Héloïse Goutte CERN Summer student program 2009 Introduction to Nuclear physics; The nucleus a complex system Héloïse Goutte CEA, DAM, DIF 1

Héloïse Goutte CERN Summer student program 2009 The nucleus : a complex system I) Some features about the nucleus discovery radius binding energy nucleon-nucleon interaction life time applications II) Modeling of the nucleus liquid drop shell model mean field III) Examples of recent studies exotic nuclei isomers shape coexistence super heavy IV) Toward a microscopic description of the fission process 2

Héloïse Goutte CERN Summer student program 2009 Some features about the nucleus: summary A nucleus is made of Z protons and N neutrons (the nucleons). A nucleus is characterized by its mass number A = N + Z and its atomic number Z. It is written A X. ATOMNUCLEUSNUCLEON (10 -9 m)( m)( m) A nucleus is almost times smaller than an atom 3

Héloïse Goutte CERN Summer student program 2009 R = 1.25 x A 1/3 (fm) Nuclear radius The radius increases with A 1/3  The volume increases with the number of particles R(fm) A 1/3 4

Héloïse Goutte CERN Summer student program 2009 The nucleon-nucleon interaction Proton and neutron interact through the strong interaction. The strong interaction is very intense of short range Proton –neutron interaction :Vpn > Vnn et Vpp  Vnn V (MeV) r(fm) The nuclear interaction is stabilizing the nucleus PLUS Coulomb interaction between protons (repulsive) 5

Héloïse Goutte CERN Summer student program 2009 Binding energy M(A,Z) = N M n + Z M p – B(A,Z) B(A,Z) : binding energy Stable bound system for B > 0 (its mass is lower than the mass of its components) Unstable systems : they transform into more stable nuclei Mass of a given nucleus : Exponential decay Half –life T defined as the time for which the number of remaining nuclei is half of its the initial value. 6

Héloïse Goutte CERN Summer student program 2009 ---- +,+,+,+,  p n Neutrons Protons Different types of radioactivity 7

Héloïse Goutte CERN Summer student program

How do we experimentally study a nucleus ? (e -,e +, p,n, heavy ions, …) I ) Elastic and inelastic scattering II ) Gamma spectroscopy The level scheme: the barcode of a nucleus 9

Héloïse Goutte CERN Summer student program 2009 II) Modeling of the nucleus 10

Héloïse Goutte CERN Summer student program 2009 The nucleus : a liquid drop ? 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 11

Héloïse Goutte CERN Summer student program 2009 The liquid drop model * Model developed by Von Weizsacker and N. Bohr (1937) It has been first developed to describe the nuclear fission. * The nucleus is represented by a charged liquid drop. * 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. * The binding energy of the nuclei is described by the Bethe Weizsacker formula 12

Héloïse Goutte CERN Summer student program 2009 Mass formula of Bethe and Weizsäcker Binding energy for a liquid drop a v = volume term MeV a s = surface term MeV a c = coulomb term MeV a a = asymmetry MeV  = pairing term Parameters adjusted to experimental results (for e-e nuclei) (for o-o nuclei) 13

Héloïse Goutte CERN Summer student program 2009 Problems with the liquid drop model 1) Nuclear radii Evolution of mean square radii with respect to 198 Hg 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 A 2/3 law Fig. from 14

Héloïse Goutte CERN Summer student program 2009 Halo nuclei I. Tanihata et al., PRL 55 (1985) 2676 I. Tanihata and R. Kanungo, CR Physique (2003)

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

Héloïse Goutte CERN Summer student program 2009 Two neutron separation energy S 2n 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. S 2n : energy needed to remove 2 neutrons to a given nucleus (N,Z) S 2n =B(N,Z)-B(N-2,Z) 17

Héloïse Goutte CERN Summer student program ) Fission fragment distributions Liquid drop : only symmetric fission Proton number Neutron number Experimental Results : K-H Schmidt et al., Nucl. Phys. A665 (2000) 221 A heavy and a light fragments = asymmetric fission Two identical fragments = symmetric fission 18

Héloïse Goutte CERN Summer student program 2009 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» There are «magic numbers» 2, 8, 20, 28, 50, 82, 126 and so «magic» and «doubly magic» nuclei The nucleus is not a liquid drop : structure effects 19

Héloïse Goutte CERN Summer student program 2009 Microscopic description of the atomic nucleus Nucleus = N nucleons in strong interaction Nucleon-Nucleon force unknown No complete derivation from the QCD 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 Different forces used depending on the method chosen to solve the Many-body problem 20

Héloïse Goutte CERN Summer student program 2009 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

Héloïse Goutte CERN Summer student program 2009 The Goeppert Mayer shell model * Model developped 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 which we assume to have a spherical symmetry”, M. Goeppert Mayer, Nobel Conference

Héloïse Goutte CERN Summer student program 2009 Nuclear potential Nuclear potential deduced from exp : Wood Saxon potential or square well or harmonic oscillator Thanks to E. Gallichet 23

Héloïse Goutte CERN Summer student program 2009 Quantum numbers Quantum numbers characterizing the Nucleon states: The principal quantum number N The radial quantum number n The azimutal quantum number l The spin s (s = ± ½) N= 2(n-1) +l Introduction of the angular momentum j= l ± 1/2 24

Héloïse Goutte CERN Summer student program 2009 Single particle levels in the shell model Magic Numbers j = l +/- 1/2 Isotropic Square well (n,l) Isotropic square well + spin-orbit (n,l,j) 1s 1p 2f 1g 1f 2s 1d 2d 3p 3s 2p 1h 0   1   2   3   4   6   5   s 1/2 -1p 3/2 -1p 1/2 +1d 5/2 +2s 1/2 +1d 3/2 +3s 1/2 -1h 11/2 +1g 7/2 +2d 5/2 -1h 9/2 +1g 9/2 -1f 7/2 -2p 3/2 -1f 5/2 -1p 1/2 -2f 7/2 +1d 3/2 -1i 13/2 -2f 5/2 -3p 3/2 -3p 1/2 -2g 9/2 -3d 5/

Héloïse Goutte CERN Summer student program 2009 Beyond this “independent particle shell model” Satisfying results for magic nuclei : ground state and low lying excited states Problems : Neglect of collective deformation, vibration, rotation 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

Héloïse Goutte CERN Summer student program 2009 The shell model space 27

Héloïse Goutte CERN Summer student program

Héloïse Goutte CERN Summer student program 2009 The self consistent 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 Self consistent mean field : the mean field is not fixed. It depends on the configuration. No inert core 29

Héloïse Goutte CERN Summer student program 2009 The self consistent mean field approach The Hartree Fock method The basis ingredient is the Hamiltonian which governs the dynamics of the individual nucleons (equivalent to the total energy in classical physics) Effective force Orbitals are obtained by minimizing the total energy of the nucleus Wave function = antisymmetrized product of A orbitals of the nucleons with 30

Héloïse Goutte CERN Summer student program 2009 The phenomenological effective finite-range Gogny force 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 back 31

Héloïse Goutte CERN Summer student program 2009 The Hartree Fock equations Hartree-Fock equations (A set of coupled Schrodinger equations) Hartree-Fock potential Single particle wave functions Self consistent mean field : the Hartree Fock potential depends on the solutions (the single particle wave functions) -> Resolution by iteration 32

Héloïse Goutte CERN Summer student program 2009 Trial single particle wave function Calculation of the HF potential Resolution of the HF equations New wave functions Test of the convergence Calculations of the properties of the nucleus in its ground state Effective interaction Resolution of the Hartree Fock equations 33

Héloïse Goutte CERN Summer student program 2009 Deformation We can “measure” nuclear deformations as the mean values of the mutipole operators If we consider the isoscalar axial quadrupole operator We find that: Most ot the nuclei are deformed in their ground state Magic nuclei are spherical Spherical Harmonic 34

Héloïse Goutte CERN Summer student program 2009 g.s deformation predicted with HFB using the Gogny force Results 35

Héloïse Goutte CERN Summer student program 2009 Constraints Hartree-Fock-Bogoliubov calculations We can impose collective deformations and test the response of the nuclei with Where ’s are Lagrange parameters 36

Héloïse Goutte CERN Summer student program 2009 Potential energy curves Energy (MeV) Quadrupole Deformation Example : fission barrier Ground state Isomeric well First and second barriers 37

Héloïse Goutte CERN Summer student program 2009 What are the most commonly used constraints ? What are the problems with this deformation ? 38

Héloïse Goutte CERN Summer student program 2009 Potential energy landscapes Deformations pertinent for fission: Elongation Asymmetry … 39

Héloïse Goutte CERN Summer student program 2009 Evolution of s.p. states with deformation Deformation Energy (MeV) New gaps 40

Héloïse Goutte CERN Summer student program Sm From microscopic calculations 41

Héloïse Goutte CERN Summer student program 2009 Beyond the 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

Héloïse Goutte CERN Summer student program 2009 Results from beyond mean field calculations 43

Héloïse Goutte CERN Summer student program 2009 The nuclear shape : a pertinent information ? Spherical nuclei «vibrational» spectrum «vibrational» spectrum Deformed nuclei«rotational» spectrum

Héloïse Goutte CERN Summer student program 2009 Two examples of spectra 148 Sm 160 Gd 45

Héloïse Goutte CERN Summer student program 2009 Sm Change of the shape of the nuclei along an isotopic chain 46

Héloïse Goutte CERN Summer student program 2009 Angular velocity of a rotating nucleus so With To compare with a wash machine: 1300 tpm For a rotating nucleus, the energy of a level is given by* : With J the moment of inertia We also have * Mécanique quantique by C. Cohen-Tannoudji, B. Diu, F. Laloe) 47

Héloïse Goutte CERN Summer student program 2009 Modeling of 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 and the mean field Many nuclei are found deformed in their ground states The spectroscopy strongly depends on the deformation 48