Nuclear Physics

1
Chapter 30

• Nuclear Physics

2
Milestones in the Development of Nuclear Physics

• 1896 the birth of nuclear physics
• Becquerel discovered radioactivity in uranium
  compounds
• Rutherford showed the radiation had three types
• Alpha (He nucleus)
• Beta (electrons)
• Gamma (high-energy photons)

3
More Milestones

• 1911 Rutherford, Geiger and Marsden performed
  scattering experiments
• Established the nucleus could be treated as a
  point mass and a point charge
• Most of the atomic mass was contained in the
  nucleus
• Nuclear force was a new type of force

4
Some Properties of Nuclei

• All nuclei are composed of protons and neutrons
• Exception is ordinary hydrogen with just a proton
• The atomic number, Z, equals the number of
  protons in the nucleus
• Sometimes called the charge number
• The neutron number, N, is the number of neutrons
  in the nucleus

5
More Properties of Nuclei

• The mass number, A, is the number of nucleons in
  the nucleus
• A Z N
• Nucleon is a generic term used to refer to either
  a proton or a neutron
• The mass number is not the same as the mass

6
Symbolism

• X is the chemical symbol of the element
• Example
• Mass number is 56
• Atomic number is 26
• Contains 26 protons
• Contains 30 (56-26) neutrons
• The Z may be omitted since the element can be
  used to determine Z

7
More Properties

• The nuclei of all atoms of a particular element
  must contain the same number of protons
• They may contain varying numbers of neutrons
• Isotopes of an element have the same Z but
  differing N and A values
• The natural abundance of isotopes can vary
• Example

8
Charge

• The proton has a single positive charge, e
• The electron has a single negative charge, -e
• The neutron has no charge
• Makes it difficult to detect
• e 1.60217733 x 10-19 C

9
Mass

• It is convenient to use atomic mass units, u, to
  express masses
• 1 u 1.660539 x 10-27 kg
• Based on definition that the mass of one atom of
  C-12 is exactly 12 u
• Mass can also be expressed in MeV/c2
• From ER m c2
• 1 u 931.494 MeV/c2

10
Some Masses in Various Units
11
The Size of the Nucleus

• First investigated by Rutherford in scattering
  experiments
• He found an expression for how close an alpha
  particle moving toward the nucleus can come
  before being turned around by the Coulomb force
• From Conservation of Energy, the kinetic energy
  of the particle must be completely converted to
  potential energy

12
Size of the Nucleus, cont

• d is called the distance of closest approach
• d gives an upper limit for the size of the
  nucleus
• Rutherford determined that
• For gold, he found d 3.2 x 10-14 m
• For silver, he found d 2 x 10-14 m

13
More About Size

• Rutherford concluded that the positive charge of
  the atom was concentrated in a sphere whose
  radius was no larger than about 10-14 m
• He called this sphere the nucleus
• These small lengths are often expressed in
  femtometers where 1 fm 10-15 m
• Also called a fermi

14
Size of Nucleus, Final

• Since the time of Rutherford, many other
  experiments have concluded the following
• Most nuclei are approximately spherical
• Average radius is
• ro 1.2 x 10-15 m
• A is the mass number

15
Density of Nuclei

• The volume of the nucleus (assumed to be
  spherical) is directly proportional to the total
  number of nucleons
• This suggests that all nuclei have nearly the
  same density
• Since r3 would be proportional to A
• Nucleons combine to form a nucleus as though they
  were tightly packed spheres

16
Nuclear Stability

• There are very large repulsive electrostatic
  forces between protons
• These forces should cause the nucleus to fly
  apart
• The nuclei are stable because of the presence of
  another, short-range force, called the nuclear
  force
• This is an attractive force that acts between all
  nuclear particles
• The nuclear attractive force is stronger than the
  Coulomb repulsive force at the short ranges
  within the nucleus

17
Features of the Nuclear Force

• Attractive force that acts between all nuclear
  particles
• It is the strongest force in nature
• Very short range
• It falls to zero when the separation between
  particles exceeds about several fermis
• Independent of charge
• The nuclear force on p-p, p-n, n-n are all the
  same
• Does not affect electrons

18
Nuclear Stability, cont

• Light nuclei are most stable if N Z
• Heavy nuclei are most stable when N gt Z
• Above about Z 20
• As the number of protons increases, the Coulomb
  force increases and so more neutrons are needed
  to keep the nucleus stable
• No nuclei are stable when Z gt 83

19
Magic Numbers

• Most stable nuclei have even numbers of A
• Certain values of N and Z correspond to unusually
  high stability
• These values are called magic numbers
• N or Z 2, 8, 20, 28, 50, 82, 126
• For example, He has N 2 and Z 2 and is very
  stable

20
Nuclear Spin

• Protons and neutrons have intrinsic angular
  momentum called spin
• A nucleus has a net intrinsic angular momentum
  that arises from the individual spins of the
  protons and neutrons
• This angular momentum must obey the same quantum
  rules as orbital angular momentum

21
Nuclear Angular Momentum

• The magnitude of the nuclear angular momentum is
  due to the combination of all nucleons
• It is equal to
• I is called the nuclear spin quantum number
• It may be an integer or half-integer

22
Possible Orientations of Nuclear Spin

• Shown is a vector model giving possible
  orientations of the spin and its projection on
  the z axis
• This is for the case where I 3/2

23
Nuclear Magneton

• The nuclear angular momentum has a nuclear
  magnetic moment associated with it
• The magnetic moment is measured in terms of the
  nuclear magneton, mn
• Note that mn is smaller than mB by a factor of
  about 2000, due to the difference in mass between
  the electron and the proton

24
Nuclear Magnetic Moment, final

• The magnetic moment of a free proton is 2.792 8mn
• No general theory of nuclear magnetism explains
  this value
• The neutron also has a magnetic moment, even
  though it has no charge
• The magnetic moment of the neutron is
  1.913 5mn
• The negative sign indicates the neutrons
  magnetic moment is opposite its spin angular
  momentum

25
Nuclear Magnetic Resonance (NMR)

• Because the direction of the magnetic moment for
  a particle is quantized, the energies of the
  particle are also quantized
• The spin vector cannot align exactly with the
  direction of the magnetic field
• This gives the extreme values of the energy as
  mzB
• mz is the z component of the magnetic moment

26
NMR, cont

• These states are often called spin states
• It is possible to observe transitions between two
  spin states using NMR
• The magnetic field splits the states

27
NMR, final

• The sample is irradiated with electromagnetic
  waves in the radio range of the em spectrum
• The frequency of the radio waves is adjusted so
  that the photon energy matches the separation
  energy between spin states
• There is a net absorption of energy in the system
  which is detected by experimental control and
  measuring systems

28
MRI

• An MRI (Magnetic Resonance Imaging) is based on
  NMR
• Because of variations in an external field,
  protons in different parts of the body have
  different splittings in energy between spin
  states
• The resonance signal can provide information
  about the positions of the protons

29
Binding Energy

• The total rest energy of the bound system (the
  nucleus) is less than the combined rest energy of
  the separated nucleons
• This difference in energy is called the binding
  energy of the nucleus
• It can be thought of as the amount of energy you
  need to add to the nucleus to break it apart into
  its components

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Binding Energy, cont

• The binding energy can be calculated
• The masses are expressed in atomic mass units
• M(H) is the atomic mass of the neutral hydrogen
  atom
• mn is the mass of the neutron
• M(X) is the mass of that isotope

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Binding Energy per Nucleon
32
Notes from the Graph

• The curve peaks in the vicinity of A 60
• Nuclei with mass numbers greater than or less
  than 60 are not as strongly bound as those near
  the middle of the periodic table
• The binding energy is about 8 MeV per nucleon for
  nuclei with A gt 50
• This suggests that the nuclear force saturates
• A particular nucleon can interact with only a
  limited number of other nucleons

33
Marie Curie

• 1867 1934
• Shared Nobel Prize in 1903 for studies in
  radioactive substances
• Prize in physics
• Shared with Pierre Curie and Becquerel
• Won Noble Prize in 1911 for discovery of radium
  and polonium
• Prize in chemistry

34
Radioactivity

• Radioactivity is the spontaneous emission of
  radiation
• Discovered by Becquerel in 1896
• Many experiments were conducted by Becquerel and
  the Curies
• Experiments suggested that radioactivity was the
  result of the decay, or disintegration, of
  unstable nuclei

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Radioactivity Types

• Three types of radiation can be emitted
• Alpha particles
• The particles are 4He nuclei
• Beta particles
• The particles are either electrons or positrons
• A positron is the antiparticle of the electron
• It is similar to the electron except its charge
  is e
• Gamma rays
• The rays are high energy photons

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Distinguishing Types of Radiation

• The gamma particles carry no charge
• The alpha particles are deflected upward
• The beta particles are deflected downward
• A positron would be deflected upward, but would
  follow a different trajectory than the a due to
  its mass

37
Penetrating Ability of Particles

• Alpha particles
• Barely penetrate a piece of paper
• Beta particles
• Can penetrate a few mm of aluminum
• Gamma rays
• Can penetrate several cm of lead

38
The Decay Constant

• The rate at which a decay process occurs is
  proportional to the radioactive nuclei present in
  the sample
• ? is called the decay constant and determines
  the rate at which the material will decay
• N is the number of undecayed radioactive nuclei
  present
• No is the number of undecayed nuclei at time t0

39
Decay Rate

• The decay rate, R, of a sample is defined as the
  number of decays per second
• Ro Nol is the decay rate at t o
• The decay rate is often referred to as the
  activity of the sample

40
Decay Curve

• The decay curve follows the equation N No e- ?t
  
• The half-life is also a useful parameter
• The half-life is defined as the time interval
  during which half of a given number of
  radioactive nuclei decay

41
Decay Processes

• The blue circles are the stable nuclei seen
  before
• Above the line the nuclei are neutron rich and
  undergo beta decay (red)
• Just below the line are proton rich nuclei that
  undergo beta (positron) emission or electron
  capture (green)
• Farther below the line the nuclei are very proton
  rich and undergo alpha decay (yellow)

42
Alpha Decay

• When a nucleus emits an alpha particle it loses
  two protons and two neutrons
• N decreases by 2
• Z decreases by 2
• A decreases by 4
• Symbolically
• X is called the parent nucleus
• Y is called the daughter nucleus

43
Decay General Rules

• When one element changes into another element,
  the process is called spontaneous decay or
  transmutation
• The sum of the mass numbers, A, must be the same
  on both sides of the equation
• The sum of the atomic numbers, Z, must be the
  same on both sides of the equation
• The total energy and total momentum of the system
  must be conserved in the decay

44
Disintegration Energy

• The disintegration energy, Q of a system is
  defined as
• Q (Mx My Ma) c2
• The disintegration energy appears in the form of
  kinetic energy in the daughter nucleus and the
  alpha particle
• It is sometimes referred to as the Q value of the
  nuclear decay

45
Alpha Decay Example

• Decay of 226 Ra
• If the parent is at rest before the decay, the
  total kinetic energy of the products is 4.87 MeV
• In general, less massive particles carry off more
  of the kinetic energy

46
Alpha Decay, Notes

• Experimental observations of alpha-particle
  energies show a number of discrete energies
  instead of a single value
• The daughter nucleus may be left in an excited
  quantum state
• So, not all of the energy is available as kinetic
  energy
• A negative Q value indicates that such a proposed
  decay does not occur spontaneously

47
Alpha Decay, Mechanism

• In alpha decay, the alpha particle tunnels
  through a barrier
• For higher energy particles, the barrier is
  narrower and the probability is higher for
  tunneling across
• This higher probability translates into a shorter
  half-life of the parent

48
Beta Decay

• During beta decay, the daughter nucleus has the
  same number of nucleons as the parent, but the
  atomic number is changed by one
• Symbolically
• Beta decay is not completely described by these
  equations

49
Beta Decay, cont

• The emission of the electron or positron is from
  the nucleus
• The nucleus contains protons and neutrons
• The process occurs when a neutron is transformed
  into a proton or a proton changes into a neutron
• The electron or positron is created in the
  process of the decay
• Energy must be conserved

50
Beta Decay Particle Energy

• The energy released in the decay process should
  almost all go to kinetic energy of the b particle
• Since the decaying nuclei all have the same rest
  mass, the Q value should be the same for all
  decays
• Experiments showed a range in the amount of
  kinetic energy of the emitted particles

51
Neutrino

• To account for this missing energy and
  momentum, in 1930 Pauli proposed the existence of
  another particle
• Enrico Fermi later named this particle the
  neutrino
• Properties of the neutrino
• Zero electrical charge
• Mass much smaller than the electron, probably not
  zero but less than 2.8 eV/c2
• Spin of ½
• Very weak interaction with matter and so is
  difficult to detect

52
Beta Decay Completed

• Symbolically
• ? is the symbol for the neutrino
• is the symbol for the antineutrino
• To summarize, in beta decay, the following pairs
  of particles are emitted
• An electron and an antineutrino
• A positron and a neutrino

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Beta Decay Examples
54
Beta Decay, Final Notes

• The fundamental process of e- decay is a neutron
  changing into a proton, an electron and an
  antineutrino
• In e, the proton changes into a neutron,
  positron and neutrino
• This can only occur within a nucleus
• It cannot occur for an isolated proton since its
  mass is less than the mass of the neutron

55
Electron Capture

• Electron capture is a process that competes with
  e decay
• In this case, a parent nucleus captures one of
  its own orbital electrons and emits a neutrino
• In most cases, a K shell electron is captured, a
  process often referred to as K capture

56
Carbon Dating

• Beta decay of C-14 is used in dating organic
  materials
• The process depends on the ratio of C-14 to C-12
  in the atmosphere which is relatively constant
• When an organism dies, the ratio decreases as a
  result of the beta decay of the C-14

57
Enrico Fermi

• 1901 1954
• Awarded Nobel Prize in physics in 1938 for
  producing transunranic elements and discovery of
  nuclear reactions produced by slow neutrons
• Other contributions include
• Theory of beta decay
• Free electron theory of metals
• Development of first fission reactor (1942)

58
Gamma Decay

• Gamma rays are given off when an excited nucleus
  decays to a lower energy state
• The decay occurs by emitting a high-energy photon
• The X indicates a nucleus in an excited state

59
Gamma Decay Example

• Example of a decay sequence
• The first decay is a beta emission
• The second step is a gamma emission
• Gamma emission doesnt change Z, N, or A
• The emitted photon has an energy of hƒ equal to
  DE between the two nuclear energy levels

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Summary of Decays
61
Nuclear Reactions

• Structure of nuclei can be changed by bombarding
  them with energetic particles
• The changes are called nuclear reactions
• As with nuclear decays, the atomic numbers and
  mass numbers must balance on both sides of the
  equation

62
Nuclear Reactions, cont

• A target nucleus, X, is bombarded by a particle
  a, resulting in a daughter nucleus Y and an
  outgoing particle b
• a X Y b
• The reaction energy Q is defined as the total
  change in mass-energy resulting from the reaction
• Q (Ma MX MY Mb) c2

63
Q Values for Reactions

• The Q value determines the type of reaction
• An exothermic reaction
• There is a mass loss in the reaction
• There is a release of energy
• Q is positive
• An endothermic reaction
• There is a gain of mass in the reaction
• Energy is needed, in the form of kinetic energy
  of the incoming particles
• Q is negative
• The minimum energy necessary for the reaction to
  occur is called the threshold energy

64
Nuclear Reactions, final

• If a and b are identical, so that X and Y are
  also necessarily identical, the reaction is
  called a scattering event
• If the kinetic energy before the event is the
  same as after, it is classified as elastic
  scattering
• If the kinetic energies before and after are not
  the same, it is an inelastic scattering

65
Conservation Rules for Nuclear Reactions

• The following must be conserved in any nuclear
  reaction
• Energy
• Momentum
• Total charge
• Total number of nucleons

66
Types of Nuclear Reactions

• One important feature of nuclear reactions is
  that much more energy is released than with
  normal chemical reactions
• Two types of reactions can occur
• Fission
• Fusion

67
Nuclear Fission

• A heavy nucleus splits into two smaller nuclei
• A fissionable nucleus (X) absorbs a slowly moving
  neutron (a) and splits into two smaller nuclei
  (Y1 and Y2), releasing energy and multiple
  neutrons (several particles b)
• With no means of control, a chain reaction
  explosion occurs
• With controls, the fission process is used in
  nuclear power generating plants

68
Chain Reaction Diagram
69
Nuclear Fusion

• Nuclear fusion occurs when two light nuclei
  combine to form a heavier nucleus
• This is difficult since the nuclei must overcome
  the Coulomb repulsion before they are close
  enough to fuse
• One way to do this is to cause them to move with
  high kinetic energy by raising them to a high
  temperature
• Also need a high density

70
Fusion in the Sun

• The centers of stars have high enough
  temperatures and densities to generate energy
  through fusion
• The Sun fuses hydrogen
• Some stars fuse heavier elements
• Reactions in cool stars (T lt 15 x 106 K) take
  place through the proton-proton cycle
• In stars with hotter cores (T gt 15 x 106 K), the
  carbon-cycle dominates

71
Proton-Proton Cycle

• The proton-proton cycle is a series of three
  nuclear reactions believed to operate in the Sun
• The net result is the joining of four protons to
  form a helium-4 nucleus
• Energy liberated is primarily in the form of em
  radiation (gamma rays), positrons and neutrinos

72
The Carbon Cycle

• Hydrogen nuclei can fuse into nuclei heavier than
  helium
• Multiple steps lead to the release of energy
• The net effect is that four hydrogen nuclei
  combine to form a helium-4 nucleus
• The Carbon-12 is returned at the end, it acts as
  a catalyst

73
Energy in a Star

• Depending on its mass, a star transforms energy
  at the rate of 1023 to 1033 W
• The energy from the core is transferred outward
  through the surrounding layers
• Neutrinos carry energy directly through these
  layers to space
• Energy carried by photons is absorbed by the
  gasses in the layer outside the core and
  gradually works it way to the surface by
  convection
• This energy is emitted from the surface in the
  form of em radiation, mainly infrared, visible
  and ultraviolet
• The star is stable as long as its supply of
  hydrogen in the core lasts