Status of the Density Functional Theory in nuclei 2 issues: Many body problem What is the nucleon-nucleon interaction? Specific Features: Fermionic system with 2 types (n and p): alpha Non elementary particles: implies 3 body (polarisation) E. Khan
DFT / not DFT Advantages compared to other models (such as the shell model) Simple, including numerically : relevant interpretation ( Mean field with tensor force and shell structure of exotic nuclei, T. Otsuka et al., PRL97(2006)162501) No core : study of the whole nuclear chart Description of high energy excited states (giant resonances) Matter and transiton densities: spatial insight of the nucleus Drawbacks Treatment of correlations beyond the mean field is heavy More important discrepancy with the data (especially for excitation spectrum)
Outline 1. Method : DFT, independent particles 2. State of the art : pairing, deformation, excitations 3. Outlooks NB : show important points of demo, not technical details
1) Method
The least action principle (I) A physical state of a system is characterised by an action (J.s) which is minimal Variationnal principle : variation of the action S around its minimum is zero Numerous applications : mathematics, mecanics, optics, quantum physics, Fermat (XVII eme ) Maupertuis (XVIII eme ) Lagrange (XVIII eme XIX eme ) Feynman (XX eme )
The least action principle (II) Nuclear Hamiltonian: Action: Stationnarity of S ( S=0) for any variation of < (t) Schrödinger equation Reformulation of the starting point: Energy density functionnal Hohenberg-Kohn theorem: existence
The Hohenberg-Kohn (HK) theorem (Chemistry Nobel 98) There exists an energy functionnal E[ ] which depends on the (local) density. It allows to exactly predict ground state observables (solves the many body problem) Knowledge of this functional in nuclei? HK states the existence of a functional for a given state, not an universal functional for the nuclear chart In nuclear physics coefficients in E[ ] are adjusted on radii, masses, : takes into account correlations beyond mean field. Nuclei: symmetry restoration (broken in self-bound systems) Kohn-Sham = method to calculate knowing E[ ]
Independent particles Application of the least action principle to the many body problem: nuclear physics (~1970) Slater determinant Justification: nucleus is a quantum liquid (range and intensity of strong interaction) nucleons are good independent particles (B. Mottelson ~ 2000)
Time Dependent Hartree-Fock (TDHF) Variation: i* (t) i* (t) + i* (t) A coupled equations : (self-consistent) mean field: In practice : - treat V NL quasi-locally : Skyrme, Gogny - interactions fitted on nuclei properties : radii, energies, etc. correlations beyond HF - LDA from infinite nuclear matter? L
TDHF properties Self consistent Minimum of the functionnal : static HF (stationnarity) HF Fusion, fission, compound nucleus, damping, Numerically heavy, tunnel effect, interpretation of
Milestones Brueckner-HF : HF calculation with the bare nucleon-nucleon interaction renormalised by the nuclear medium (G matrix). Poor description of exp masses (B/A ~ 5 MeV, Coester line) HF: no suitable phenomenologic interaction able to describe masses and radii (1960) Breakthrough: Skyrme HF (Brink,Vautherin (1972)) Gogny HF (1975) Relativistic DFT RMF (1990,V L ) and RHF (2006) N.B : the 3 above has the best agreement with the data Now, in progress: V lowk = renormalised bare interaction to be used in HF, Bare N n LO potentials : Effective Field Theories (Weinberg, 1990)
Skyrme: a genuine effective approach What happens if you Fourier transform Taylor series? o) The mean field is similar to the density: WS ~ (r) i) The nucleon-nucleon interaction is of (very) short range ii) Momentum expansion of the interaction iii) Fourier transform iv) Add a LS term, a tensor term, and a density dependent term (EDF)
W.A. Richter, B.A. Brown, Phys. Rev. C 67, 034317 (2003)
Gogny: a stable parameterisation The momentum expansion is replaced by two Gaussians, to describe the short range (~1fm). Website HFB-Gogny: http://www-phynu.cea.fr/science_en_ligne/carte_potentiels_microscopiques/carte_potentiel_nucleaire.htm
Why a relativistic DFT? Interaction mediated by (4) mesons ; link with effective QCD? Nucleon is non-relativistic : E c ~ 30 MeV But E pot = balance between répulsion (150 MeV) and attraction (-200 MeV) of the nucleon-nucleon interaction correction of few % at these energies have an impact on B/A Importance of the LS term in nuclear physics: relativistic origin Complementary from non relativistic EDF (Ex: Kinf) V (MeV) r (fm)
Relativistic DFT Interaction described by meson exchange : (tensor) Lagrangian imposed by Lorentz invariance nucleon : meson : with : interaction :
Relativistic DFT Least action principle Dirac Eq. Klein-Gordon Eq. Fermi states Dirac states
What are the important terms of the EDF? In the EDF several terms (coming or not from the nucleon-nucleon interaction): V=kin + central + LS + tensor (interaction between 2 magnetic dipoles) At first glance all 4 have to be considered, not only tensor in shell-structure evolution Magic Numbers in Exotic Nuclei and Spin-Isospin Properties of the NN Interaction T. Otsuka et al Phys. Rev. Lett. 87, 082502 (2001) Evolution of Nuclear Shells due to the Tensor Force T. Otsuka et al Phys. Rev. Lett. 95, 232502 (2005) Three-body forces and the limit of oxygen isotopes T. Otsuka et al. arxiv:0908.2607
2) State of the art
Nuclear superfluidity There are only 5 stable odd-odd nuclei ( 1 H, 6 Li, 10 B, 14 N, 50 V) DFT: keep the independent particle approach : independent quasiparticle BCS < HFB Vortices expected in rotating nuclear system: neutrons stars (E*~ 100 MeV in 150 Sm) Time reversal invariance preserved: not accurate to predict odd masses Pairing phase transition : Finite temperature HFB n (MeV) E. Khan, Nguyen Van Giai, N. Sandulescu Nuclear Physics A, 789 (2007) 94 T (MeV)
The pairing density and the EDF Mean values depend on ρ=<ψ + (r)ψ(r)> particle density κ=<ψ(r)ψ(r)> (κ=<ψ + (r)ψ + (r)>) pairing densities EDF 86 Ni Pairing is not only in the surface of the nucleus EDF: terms dependent on and terms dependent on can be generated by different interactions: ph channel and pp channel Ex : Skyrme for ph and surface delta interaction for pp
Superfluid/normal phase transition DFT models convenient to investigate temperature effects: Fermi-Dirac factors in HFB eq. Experimentaly: heat capacity determined by level density measurements (inelastic scattering of 3 He at 15 MeV/A (Oslo group) FT-HFB Exp. A. Schiller at al, Phys Rev. C 63, 021306R (2001) E. Khan, Nguyen Van Giai, N. Sandulescu Nuclear Physics A, 789 (2007) 94
Deformed HFB with Skyrme interaction 2n<->2p interaction in even-even nuclei : Status on Skyrme DFT N N Good description on the nuclear chart (~100 kev discrepancy, but Pb) M. Stoitsov, R. B. Cakirli, R. F. Casten, W. Nazarewicz, and W. Satuła Phys. Rev. Lett. 98, 132502 (2007)
Towards the drip line: continuum effects? Single particle states Single quasiparticle states 0-50 50 2 0 E ( ) 2 2 HF HFB
Continuum HFB The continuum treatment has a small effect on masses and radii close to the drip line M. Grasso, N. Sandulescu, Nguyen Van Giai, and R. J. Liotta, Phys. Rev. C 64, 064321 (2001)
Status on Gogny DFT Vanishing of Z=50,82 for n rich nuclei MEM at Z=82, N=126 J. -P. Delaroche, M. Girod, J. Libert, H. Goutte, S. Hilaire, S. Péru, N. Pillet, and G. F. Bertsch Phys. Rev. C 81, 014303 (2010)
Relativistic DFT RMF: exchange term neglected: ~ simulated by the fit of the EDF Dirac sea neglected Deformed RHB (NL3+D1S): good description of the proton dripline D. Vretenar, A. V. Afanasjev, G. A. Lalazissis, and P. Ring, Physics Reports 409,101 (2005).
RMF with density dependence EDF: allow for density dependence in the meson-nucleon coupling constant Improves the description of masses : rms <900 kev on 200 nuclei G. A. Lalazissis, T. Nikšić, D. Vretenar, and P. Ring, Phys. Rev. C 71, 024312 (2005)
Deformed Relativistic HFB The exchange term can be successfully included with density dependent Lagrangians: RHF J.P Ebran, PhD thesis
Deformed Relativistic HFB Proton density Neutron density 22 Ne J.P Ebran, PhD thesis
Excited states in the DFT: GCM or RPA? GCM (~5DCH): mixes the HF solutions with various deformation to obtain the lowest energy states. Adapted for low E and low J states (does not take into account 1p-1h configurations) and for quadrupolar correlations J. -P. Delaroche, M. Girod, J. Libert, H. Goutte, S. Hilaire, S. Péru, N. Pillet, and G. F. Bertsch Phys. Rev. C 81, 014303 (2010) RPA: Mixes the 1p-1h configuration on a single HF solution. Adapted for collective states, at low or high E (giant resonances)
RPA: the linear response theory External oscillating field: TDHF: ext ext First order: ext N.B. 1) 2) Excited states are a superposition of particle-hole excitations.
Small amplitude perturbations RPA HF
Consistent RPA Small amplitudes perturbation (RPA) in the DFT framework : residual interaction (beyond mean field) V Res 1975 : first calculation with the same EDF for HF and V res Advantage - EDF is the only parameter constrain it with excited states - symmetry restoration G.F. Bertsch and S.F. Tsai, Phys. Rept. C18 (1975) 125
The Quasiparticle-RPA (QRPA) Excitation and pairing Method known since ~40 years in nuclear physics Strong peak of activity since year 2000. Why? Study of nuclear transition of the whole nuclear chart (isotopic chain, open shell, drip-line, ) N,Z N+2,Z N+1,Z-1 E*, S(E*) inelastic cross section Pairing vibrations, 2n transfer cross sections half-life, GT strength, charge exchange cross section
QRPA residual interaction EDF: res V V V res V ph,ph pp,ph hh,ph V V V ph,pp pp,pp hh,pp V V V ph,hh pp,hh hh,hh
RPA equation Perturbation of the density : ext Response function TDHF ext ext Bethe-Salpeter equation
Skyrme QRPA Low energy states 32 Mg 30 Ne 36 S 34 Si 38 Ar M. Yamagami and Nguyen Van Giai, Phys. Rev. C 69, 034301 (2004)
Spatial insight Transition densities N=14 shell closure E. Becheva et al, Phys. Rev. Lett. 96, 012501, (2006)
Giant resonances in exotic nuclei Lorentzian (Hybrid) Microscopic Pygmy resonance S. Goriely, E. Khan, M. Samyn, Nuclear Physics A739, (2004) 331
DFT & astrophysics QRPA/Hybrid micro/pheno comparison 10 Sn(MeV) 0 L=2 QRPA HFB Supergiant resonances E. Khan et al.
Continuum effects on excited states Continuum has an effect on giant resonances, not on low lying states, even close to the drip line E. Khan, N. Sandulescu, M. Grasso, and Nguyen Van Giai, Phys. Rev. C 66, 024309 (2002)
Pairing effects on excited states Low lying states are senstive to the pairing interaction which is not well characterised: surface, volume, D1S wo density dep? 22 O excitation energy spectrum Pairing vibrations: 124 Sn+2n Surface pairing Mixed pairing E. Khan, M. Grasso, and J. Margueron, Phys. Rev. C 80, 044328 (2009)
Gogny Deformed-QRPA S. Péru and H. Goutte, Phys. Rev. C 77, 044313 (2008)
Relativistic RPA E (MeV) Dirac sea has to be included for completeness
Relativistic Deformed-QRPA The pygmy mode is quenched by the deformation because of the reduction of the n skin D. Peña Arteaga, E. Khan, and P. Ring, Phys. Rev. C 79, 034311 (2009)
3) Outlooks
Interdisciplinary Metallic clusters : Several hubdreds of atoms in a cluster : N delocalised valence electrons In an N ion potential Jellium model Fermionic atoms trap : 6 Li QRPA M. Grasso, E. Khan, and M. Urban Phys. Rev. A 72, 043617 (2005)
Summary DFT is practiced since ~40 years in nuclear physics: Skyrme- HF(1972), RPA (1975), QRPA (1995) The explicit link with an (universal) EDF is looked for since few years Recent developpements (RHFB, QRPA, GCM) The tensor term is not the only relevant one for shell structure evolution. DFT allow this investigation Continuum treatment affects giant resonances close to the drip line, but not the other observables GCM low E, J RPA: collective modes, high E possible
Questions Is there a universal EDF on the nuclear chart? What the the pairing EDF? How to relate with bare (EFT) interactions, V lowk? How to establish the DFT framework for nuclei? Rms limit to microscopic DFT mass formula? 600-800 kev (Skyrme, Gogny). Relativistic in progress