Introduction to Nuclear Physics 2/3 S.PÉRU

The nucleus : a complex system I) Some features about the nucleus discovery radius, shape binding energy nucleon-nucleon interaction stability and life time nuclear reactions applications II) Modeling of the nucleus liquid drop shell model mean field III) Examples of recent studies figures of merit of mean field approaches exotic nuclei isomers shape coexistence IV) Toward a microscopic description of the fission process

Modeling of the nucleus nucleus = A nucleons in interaction 2 challenges Nuclear Interaction inside the nuclei (unknown) N body formalism The liquid drop model : global view of the nucleus associated to a quantum liquid. The Shell Model : each nucleon is independent in a attractive potential. « Microscopic » methods ~ Hartree-Fock , BCS ,Hartree-Fock-Bogoliubov : The nuclear structure is described within the assumption that each nucleon is interacting with an average field generated by all the other nucleons.

The nucleus and its features, radii, and binding energies have many The nucleus is a charged quantum liquid. Quantum : The wave length of the nucleons is large enough with respect to the size of the nucleons to vanish trajectory and position meaning. Liquid : Inside the nucleus nucleons are like water molecules. They roll “ones over ones” without going outside the “container”. The nucleus and its features, radii, and binding energies have many similarities with a liquid drop :  The volume of a drop is proportional to its number of molecules.  There are no long range correlations between molecules in a drop. -> Each molecule is only sensitive to the neighboring molecules. -> Description of the nucleus in term of a model of a charged liquid drop

The liquid drop Model developed by Von Weizsacker and N. Bohr (1937) It has been first developed to describe the nuclear fission. The model has been used to predict the main properties of the nuclei such as: * nuclear radii, * nuclear masses and binding energies, * decay out, * fission. Equilibrium shape spherical Density ρ0 Volume ≈ A R= r0 A 1/3 r r R R The binding energy of the nuclei is described by the Bethe-Weiszaker formula

Bethe and Weizsacker formula Paring terms + even-even - odd-odd 0 odd-even and even-odd Binding term : volume av Unbinding terms : Surface as , Coulomb ac , Asymmetry aa

Binding energy per nucleon

Problems with the liquid drop model 1) Fission fragment distributions A heavy and a light fragments = asymmetric fission Liquid drop : only symmetric fission Proton number Neutron number Experimental Results : K-H Schmidt et al., Nucl. Phys. A665 (2000) 221 Two identical fragments = symmetric fission 18

Problems with the liquid drop model 2) Nuclear radii Evolution of mean square radii with respect to 198Hg as a function of neutron number. Light isotopes are unstable nuclei produced at CERN by use of the ISOLDE apparatus. -> some nuclei away from the A2/3 law Fig. from http://ipnweb.in2p3.fr/recherche 14

Problems with the liquid drop model Halo nuclei I. Tanihata et al., PRL 55 (1985) 2676 I. Tanihata and R. Kanungo, CR Physique (2003) 437 15

Fig. from L. Valentin, Physique subatomique, Hermann 1982 3) Nuclear masses Difference in MeV between experimental masses and masses calculated with the liquid drop formula as a function of the neutron number Nuclear shell effects E (MeV) Neutron number Existence of magic numbers : 8, 20 , 28, 50, 82, 126 Fig. from L. Valentin, Physique subatomique, Hermann 1982 16

Two neutron separation energy S2n S2n : energy needed to remove 2 neutrons to a given nucleus (N,Z) S2n=B(N,Z)-B(N-2,Z) For most nuclei, the 2n separation energies are smooth functions of particle numbers apart from discontinuities for magic nuclei Magic nuclei have increased particle stability and require a larger energy to extract particles. 17

The nucleus is not a liquid drop : Shell effects There are many « structure effects » in nuclei, that can not be reproduced by macroscopic approaches like the liquid drop model. -> need for microscopic approaches, for which the fundamental ingredients are the nucleons and the interaction between them There are «magic numbers» 2, 8, 20, 28, 50, 82, 126 and so «magic» and «doubly magic» nuclei 19

Nucleus = N nucleons in strong interaction Microscopic description of the atomic nucleus Nucleus = N nucleons in strong interaction The many-body problem (the behavior of each nucleon influences the others) Can be solved exactly for N < 4 For N >> 10 : approximations Shell Model only a small number of particles are active Approaches based on the Mean Field no inert core but not all the correlations between particles are taken into account Nucleon-Nucleon force unknown Different effective forces used Depending on the method chosen to solve the many-body problem 20

Quantum mechanics Nucleons are quantum objects : Only some values of the energy are available : a discrete number of states Nucleons are fermions : Two nucleons can not occupy the same quantum state : the Pauli principle  Neutrons Protons 21

Shell Model Model developed by M. Goeppert Mayer in 1948 : The shell model of the nucleus describes the nucleons in the nucleus in the same way as electrons in the atom. “In analogy with atomic structure one may postulate that in the nucleus the nucleons move fairly independently in individual orbits in an average potential …” , M. Goeppert Mayer, Nobel Conference 1963.

Shell Model U (r ) Schrödinger equation Energy (MeV) U (r ) R r (fm) Schrödinger equation Wave function φ and energy ε for each nucleon Wave function ψ and nuclear binding energy E R Features of the nucleus in his ground state and his excited ones r

Shell Model : potential Nuclear potential deduced from exp : Wood Saxon potential or square well harmonic oscillator

Shell Model : potential spin orbit effect

Shell Model : how describe the ground state ? 126 -1i13/2 -3p1/2 -3p3/2 -2f5/2 +1s1/2 -1p3/2 -1p1/2 2 8 Ex: Z=10 +1d5/2 +2s1/2 +1d3/2 20 Ex: N=20 -2f7/2 -1h9/2 82 -1h11/2 +3s1/2 +1d3/2 +1g7/2 +2d5/2 50 +1g9/2 -1p1/2 -1f5/2 -2p3/2 28 -1f7/2 20 +1d3/2 +2s1/2 +1d5/2 8 -1p1/2 -1p3/2 2 +1s1/2 For a nucleus with A nucleons you fill the A lowest energy levels, and the energy is the sum of the energy of the individual levels

Shell Model : how describe excited states ? -1p3/2 -1p1/2 2 8 Ex: Z=10 +1d5/2 +2s1/2 +1d3/2 20 Ex: N=20 -1f7/2 Ground state +1s1/2 -1p3/2 -1p1/2 2 8 Ex: Z=10 +1d5/2 +2s1/2 +1d3/2 20 Ex: N=20 -1f7/2 Excited state : you make a particle-hole excitation. You promote one particle to a higher energy level

Beyond this “independent particle Shell Model Satisfying results for magic nuclei : ground state and low lying excited states Problems : Neglect of collective excitations Same potential for all the nucleons and for all the configurations Independent particles Improved shell model (currently used): The particles are not independent : due to their interactions with the other particles they do not occupy a given orbital but a sum of configurations having a different probability. -> definition of a valence space where the particles are active 26

Beyond this “independent particle Shell Model proton neutron 26

many nucleons  effective interaction Mean field approach Main assumption each particle is interacting with an average field generated by all the other particles : the mean field. The mean field is built from the individual excitations between the nucleons. Nuclear interaction 2 nucleons bare force many nucleons  effective interaction Neptune FG Feffective Soleil Flibre Uranus Self consistent mean field : the mean field is not fixed. It depends on the configuration. No inert core

Jacques Dechargé Jacques Dechargé

Finite range for pairing treatment The phenomenological effective finite-range Gogny force P : isospin exchange operator P : spin exchange operator Finite range central term Density dependent term Spin orbit term Coulomb term Finite range for pairing treatment

Mean field approach : Hatree-Fock method Hartree-Fock equations (A set of coupled Schrodinger equations) For more formalism see “The nuclear many body problem”, P. Ring and P. Schuck Single particle wave functions Hartree-Fock potential Self consistent mean field : the Hartree Fock potential depends on the solutions (the single particle wave functions) -> Resolution by iteration

Resolution of the Hatree-Fock equations Trial single particle wave function Effective interaction Calculation of the HF potential Resolution of the HF equations New wave functions Test of the convergence Jacques Dechargé Calculations of the properties of the nucleus in its ground state

Hatree-Fock method : deformation We can “measure” nuclear deformations as the mean values of the mutipole operators Spherical Harmonic If we consider the isoscalar axial quadrupole operator We find that: Most of the nuclei are deformed in their ground state Magic nuclei are spherical http://www-phynu.cea.fr 34

Constraints Hatree-Fock-bogoliubov calculations We can impose collective deformations and test the response of the nuclei: with Where ’s are Lagrange parameters. 36

Constraints Hatree-Fock-bogoliubov calculations : results g.s deformation predicted with HFB using the Gogny force http://www-phynu.cea.fr 35

Constraints Hatree-Fock-bogoliubov calculations : What are the most commonly used constraints ? What are the problems with this deformation ? 38

Deformations pertinent for fission: Constraints Hatree-Fock-bogoliubov calculations : Potential energy landscapes Deformations pertinent for fission: Elongation Asymmetry … 39

Evolution of s.p. states with deformation 154Sm New gaps

Hatree-Fock-bogoliubov calculations with blocking Particle-hole excitations  one (or two, three, ..) quasi-particles curves

Beyond mean field … Introduction of more correlations : two types of approaches Random Phase Approximation (RPA) Coupling between HFB ground state and particle hole excitations Generator coordinate Method (GCM) Introduction of large amplitude correlations Give access to a correlated ground state and to the excited states Individual excitations and collective states 42

Beyond mean field … with GCM (GCM+GOA 2 vibr. + 3 rot.) = 5 Dimension Collective Hamiltonian 5DCH

Beyond mean field … with RPA Monopole Dipole S. Péru, J.F. Berger, and P.F. Bortignon, Eur. Phys. Jour. A 26, 25-32 (2005)

The nuclear shape : spectrum ? «vibrational» spectrum Spherical nuclei 6+ 4+ 2+ 0+ Deformed nuclei «rotational» spectrum 0+ 4+ 6+ 2+ 44

Angular velocity of a rotating nucleus For a rotating nucleus, the energy of a level is given by* : With J the moment of inertia We also have so With To compare with a wash machine: 1300 tpm * Mécanique quantique by C. Cohen-Tannoudji, B. Diu, F. Laloe 47

Modeling the nuclei: Summary Macroscopic description of a nucleus : the liquid drop model Microscopic description needed: the basic ingredients are the nucleons and the interaction between them. Different microscopic approaches : the shell model, the mean field and beyond Many nuclei are found deformed in their ground states The spectroscopy strongly depends on the deformation 48