 114 Sn 146 Gd experiment AV18α

29 Missing Pieces long-range correlations Ab Initio Strategy improve many-body states such that long-range correlations are included many-body perturbation theory (MPT), configuration interaction (CI), coupled-cluster (CC),...

30 Long-Range Correlations many-body perturbation theory: second-order energy shift gives estimate for influence of long-range correlations E (2) = 1 4 occu. i,j unoccu. a,b φ a φ b VUCOM φi φ j 2 ɛ a + ɛ b ɛ i ɛ j E/A [MeV] preliminary He 16 O 24 O 34 Si 40 Ca 48 Ni 68 Ni 88 Sr 100 Sn 132 Sn 208 Pb 48 Ca 56 Ni 78 Ni 90 Zr 114 Sn 146 Gd experiment HF HF+MPT2

31 Missing Pieces long-range correlations genuine three-body forces three-body cluster contributions Pragmatic Approach phenomenological two-body correction δv c+p+ls = v 1 (r) + q v qq (r) q + v LS (r) L S Gaussian radial dependencies with fixed ranges strengths used as fit parameters (fitted to 4 He, 16 O, 24 O, 40 Ca, 48 Ca, 48 Ni, 90 Zr)

32 Correlated Argonne V18 + Correction 0 E/A [MeV] Rch [fm] He 16 O 24 O 34 Si 40 Ca 48 Ni 68 Ni 88 Sr 100 Sn 132 Sn 208 Pb 48 Ca 56 Ni 78 Ni 90 Zr 114 Sn 146 Gd experiment AV18α AV18α + δv c+p+ls

33 Charge Distributions O 48 Ca 90 Zr [fm 3 ] Ca 88 Sr 208 Pb [fm 3 ] r [fm] r [fm] r [fm] experiment HF with AV18α + δv c+p+ls

34 Application II Fermionic Molecular Dynamics (FMD)

35 UCOM-FMD Approach Gaussian Single-Particle States q = n ν=1 c ν aν, b ν χν mt x aν, b ν = exp [ ( x b ν ) 2 ] 2 a ν a ν : complex width χ ν : spin orientation b ν : mean position & momentum Slater Determinant Q = A ( q1 q2 qa ) Correlated Hamiltonian Variation Q Ĥ Tcm Q Q Q min Diagonalization in sub-space spanned by several non-orthogonal Slater determinants Qi Ĥ = T + V UCOM [+δv c+p+ls ]

36 Variation: Chart of Nuclei Z He3 He4 He5 He6 He7 He8 H2 one Gaussian per nucleon Si22 O13 O14 O15 O16 O17 O18 O19 O20 O21 O22 O23 O24 N12 N13 N14 N15 N16 N17 N18 N19 N20 N21 N22 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 B8 B9 B10 B11 B12 B13 B14 B15 B16 B17 Be7 Be8 Be9 Be10Be11Be12Be13Be14 Li5 Li6 Li7 Li8 Li9 Li10 Li11 P27 P28 P29 P30 P31 P32 P33 P34 P35 P36 P37 P38 P39 P40 P41 Si24 Si25 Si26 Si27 Si28 Si29 Si30 Si31 Si32 Si33 Si34 Si35 Si36 Si37 Si38 Si39 Al24 Al25 Al26 Al27 Al28 Al29 Al30 Al31 Al32 Al33 Al34 Al35 Al36 Mg20Mg21Mg22Mg23Mg24Mg25Mg26Mg27Mg28Mg29Mg30Mg31Mg32Mg33Mg34 Na20Na21Na22Na23Na24Na25Na26Na27Na28Na29Na30Na31 Ne17Ne18Ne19Ne20Ne21Ne22Ne23Ne24Ne25Ne26Ne27Ne28 F16 F17 F18 F19 F20 F21 F22 F23 F24 F25 F26 Ni48 Cr46 Cr47 Cr48 Cr49 Cr50 Cr51 Cr52 Cr53 Cr54 Cr55 Cr56 Cr57 V45 V46 V47 V48 V49 V50 V51 V52 V53 V54 V55 V56 Ti41 Ti42 Ti43 Ti44 Ti45 Ti46 Ti47 Ti48 Ti49 Ti50 Ti51 Ti52 Ti53 Ti54 Ti55 Sc39Sc40Sc41Sc42Sc43Sc44Sc45Sc46Sc47Sc48Sc49Sc50Sc51Sc52Sc53Sc54 Ca36Ca37Ca38Ca39Ca40Ca41Ca42Ca43Ca44Ca45Ca46Ca47Ca48Ca49Ca50Ca51Ca52Ca53Ca54 K35 K36 K37 K38 K39 K40 K41 K42 K43 K44 K45 K46 K47 K48 K49 K50 Ar32 Ar33 Ar34 Ar35 Ar36 Ar37 Ar38 Ar39 Ar40 Ar41 Ar42 Ar43 Ar44 Ar45 Ar46 Ar47 Cl31 Cl32 Cl33 Cl34 Cl35 Cl36 Cl37 Cl38 Cl39 Cl40 Cl41 Cl42 Cl43 Cl44 Cl45 S29 S30 S31 S32 S33 S34 S35 S36 S37 S38 S39 S40 S41 S42 S43 S Ni54 Ni55 Ni56 Ni57 Ni58 Ni59 Ni60 Ni61 Co53Co54Co55Co56Co57Co58Co59Co60 Fe50Fe51Fe52Fe53Fe54Fe55Fe56Fe57Fe58Fe59 Mn49Mn50Mn51Mn52Mn53Mn54Mn55Mn56Mn57Mn58 B8 B9 B10 B11 B12 B13 B14 B15 B16 B17 Be7 Be8 Be9 Be10 Be11 Be12 Be13 Be14 Li5 Li6 Li7 Li8 Li9 Li10 Li11 (E E exp )/A [MeV] H3 He3 He4 He5 He6 He7 He8 2 H2 H N C9 Na20 Na21 Na22 Na23 Na24 Na25 Na26 Na27 Ne17 Ne18 Ne19 Ne20 Ne21 Ne22 Ne23 Ne24 Ne25 Ne26 Ne27 Ne28 F16 F17 F18 F19 F20 F21 F22 F23 O13 O14 O15 O16 O17 O18 O19 O20 O21 O22 O23 O24 N12 N13 N14 N15 N16 N17 N18 N19 N20 N21 N22 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C Mg20Mg21Mg22Mg23Mg24Mg25Mg26Mg27Mg28Mg29Mg30Mg31Mg32Mg33Mg34 Na31 AV18α + δv c+p+ls two Gaussians per nucleon

37 Intrinsic One-Body Density Distributions 4 He 16 O 40 Ca ρ (1) ( x) [ρ0] capable of describing spherical shell-model as well as intrinsically 9 Be deformed 20 Ne and α-cluster states 24 Mg

38 Beyond Simple Variation Projection after Variation (PAV) 16 O restore inversion and rotational symmetry by angular momentum projection Variation after Projection (VAP) find energy minimum within parameter space of parity and angular momentum projected states implementation via generator coordinate method (constraints on multipole moments) Multi-Configuration diagonalization within a set of different Slater determinants

39 Helium Isotopes: Energies & Radii MeV fm Binding energies He4 He5 He6 He7 He8 Matter & charge radii ch 0 ch 0 ch ch 0 ch He4 He5 He6 He7 He8 0 0 ch ch PAV Π Multi-Config Experiment 0 0 ch 0 0

40 Helium Isotopes: Density Profiles 1 4 He 1 6 He 1 8 He ρ(r) [ρ 0 ] total neutron proton ρ(r) [ρ 0 ] total neutron proton ρ(r) [ρ 0 ] total neutron proton r [fm] r [fm] r [fm] neutron halo

41 Structure of 12 C E MeV C Variation PAV Π Multi-Config Experiment E [ MeV] R ch [ fm] B(E2) [e 2 fm 4 ] V/PAV VAP α-cluster PAV π VAP Multi-Config Experiment ± 3.3 z [fm] z [fm] V/PAV V π /PAV π z [fm] y [fm] Multi-Config y [fm] y [fm] z [fm] y [fm] VAPα y [fm] VAP x [fm] x [fm] y [fm]

42 Structure of 12 C Hoyle State E MeV C z [fm] z [fm] = y [fm] = Multi-Config 4 Multi-Config 14 Experiment Multi-Config Experiment E [ MeV] R ch [ fm] B(E2, ) [e2 fm 4 ] ± 3.3 M(E0, ) [ fm2 ] ± 0.2 z [fm] y [fm] = y [fm] 2

43 Conclusions exciting times for nuclear structure physics! realistic NN-potentials & ab initio calculations systematic schemes to derive effective (correlated / low-momentum) interactions innovative ways to treat the many-body problem unified description of nuclear structure across the whole nuclear chart is within reach

44 Epilogue thanks to my group & my collaborators A. Ehrlich, H. Hergert, N. Paar, P. Papakonstantinou, Technical University Darmstadt H. Feldmeier, T. Neff Gesellschaft für Schwerionenforschung (GSI) supported by DFG through SFB 634 Nuclear Structure and Nuclear Astrophysics...