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 / 2

2

B( Z , A)  aV A  aS A

2/3

 aC Z A 2

1 / 3

 aA

A

 B5

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-orbitcoupling 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   pi Z e2 Hi     Vi j ri  rj  2 m e 4 π ε 0 ri i  j Coulombfeld des Atomkerns

 Vri 

Bewegung im mittleren Potential der übrigen 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 parametervalues)

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-particleorbitals: 3-dimensional harmonic oscillator

• Spectroscopic notation

Principal quantum number number Orbital angular momentum quantum number

occupation numbers wrong sequence

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 WoodsSaxon 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 1909

Maria Goeppert was born on June 28th in Kattowitz 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 at all 1972 On February 20th, she died of a stroke

Reminder: spin-orbit-coupling • Classic view: electron moves arround the core, thereby changing the view of the electron spin s to the relative position of the core • Moving charges generate magnetic field, thus sees spin of the electron intrinsic magnetic field, for which there are two alignment options • Classical and exact quantum mechanical consideration provides

• D

add vectorially to total angular momentum called

The absolut value is Important

and

andare determined from a rule for • Possible orientations of each other calculating the quantum number j of the quantum numbers

and

The important for the coupling energy scalar product calculated as

• Reminder: spin-orbit potential in the atomic shell

(Thomas-Präzession) • Spin-orbit potential in nuclear physics • Try it with an analog Ansatz

 Falsches Vorzeichen und viel zu klein(um ca. Faktor 20)

Nuclear shell model with spin-orbitcoupling Problem: Nucleons have a similar size as the core itself. How are welldefined paths without nucleon-nucleon collisions? Antwort: When energy is transferred in a collision, the nucleons must occupy different orbitals (higher and lower). All nearby low-lying states, however, are occupied. Therefore, the nucleons in the ground state must move without collision within the core. Spin-orbit coupling in the effective potential

expected value

Nuclear shell model with spin-orbitcoupling • Important extension of the shell model potential Goeppert-Mayer and Jensen (both Nobel price 1963)

• Total angular momentum of a nucleon in the nucleus: Eigenvalues ​of the spin-orbit operator:

• Spin-orbit splitting of the single particle: • Single-particle wave function with spin-orbit coupling:

The spin-orbit potential in the nucleus • It turns out that is exactly how negative, Therefore the states with are energetically higher then these with . • Number of allowed combinations of n, l for fixed j yields maximum number of nucleons in the state . • There exists for every value of j as in the atomic shell 2j + 1 possible energieent-degenerated direction settings, yields the nucleon below • Energetic ordering and absolute energies of the levels are only produced from extensive calculation with the wave functions given above.

Explanation of the magic numbers • Spin-orbit potential in nuclear physics is unusually high • Explanation of the empirically observed magic numbers

• In particular: magic number 28 is only possible with strong spin-orbit coupling

Explanation of the magic numbers • the use of the Wood-Saxon potential provides a shift and splitting of the terms of the oscillator potential • taking into account the spin-orbit coupling provides a further breakdown and creates gaps in the term scheme to match the magic numbers • the spins and orbital angular momenta of the nucleons in a fully filled shell coupling to zero. Angular momentum and magnetic moment of the nucleus are then determined solely by lightnucleon (or the nucleon-hole)

• Applying the resulting sequence level with occupation numbers obtained experimentally observed magic numbers as the sum of all the protons and neutrons that can occupy all levels to , at the very large energy gap to the next higher level occurs (energy gaps hatched) • Fully occupied shell (i.e. state with particles of given total angular momentum • momentum quantum number j) has Nuclear spin = total angular momentum momentum , since alle 2j+1 Substates with are occupied Closed-shell nuclei must be spherically symmetric, the nuclear spin I = 0 have and can not have a quadrupole moment → deviation for heavier nuclei

Maria Goeppert-Mayer

Hans Jensen 1907-1973

Otto Haxel 1909-1998

Hans Suess 1909-1993

Conclusion: Spin-Orbit-coupling in the nucleus • Approach involves the assumption that the spin-orbit potential at the core surface is dominant • Strong spin-orbit coupling (coupling of the spin of a nucleon with its own orbital angular momentum due to the nuclear potential) – selective shift of energy levels to higher or lower values – Gaps in the energy level scheme which occur exactly at the points corresponding to the magic numbers

• States with higher total angular momentum j are lower in energy than the small total angular momentum, i. e. – P3/2-states are lower, than the P1/2-states, opposite to the atomic shell – For construction of the cores those states with higher angular momentum are occupied first – Magnetic moments are each determined by unpaired nucleons

Experiments show the specific numbers of nucleons (2, 8, 20, 28, 50, 82, 126,...) Are stably bonded. A central potential can not explain it. If we add a spin-orbit potential, this provides an explanation.

The spin-orbit potential splits the j = l + 1/2 Configuration of the l-1/2 Configuration from

By adjusting the strength of the interference by the spin-orbit coupling and Mayer were Jensen is able to explain the magic numbers

• Example:  By scattering phase decomposition for scattering of neutrons by reveals that energetically lowest scattering state has angular momentum 3/2 and next higher has angular momentum ½  Since the 1s shell is occupied in , the neutron must be scattered at pstate with l = 1. By spin-orbit coupling is the l = 1 state energetically lower at j = 1 + 1/2 = 3/2 and higher at j = 1-1 / 2 = 1/2 splits.  Experiment indicates that term follow is exactly the reverse of the fine structure splitting in the atomic shell

Discussion of the empirical data for nuclei with odd A=N+Z Different values ​of N and Z (Ground states of nuclei)

N or Z = 1 odd -The odd-odd nucleus of deuterium has magnetic moment of 0.85761, differs by 2.5% from the magnetic moment of the proton and the neutron. Explained by mixing a d-state with s-state (→magic Number 2)

N or Z odd from 2 to 8 p-state is filled  Due to strong spin-orbit coupling the P3 / 2 state is lower than the P1 / 2 state For N or Z = 3 and 5 ground-state spin and magnetic moment corresponding to P3 / 2, N or Z = 7 to P1 / 2 (→ magic number 8)

Z or N is odd from 8 to 20  States in this shell are in the order d5 / 2, s1 / 2, d3 / 2, Z or N=13  with 5p or 5n or a hole in the d5 / 2 state, one finds as possible condition only d5 / 2 state  The positive quadrupole moment of shows that the shell is more than half filled  At 15 the S1 / 2 state of odd number of protons is filled

N or Z is odd from 8 to 20

N or Z is odd from 20 to 28  The only state in this region is f7/2

N or Z is odd from 28 to 50  28 is the Spin-orbit coupling crucial one. The states will be filled in the order p3/2, f5/2, p1/2, g9/2

N or Z is odd from 28 to 50

N or Z is odd from 50 to 82  Until now the nuclei with odd Z and odd N behaved equal, this will be different now  For neutrons we have the occupation order  For Protons we have the occupation order

 We have Barium with N=81, A=137, therefore for Thallium with Z=81, we have A=203 and 205  Coulomb-energy plays an important rule

N or Z is odd from 50 to 82

N or Z is odd from 50 und 82

N or Z is odd from 82 to 126

N or Z is odd from 82 to 126

Comparison of the deviation of the magnetic moments of nuclei with N and Z odd

“This was wonderful. I liked the mathematics in it… Mathematics began to seem too much like puzzle solving… Physics is puzzle solving, too, but of puzzles created by nature, not by the mind of man… Physics was the challenge.” Maria Goeppert-Mayer

References •

• • • • • • • • • • • • •

Maria Goeppert-Mayer, H. Jensen: Elementary Theory of Nuclear Shell Strukture, Wiley New York 1955 W. Demtröder: Experimentalphysik 4, Kern-,Teilchen- und Astrophysik, Springer 1998 K. Bethge, G. Walter, B. Wiedemann: Kernphysik, Springer 2008 T. Mayer-Kuckuk: Kernphysik, Teubner 1994 gmf-lectures5: Nucleons and Nuclei Grace Ross: Mary Goeppert Mayer, Interdisciplinary Symposium J. Bleck-Neuhaus: Elementare Teilchen – Moderne Physik von den Atomen bis zum Standardmodell, Springer Verlag B. Povh, K. Rith, C. Scholz, F. Zetsche: Teilchen und Kerne, Springer-Verlag D. Griffith, Introduction to Elementary Particles, Verlag Wiley-VCH F. Halzen, A. D. Martin: Quarks &Leptons, Verlag J. Wiley C. Grupen: Teilchendetektoren, BI Wissenschaftsverlag W. R. Leo: Techniques for Nuclear and Particle Physics Experiments, Springer Verlag E. Bodenstedt: Experimente der Kernphysik und ihre Deutung, BI Wissenschaftsverlag 1978 P.A. Tipler, R. A. Llewellyn, Moderne Physik, Oldenburg Verlag 2002