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Maria Goeppert-Mayer Nuclear Models and Magic Numbers Nobel Price 1963

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Maria Goeppert-Mayer Nuclear Models and Magic Numbers Nobel Price 1963 Maria Goeppert-Mayer Nuclear Models and Magic Numbers Nobel Price 1963 Justus-Liebig-Universität Giessen Dr. Frank Morherr Table of Contents • Liquid drop model of the nucleus • Bethe-Weizsäcker-Formula (semi-empirical mass formula) • Thomas-Fermi-Model of the atomic nucleus • What are magic numbers? • Nuclear shell model without spin-orbit-coupling • Maria Goeppert-Mayer • Reminder: spin-orbit-coupling • Nuclear shell model with spin-orbit-coupling • Explanation of the magic numbers • Conclusion: Spin-orbit-coupling in the nucleus • Discussion of the empirical Data Nuclear Models • Cores are complex many-body systems of interacting nucleons • a universal, all core properties descriptive theory does not yet exist • Development of phenomenological models for certain properties Liquid drop model of the nucleus Core in close analogy to the charged liquid droplets (quasi-classical), nucleons move strongly correlated incompressible liquid Thomas-Fermi-model of the atomic nucleus Nucleons move independently in a resulting nuclear potential well depth of the quantum statistics of a Fermi gas Nuclear shell model: Nucleons move fully quantum mechanical (Schrödinger equation) potential with strong spin-orbit term, → magic numbers, spin, parity Liquid drop model of the nucleus • Large number of nucleons in heavy nuclei justify that composite cores behave similarly as liquid droplets, are held together in the water molecules, but nevertheless perform movements. • Binding energies are related per nucleon • Values ​​can be represented as a function of the mass number Liquid drop model of the nucleus • Binding energy per nucleon reaches its highest value at 8 MeV / u in the mass range 55-60 u. • Behavior represents the saturation of nuclear forces. Attractive force only reaches the next nucleon neighbours • First model that describes these facts, dates of Albrecht Bethe and Carl Friedrich von Weizsäcker (1935): Not all nucleons in the nucleus experience the same forces. The particles at the surface have fewer neighbors. Bounding there is not so strong repulsive effect of the Coulomb force between the bounded Protons Asymmetry in the number of protons and neutrons reduces the binding energy, is particularly evident in heavy nuclei noticeable Pairing forces between the same art of nucleons may enhance binding slightly Bethe-Weizsäcker-Formula Bethe-Weizsäcker-Formula Derivation of ac from electrostatics: Usual representation of the Bethe-Weizsäcker formula: Z A/ 22 B(Z, A) a A a A2/ 3 a Z 2 A1/ 3 a B V S C A A 5 Droplet model can not explain the abundances of the elements and the magic numbers → nuclear shell model Thomas-Fermi-model of the atomic nucleus • The droplet model is empirical and granted little insight into the structure of the atomic nucleus → for understanding the properties of nuclei other physical models are required • In the Fermi-gas model, the forces of all the surrounding nucleons lift practically, so that move the protons and neutrons quasi-free in a sphere of radius a/2 • Because of the independence of the two types of particles (isospin) is one of two potential wells, which differ in depth, because the protons repel each other • Each box is filled up to the Fermi energy • The Fermi energies depend only on the density of the neutrons or Protons in the nucleus (Herleitung unten) Thomas-Fermi-model of the atomic nucleus Mass formula in extended form takes into account the thickness of the surface layer of the core Fermi distribution of charge density radial charge distribution of some atomic nuclei Fermi-Gas-Modell Konsequenzen with independent of the size of the core • heavy nuclei must have a higher density and smaller distances between the energy levels • the neutron-pot is deeper than the proton pot, because protons repel each other → is the same Fermi energy • the relative difference (N-Z)/A is greater in heavy nuclei, because the increasing Coulomb repulsion can increase the distance What are magic numbers? • Magic numbers are in nuclear physics certain neutron and proton numbers in atomic nuclei, in which the ground state of the core higher stability is observed than in neighboring nuclides →known as magic nuclei • magic nuclei have a particularly high separation energy for a single nucleon • magic numbers explained by the shell model of nuclear physics. • Natural islands of stability in atomic numbers above occurring elements are predicted. Nuclear shell model without spin-orbit- coupling Notes on the shell structure of the atomic Nucleus Magic numbers • Nuclei with magic proton number Z or neutron number N are more stable than other nuclei in the neighbourhood of the Table of nucleids • In the Neighbourhood of these magic proton or neutron numbers there are very many isotops Example: There are 6 (stable) nuclei with N = 50 and 7 nuclei with N = 82 There are 10 naturally occurring isotopes of Sn (Z = 50) • Double magic nuclei (Z and N magical) are exceptionally “stable” (in comparison to their environment): Examples: Assumptions of the shell model • Each nucleon moves in an average potential field U(r) that is generated by the interaction with all other nucleons • The occupation of the discrete quantum states (orbitals) in the shell model-potential according to the rules of the Pauli principle • Ansatz: potential U (r) is proportional to the density ρ(r) of the nucleons Hamiltonian of the nucleus in the shell model • Starting point: nuclear Hamiltonian • The average single-particle shell model potential U: Shell model Hamiltonian Residual interaction: small • Optimization of the shell model potential: Hartree-Fock method • Schrödinger-equation: Hartree-Fock 2 2 pi Ze Hi Vi j ri rj 2me 4πε0 ri i j Bewegung im mittleren Coulombfeld Potential der übrigen Vri des Atomkerns Elektronen Schrödinger equation in spherical coordinates Solution through the product ansatz: Equation for the radial function R(r) Phenomenological nuclear potentials in the shell model • 3-dimensional harmonic oscillator • energy-eigenvalues: degree of degeneracy : • Woods-Saxon-potential: (typical parameter- values) Wavefunction • Isospin-symmetry: proton and neutron as isospin-dublet Orbit • Pauli-principle: Wave function is totally antisymmetric under exchange of two nucleons • Solution of the one-particle-Schrödinger-equation • Antisymmetrized product wave function of the nucleus: (Slater-determinant) • Energy • Spectrum of the one-particle- orbitals: 3-dimensional harmonic oscillator occupation numbers wrong • Spectroscopic notation sequence Principal quantum number number Orbital angular momentum quantum number Problem of the nuclear shell model • calculations with harmonic oscillator potential only reproduce the magic numbers until 20 • also, model calculations with rectangular and Woods-Saxon potential can not reproduce the magic numbers greater than 20 Improvement of the model: coupling of orbital angular momentum and spin (1949: Maria Goeppert-Mayer, Hans D. Jensen → Nobel Prize for Physics 1963) Splitting of the energy levels corresponding to the fine structure splitting of the electron states in the atomic shell, but is significantly stronger as a result of nuclear forces Splitting can be greater than the difference in energy between two shells Both square-well potential, as well as harmonic oscillator with parabolic potential and bill with Woods- Saxon potential in incorrect sequence • Correct shell closures since 1949 independently Haxel, Jensen, and Suess other hand of Maria Goeppert-Mayer found • Nuclear forces cause spin-orbit interaction of such strength that they determined term follow critical • In contrast to the atomic shell is at the core spin-orbit coupling energy in the same order as the term distances Conclusion: • Solutions of the Schrödinger equation resulting energy levels, which can only explain the magic numbers 2, 8, 20 as shell closures. • With a larger number of nucleons in the nucleus other than the magic numbers arise. • The oscillator potential supplies for all levels constant distances. • The correct shell closures were found independently in 1949 by Hans Jensen and Maria Goeppert-Mayer et .Al. • Essential idea: • Analogy to the atomic shell, in the based on electromagnetic interaction spin-orbit coupling of the electron plays role → Splitting of spectrallines (fine structure) Introduction of just such a spin-orbit coupling for the strong interaction of the nucleons Maria Goeppert-Mayer 1906 Maria Goeppert was born on June 28th in Kattowitz 1909 Moves with her ​​parents in Göttingen, stronghold of Mathematics and Physics, neighbor David Hilbert 1921 Maria finished the “elementary school” (Volkschule of 8 Years) 1923 Maria is an External at a boys' high school in Hannover and makes her matura 1924 Studies at the University, initially for Mathematics 1930 Maria married in the spring the American Joseph Mayer 1930 Maria writes with Max Born's supervision of their dissertation 1930 Maria research independently and for her own in different fields 1941 Maria gets a part-time position as a science teacher at the College in Bronxville 1943 – 46 She is recruited to separate uranium-235 of the more stable uranium-238 (Manhattan Project) 1946 Maria and her husband moves to Chicago 1950 From April on she begins investigations in theoretical physics in the field of atomic nuclei 1963 Hans Jensen and Maria Goeppert-Mayer are honored for their "discovery of the shell structure of the core" with the Nobel Prize for Physics, the first woman in theoretical physics and the second woman after Marie Curie
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