Modern description of nuclei far from stability

1. Modern description of nucleifar from stability Covariant density functional theoryof the dynamicsof nuclei far from stability Barcelona, Dec. 10, 2007 Peter Ring Universidad Autónoma de Madrid Technische Universität München Istanbul, July 2/3, 2008 Peter Ring Technische Universität München Summer School IV on Nuclear Collective Dynamics

2. Content Content II -------------------- Motivation Density Functional Theory The Nuclear Density Functional Covariant Density Functional Ground state properties Nuclear dynamics and excitations Outlook Summer School IV on Nuclear Collective Dynamics

3. H Fe Au Pb U neutron number N proton number Z neutron number N Summer School IV on Nuclear Collective Dynamics

4. magic numbers 2 8 20 28 50 82 126 168 ? Summer School IV on Nuclear Collective Dynamics

5. Forces acting in the nucleus: the Coulomb force repels the protons the strong interaction ("nuclear force") causes binding is stronger for pn-systems than nn-systems neutrons alone form no bound states exception: neutronen stars (gravitation!) e the weak interaction causes β-decay: n p ν - Summer School IV on Nuclear Collective Dynamics

6. distance < 0.5 fm ? the nucleon-nucleon interaction: distance > 1 fm attractive π-meson 1 fm repulsive three-body forces ? Summer School IV on Nuclear Collective Dynamics

7. H reactor energy fusion fission U He Fe A binding energy per particle sun energy B 8 (MeV) particle number Summer School IV on Nuclear Collective Dynamics

8. β+ β- N-Z β+decay β- decay Summer School IV on Nuclear Collective Dynamics

9. β+ β- N-Z Summer School IV on Nuclear Collective Dynamics

10. the nuclear density: ρ(r) simplified representation: ρ r ρ=1.6 nucleons/fm3 Summer School IV on Nuclear Collective Dynamics

11. ρ ρ r r small neutron excess large neutron excess proton and neutron densities or? p n ρ ρ p n r r Summer School IV on Nuclear Collective Dynamics

12. Nuclei far from stability: what can we learn? • the origin of more than half of the elements with Z>30 • constraints on effective nuclear interactions • evolution of shell structure • reduction of the spin-orbit interaction • properties of weakly-bound and open quantum systems • exotic modes of collective excitations • (pygmy, toroidal resonances) • possible new forms of nuclei (molecular states, • bubble nuclei, neutron droplets...) • asymmetric nuclear matter equation of state and • the link to neutron stars • - applications in astrophysics Summer School IV on Nuclear Collective Dynamics

13. Au Fe Abundancies of elements in the solar system Summer School IV on Nuclear Collective Dynamics

14. e- n (N,Z+1) (N,Z) (N+1,Z) synthesis of heavy elements beyond Fe neutron capture and successive β-decay: Z+1 Z N N+1 Summer School IV on Nuclear Collective Dynamics

15. Study of Nucleosynthesis r process Summer School IV on Nuclear Collective Dynamics

16. nuclear masses (bindung energies – Q-values) equation of state (EOS) of nuclear matter: E(ρ) isospin dependence E(ρp, ρn) nuclear matrix elements (life times of β-decay ..) cross section for neutron or electron capture …. fission probabilities cross sections for neutrino reactions ….. ….. What do the astrophysicists need ? Summer School IV on Nuclear Collective Dynamics

17. QCD NN- forces in the vacuum effective forces in the nucleus nuclei and QCD? Scales: 1 GeV 100 keV Summer School IV on Nuclear Collective Dynamics

18. Content Content II -------------------- Motivation Density Functional Theory The Nuclear Density Functional Covariant Density Functional Ground state properties Nuclear dynamics and excitations Outlook Summer School IV on Nuclear Collective Dynamics

19. density functional theory: theorem of Hohenberg und Kohn: Hohenberg Kohn Summer School IV on Nuclear Collective Dynamics

20. Many-body system in an external field U(r): We consider now a realistic manybody system in an external fieldU(r) and a two-body interactionV(ri,rk).The total energy Etot of the system depends onU(r). It is afunctional of U(r): in the same way we obtain the density: Inverting this relation we can introduce aLegendre transformation replacing the independent functionU(r)by the densityρ(r): Summer School IV on Nuclear Collective Dynamics

21. Decomposition of HK-functional In practical applications the functional EHK[ρ] is decomposed into three parts: The Hartree term is simple: The non interacting part: The exchange-correlation part is the rest: Excis less important and often approximated, but for modern calculations it plays a essential rule. Summer School IV on Nuclear Collective Dynamics

22. Thomas Fermi approximation: Thomas Fermi Thomas and Fermi used the local density approximation (LDA) in order to get an analytical expression for the non-interacting term. They calculated the kinetic energy density of a homogeneous system with constant density ρ where γ is the spin/isospin degeneracy. Using this expression at the local density they find: This is not very good (molecules are never bound) and therefore one added later on gradient terms containing ∇ρ and Δρ. This method is called Extended Thomas Fermi (ETF) theory. However, these are all asymptotic expansions and one always ends up with semi-classical approximations. Shell effects are never included. Summer School IV on Nuclear Collective Dynamics

23. Example for Thomas-Fermi approximation: exact Thomas-Fermi appr. Summer School IV on Nuclear Collective Dynamics

24. Kohn-Sham theory: Kohn-Sham theory In order to reproduce shell structureKohn and Sham introduced a single particle potential Veff(r), which is defined by the condition, that after the solution of the single particle eigenvalue problem the density obtained as is the exact density Obviously to each density ρ(r) there exist such a potential Veff(r). The non interacting part of the energy functional is given by: and obviously we have: Summer School IV on Nuclear Collective Dynamics

25. limitations of exact density functionals: in practice formally exact Hohenberg-Kohn:Kohn-Sham:Skyrme:Gogny: no shell effectsno l•s,no pairingno config.mixing generalized mean field: no configuration mixing, no two-body correlations local density: kinetic energy density: pairing density: twobody density: Summer School IV on Nuclear Collective Dynamics

26. Content Content II -------------------- Motivation Density Functional Theory The Nuclear Density Functional Covariant Density Functional Ground state properties Nuclear dynamics and excitations Outlook Summer School IV on Nuclear Collective Dynamics

27. Density functional theory in nuclei: Density functional theory 1) The interaction is not well known and very strong 2) More degrees of freedom: spin, isospin, relativistic, pairing 3) Nuclei are selfbound systems. The exact density is a constant. ρ(r) = const Hohenberg-Kohn theorem is true, but useless 4) ρ(r) has to be replaced by the intrinsic density: 5) Density functional theory in nuclei is probably not exact, but a very good approximation. Summer School IV on Nuclear Collective Dynamics

28. Mean field: Eigenfunctions: Interaction: Density functional theory in nuclei D.Brink D.Vauterin Skyrme Slater determinant density matrix Extensions: Pairing correlations, Covariance Relativistic Hartree Bogoliubov (RHB) Summer School IV on Nuclear Collective Dynamics

29. the nuclear energy functional is so far phenomenological and not connected to any NN-interaction. it is expressed in terms of powers and gradients of the nuclear ground state density using the principles of symmetry and simplicity The remaining parameters are adjusted to characteristic properties of nuclear matter and finite nuclei General properties of self-consistent mean field theories: Virtues: (i) the intuitive interpretation of mean fields results in terms of intrinsic shapes and of shells with single particle states (ii) the full model space is used: no distinction between core and valence nucleons, no need for effective charges (iii) the functional is universal: it can be applied to all nuclei throughout the periodic chart, light and heavy, spherical and deformed Summer School IV on Nuclear Collective Dynamics

30. Content Content II -------------------- Motivation Density Functional Theory The Nuclear Density Functional Covariant Density Functional Ground state properties Nuclear dynamics and excitations Outlook Summer School IV on Nuclear Collective Dynamics

31. Dirac equation in atoms: Dirac equation Coulomb potential:(r) with magnetic field: magnetic potential: (r) Summer School IV on Nuclear Collective Dynamics

32. Dirac equation in nuclei: Dirac equation scalar potential vector potential (time-like) vector potential (space-like) vector space-like corresponds to magnetic potential (nuclear magnetism) is time-odd and vanishes in the ground state of even-even systems Summer School IV on Nuclear Collective Dynamics

33. Relativistic potentials continuum V-S ≈ 50 MeV Fermi sea 2m* ≈ 1200 MeV 2m ≈ 1800 MeV Dirac sea V+S ≈ 700 MeV Summer School IV on Nuclear Collective Dynamics

34. for Elimination of small components: (ε→ m+ε) Summer School IV on Nuclear Collective Dynamics

35. Why covariant ? • no relativistic kinematic necessary: • non-relativistic DFT works well • 3) technical problems: • no harmonic oscillator • no exact soluble models • double dimension • huge cancellations V-S • no variational method • conceptual problems: • treatment of Dirac sea • no well defined many-body theory Summer School IV on Nuclear Collective Dynamics

36. Why covariant Why covariant? • Large spin-orbit splitting in nuclei • Large fields V≈350 MeV , S≈-400 MeV • Success of Relativistic Brueckner • Success of intermediate energy proton scatt. • relativistic saturation mechanism • consistenttreatment of time-odd fields • Pseudo-spin Symmetry • Connection to underlying theories ? • As many symmetries as possible Coester-line Summer School IV on Nuclear Collective Dynamics

37. Walecka model Walecka model Nucleons are coupled by exchange of mesons through an effective Lagrangian (EFT) (J,T)=(1-,0) (J,T)=(0+,0) (J,T)=(1-,1) Rho-meson: isovector field Omega-meson: short-range repulsive Sigma-meson: attractive scalar field Summer School IV on Nuclear Collective Dynamics

38. Lagrangian Lagrangian density free Dirac particle free meson fields free photon field interaction terms Parameter: meson masses: mσ, mω, mρ meson couplings: gσ, gω, gρ interaction terms Summer School IV on Nuclear Collective Dynamics

39. Equations of motion equations of motion for the nucleons we find theDirac equation No-sea approxim. ! for the mesons we find theKlein-Gordon equation Summer School IV on Nuclear Collective Dynamics

40. No-sea approxim. ! Static limit (with time reversal invariance) static limit for the nucleons we find thestaticDirac equation for the mesons we find theHelmholtz equations Summer School IV on Nuclear Collective Dynamics

41. Relativistic saturation mechanism: We consider only the σ-field, the origin of attraction its source is the scalar density for high densities, when the collapse is close, the Dirac gap ≈2m* decreases, the small components fi of the wave functions increase and reduce the scalar density, i.e. the source of the σ-field, and therefore also scalar attraction. In the non-relativistic case, Hartree with Yukawa forces would lead to collapse Summer School IV on Nuclear Collective Dynamics

42. Equation of state (EOS): EOS-Walecka σω-model J.D. Walecka, Ann.Phys. (NY) 83, (1974) 491 Summer School IV on Nuclear Collective Dynamics

43. Effective density dependence: Density dependence non-linear potential: NL1,NL3.. Boguta and Bodmer, NPA 431, 3408 (1977) density dependent coupling constants: R.Brockmann and H.Toki, PRL68, 3408 (1992) S.Typel and H.H.Wolter, NPA656, 331 (1999) T. Niksic, D. Vretenar, P. Finelli, and P. Ring, PRC 56 (2002) 024306 g  g(r(r)) DD-ME1,DD-ME2 Summer School IV on Nuclear Collective Dynamics

44. Point-Coupling Models Point-coupling model σ ω δ ρ J=1, T=1 J=0, T=1 J=1, T=0 J=0, T=0 Manakos and Mannel, Z.Phys.330, 223 (1988) Bürvenich, Madland, Maruhn, Reinhard, PRC 65, 044308 (2002) Summer School IV on Nuclear Collective Dynamics

45. Lagrangian Lagrangian density for point coupling free Dirac particle interaction terms interaction terms Parameter: point couplings: Gσ, Gω,Gδ , Gρ, derivative terms: Dσ photon field Summer School IV on Nuclear Collective Dynamics

46. Content Content II -------------------- Motivation Density Functional Theory The Nuclear Density Functional Covariant Density Functional Ground state properties Nuclear dynamics and excitations Outlook Summer School IV on Nuclear Collective Dynamics

47. Nuclear matter equation of state EOS for DD-ME2 Neutron Matter Summer School IV on Nuclear Collective Dynamics

48. Symmetry energy Symmetry energy saturation density empirical values: 30 MeV£a4£34 MeV 2 MeV/fm3< p0 < 4 MeV/fm3 -200 MeV< DK0 < -50 MeV Lombardo Summer School IV on Nuclear Collective Dynamics

49. Summer School IV on Nuclear Collective Dynamics

50. Conclusions part I: Conclusions I • density functional theory is in principle exact • microscopic derivation of E(ρ) very difficult • Lorentz symmetry gives essential constraints • - large spin orbit splitting • - relativistic saturation • - unified theory of time-odd fields • 4) in realistic nuclei one needs a density dependence • - non-linear coupling of mesons • - density dependent coupling-parameters • 5) modern parameter sets (7 parameter) provide • excellent description of ground state properties • - binding energies (1 ‰) • - radii (1 %) • - deformation parameters • 6) pairing effects are non-relativisitic Summer School IV on Nuclear Collective Dynamics