Download What are magic numbers? - Justus-Liebig

Maria Goeppert-Mayer
Nuclear Models and
Magic Numbers
Nobel Price 1963
Justus-Liebig-Universität Giessen
Dr. Frank Morherr
Table of Contents
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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
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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