Von Schalen, Clustern und Halos

Transcription

1 Von Schalen, Clustern und Halos - Neue Konzepte zur Lösung des nuklearen Vielteilchenproblems - Hans Feldmeier GSI Darmstadt Thomas Neff MSU East Lansing He Robert Roth TU Darmstadt ρ(r) [ρ ] total neutron proton Physik Kolloquium Dresden. Mai 6

2 Von der QCD zur Kernstruktur mehr Auflösung/fundamentaler Quantum Chromo Dynamics Kernstruktur Atomkerne Wenig-Nukleonen-Systeme Nukleon-Nukleon-Potential Hadronenstruktur Quarks & Gluonen Deconfinement Physik-Kolloquium Dresden -. Mai 6

3 Von der QCD zur Kernstruktur mehr Auflösung/fundamentaler Quantum Chromo Dynamics Kernstruktur löse das wechselwirkende nukleare Vielteilchenproblem konstruiere realistische Nukleon-Nukleon Wechselwirkung aus der QCD Physik-Kolloquium Dresden -. Mai 6

4 Potential und Größe des Protons Proton Ladungsradius r e.86) fm Proton Ladungsverteilung und S=, T=1 Potential Proton Durchmesser nicht klein gegen Reichweite der Wechselwirkung V [MeV] 1 AV18 Halbdichte-Überlapp bei maximaler Anziehung, Ausläufer berühren sich noch bei mittlerem NN-Abstand in Kernmaterie V NN nicht elementar ähnlich wie Atom-Atom-Potential erwartet Dreikörperkräfte -1 v c (r) Physik-Kolloquium Dresden -. Mai 6

5 Realistic Nucleon-Nucleon Interaction Realistic NN Potential NN scattering data S+1 L J describes phase shifts and deuteron properties central, spin-orbit, tensor, momentum dependent V NN(r 1, p 1,σ 1,σ,τ 1,τ ) Phase Shift (deg) 7 1 S 1 3 Lab. Energy (MeV) Phase Shift (deg) 1 3 P Lab. Energy (MeV) e.g. different realistic NN interactions describe scattering data chiral effective field theory (pion exchange & contact terms) Bonn potential (pion & heavier meson exchange) Argonne potential (pion exchange & phenomenological short range terms) Phase Shift (deg) Phase Shift (deg) 1 P Lab. Energy (MeV) 3 S Lab. Energy (MeV) Phase Shift (deg) Phase Shift (deg) P Lab. Energy (MeV) 3 D Lab. Energy (MeV) Physik-Kolloquium Dresden -. Mai 6

6 Ab Initio Nuclear Structure Calculations describe basic properties of nuclear many-body system in terms of a realistic nucleon-nucleon interaction H and a many-body state ˆΨ r1,σ 1,τ 1 ; r,σ,τ ;...;r A,σ A,τ A ˆΨ Aim solve H ˆΨ n = En ˆΨ n by means of Green Function Monte Carlo Method (GFMC) Physik-Kolloquium Dresden -. Mai 6

7 GFMC calculations ( exact numerical solution) Energy (MeV) He 6 He Li Li / / 7/ 1/ 3/ / / 7/ 1/ 3/ 8 He Argonne v 18 With Illinois- GFMC Calculations June AV18 8 Be Be 9/ (/ ) 7/ (3/ ) 1/ / 3/ + ( + ) Be 1 B3 + genuine three-body forces (IL) needed for -1 description of real nuclei IL Exp 1 C + 1 C results are preliminary. Wiringa, Pieper, PRL 89 () 181 Physik-Kolloquium Dresden -. Mai 6

8 GFMC calculations ( exact numerical solution) Energy (MeV) He 6 He Li Li / / 7/ 1/ 3/ / / 7/ 1/ 3/ 8 He Argonne v 18 With Illinois- GFMC Calculations June Be Be 9/ (/ ) 7/ (3/ ) 1/ / 3/ + ( + ) Be 1 B3 + 7 CPU-hours -8 AV18 genuine three-body forces (IL) needed for -1 description of real nuclei IL Exp 1 C + 1 C results are preliminary. Wiringa, Pieper, PRL 89 () 181 Physik-Kolloquium Dresden -. Mai 6

9 Ab Initio Kernstruktur GFMC Ergebnis: r1,σ 1,τ 1 ; r,σ,τ ;...;r A,σ A,τ A ˆΨ n ist sehr kompliziert für realistische Nukleon-Nukleon-Potentiale. WARUM? Physik-Kolloquium Dresden -. Mai 6

10 Inhalt Wo liegt das Problem? Unitary Correlation Operator Method realistischer korrelierter Hamiltonian Anwendungen: No-Core Schalenmodell Fermionische Molekular Dynamik Zusammenfassung und Ausblick Physik-Kolloquium Dresden -. Mai 6

11 Realistic NN-potential VNN [MeV] V NN (r 1, p 1 =,σ 1,σ, T=) AV18 1 r 1-1 r 1 V NN repulsive at small distances strong short-range central correlations nucleons cannot get closer than.6 fm V NN depends strongly on orientation ofσ 1,σ with respect to r 1 tensor correlations protons and neutrons want to align their spins with r r 1 [fm] Argonne potential Problem: Slater determinants cannot describe these correlations Physik-Kolloquium Dresden -. Mai 6

12 Realistic NN-potential VNN [MeV] V NN (r 1, p 1 =,σ 1,σ, T=) AV18 1 r 1-1 r 1 V NN repulsive at small distances strong short-range central correlations nucleons cannot get closer than.6 fm V NN depends strongly on orientation ofσ 1,σ with respect to r 1 tensor correlations protons and neutrons want to align their spins with r r 1 [fm] Argonne potential Problem: Slater determinants cannot describe these correlations Solution: include short-range correlations by unitary transformation (UCOM) ˆΨ = C Ψ = C Ω C r Ψ C r central correlator shifts nucleons out of repulsive core C Ω tensor correlator aligns spins along r 1 Physik-Kolloquium Dresden -. Mai 6

13 Unitary Correlation Operator Method (UCOM) Correlation Operator introduce short-range correlations by means of a unitary transformation with respect to the relative coordinates of all pairs C = exp[ i G ]=exp [ ] i g i j i< j (1) G = G C C = 1 Correlated States ˆΨ = C Ψ Correlated Operators Ô = C 1 O C ˆΨ O ˆΨ = Ψ C 1 O C Ψ = Ψ Ô Ψ Physik-Kolloquium Dresden -. Mai 6

14 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 [ s(r ) q r+ qr s(r ) ] q r = 1 [ 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 ϑ(r)[ (σ 1 q Ω)(σ r)+(r q Ω) ] q Ω = q r r q r s(r) andϑ(r) for given potential determined in the two-body system Physik-Kolloquium Dresden -. Mai 6

15 Correlated States: The Deuteron. r φ.1.1. L= 1 3 Physik-Kolloquium Dresden -. Mai 6

16 Correlated States: The Deuteron.1 L=. r C r φ r φ L= central correlations [fm] s(r) Physik-Kolloquium Dresden -. Mai 6

17 Correlated States: The Deuteron.1 L=. r C r φ r φ. r C ΩC r φ L= 1 3 L= L= 1 3 central correlations tensor correlations [fm] constraint on range of tensor correlator. s(r) 1 3 ϑ(r) 1 3 Physik-Kolloquium Dresden -. Mai 6

18 Correlations in Nuclei two-body densities z [fm] x [fm] x [fm] x [fm] ρ () C r ~ ~ S,T (r 1 r ) S= 1, M S = 1, T= C Ω radial correlator shifts density out of the repulsive core tensor correlator aligns density with spin orientation both radial and tensor correlations are essential for binding energies [A MeV] 6 He 16 O Ca C r C Ω T V H 6 Physik-Kolloquium Dresden -. Mai 6

19 Correlated Interaction V UCOM [] [] [3] [3] [3] Ĥ = Ĥ = T T + ( ˆT V UCOM + ˆV )+( + ˆT V + UCOM ˆV )+ + closed operator expression for the correlated interaction V UCOM in two-body approximation correlated interaction and original NN-potential are phase shift equivalent by construction unitary transformation results in a pre-diagonalisation of Hamiltonian Physik-Kolloquium Dresden -. Mai 6

20 Anwendung 1 No-Core Shell Model

21 Benchmarking V UCOM for 1 B: large-scale NCSM calculations troughout the p-shell in progress 7 6 V UCOM Exp + E E3+ [MeV] ω ω ω 6 ω 8 ω ω=18mev calculations by Petr Navrátil preliminary Physik-Kolloquium Dresden -. Mai 6

22 Benchmarking V UCOM for 1 B: large-scale NCSM calculations troughout the p-shell in progress E E3+ [MeV] 7 V UCOM Exp CDBonn Chiral V6 UCOM gives correct level ordering without any NNN interaction NN NN+NNN ω ω ω 6 ω 8 ω 8 ω -6.8 ω=18mev -6.3 calculations by Petr Navrátil preliminary 6 ω Physik-Kolloquium Dresden -. Mai 6

23 Anwendung Fermionic Molecular Dynamics

24 Hilbert Space: Fermionic Molecular Dynamics Fermionic Slater determinant Q ( =A q 1 ) q A Antisymmetrization antisymmetrized A-body state Molecular single-particle states x q = c i exp { (x b i) } χ i ξ a i i Gaussian wave-packets in phase-space, spin is free, isospin is fixed Hilbert space contains shell-model, clusters, halos Dynamics in Hilbert space spanned by one or several non-orthogonal Q (a) Ψ = ψ a Q (a) a variational principle Q (a) ={q (a) ν,ν=1 A},ψ a Physik-Kolloquium Dresden -. Mai 6

25 NN Interaction Effective Correction to the Interaction Effective two-body interaction correlated two-body interaction Ĥ = C H C is lacking three-body forces instead of three-body force use additional momentumdependent and spin-orbit two-body correction term fit correction term to binding energies and radii of closed-shell nuclei altogether a 1% correction to the ab-initio two-body potential 16 O Ca O 8 Ca ρ (1) (r) [ρ] 16 O and Ca are not closed shell nuclei! Physik-Kolloquium Dresden -. Mai 6

26 FMD - Projection, Variation after Proj., Multiconfiguration PAV VAP Radius and Quadrupole Moment as Generator Coordinates r charge [fm] Q [fm ] B(E) [e fm ] PAV VAP Multiconfig Multiconfig [MeV] Be 6 y [fm] 6 6 x [fm] y [fm] y [fm] 6 x [fm] y [fm] x [fm] 6 x [fm] x [fm] y [fm] Variation PAV VAP Multiconf Exp Physik-Kolloquium Dresden -. Mai 6

27 FMD - Variation, PAV π, Multiconfig. V/PAV PAV π Multiconfig() C 6 6 E MeV E [MeV] r charge [fm] B(E) [e fm ] V/PAV PAV π Multiconfig() Multiconfig(1) Exp ± C Multi-Config Multi-Config 1 Experiment 1 3 Physik-Kolloquium Dresden -. Mai 6

28 Structure of the Carbon Hoyle State E MeV C 1 3 z [fm] z [fm] z [fm] + = y [fm] + =.71 - y [fm] + =. - 9 Multi-Config Multi-Config 1 Experiment - y [fm] Physik-Kolloquium Dresden -. Mai 6

29 .1.1 Helium Isotopes He He He He z [fm] ρ(r) [ρ ] total neutron proton z [fm] ρ(r) [ρ ] total neutron proton - x [fm] x [fm] He He He He z [fm] ρ(r) [ρ ] total neutron proton z [fm] ρ(r) [ρ ] total neutron proton - x [fm] x [fm] z [fm] He ρ(r) [ρ ] total neutron proton 6 He soft-dipole mode neutrons are oscillating againstα-core - x [fm] Physik-Kolloquium Dresden -. Mai 6

30 Helium Isotopes MeV Binding energies 3 He He He6 He7 He8 Matter radii PAV Multiconf Exp fm He He He6 He7 He8 Exp: Ozawa,Suzuki,Tanihata, NPA693(1)3; Raman,Nestor,Tikkanen, Atomic Data and Nucl. Data Tables 78(1)1 Physik-Kolloquium Dresden -. Mai 6

31 Helium Isotopes MeV fm Binding energies zero-point oscillations of the He soft-dipole mode essential for He description ofhe6 binding energieshe7 He8 and radii Matter radii PAV Multiconf Exp 1. He He He6 He7 He8 Exp: Ozawa,Suzuki,Tanihata, NPA693(1)3; Raman,Nestor,Tikkanen, Atomic Data and Nucl. Data Tables 78(1)1 Physik-Kolloquium Dresden -. Mai 6

32 1.. Beryllium Isotopes 7 Be - p 7 Be - n x [fm] Be - p Physik-Kolloquium Dresden -. Mai z [fm] x [fm] Be - n.1.1 z [fm] x [fm] Be - p - 9 x [fm] Be - n z [fm] x [fm] Be - p - 1 x [fm] Be - n z [fm] x [fm] - x [fm] Be - p - 1 x [fm] Be - p - 13 x [fm] Be - p 11 Be - n - 1 x [fm] Be - n x [fm] Be - n x [fm] Be - p - 1 x [fm] Be - n x [fm] - x [fm] cluster structure changes with addition of neutrons z [fm] z [fm] z [fm] z [fm]

33 Beryllium Isotopes 3 Binding energies PAV Multiconf Exp MeV fm Be7 Be8 Be9 Be1 Be11 Be1 Be13 Be1 Matter radii Be7 Be8 Be9 Be1 Be11 Be1 Be13 Be Exp: Ozawa,Suzuki,Tanihata, NPA693(1)3; Raman,Nestor,Tikkanen, Atomic Data and Nucl. Data Tables 78(1)1 Physik-Kolloquium Dresden -. Mai 6

34 Mikroskopische Kern-Kern-Potentiale Vielteilchenzustände: Fermionische Molekulardynamik (FMD) Effektive NN-Wechselwirkung: abgeleitet von der realistischen Argonne-V18 NN-Wechselwirkung (UCOM) 16 O - 16 O O - O O O O ρ [fm -3 ] 1-3 O - O Massendichten Physik-Kolloquium Dresden -. Mai 6

35 Potentiale & Astrophysikalische S-Faktoren V eff (r) + V Coul (r) [MeV] O- 16 O O- O O- O Mikroskopisch berechnete Kern-Kern-Potentiale 1 1 Astrophysikalischer S-Faktor S (E)=σ fusion (E) E e πη η=sommerfeld-parameter S-factor [MeV barn] O- 16 O O- O O- O E cm [MeV] Physik-Kolloquium Dresden -. Mai 6

36 Outlook: Resonances & Scattering in FMD E [MeV] ev FMD Exp. Frozen Frozen PAV Π Exp. 7 collective coordinate representation as tool for the description of continuum states in FMD He α 3 He α 7 Be 7 Be 7 Be 1 3 δ(e) [deg] He-α 3/ + / + first steps towards fully microscopic and consistent description of structure and reactions -1 Frozen+PAV π 1/ E [MeV] Physik-Kolloquium Dresden -. Mai 6

37 Zusammenfassung und Ausblick Neue Ära der Kernstruktur Unitary Correlation Operator Method beschreibt kurzreichweitige radiale und tensorielle Korrelationen UCOM erzeugt ab-initio korrelierte Wechselwirkung Ĥ für Vielteilchenmethoden wie HF, Shalenmodell, FMD Observablen müssen und können auch korreliert werden no-core Schalenmodell Rechnungen für leichte Kerne FMD Rechnungen mit demselben Ĥ C +Ĥ corr für 3 A 6 Ein mikroskopisches Modell für: Bindungsenergien, Radien, Spektren, Übergänge, Kontinuum, Resonanzen, Reaktionen M S = M S = 1 16 O ab initio Idee weiter verfolgen Vorhersagekraft Vorhersagen für viele exotische leichte Kerne (vor Messung) Ausblick Korrelierte Übergänge: M1, β-zerfall, quenching Ab-initio korrelierte WW in HF, RPA oder ähnlichem für A>6 Langreichweitige Tensorkorr. 3-Teilchenkräfte Physik-Kolloquium Dresden -. Mai 6

38 Epilogue thanks to my group & my collaborators A. Cribeiro, K. Langanke Gesellschaft für Schwerionenforschung (GSI) T. Neff NSCL, Michigan State University R. Roth, H. Hergert, N. Paar, P. Papakonstantinou, A. Zapp Institut für Kernphysik, TU Darmstadt supported by the DFG through SFB 63 Nuclear Structure, Nuclear Astrophysics and Fundamental Experiments... Physik-Kolloquium Dresden -. Mai 6

39 How to improve? Projection After Variation (PAV) mean-field may break symmetries of Hamiltonian restore reflection and rotational symmetry by parity and angular-momentum projection P J π MK K Q J H P π KK Q c K = EK Jπ Q J P π KK Q c K K Variation After Projection (VAP) effect of projection can be large perform VAP applying constraints on radius, dipole moment, quadrupole moment or octupole moment and minimize the energy in the projected energy surface Multiconfiguration Calculations diagonalize Hamiltonian in a set of projected intrinsic states { J P π KK Q (a) }, a=1,, N Physik-Kolloquium Dresden -. Mai 6

40 Unitary Correlation Operator Method, UCOM Two-Body Correlations two-body generator C = e ig, G = G = i< j g i j Correlator C should conserve translational, rotational and Galilei invariance short ranged Cluster Expansion correlated operators  = C A C are no longer operators with definite particle number decompose correlated operator into irreducible k-body operators  = C A C =  [1] +  [] +  [3] + Two-Body Approximation ˆT C = ˆT [1] + ˆT [], ˆV C [] = ˆV correlation range should be smaller than mean distance of nucleons (to avoid 3-body terms) Physik-Kolloquium Dresden -. Mai 6

41 Unitary Correlation Operator Method, UCOM Two-Body Correlations two-body generator C = e ig, G = G = Cluster Expansion i< j g i j correlated operators  = C A C are no longer operators with definite particle number decompose correlated operator into irreducible k-body operators  = C A C =  [1] +  [] +  [3] + Two-Body Approximation ˆT C = ˆT [1] + ˆT [], ˆV C [] = ˆV Correlator C should conserve translational, rotational and Galilei invariance short ranged Spin-Isospin Dependence nuclear interaction strongly depends on spin and isospin V = v S T Π S T S,T different correlations in the respective channels g = g S T Π S T S,T correlated interaction in two-body space ˆV = ( igs T e v S T e ig ) S T Π S T S,T correlation range should be smaller than mean distance of nucleons (to avoid 3-body terms) Physik-Kolloquium Dresden -. Mai 6

42 Radial and Tensor Correlations r p r p p Ω C = C Ω C r = e i G Ω e i G r p r = 1 { r r p=p r + p Ω ( r r p) + ( ) } p r r r r, pω = 1 r { l r r r r l} Radial Correlator G r= 1 } { p rs(r )+ s(r )pr probability density shifted out of the repulsive core S=, T=1 AV ρ () (r) V [MeV] 1 AV18 [fm 3 ] ˆρ () (r) -1 v c (r) Manfred Ristig Z.Physik 199 (1967) 3 Physik-Kolloquium Dresden -. Mai 6

43 Radial and Tensor Correlations r p r p p Ω C = C Ω C r = e i G Ω e i G r p r = 1 { r r p=p r + p Ω ( r r p) + ( ) } p r r r r, pω = 1 r { l r r r r l} Radial Correlator G r= 1 } { p rs(r )+ s(r )pr probability density shifted out of the repulsive core [fm 3 ] S=, T= ρ () (r) ˆρ () (r) V [MeV] AV Manfred Ristig Z.Physik 199 (1967) 3 v c (r) AV Tensor Correlations G Ω=ϑ(r ) 3 { (σ 1 p Ω)(σ r )+(σ 1 r )(σ p Ω) } tensor force admixes other angular momenta [fm 3 ] S=1, T= ρ () (r) ˆρ () (r) V [MeV] ˆρ () L= () (r)ˆρ L= (r) AV v c (r) v t (r) AV Physik-Kolloquium Dresden -. Mai 6

44 Radial and Tensor Correlations r p r p p Ω C = C Ω C r = e i G Ω e i G r p r = 1 { r r p=p r + p Ω ( r r p) + ( ) } p r r r r, pω = 1 r { l r r r r l} Radial Correlator G r= 1 } { p rs(r )+ s(r )pr probability density shifted out of the repulsive core [fm 3 ] S=, T= ρ () (r) ˆρ () (r) V [MeV] AV Manfred Ristig Z.Physik 199 (1967) 3 v c (r) AV Tensor Correlations G Ω=ϑ(r ) 3 { (σ 1 p Ω)(σ r )+(σ 1 r )(σ p Ω) } tensor force admixes other angular momenta [fm 3 ] S=1, T= ρ () (r) ˆρ () (r) V [MeV] ˆρ () L= () (r)ˆρ L= (r) AV v c (r) v t (r) AV Physik-Kolloquium Dresden -. Mai 6

45 AV8 Interaction No-Core Shell Model Calculations preliminary He - α Quasi-exact calculations for light nuclei possible -1 E [MeV] -1 - exact result from PRC6 (1) 1 use no-core shell model code from Petr Navratil (LLNL) experiment Ω [MeV] neglected 3-body correlated terms same order as genuine 3N interactions more investigations needed test of two-body approximation Physik-Kolloquium Dresden -. Mai 6

46 MTV Interaction No-Core Shell Model Calculations 3 He He correlated - correlated - -1 E [MeV] E [MeV] Ω [MeV] bare E [MeV] E [MeV] Ω [MeV] 1 18 bare Ω [MeV] Ω [MeV] exact results from PRC (199) 88 only radial correlations use no-core shell model code from Pétr Navratil (LLNL) Physik-Kolloquium Dresden -. Mai 6

47 Other Observables Nucleon Momentum Distributions Bonn-A Argonne V O O.1.1 ˆn(k)/A [fm 3 ] central+ tensor ˆn(k)/A [fm 3 ] k [fm 1 ] central.1 VMC LDA 1 3 k [fm 1 ] correlations induce high-momentum components contributions of tensor correlations very big different correlator ranges relevant especially at the fermi surface Physik-Kolloquium Dresden -. Mai 6

48 Interaction in Momentum Space klm Ĥ [] k l m = i l l M d 3 x Y lm (ˆx) j l(kx) x Ĥ [] x j l (k x)y l m (ˆx) 1 S channel 1 3 S 1 channel 1 k Ĥ [] k, k V k [fm] -1 - uncorrelated Bonn A -3 AV k [fm 1 ] correlated k Ĥ [] k, k V k [fm] α β γ Bonn A AV k [fm 1 ] unique effective potential identical to V lowk Kuo, Schwenk, nucl-th/181 V lowk CutoffΛ=1.. fm 1 Physik-Kolloquium Dresden -. Mai 6

49 AV18 Interaction in Momentum Space Off-diagonal Matrix Elements 1S ; k V 1 S ; k 3 k [fm 1 ] k [fm 1 ] 3S 1; k V 3 D 1; k 3 k [fm 1 ] k [fm 1 ] pre-diagonalization bare potential 1S ; k Ĥ [] 1 S ; k 3 k [fm 1 ] k [fm 1 ] 3S 1; k Ĥ [] 3 D 1; k 3 k [fm 1 ] k [fm 1 ] correlated interaction Physik-Kolloquium Dresden -. Mai 6

50 AV8 Interaction No-Core Shell Model Calculations preliminary 3 He Increasing range of tensor correlator E [MeV] Ω [MeV] Ω [MeV] Ω [MeV] Ω [MeV] He E [MeV] Ω [MeV] Ω [MeV] Ω [MeV] Ω [MeV] Physik-Kolloquium Dresden -. Mai 6

51 11 B ( 3 He, t) 11 C Gamov-Teller transitions 6 11 B 1..8 Experiment 6 11 C 6 transition: C 1 Ω στ + C Ω up to now only:στ + NCSM: Navrátil, Ormand no core shell model with 3-body force, PRC 68(3) third 3/ missing FMD with configuration mixing B(GT) B(GT) B(GT) / - 1/ - / - 3/ - 1 NCSM 1 E ex [MeV] 3/ - 1/ - / - 3/ - 1 FMD 1 E ex [MeV] 3/ - 1/ - / - 3/ - 3/ - / - / - preliminary E ex [MeV] 3/ - / - Exp.: Y. Fujita, P. von Brentano et al. Phys. Rev. C 7, 1136(R) () Physik-Kolloquium Dresden -. Mai 6

52 16 C PAV only Variation E b [MeV] r charge [fm] r matte C C PAV 1g PAV y [fm] ρ(r) [ρ ] total neutron proton Exp ±.3.76±.6 - x [fm] 6 8 E + [MeV] B(E) [e fm ] PAV y [fm] C - p y [fm] C - n.1 Exp ±.9 Global Best Fit ±1 1 Raman et al, Atomic Data and Nuclear Data Tables 78 (1) x [fm] - x [fm] calculated B(E) consistent with anomalously long lifetime of + state measured at RIKEN Imai et al, PRL in print Physik-Kolloquium Dresden -. Mai 6

53 FMD, Variation Nuclear Chart H H3 1 Gaussian per single-particle state He3 He He He6 He7 He8 Si O13 O1 O1 O16 O17 O18 O19 O O1 O O3 O N1 N13 N1 N1 N16 N17 N18 N19 N N1 N C9 C1 C11 C1 C13 C1 C1 C16 C17 C18 C19 C B8 B9 B1 B11 B1 B13 B1 B1 B16 B17 Be7 Be8 Be9 Be1 Be11 Be1 Be13 Be1 Li Li6 Li7 Li8 Li9 Li1 Li11 Al Al Al6 Al7 Al8 Al9 Al3 Al31 Al3 Al33 Al3 Al3 Al36 MgMg1MgMg3MgMgMg6Mg7Mg8Mg9Mg3Mg31Mg3Mg33Mg3 Na Na1 Na Na3 Na Na Na6 Na7 Na8 Na9 Na3 Na31 Ne17 Ne18 Ne19 Ne Ne1 Ne Ne3 Ne Ne Ne6 Ne7 Ne8 F16 F17 F18 F19 F F1 F F3 F F F6 Ni8 Ca36 Ca37 Ca38 Ca39 Ca Ca1 Ca Ca3 Ca Ca Ca6 Ca7 Ca8 Ca9 Ca Ca1 Ca K3 K36 K37 K38 K39 K K1 K K3 K K K6 K7 K8 K9 K Ar3 Ar33 Ar3 Ar3 Ar36 Ar37 Ar38 Ar39 Ar Ar1 Ar Ar3 Ar Ar Ar6 Ar7 Cl31 Cl3 Cl33 Cl3 Cl3 Cl36 Cl37 Cl38 Cl39 Cl Cl1 Cl Cl3 Cl Cl S9 S3 S31 S3 S33 S3 S3 S36 S37 S38 S39 S S1 S S3 S P7 P8 P9 P3 P31 P3 P33 P3 P3 P36 P37 P38 P39 P P1 Si Si Si6 Si7 Si8 Si9 Si3 Si31 Si3 Si33 Si3 Si3 Si36 Si37 Si38 Si39 Ni Ni Ni6 Ni7 Ni8 Ni9 Ni6 Ni61 Co3 Co Co Co6 Co7 Co8 Co9 Co6 Fe Fe1 Fe Fe3 Fe Fe Fe6 Fe7 Fe8 Fe9 Mn9MnMn1MnMn3MnMnMn6Mn7Mn8 Cr6 Cr7 Cr8 Cr9 Cr Cr1 Cr Cr3 Cr Cr Cr6 Cr7 V V6 V7 V8 V9 V V1 V V3 V V V6 Ti1 Ti Ti3 Ti Ti Ti6 Ti7 Ti8 Ti9 Ti Ti1 Ti Ti3 Ti Ti Sc39 Sc Sc1 Sc Sc3 Sc Sc Sc6 Sc7 Sc8 Sc9 Sc Sc1 Sc Sc Ca correlated AV18+ Ĥ corr (E E exp )/A [MeV] Physik-Kolloquium Dresden -. Mai 6

54 FMD, Variation Nuclear Chart H Li Li6 Li7 Li8 Li9 Li1 Li11 He3 He He He6 He7 He8 H3 1 Gaussian per single-particle state Si Ne17 Ne18 Ne19 Ne Ne1 Ne Ne3 Ne Ne Ne6 Ne7 Ne8 O13 O1 O1 O16 O17 O18 O19 O O1 O O3 O N1 N13 N1 N1 N16 N17 N18 N19 N N1 N C9 C1 C11 C1 C13 C1 C1 C16 C17 C18 C19 C B8 B9 B1 B11 B1 B13 B1 B1 B16 B17 Be7 Be8 Be9 Be1 Be11 Be1 Be13 Be1 P7 P8 P9 P3 P31 P3 P33 P3 P3 P36 P37 P38 P39 P P1 Si Si Si6 Si7 Si8 Si9 Si3 Si31 Si3 Si33 Si3 Si3 Si36 Si37 Si38 Si39 Al Al Al6 Al7 Al8 Al9 Al3 Al31 Al3 Al33 Al3 Al3 Al36 MgMg1MgMg3MgMgMg6Mg7Mg8Mg9Mg3Mg31Mg3Mg33Mg3 Na Na1 Na Na3 Na Na Na6 Na7 Na8 Na9 Na3 Na31 F16 F17 F18 F19 F F1 F F3 F F F6 Ni8 Cr6 Cr7 Cr8 Cr9 Cr Cr1 Cr Cr3 Cr Cr Cr6 Cr7 V V6 V7 V8 V9 V V1 V V3 V V V6 Ti1 Ti Ti3 Ti Ti Ti6 Ti7 Ti8 Ti9 Ti Ti1 Ti Ti3 Ti Ti Sc39 Sc Sc1 Sc Sc3 Sc Sc Sc6 Sc7 Sc8 Sc9 Sc Sc1 Sc Ca36 Ca37 Ca38 Ca39 Ca Ca1 Ca Ca3 Ca Ca Ca6 Ca7 Ca8 Ca9 Ca Ca1 Ca K3 K36 K37 K38 K39 K K1 K K3 K K K6 K7 K8 K9 K Ar3 Ar33 Ar3 Ar3 Ar36 Ar37 Ar38 Ar39 Ar Ar1 Ar Ar3 Ar Ar Ar6 Ar7 Cl31 Cl3 Cl33 Cl3 Cl3 Cl36 Cl37 Cl38 Cl39 Cl Cl1 Cl Cl3 Cl Cl S9 S3 S31 S3 S33 S3 S3 S36 S37 S38 S39 S S1 S S3 S Ni Ni Ni6 Ni7 Ni8 Ni9 Ni6 Ni61 Co3 Co Co Co6 Co7 Co8 Co9 Co6 Fe Fe1 Fe Fe3 Fe Fe Fe6 Fe7 Fe8 Fe9 Mn9MnMn1MnMn3MnMnMn6Mn7Mn H H3 Sc Ca Be7 Be8 Be9 Be1Be11Be1Be13Be1 Li Li6 Li7 Li8 Li9 Li1 Li11 He3 He He He6 He7 He8 correlated AV18+ Ĥ corr F16 F17 F18 F19 F F1 F F3 C9 C1 C11 C1 C13 C1 C1 C16 C17 C18 C19 C B8 B9 B1 B11 B1 B13 B1 B1 B16 B17 Si8 Si9 Si3 Al7 MgMg1MgMg3MgMgMg6Mg7Mg8Mg9Mg3Mg31Mg3Mg33Mg3 NaNa1NaNa3NaNaNa6Na7 Ne17Ne18Ne19NeNe1NeNe3NeNeNe6Ne7Ne8 O13 O1 O1 O16 O17 O18 O19 O O1 O O3 O N1 N13 N1 N1 N16 N17 N18 N19 N N1 N (E E exp )/A [MeV] Na31 Gaussians per single-particle state Physik-Kolloquium Dresden -. Mai 6

55 Nuclear Degrees of Freedom Physik-Kolloquium Dresden -. Mai 6

56 Nuclear Degrees of Freedom σ j σ i r j r i cm-coordinates and spins Physik-Kolloquium Dresden -. Mai 6