4 November Master 2 APIM. Le problème à N corps nucléaire: structure nucléaire

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1 4 November Master 2 APIM Le problème à N corps nucléaire: structure nucléaire

2 The atomic nucleus is a self-bound quantum many-body (manynucleon) system

3 Rich phenomenology for nuclei Mean field

4 Which particles? Which are the degrees of freedom taken into account in the nuclear manybody problem? Which is the interaction? STRONG INTERACTION

5 D. Lacroix Lecture. École Joliot-Curie 2009, Lacanau, France

6 Deriving the nucleon-nucleon interaction from the underlying theory of QCD is extremely difficult (so far almost no quantitative results). Three main possible directions to deal with the nuclear interaction in low-energy nuclear physics are: Realistic interactions Effective field theories (deep links with QCD) Phenomenological interactions

7 Nuclear physics and the strong interaction Chadwick discovers the neutron (he measures its mass) in 1932 (Nobel Prize in 1935): the two types of nucleons are thus known How do the nucleons interact among themselves? The idea at the basis of Yukawa hypothesis: In the case of the electromagnetic interaction the mediator of the force is the photon (zero mass). Quantum electrodynamics: at the beginning of the 30s, it is already formulated.

8 Yukawa hypothesis Yukawa introduces a particle that he calls U. He suggests that this particle is the mediator of the nuclear strong interaction m U 200 m e ; m p 1800 m e

9 15 years later the formulation of the hypothesis In 1947 and 1948 the Yukawa particle is identified with the meson π: pseudoscalar (negative parity), spin 0 (the two quarks have opposite spin). The mass is ~ 140 MeV/c 2 - Observed in cosmic rays - Produced at Berkeley π mesons interact strongly with nucleons and have the correct mass that leads to the correct range of the nuclear force 1949: NOBEL PRIZE FOR YUKAWA!!

10 Yukawa potential g -> coupling constant m π -> mass of pion Range Compton wave length of pion:

11 60s: heavier mesons are discovered ω and ρ mesons: mass 800 MeV

12 Realistic interactions The earliest realistic potentials in the 60s: one-pion exchange potentials with adjustable parameters + a repulsive hard core (to describe high-energy scattering data) 70s - 80s: Paris and Bonn potentials 90s: High-precision realistic potentials (starting from the 90s) : CD-Bonn, Argonne V18, Nijmegen I et II. High-precision adjustments (around 50 parameters) are done to reproduce phase shifts (free nucleonnucleon scattering) and spectroscopy of nuclei with a small number of nucleons

13 The adjustment of the parameters Scattering amplitude Differential cross section Total cross section

14 Long range (pion) Intermediate range (more or less phenomenological +, eventually, exchange of mesons) Short range: hard core (phenomenological +, eventually, the exchange of mesons)

15 Problem of the hard core: motivation to use phenomenological interactions (calculations with more sophisticated models become easier) Two types: 1) zero range (contact interaction) (Skyrme) 2) finite range (Gogny)

16 The simple idea Construction of an effective phenomenological force that is chosen to reproduce (at the mean-field level) global properties of some selected nuclei (binding energies and radii) and properties related to the EoS of nuclear matter (first of all saturation point) Form and parameters to choose For instance, to have in-medium effects (Pauli principle), we need the density-dependent term Correlations are contained in the interaction in an effective way.

17 Finite-range: Gogny interaction It has been introduced from a realistic G matrix Standard form: Finite-range (gaussian form) Density dependent (zero-range) First parametrization: D1 Déchargé, Gogny, PRC 21, 1568 (1980) Spin-orbit (zero-range) Most used: D1S (fit also on fission barriers) Berger, Girod, Gogny, NPA 502, 85c (1989) A recent modification (towards finite-range in all terms): D1N Chappert, Girod, Hilaire, Phis. Lett. B 668 (2008), 420 CEA Bruyères-lechâtel

18 and -> 14 parameters acting on the right acting on the left Spin and isospin exchange operators Isospin matrices Spin matrices

19 Zero-range: Skyrme interaction Standard form: 10 parameters central non local density dependent spin-orbit T.H.R. Skyrme, Phil. Mag. 1, 1043 (1956), Nucl. Phys. 9, 615 (1959) First applications: Vautherin, Brink, PRC 5, 626 (1972) IPN Orsay

20 Advantage of using phenomenological interactions. Which nuclei can be treated?

21 Mean field for ground state nuclear structure (HF, HFB,..) RPA and QRPA for small amplitude oscillations Beyond small amplitude oscillations: timedependent mean field for dynamics (TDHF, TDHFB, ) Beyond-mean - field models (correlations): GCM, particle-vibration coupling, variational multiparticlemultihole configuration mixing, extensions of RPA, Second RPA,

22 Links with Quantum Chromodynamics (70s, Standard Model) Internal degrees of freedom of the nucleon

23 Links with QCD Lagrangian (chiral symmetry)

24 Djalali Lecture. Ecole Joliot- Curie 2009

25 The symmetry is spontaneously broken Djalali Lecture. Ecole Joliot- Curie 2009 All nucleons have positive parity Goldstone boson (pion!!! Links with realistic interactions) but the pion has a non-zero mass

26 and the pion has non-zero mass because the symmetry is explicitly broken non-zero mass of quarks!! Broken symmetry scale: 1 GeV (see quark masses)

27 Effective field theories: choice of the relevant degrees of freedom at the scales we are interested in Leading order (LO) Next-to-leading order (NLO) q ~ M l << M h q -> soft scale It diverges at large q. Introduce a cut-off Λ Hard scale => 1 GeV, chiral symmetry breaking scale

28 A natural hierarchy (that is intuitive) can be formally established between 2-body, 3-body,,N-body forces Q=q/Λ Leading order Next-to-leading order Next-to-next-to-leading order

29 Nuclear matter Ideal and infinite system composed by nucleons: Symmetric matter (protons and neutrons with equal densities) Asymmetric matter Neutron matter Nuclei Neutron stars

30 After evolution (thermonuclear reactions) of massive stars (M 8 M ) ( 10 7 years) -> supernova explosion. Slow cooling for millions of years. Typical values: R 10 km M 1.4 M Saturation density ρ fm -3 -> g/cm 3 EXOTIC NUCLEI

31 Mass measurements of neutron stars in binary systems Mass (M )

32 Equation of state (EOS): E/A as a function of the density Equilibrium point: saturation density Symmetric matter From symmetric to neutron matter. Y p =Z/A Isospin effects and density dependence No equilibrium point in pure neutron matter

33 Equation of State with different models Symmetric nuclear matter Pure neutron matter stiff soft

34 48 Ca density profiles. Spherical nucleus. Hartree-Fock calculations with a Skyrme interaction Neutron density Proton density Density R (fm) R (fm) N = 28 Z = 20

35 Rich phenomenology for nuclei Mean field

36 Light nuclei

37 Some properties of light nuclei

38 Borromean nuclei Borromean Nuclei: 3-body systems. Example: 11 Li = 9 Li+n+n Generalization to n-body systems. Example : 10 C = 4 He+ 4 He+p+p Mixing of complex correlations

39 Halo densities

40 Some applications of meanfield-based models. The basic method for many-body systems

41 Ground state of nuclei. Mean field: individual degrees of freedom Masses Separation energies Drip lines? Densities Single particle spectra Energies and occupation probabilities?

42 Binding energy (top) and twoneutron separation energy (bottom) Mean field calculations (HFB) with a Skyrme force DRIP LINE? Strongly model dependent!!!

43 48 Ca density profiles. Spherical nucleus. Hartree-Fock with Skyrme Neutron density Proton density Density R (fm) R (fm)

44 Last neutron single-particle Hartree-Fock Ca isotopes states Can we predict singleparticle 1f5/2 2p1/2 energies and spectroscopic factors? We need to go beyond mean field (see next Lecture). Grasso, Yoshida, Sandulescu, Van Giai, PRC 74, (2006) Why?

45 Single-particle and collective degrees of freedom couple: beyond mean field

46 Effects of particlevibration coupling on the single-particle spectrum Neutron states in 208 Pb Bernard, Van Giai, Nucl. Phys. A 348 (1980), 75 These correlations also affect the excited states

47 Particle-vibration coupling E. Litvinova, et al. 208 Pb 132 Sn

48 At mean field level we can include pairing correlations to treat superfluid nuclei (BCS superfluidity -> Cooper pairs) Energy gap in excitation spectra Odd-even effect in binding energy Moments of inertia

49 Odd-even effect No isotopes (Z=102) Duguet et al. arxiv:nucl-th/ v1 Sn = E (Z,N)-E(Z,N-1)

50 Excitations. Collective degrees of freedom

51 Excitation modes (small amplitudes) Quadrupole mode QRPA in particle-hole channel Response function for 22 O Two-neutron 0 + addition mode QRPA in particle-particle channel Response function for 124 Sn Pair transfer mode Khan, Grasso, and Margueron Khan, Grasso, Margueron, submitted PRC Khan, Sandulescu, Grasso, and Van Giai, PRC 66, (2002)

52 Dynamics. Beyond small amplitude oscillations. Time dependent approaches for the dynamics Collision Dynamics of Two 238 U Atomic Nuclei Golabek and Simenel, PRL 103, (2009) Time-dependent HF

53 Nuclear structure / Astrophysics Neutron stars. Very exotic nuclear systems

54 Picture of the crust of a neutron star Baym, Bethe, Pethick, NPA 175 (1971), 225 Inner crust: ρ 0 ρ 0.5 ρ 0 Saturation density ρ 0 = 0.16 fm g/cm 3 Inner crust: crystal of nuclear clusters in an electron sea and in a gas of superfluid neutrons Drip point for neutrons g/cm 3 Outer crust: crystal of nuclei in an electron sea ρ 0.5 ρ 0 ρ β-stability condition -> µ e = µ n - µ p

55 Wigner-Seitz cell model

56 J.W. Negele and D. Vautherin, Nucl. Phys. A 207, 298 (1973)