Chapters 56: Strong Force

1
Chapters 5-6 Strong Force Nuclear Structure

• Abby Bickley
• University of Colorado
• NCSS 99

Additional References Choppin (CLR),
Radiochemistry and Nuclear Chemistry, 2nd
Edition, Chapter 11 Friedlander (FKMM), Nuclear
and Radiochemistry, 3rd Edition, Chapter 10
2
Particle Classifications

• Fermions
• Spin 1/2 particles
• Obey Pauli Exclusion Principle
• No more than one fermion can occupy the same
  quantum state
• Antisymmetric wave functions
• Two classifications
• Hadrons - interact via strong nuclear force
  composed of quarks neutrons, protons, etc.
• Leptons - fundamental particles, ie no
  substructure electron, muon, tau, neutrino.
• Bosons
• Integer spins
• Obey Bose-Einstein Statistics
• Any number of bosons can share the same quantum
  state
• Symmetric wave functions
• Force mediators- Photons, gluons, etc.

3
Fundamental Forces

• Virtual particles - exchange particles that exist
  for a short period of time to convey force
  existence must obey Heisenberg Uncertainty
  Principle

4
Heisenbergs Uncertainty Principle

• Conservation of energy can only be violated by ?E
  and ?t as long as the uncertainty principle
  remains valid.
• How far can an exchange particle travel without
  violating HUP?
• Assume exchange particle moves at the speed of
  light, c.
• Insert HUP for ?t and Emc2

5
Problem

• If the effective range of the weak force is 10-18
  m, what is the maximum allowed mass of the
  exchange particle?
• Answer (GeV/c2)

6
Problem

• If the effective range of the weak force is 10-18
  m, what is the maximum allowed mass of the
  exchange particle?
• Answer (GeV/c2)

7
Properties of the Strong Force

• Range less than nuclear radius, lt1.4fm
• Attractive on the distance scale of 1fm,
  overcomes coulombic force to hold charged protons
  together
• Repulsive on the distance scale of lt0.5fm
• Charge Independent interaction is independent of
  nucleon electrical charge, ie p-n p-p, n-n

8
Charge Independence
9
The Nucleus

• As chemists what do we already know about the
  nucleus of an atom?

10
The Nucleus

• As chemists what do we already know about the
  nucleus of an atom?
• Composed of protons and neutrons
• Carries an electric charge equivalent to the
  number of protons atomic number of the element
• Protons and neutrons within nucleus held together
  by the strong force
• Any model of nuclear structure must account for
  both Coulombic repulsion of protons and Strong
  force attraction between nucleons

11
Two Nucleon Systems

• Combinatorics gives us three possible states, but
  only one occurs in nature!
• nn
• Unbound and comes apart easily
• Free neutrons decay on the time scale of 10min
• pp
• More unstable than nn due to Coulomb repulsion
• pn - deuteron
• Stable and naturally occuring
• Spins of n and p align parallel in ground state
  configuration
• Non-spherical structure

12
Chart of the Nuclides
13
Empirical Observations

• Chart of the nuclides
• 275 stable nuclei
• 60 even-even
• 40 even-odd or odd-even
• Only 5 stable odd-odd nuclei
• 21 H, 63Li, 105B, 147N, 5023Va (could have large
  t1/2)
• Nuclei with an even number of protons have a
  large number of stable isotopes
• Even protons Odd protons
• 50Sn10 (isotopes) 47Ag 2 (isotopes)
• 48Cd 8 51Sb2
• 52Te 8 45Rh1
• 49In1
• 53I 1
• Roughly equal numbers of stable even-odd and
  odd-even nuclei

14
Implications for Nuclear Models I

• Proton-proton and neutron-neutron pairing must
  result in energy stabilization of bound state
  nuclei
• Pairing of protons with protons and neutrons with
  neutrons results in the same degree of
  stabilization
• Pairing of protons with neutrons does not occur
  (nor translate into stabilization)

15
Problem

• Which nucleus is stable?

p n
p n
p n
12B
12C
12N
16
Problem

• Which nucleus is stable?

17
Chart of the nuclides

• Light elements N/Z 1
• Heavy elements N/Z ? 1.6
• Implies simple pairing not sufficient for
  stability
• Neutron Rich (NgtZ)
• NgtZ nucleus will ?- decay to stability
• NgtgtZ neutron drip line
• Proton Rich (NltZ)
• NltZ nucleus will ? decay or electron capture to
  achieve stability
• NltltZ proton drip line
• (very rare)

Friedlander, Nuclear and Radiochemistry, 3rd
Edition, 1981.
18
Implications for Nuclear Models II

• 4. Pairing not sufficient to achieve stability
• Why?
• Coulomb repulsion of protons grows with Z2
• Nuclear attractive force must compensate ? all
  stable nuclei with Z gt 20 contain more neutrons
  than protons

19
General Nuclear Properties I

• Binding energy per nucleon constant for all
  stable nuclei
• Implies all nucleons in the nucleus do not
  interact with one another
• If they did the BE per nucleon would be
  proportional to the mass number

8.9
MeV/u
7.4
Mass Number
Friedlander, Nuclear and Radiochemistry, 3rd
Edition, 1981.
20
General Nuclear Properties II

• Nuclear radius is proportional to the cube root
  of the mass
• r r0 A1/3 Eq. 2
• Experimental studies show uniform distribution
  of the charge and mass throughout the volume of
  the nucleus

dl skin thickness
Rl Half density radius
Friedlander, Nuclear and Radiochemistry, 3rd
Edition, 1981.
21
Simple Nuclear Potential Well

• Potential energy of nucleons as they approach the
  nucleus
• Neutron - feels no effective force until reaches
  surface of nucleus constant attractive force at
  interior of nucleus.
• Proton - coulombic repulsion as approaches
  nucleus constant attractive force at interior of
  nucleus coulombic repulsion from other protons
  decreases depth of potential well relative to
  neutron well.

22
Liquid Drop Model (1935)

• Treats nucleus as a statistical assembly of
  neutrons and protons with an effective surface
  tension - similar to a drop of liquid
• Rationale
• Volume of nucleus ? number of nucleons
• Implies nuclear matter is incompressible
• Binding energy of nucleus ? number of nucleons
• Implies nuclear force must have a saturation
  character, ie each nucleon only interacts with
  nearest neighbors
• Mathematical Representation
• Treats binding energy as sum of volume, surface
  and Coulomb energies

23
Liquid Drop Model Components I
c1 15.677 MeV, c2 18.56 MeV, c3 0.717 MeV,
c4 1.211 MeV, k 1.79

• Volume Energy
• Binding energy of nucleus ? number of nucleons
• Correction factor accounts for symmetry energy
  (for a given A the binding energy due to only
  nuclear forces is greatest for nuclei with equal
  numbers of protons and neutrons)
• Surface Energy
• Nucleon at surface are unsaturated ? reduce
  binding energy ? surface area
• Surface-to-volume ratio decreases with increasing
  nuclear size ? term is less important for large
  nuclei

24
Liquid Drop Model Components II
Coulomb Energy
Pairing Energy
c1 15.677 MeV, c2 18.56 MeV, c3 0.717 MeV,
c4 1.211 MeV, k 1.79

• Coulomb Energy
• Electrostatic energy due to Coulomb repulsion
  between protons
• Correction factor accounts for diffuse boundary
  of nucleus (accounts for skin thickness of
  nucleus)
• Pairing Energy
• Accounts for added stability due to nucleon
  pairing
• Even-even ? 11/A1/2
• Even-odd odd-even ? 0
• Odd-odd ? -11/A1/2

25
Problem

• Using the binding energy equation for the liquid
  drop model, calculate the binding energy per
  nucleon for 15N and 148Gd.
• Compare these results with those obtained by
  calculating the binding energy per nucleon from
  the atomic mass and the masses of the constituent
  nucleons.

26
Problem Answers

• 15N 6.87 MeV/nucleon
• 148Gd 8.88 MeV/nucleon
• 15N 7.699 MeV/nucleon
• 148Gd 8.25 MeV/nucleon

27
Mass Parabolas

• Represent mass of atom as difference between sum
  of constituents and total binding energy
• Substitute binding energy equation for EB and
  group terms by power of Z
• For a given number of nucleons (A) f1, f2 and f3
  are constants
• Functional form represents a mass-energy parabola
• Single parabola for odd A nuclei (? 0)
• Double parabola for even A nuclei (? 11/A1/2)

28
Mass Parabolas Example 1A 75 or 157

• Parabola Vertex
• ZA-f2 / 2f1
• Minimum mass Maximum EB
• Used to find mass and EB difference between
  isobars
• Nuclear charge of minimum mass is derivative of M
  Eqn gt not necessarily integral
• Comparison of Z 75 and Z 157
• Valley of stability broadens with increasing A
• For a given value of odd-A only one stable
  nuclide exists
• In odd-A isobaric decay chains the ?-decay energy
  increases monotonically

Friedlander, Nuclear and Radiochemistry, 3rd
Edition, 1981.
29
Mass Parabolas Example 2A 156

• Eqn results in two mass parabola for a given even
  value of A
• For a given value of even-A their exist 2 (or 3)
  stable nuclides
• In this figure both 156Gd and 156Dy are stable
• In even-A isobaric decay chains the ?-decay
  energies alternate between small and large values
• This model successfully reproduces experimentally
  observed energy levels
• BUT.

Friedlander, Nuclear and Radiochemistry, 3rd
Edition, 1981.
30
Problem

• Find the nuclear charge (ZA) corresponding to the
  maximum binding energy for
• A 157, 156 and 75
• To which isotopes do these values correspond?
• Compare your results with the mass parabolas on
  slides 28 29.

31
Problem Answers

• Find the nuclear charge (ZA) corresponding to the
  maximum binding energy for
• A 157, 156 and 75
• ZA 64.69, 64.32, 33.13
• To which isotopes do these values most closely
  correspond?
• 15765Tb, 15664Gd, 7533As
• Compare your results with the mass parabolas on
  slides 28 29.

32
Fermi Gas Model

• Model emphasizes free particle character of
  nucleons allows average behavior of lg nuclei
  to be described by thermodynamics
• Assume nucleus is composed of a degenerate Fermi
  gas of p n
• Degenerate - particles occupy lowest possible
  energy states
• Fermi gas - all particles obey Pauli Principle
• Fermi Wavenumber - highest state occupied by
  nucleons
• Fermi Energy - gas is characterized by kinetic
  energy of highest filled state
• When ngtp then must calculate Fermi energies of
  neutrons and protons separately

33
Fermi Gas Potential Well
Excited states

• EF,p - Fermi energy of proton
• EF,n - Fermi energy of neutron
• EC - coulombic energy
• B - binding energy
• U0 - depth of potential well
• Fermi level - uppermost filled energy level,
  approximately -8MeV.

34
Problem

• What is the average Fermi energy of a neutron in
  a 208Pb nucleus?

35
Problem

• What is the average Fermi energy of a neutron in
  a 208Pb nucleus?

36
Magic Numbers

• Nuclides with magic numbers of protons and/or
  neutrons exhibit an unusual degree of stability
• 2, 8, 20, 28, 50, 82, 126
• Suggestive of closed shells as observed in atomic
  orbitals
• Analogous to noble gases
• Much empirical evidence was amassed before a
  model capable of explaining this phenomenon was
  proposed
• Result Shell Model

Friedlander, Nuclear and Radiochemistry, 3rd
Edition, 1981.
37
Atomic Orbitals History

• Plum Pudding Model (Thomson, 1897)
• Each atom has an integral number of electrons
  whose charge is exactly balanced by a jelly-like
  fluid of positive charge
• Nuclear Model (Rutherford, 1911)
• Electrons arranged around a small massive core of
  protons and neutrons (added later)
• Planetary Atomic Orbitals (Bohr, 1913)
• Assume electrons move in a circular orbit of a
  given radius around a fixed nucleus
• Assume quantized energy levels to account for
  observed atomic spectra
• Fails for multi-electron systems
• Schrodinger Equation (1925)
• Express electron as a probability distribution in
  the form of a standing wave function

38
Atomic Orbitals

• Schrodinger equation solution reveals quantum
  numbers
• n principal, describes energy level
• l angular momentum, 0?n-1 (s,p,d,f,g,h)
• m magnetic, - l ? l, describes behavior of atom
  in external B field
• ms spin, -1/2 or 1/2
• Pauli Exclusion Principle e-s are fermions ? no
  two e-s can have the same set of quantum numbers
• Hunds Rule when electrons are added to orbitals
  of equal energy a single electron enters each
  orbital before a second enters any orbital the
  spins remain parallel if possible.
• Example C 1s22s22px12py1

1s
39
Shell Structure of NucleusHistorical Evolution

• Throughout 1930s and early 1940s evidence of
  deviation from liquid drop model accumulates
• 1949 Mayer Jenson
• Independently propose single-particle orbits
• Long mean free path of nucleons within nucleus
  supports model of independent movement of
  nucleons
• Using harmonic oscillator model can fill first
  three levels before results deviate from
  experiment (2,8,20 only)
• Include spin-orbit coupling to account for magic
  numbers
• Orbital angular momentum (l) and nucleon spin
  (1/2) interact
• Total angular momentum must be considered
• (l1/2) state lies at significantly lower energy
  than (l-1/2) state
• Large energy gaps appear above 28, 50, 82 126

40
Single Particle Shell Model

• Assumes nucleons are distributed in a series of
  discrete energy levels that satisfy quantum
  mechanics (analogous to atomic electrons)
• As each energy level is filled a closed shell
  forms
• Protons and neutrons fill shells and energy
  levels independently
• Mainly applicable to ground state nuclei
• Only considers motion of individual nucleons

41
Shell Model and Magic Numbers

• Magic numbers represent closed shells
• Elements in periodic table exhibit trends in
  chemical properties based on number of valence
  electrons (Noble gases2,10,18,36..)
• Nuclear properties also vary periodically based
  on outer shell nucleons

42
Pairing

• Just as electrons tend to pair up to form a
  stable bond, so do like-nucleons pairing results
  in increased stability
• Even-Z and evenN nuclides are the most abundant
  stable nuclides in nature (165/275)
• From 15O to 35Cl all odd-Z elements have one
  stable isotope while all even-Z elements have
  three
• The heaviest stable natural nuclide is 20983Bi
  (N126)
• The stable end product of all naturally occurring
  radioactive series of elements is Pb with Z82

43
Shell Model Evidence - Abundances

• The most abundantly occurring nuclides in the
  universe (terrestrial and cosmogenic) have a
  magic number of protons and/or neutrons
• Large fluctuations in natural abundances of
  elements below 19F are attributed to their use in
  thermonuclear reactions in the prestellar stage

44
Shell Model Evidence - Abundances
45
Shell Model Evidence - Stable Isotopes

• The number of stable isotopes of a given element
  is a reflection of the relative stability of that
  element. Plot of number of isobars vs N shows
  peaks at
• N 20, 28, 50, 82
• A similar effect is observed as a function of Z

46
Shell Model Evidence - Alpha Decay

• Shell Model predictions
• Nuclides with 128 neutrons gt
• short half life
• Emit energetic ?
• Nuclides with 126 neutrons gt
• Long half life
• Emit low energy ?

47
Shell Model Evidence - Beta Decay

• If product contains a magic number of protons or
  neutrons the half-life will be short and the
  energy of the emitted ? will be high

N 19
N 20
N 21
Z 21
Z 20
Z 19
48
Shell Model Evidence - Neutrons

• Neutrons do not experience Coulomb barrier ? even
  thermal neutrons (low kinetic energy) can
  penetrate the nucleus
• Inside nucleus neutron experiences attractive
  strong force and becomes bound
• To escape the nucleus a neutrons KE must be
  greater than or equal to the nuclear potential at
  the surface of the nucleus
• Observation the absorption cross section for
  1.0MeV neutrons is much lower for nuclides
  containing 20, 50, 82, 126 neutrons compared to
  those containing 19, 49, 81, 125 neutrons

49
Shell Model Evidence - Energy

• The energy needed to extract the last neutron
  from a nucleus is much higher if it happens to be
  a magic number neutron
• Energy needed to remove a neutron
• 126th neutron from 208Pb 7.38 MeV
• 127th neutron from 209Pb 3.87 MeV

50
Shell Model Evidence - Nucleon Interactions

• Every nucleon is assumed to move in its own orbit
  independent of the other nucleons, but governed
  by a common potential due to the interaction of
  all of the nucleons
• Implication in ground state nucleus
  nucleon-nucleon interactions are negligible
• Implication mean free path of ground state
  nucleon is approximately equal to the nuclear
  diameter
• Experimental data does not support this
  conclusion!!!

51
Shell Model Evidence - Nucleon Interactions

• Scattering experiments show frequent elastic
  collisions
• Implication mean free path ltlt nuclear radius
• Explanation Pauli exclusion principle prohibits
  more than two protons or neutrons from occupying
  the same orbit (protons and neutrons are
  fermions)
• Why Pauli? nucleon-nucleon collisions result in
  momentum transfer between the participants BUT
  all lower energy quantum states are filled ?
  occurrence forbidden
• Severely limits nucleon-nucleon collision rate

52
Nuclear Potential Well

• Nucleon orbit nucleon quantum state
• Similar to quantum state of valence electron
• BUT
• Nucleon feels average total effect of
  interactions of all nucleons
• Implication nuclear potential is the same for
  all nucleons
• Strong Force
• All nucleons (regardless of their electrical
  charge) attract one another
• Attractive force is short range and falls rapidly
  to zero outside of the nuclear boundary (1 fm)

53
Nuclear Potential - Protons

• Protons do experience a Coulomb barrier ? a
  proton must have kinetic energy equal or greater
  than ECoul to penetrate the nucleus
• If Eprotonlt ECoul proton will back scatter
• Inside nucleus proton experiences attractive
  strong force and becomes bound
• To escape the nucleus a protons kinetic energy
  must be greater than or equal to ECoul (in the
  absence of quantum tunneling)

54
Nuclear Potential - Neutrons

• Neutrons do not experience Coulomb barrier ? even
  thermal neutrons (low kinetic energy) can
  penetrate the nucleus
• Inside nucleus neutron experiences attractive
  strong force and becomes bound
• To escape the nucleus a neutrons kinetic energy
  must be greater than or equal to the nuclear
  potential at the surface of the nucleus

55
Nuclear Potential Well
Depth of well represents binding energy
56
Nuclear Potential Functions

• Square Well Potential
• Harmonic Oscillator Potential
• Woods-Saxon
• Exponential Potential
• Gaussian Potential
• Yukawa Potential

Note R nuclear radius r distance from center
of nucleus
57
Nuclear Potential Functions
Exact shape of well is uncertain and depends on
mathematical function assumed for the interaction
Yukawa
Exponential
Gaussian
Square Well
58
Neutron vs Proton Potential Wells
Coulomb repulsion prevents potential well from
being as deep for protons as for neutrons
59
Quantized Energy Levels

• Schrodinger Equation developed to find wave
  functions and energies of molecules also can be
  applied to the nucleus
• Choose functional form of nuclear potential well
  and solve Schrodinger Equation
• H? E ?
• Wave equation allows only certain energy states
  defined by quantum numbers
• n principal quantum number, related to total
  energy of the system
• l azimuthal (radial) quantum number, related to
  rotational motion of nucleus
• ms spin quantum number, intrinsic rotation of a
  body around its own axis

60
Angular Momentum

• Associated with the rotational motion of an
  object
• Like linear motion, rotational motion also has an
  associated momentum
• Orbital angular momentum
• pl mvrr
• Spin angular momentum
• ps ?Irot
• A vector quantity ? always has a distinct
  orientation in space

61
Magnetic Quantum Effects - Spin

• A rotating charge gives rise to a magnetic moment
  (?s).
• Electrons and protons can ? be conceptualized as
  small magnets
• Neutrons have internal charge structure and can
  also be treated as magnets
• In the absence of a B-field magnets are
  disoriented in space (can point any direction)
• In the presence of a B-field the electron, proton
  and neutron spins are oriented in specific
  directions based upon quantum mechanical rules

62
Spin Angular Momentum
No External B-field
Applied External B-field

Project spin angular momentum onto the field
axes Allowed values are units of hbar ps(z)
hbar ms
63
Spin Angular Momentum

• Quantum mechanics requires that the spin angular
  momentum of electrons, protons and neutrons must
  have the magnitude
• s is the spin quantum number
• For protons and neutrons (just like for
  electrons) spin is always ?1/2

64
Magnetic Quantum Effects - Orbitals

• The orbital movement of an atomic electron or a
  nucleon gives rise to another magnetic moment
  (?l)
• This magnetic moment also interacts with an
  external B-field in a similar manner to the spin
  magnetic moment
• Quantum mechanics governs how the orbital plane
  may be oriented in relation to the external field
• The orbital angular momentum vector (pl) can only
  be oriented such that its projection onto the
  z-axis (field axis) has values
• pl (z) hbar ml
• Where ml magnetic orbital quantum number
• ml -l, -l1, -l2.0. l-2, l-1, l

65
Orbital Angular Momentum

• Project orbital angular momentum onto the field
  axes
• Allowed values are units of hbar
• pl(z) hbar ml

66
Orbital Angular Momentum

• Quantum mechanics requires that the orbital
  angular momentum of electrons, protons and
  neutrons must have the magnitude
• l is the orbital quantum number
• Allowed values of l
• Nucleons 0 ? l ( or ) n
• Electrons 0 ? l lt (n-1)
• For nucleons (but not electrons) l can exceed n

67
Orbital Angular Momentum

• The numerical values of the orbital angular
  momentum quantum number (l) are designated by the
  familiar spectroscopic notation
• Remember l can only have positive integral
  values (including 0)

68
Quantum States
69
Energy Level Diagram
Isotropic Harmonic Oscillator Levels
70
Problem

• Using the harmonic oscillator energy levels what
  is the level ordering for
• 94Be
• 3115P
• 5927Co

71
Problem

• Using the harmonic oscillator energy levels what
  is the level ordering for
• 94Be protons (1s2 1p2)
• neutrons (1s2 1p3)
• 3115P protons (1s2 1p6 1d7 )
• neutrons (1s2 1p6 1d8 )
• 5927Co protons (1s2 1p6 1d10 2s2 1f7 )
• neutrons (1s2 1p6 1d10 2s2 1f12 )

72
Spin Orbit Coupling

• Spin and orbital angular momenta are vector
  quantities ? vector coupling occurs to form a
  resultant vector pj
• Total angular momentum
• pj pl ps
• Coupling splits degeneracy of orbital angular
  momentum states

73
Spin Orbit Coupling

• For nucleons and electrons the orbital and spin
  angular momenta add vectorially to form a
  resultant vector (pj)
• pj pl ps
• The resultant is oriented towards an external
  magnetic field so that the projections on the
  field axis are
• pj(z) hbar mj
• The magnitude of pj is
• pj hbar j(j1)1/2
• j l ? s
• j is the total angular quantum number of the
  particle

74
Total Angular Momentum

• Total angular quantum number (j) can have two
  different values for each orbital quantum number
  (l)
• However, j can only have positive values!!
• Implication when l 0, only j1/2 is allowed
• All allowed values of j are half-integers
• j 1/2, 3/2, 5/2, 7/2

75
Problem

• What are the allowed values of j for a nucleon
  with
• l 1, s 1/2

76
Problem

• What are the allowed values of j for a nucleon
  with
• l 1, s 1/2
• Answer j 1/2 or 3/2

77
Total Angular Momentum

• Example n 1, l 1, s 1/2
• Standard atomic notation
• Electron in 1p1 state
• leading 1principal quantum number
• p orbital angular quantum number
• superscript 1 spin quantum number
• Standard nuclear notation
• Nucleon in 1p1/2 state or 1p3/2 state
• But we know that spin orbit coupling splits the
  degeneracy of the 6 existing p states
• This results in a two-fold degenerate 1p1/2 state
  and a four-fold degenerate 1p3/2 state
• Which is lower in energy????

78
Level Ordering of States

• Split degenerate states with higher j are always
  more stable than those with lower j
• Energetically
• 1p1/2 gt 1p3/2
• Neutrons and protons fill levels independently

79
Problem

• Including spin orbit coupling what is the level
  ordering for
• 94Be
• 3115P
• 5927Co

80
Problem

• Including spin orbit coupling what is the level
  ordering for
• 94Be protons (1s21/2 1p23/2)
• neutrons (1s21/2 1p33/2)
• 3115P protons (1s21/2 1p43/2 1p21/2 1d65/2
  2s11/2)
• neutrons (1s21/2 1p43/2 1p21/2 1d65/2
  2s21/2)
• 5927Co protons (1s21/2 1p43/2 1p21/2 1d65/2
  2s21/2 1d43/2 1f77/2)
• neutrons (1s21/2 1p43/2 1p21/2 1d65/2 2s21/2
  1d43/2 1f87/2 2p43/2)

81
Magnetic Quantum Numbers

• For each of the angular momentum quantum numbers
  (l, s, j) there exists a magnetic analogue (ml,
  ms, mj) representing the resolved component of
  the original quantum number along the axis of the
  applied magnetic field
• Each magnetic quantum number can be derived from
  the related quantum numbers
• ms
• Magnetic spin angular momentum quantum number
• has only 2 allowed values (?s)
• For protons, neutrons and electrons s 1/2
• ms 1/2 or -1/2

82
Magnetic Quantum Numbers

• ml
• Magnetic orbital angular momentum quantum number
• Has (2l1) possible values
• Can be positive or negative integral values
• ml -l, -l1, -l2.0. l-2, l-1, l
• Example l 3 or p orbital
• (231) 7 allowed values
• ml -3, -2, -1, 0, 1, 2, 3
• mj
• Magnetic total angular momentum quantum number
• Has (2j1) possible values
• Can be positive or negative integral values
• mj -j, -j1, -j2.0. j-2, j-1, j

83
Nucleon Total Angular Momentum

• Each nucleon has an associated orbital angular
  momentum and an associated spin angular momentum
• The total angular momentum quantum number of the
  nucleon is given by
• j l s
• The total angular momentum is
• pj hbarj(j1)1/2
• The observable maximum total angular momentum is
• pj hbar x j

84
Summary Single Particle Quantum Numbers

• Schrodinger Equation Solutions (H? E?)
• Fundamental Quantum Numbers
• n - principal s - spin (1/2)
• l - orbital j - total (l ? s)
• Magnetic Quantum Numbers
• ml - magnetic orbital (-l, -l1.0. l-1, l)
• ms - magnetic spin (1/2 or -1/2)
• mj - magnetic total (-j, -j1.0.j-1, j)

85
Nucleus Total Angular Momentum

• When two or more nucleons come together to form a
  nucleus the momentum components of the individual
  particles interact to give a resultant total
  angular momentum characteristic of the nucleus
• The energy level of the nucleus as a whole is
  represented by the resultant (I)
• I is historically referred to (inappropriately)
  as the spin of the nucleus BUT do not confuse it
  with the spin quantum number of a nucleon (s)
• pI hbar I(I1)1/2
• The observable maximum value of the total nuclear
  angular momentum is pI hbar x I

86
Nucleus Total Angular Momentum

• Nucleon-nucleon coupling of the spin and orbital
  motions of the individual nucleons is not clearly
  understood
• Two limiting coupling modes exist
• LS coupling
• jj coupling (dominant)
• In reality the coupling probably lies in between
  the two models

87
LS Coupling

• Also known as Russell-Saunders coupling
• The interaction of the orbital motion of a
  nucleon with its own spin is considered to be
  weak or negligible
• The orbital motions of different nucleons
  interact strongly with each other
• The resultant total orbital angular momentum of
  the nucleus is represented by L and is the vector
  sum of the individual nucleons
• L ? li
• Allowed values of L 0, 1, 2, 3
• Common symbol notation S, P, D, F

88
LS Coupling

• The spin motions of different nucleons interact
  strongly with each other
• The resultant total spin angular momentum of the
  nucleus is represented by S and is the vector sum
  of the spins of the individual nucleons
• S ? si
• The total angular momentum of the nucleus is
  represented by I (or sometimes J)
• I L ? S (hence the name LS coupling!)
• The sign of S is determined by whether S and L
  are parallel or antiparallel

89
LS Coupling - Application

• The individual orbital and spin angular momenta
  of paired nucleons cancel each other and do not
  contribute to the total nuclear angular momentum
• Even-even nuclei have zero nuclear spin (I0)
• Even-Odd or Odd-Even
• Nuclear spin determined by the single unpaired
  nucleon
• I (even-odd) jp or jn 1/2, 3/2, 5/2, 7/2,
  9/2
• Odd-odd nuclei
• Nuclear spin determined by combination of
  unpaired nucleons
• I(odd-odd) j1 - j2 ? j1 j2
• I(odd-odd) 1, 2, 3, 4, 5.

90
LS Coupling - Application

• Steps to determine nuclear spin state
• Determine if nucleus is even-even, even-odd,
  odd-even or odd-odd
• If even-even I0
• If N and/or Z are odd continue with step 2 for
  the odd nucleon(s)
• Fill energy level diagram for odd nucleon
• If odd-odd remember to fill the levels
  independently for protons and neutrons
• Find the value of j for the energy level occupied
  by the unpaired nucleon
• For an even-odd or odd-even system this value is
  the total nuclear spin
• For an odd-odd nucleus the model can not predict
  the overall state nuclear spins can range in
  value from
• j1 - j2 to j1 j2

91
Example

• What is the nuclear spin for
• 94Be protons (1s21/2 2p23/2)
• neutrons (1s21/2 2p33/2)
• 3115P protons (1s21/2 2p43/2 1p21/2 1d65/2
  2s11/2)
• neutrons (1s21/2 2p43/2 1p21/2 1d65/2
  2s21/2)
• 5927Co protons (1s21/2 2p43/2 1p21/2 1d65/2
  2s21/2 1d43/2 1f77/2)
• neutrons (1s21/2 2p43/2 1p21/2 1d65/2 2s21/2
  1d43/2 1f87/2 2p43/2)

92
Example

• What is the nuclear spin for (I)
• 94Be protons (1s21/2 2p23/2) (0)
• neutrons (1s21/2 2p33/2) (3/2)
• 3115P protons (1s21/2 2p43/2 1p21/2 1d65/2
  2s11/2) (1/2)
• neutrons (1s21/2 2p43/2 1p21/2 1d65/2
  2s21/2) (0)
• 5927Co protons (1s21/2 2p43/2 1p21/2 1d65/2
  2s21/2 1d43/2 1f77/2) (7/2)
• neutrons (1s21/2 2p43/2 1p21/2 1d65/2 2s21/2
  1d43/2 1f87/2 2p43/2) (0)

93
jj Coupling

• Complementary to LS coupling
• Considers affect of strong spin-orbit coupling
  for individual nucleons in a nucleus
• The orbital and spin motions of the same nucleon
  may interact strongly
• Total nuclear spin (vector sum)
• I j1j2j3..
• where ji li si

94
LS vs jj Coupling

• These models represent two extremes of a coupling
  that in reality is most accurately represented as
  a continuum
• In general,
• jj coupling preferred for very heavy nuclei
• Light nuclei are a mixture

95
Nordheims Rules(spins of odd-odd nuclei)

• Only 5 stable odd-odd nuclei
• 21 H, 63Li, 105B, 147N, 5023Va (could have large
  t1/2)
• When dealing with two unpaired nucleons
  predicting the spin of the nucleus is less
  certain
• Nordheims Rules generally give the correct
  estimate

96
Nordheims Rules (Spins of odd-odd nuclei)

• Given and unpaired proton and neutron having
  orbital angular momenta l1 and l2 and total
  angular momenta j1 and j2
• If (j1 j2 l1 l2) even
• I j1 - j2
• If (j1 j2 l1 l2) odd
• I j1 j2

97
Problem

• What is the spin of the 76As nucleus?

98
Problem

• What is the spin of the 76As nucleus?
• Answer
• 76As (Z33, N43)
• 33rd proton f5/2
• 43rd neutron g9/2
• (j1 j2 l1 l2) (5/29/234) 14
• I j1 - j2 5/2 - 9/2 2

99
What is Parity?

• In physics parity is the name of the symmetry of
  interactions under spatial inversion.
• For all phenomena the principle of left-right or
  top-bottom symmetry as mirror images exits
• For any event the mirror image is also possible
  and would be governed by the same physical laws
• ie the laws of nature are invariant under
  reflection

100
Parity

• A conserved quantity in nuclear reactions
  involving the emission of photons and nucleons
• Nuclear property related to the symmetry
  properties of the wave function
• Parity is odd(-) or even() based on whether or
  not the wave function is symmetric
• To test for symmetry reverse the signs of all of
  the spatial coordinates in the function
• If resultant solution changes sign gt parity -
• ?(-x,-y,-z) -?(x,y,z)
• If resultant solution remains the same gt parity
  
• ?(x,y,z) ?(-x,-y,-z)
• Parity rules for combining wave functions
• - -
• --

101
Parity

• Orbital angular momentum states result from
  solutions of the Schrodinger equation ? they are
  wave functions with an associated parity
• Simple rules
• Even-even nuclei have a ground state () parity
• Even-odd and odd-even nuclei have a parity equal
  to that of the wave function of the unpaired
  nucleon
• P (-1)l
• Odd-odd nuclei gt parity is the product of the
  wave functions of the unpaired nucleons (slide 96)

102
Example

• What is the parity for (I)
• 94Be protons (1s21/2 2p23/2) (0)
• neutrons (1s21/2 2p33/2) (3/2)
• 3115P protons (1s21/2 2p43/2 1p21/2 1d65/2
  2s11/2) (1/2)
• neutrons (1s21/2 2p43/2 1p21/2 1d65/2
  2s21/2) (0)
• 5927Co protons (1s21/2 2p43/2 1p21/2 1d65/2
  2s21/2 1d43/2 1f77/2) (7/2)
• neutrons (1s21/2 2p43/2 1p21/2 1d65/2 2s21/2
  1d43/2 1f87/2 2p43/2) (0)

103
Example

• What is the parity for (I) (?)
• 94Be protons (1s21/2 2p23/2) (0)
• neutrons (1s21/2 2p33/2) (3/2) (-)
• 3115P protons (1s21/2 2p43/2 1p21/2 1d65/2
  2s11/2) (1/2) ()
• neutrons (1s21/2 2p43/2 1p21/2 1d65/2
  2s21/2) (0)
• 5927Co protons (1s21/2 2p43/2 1p21/2 1d65/2
  2s21/2 1d43/2 1f77/2) (7/2) (-)
• neutrons (1s21/2 2p43/2 1p21/2 1d65/2 2s21/2
  1d43/2 1f87/2 2p43/2) (0)

104
Nuclear Spin and Parity

• It is customary to represent the nuclear spin and
  parity together
• For example in the nuclide 178O the odd nucleon
  is the 9th neutron which occupies the 1d5/2 state
• Spin 5/2
• Parity
• Standard notation to say ground state of 178O has
  a spin and parity of 5/2

105
Example

• What is the standard spin and parity notation
  for
• 94Be 3/2-
• 3115P 1/2
• 5927Co 7/2-

106
Parity Conservation

• Parity is found to be conserved in all strong
  interactions involving the emission of photons
  and nucleons
• In weak interactions involving electrons,
  neutrinos and mesons there is evidence of
  non-conservation of parity
• The ?- decay of 60Co in a strong magnetic field
  at low temperature is found to emit particle
  anisotropically gt ie no symmetry
• Except in weak interactions parity is always
  conserved

107
Parity Violation
Mirror Image
isotropic
anisotropic

• Symmetric (isotropic) emission gt parity
  conserved
• Asymmetric (anisotropic) emission gt parity
  violated

108
Magnetic Total Nuclear Momentum Quantum Number

• Analogous to the magnetic quantum numbers of the
  individual nucleons, the nucleus as a whole has a
  magnetic total nuclear momentum quantum number
• Represented by mI
• Is the projection of the total nuclear momentum
  on the field axis
• mI has 2I1 allowed values
• mI -I, -(I-1).-1, 0, 1.(I-1), I

109
Magnetic Moments

• Magnetic Moment (?i)
• A nuclear parameter dependent upon the underlying
  nuclear structure
• a measure of the response of the nucleus to an
  external magnetic field
• Net effect of the motion of the protons plus the
  intrinsic spins of the protons and neutrons
• Can be measured experimentally

110
Magnetic Moments

• Magnetic Moment
• gl and gs are the gyromagnetic ratios

111
Magnetic Moments

• Application
• The strong coupling of nucleons due to filled
  orbitals and paired spins results in the
  cancellation of the spins and angular momenta
• Consequently, the magnetic moments will be small
  and strongly dependent upon the unpaired nucleons
• Even-even gt magnetic moment is 0
• Odd-even and even-odd nuclei gt use Schmidt Limit
• j l s ? l gl 1/2gs
• j l - s ? (j/j1)(l1)gl - 1/2gs
• Odd-odd gt no prediction available

112
Neutron Magnetic Moment

• The existence of a magnetic moment for the
  neutron suggests a complex structure for the
  particle
• The neutron is believed to be composed of equal
  amounts of positive and negative charges
• The negative charge is on average further from
  the spin axis than the positive charge thus
  leading to a large negative magnetic moment
• The large gs factors for the proton and neutron
  support the theory of a complex distribution of
  charge within the nucleons

113
Nucleon Magnetic Moments

• The proton and neutron differ only in the middle
  and outermost regions of the nucleus
• In the outermost diffuse region the proton is
  positively charged and the neutron is negatively
  charged
• The inner core of all nucleons is the same
• In contrast, the electron is an elementary
  particle with the same center of mass and charge

114
Nuclear Charge Distribution

• To understand the structure of the nucleus it is
  important to know how the protons are spatially
  distributed
• The presence of a single proton displaced from
  the center of the nucleus is important because it
  can result in an effective potential experienced
  beyond the walls of the nucleus
• Spherical nuclei act as magnetic monopoles
• Deformed nuclei can have the properties of
  dipoles, quadrupoles, octupoles, etc
• These electrical moments can be predicted if the
  nuclear spin (I) of the nucleus is known
• Monopole (I0), dipole (I1/2), quadrupole (I1)

(Most common)
115
Deformed Nuclei

• Liquid-drop model and shell model both assume a
  spherically symmetric nucleus
• This is a reasonable assumption for magic number
  nuclei
• But other nuclei have distorted shapes
• Magnitude of deviation from spherical is
  quantized by
• ? 2(a-c)/(ac)
• where a and c are the radius with respect to the
  z and x axes (see diagram slide 116)
• Two types of deformed nuclei depending on which
  axis is compressed
• Prolate ? gt 0
• Oblate ? lt 0
• Maximum observed value of ? ? 0.6

116
Deformed Nuclei
z
Oblate Spins on short axis
-x
y
Prolate Spins on long axis
117
Nuclear Quadrupole Moment

• Nuclei with quadrupole moments (I1) are common
• Let a nucleus with finite quadrupole moment be
  represented by an ellipsoid with the semi-axis
  (b) parallel to the symmetry axis (Z) and the
  semi-axis (a) perpendicular
• If the total charge of the nucleus (Ze) is
  assumed to be uniformly distributed throughout
  the ellipsoid, the quadrupole moment of the
  nucleus in the Z direction can be calculated
  using

a
b
Z
118
Collective Nuclear Model

• Proposed by Bohr Mottelsen in 1953
• Models nucleus as a highly compressed liquid that
  can experience internal rotations and vibrations
• Includes 4 discrete modes of collective motion
• Rotate around z-axis
• Rotate around y-axis
• Oscillate between prolate and oblate
• Vibrate along x-axis
• Used to calculate allowed vibrational and
  rotational levels between standard nuclear levels
• These new levels are equivalent to excited states
• Validity depends on whether each mode of
  collective motion can be treated independently
• Works best for strongly deformed nuclei (238U)

119
Collective Nuclear Model Levels
120
Unified Model of Deformed Nuclei

• Shell model suffers from discrepancies between
  experimental and theoretical spin states for
  certain nuclei
• Angular momentum of odd-A deformed nuclei
  contains components from deformed core and
  unpaired nucleon
• Results in a modification of the energy levels
  that changes their ordering

121
Nilsson Levels

• Nilsson calculated energy levels of odd-A nuclei
  as a function of the nuclear deformation (?)
• Each j state from the shell model is split into
  j1/2 levels and may contain a maximum of 2
  nucleons
• Some undeformed levels also reverse order
  (1f5/2?2p3/2)
• Nilsson levels predict energies, angular momenta,
  quantum numbers, etc better for deformed nuclei
  than the shell model

122
Nilsson Diagram (Z or N? 50)
123
Nilsson Diagram (50 ? Z or N? 82)
124
Problem

• Given a nuclear deformation of 0.11, find the
  spin and parity of 23Na using the shell model and
  the Nilsson diagram.
• Which state does the chart of the nuclides
  confirm exists in nature?

125
Problem

• Given a nuclear deformation of 0.11, find the
  spin and parity of 23Na using the shell model and
  the Nilsson diagram.
• Shell model 5/2
• Nilsson 3/2
• Which state does the chart of the nuclides
  confirm exists in nature? 3/2

126
Standard Model Particles

• Provides a description of the fundamental
  particles and forces that govern matter
• Quarks and leptons as identified as the
  elementary particles
• Each quark and lepton has an antimatter partner
  which is referred to as an antiquark or
  antilepton

127
The Quarks

• Standard model describes fundamental particles
  and forces
• Three families of quarks
• Spin 1/2 fermions
• Carry electric and color charge
• Exist in the bound state as hadrons
• Baryon 3 bound quarks
• Meson 2 bound quarks
• Never observed in isolation
• Naturally occur in three families

128
Elementary Particles Quarks

• We know that the nucleus of an atom is composed
  of nucleons (protons neutrons)
• But these nucleons also have a quark substructure
• Proton uud
• Neutron udd
• The antimatter equivalent to the proton is the
  antiproton (uud)
• The most common mesons are pions and kaons
• ? ud, ?- ud, K us, K- us

129
Elementary Particles Leptons

• Spin 1/2 fermions
• Point-like gt no substructure
• Never bound

130
Standard Model - Forces

• Standard model includes the forces that govern
  the interactions between matter
• Each force is conveyed by a mediating (or
  exchange) particle
• Weak force governs radioactive decay
• Strong force binds quarks in hadrons and nucleons
  in the nucleus
• Gravitational force has not yet been incorporated
  into the standard model

131
Quantum Chromodynamics

• Theory that describes the properties of the
  strong force
• Color property associated with interaction
  (analogous to electric charge)
• Every quark carries a color charge of red or
  green or blue
• Every gluon (exchange particle) also carries a
  color charge

132
Quantum Chromodynamics

• Coupling between color carriers INCREASES with
  distance
• (opposite behavior to the more familiar
  electromagnetic force)
• Confinement
• At large distances the QCD potential is large and
  confines quarks inside bound state ? it is not
  possible to separate bound quarks
• Asymptotic Freedom
• At very small distances the QCD potential is weak
  and quarks behave as if they are unbound

133
What is confinement?
Confinement
QCD Potential

• QCD is a confining gauge theory,with an
  effective potential.
• Energy required to separate quarks is greater
  than the pair rest mass.
• No one has ever seen a free quark.

• QCD potential between color carriers increases
  linearly with distance

134
What is confinement?
Confinement
QCD Potential

• QCD is a confining gauge theory,with an
  effective potential.
• Energy required to separate quarks is greater
  than the pair rest mass.
• No one has ever seen a free quark.

• QCD potential between color carriers increases
  linearly with distance

135
What is asymptotic freedom?
QCD Potential

• QCD potential weakens at small distances (or lg
  momentum transfers Q)
• This allows perturbative calculations to be used
  to predict the system behavior

136
What is asymptotic freedom?
QCD Potential

• QCD potential weakens at small distances (or lg
  momentum transfers Q)
• This allows perturbative calculations to be used
  to predict the system behavior

137
Asymptotic Freedom

• Quarks behave as if they are unbound or free when
  separated by only very small distances
• Theory tells us that it might be possible to
  achieve this state in systems of extreme
  temperature and/or density
• It is this deconfined state that is known as the
  Quark Gluon Plasma (QGP)
• Conceptually the QGP can be visualized as a soup
  of freely moving quarks and gluons

138
Isospin