UWC Beyond mean-field: present and future Mean-field and Beyond for low-energy nuclear spectroscopy Present and Future The long term goal Possible alternative The ingredients Applications to even-even nuclei Very recent extension to odd nuclei Perspectives 17/01/2019 UWC Beyond mean-field: present and future York

A few points on which I will stop to give more details The Hartree Fock method: principle and realisations; Pairing correlations Breaking of symmetries, why, how and which? Beyond mean-field: the different meanings Correlations, what does it mean, can one measure experimentally correlations? Nucleon-nucleon interaction: realistic, effective, energy density functional…. The code Ev8, solution of mean-field equations on a mesh 17/01/2019 UWC Beyond mean-field: present and future

The ultimate (unreachable?) goal To set up a method for nuclear spectroscopy To describe spectra and transition probabilities Applicable to all nuclei, from the lightest to the heaviest With a single phenomenological input, the modelling of the strong force 17/01/2019 UWC Beyond mean-field: present and future

The main alternative: the shell model Successful in the description of many nuclei up to mass 100 Its range of applicability is extending What we want to avoid: Only valence particles (inert core, a few active shells) Finely tuned interactions Effective charges Applicable up to the heaviest nuclei 17/01/2019 UWC Beyond mean-field: present and future

UWC Beyond mean-field: present and future 25Mg 3 lowest states are 0+ 17/01/2019 UWC Beyond mean-field: present and future

Beyond mean-field methods Starting point: mean-field wave functions: Hartree-Fock + pairing correlations Effective nucleon-nucleon interaction (EDF or an interaction or relativistic) Pairing correlations treated at the BCS or Bogoliubov level with the same or a different interaction Constraints on the shape of the nucleus introduced in the mean-field equations. One determines the best HFB wave function for each shape H -> H – lN N – lZ Z – l1q1 – l2q2 The wave function has the form: Variational parameters: those of the basis describing the individual wave functions and the v’s 17/01/2019 UWC Beyond mean-field: present and future

The Shrödinger equation is equivalent to a wave function variational principle: minimize < | H | Y> under the constraint <  | Y> =1: One-body Two-body Hartree-Fock method: the ground state wave function is a Slater determinant.

The Hartree-Fock method Wave-function of a many particle system= Slater determinant The particles interact trough a 2-body interaction v(r1-r2) They are also confined by a central potential vext(r). The total energy is: direct exchange

Minimize the energy with respect to fi Minimize the energy with respect to fi * with a constraint on norm conservation: One defines the one-body diagonal and non diagonal densities:

One rewrites the HF equations as a function of these densities: where The first line is easy: problem in a potential. The second line is complicate: non local exchange term. HF single particle energy: 2-body matrix element between i and all other j Total energy: no double counting!

Mean-field Methods Based on an “effective interaction” or a “density functional” The (small number of) parameters of the effective interaction are fixed by general considerations (no local adjustments) Pairing correlations are included at the BCS or better HFB level Full self-consistency No restrictions to a few shells, mean-field equations are solved as precisely as one wishes. Spherical and deformed nuclei are treated on the same footing, no “parametric deformation”

Deformation of the nucleus introduced by a Lagrange multiplier: by varying l, one obtains solutions for different deformations Missing ingredient: pairing correlations (superconductivity) They can be introduced using the BCS theory: single particle states are occupied with a probability v2 between 1 and 0 nucleons are grouped in pairs of opposite spin projections The nuclear density becomes: HF equations with modified densities + BCS equations to determine the occupations Total wave function with only the right mean particle number!

Argonne Beyond mean-field: present and future 208Pb 180Hg 202Rn 170Hf Mean-field energy curves (b2 proportional to Q) Argonne Beyond mean-field: present and future 17/01/2019 York

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Skyrme HFB Deformation properties of super-heavies

The shell model: a schematic view All upper shells neglected Active space: Inactive core 40Ca

Active space 2-body matrix elements between all pairs of states Diagonalization of a big matrix in the active space Starting point of the model: nucleons in a potential well. Problem of the model: active space grows very quickly. Respect always symmetries of the 2-body hamiltonian: work in the laboratory frame of reference

Mean-field methods Construct all the orbitals from the mean 2-body interaction of a particle with the others All orbitals are active! Caution: not true single particle levels Break symmetries and work in a frame of reference intrinsic to the nucleus!

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for super heavy elements (only even-even) Skyrme HFB Qa for isotopic chains for super heavy elements (only even-even) Cwiok, Heenen, Nazarewicz Nature 2005

Argonne Beyond mean-field: present and future Plus and minus of the mean-field approach: Plus: Starts from an effective interaction: generality Can describe any kind of shapes (from ground state to fission) Cranking (or qp excitations) well justified for deformed nuclei Minus: Valid only for energy (variational) and one-body operators Breaking of symmetries (no direct determination of transitions) Soft nuclei? Shape coexistence? Effects of correlations beyond mean-field on masses? 17/01/2019 Argonne Beyond mean-field: present and future York

UWC Beyond mean-field: present and future Restoration of symmetry: A deformed wave function breaks the rotational invariance: Not an eigenstate of J2 and Jz. Nuclear density Any rotated density would also be a solution of the HFB equations, with the same energy. Symmetry restoration: mixing of all these degenerate wave functions with appropriate weights to have good J2 and Jz 17/01/2019 UWC Beyond mean-field: present and future

Projection on particle number Projection operator: To check that it extracts an eigenstate of particle number, one applies it to |BCS> = Sn cn |n> Effect on a BCS wave function: vk replaced by eif vk 17/01/2019 UWC Beyond mean-field: present and future

UWC Beyond mean-field: present and future Configuration mixing Mixing of projected mean-field wave functions as a function of shape variable(s) The coefficients f are determined by varying the energy: They are solution of the Hill Wheeler and Griffin equation: 17/01/2019 UWC Beyond mean-field: present and future

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16O: mean-field and projection

Single particle levels

Configuration mixing: Energies Wave functions

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Shape coexistence York April 2013 Mean field Mean field projected on J=0 Bars in red: 0+ states obtained after configuration mixing 17/01/2019 Shape coexistence York April 2013 York

UWC Beyond mean-field: present and future 186Pb exp. cal. conf. mix of mean field states (Skyrme intera.SLy6) M. Bender et al. PRC 69 (2004), 064303 & privat com. 16+ E/MeV E/MeV (14+) 8+ pro 14+ pro 242 WU (12+) 3 3 12+ (10+) 2+ 6+ β = 0.30(2) β = 0.32 10+ (8+) 205 WU 2 8+ 2 4+ (6+) 8+ 543 WU 6+ 152 WU 246(42) WU (4+) 6+ 2+ 480 WU 438(98)WU 56 WU 4+ (2+) 1 4+ 1 0+ 480(90)WU 2+ 398 WU 0+ 2+ 9.4 WU 0+ 0+ 5.5 (15)WU 1.6 WU obl obl 0+ 0+ sph sph 17/01/2019 UWC Beyond mean-field: present and future

Application to neutron deficient nuclei around Pb 17/01/2019 UWC Beyond mean-field: present and future J. Yao, M. Bender and PHH, PRC 87 034322 (2013)

Systematic calculation of ground state properties “shell quenching” M. Bender, G. Bertsch, PHH PRC78 054312 (2008) 17/01/2019 UWC Beyond mean-field: present and future

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UWC Beyond mean-field: present and future Quadrupole and octupole deformations 17/01/2019 UWC Beyond mean-field: present and future

UWC Beyond mean-field: present and future Size of the basis: Points in the triaxial plane every 40 fm2 (around 30 points) 604 1qp states of positive parity 222 1qp states of negative parity Selection on energy between all these states, finally 100 and 60 states selected for projection and configuration mixing. After K-mixing, each of these 1 qp states can generate several states for each J-value. Final dimension of the bases: 226 for 5/2+ 149 for 3/2+ 106 for 3/2- - Accuracy around 20 keV 17/01/2019 UWC Beyond mean-field: present and future

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UWC Beyond mean-field: present and future Very heavy calculations (but mainy gains are possible) Crucial need to improve the interaction Same on cranked states… Unique interaction for all nuclei Will be applicable to any nucleus No core, positive and negative parity states described in a single calculation No effective charges. Different “philosophy than the UNEDF project: some correlations must be introduced explicitely 17/01/2019 UWC Beyond mean-field: present and future

UWC Beyond mean-field: present and future First developments by Paul Bonche (Saclay), Hubert Flocard (Orsay) and PHH (Brussels) with several collaborations (M. Weiss, J. Dobaczeswki, G. Bertsch…) Actual developments by M. Bender (Bordeaux), T. Duguet (Saclay), K. Bennaceur (Lyon) and several Ph D students B. Bally (Bordeaux), W. Ryssens (Brussels), R. Jodon (Lyon) and post docs V. Hellemans, J. Yao, K. Washyiama, S. Baroni (Brussels) and B. Avez (Bordeaux) 17/01/2019 UWC Beyond mean-field: present and future

UWC Beyond mean-field: present and future Fission barriers: Influence of the surface tension 17/01/2019 UWC Beyond mean-field: present and future