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 = x C

9. Mass It is convenient to use atomic mass units, u, to express masses
1 u = x 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 = 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 m
For silver, he found d = 2 x 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 m
He called this sphere the nucleus
These small lengths are often expressed in femtometers where 1 fm = 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 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 > 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 > 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 mn
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 – mn
The negative sign indicates the neutron’s 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

30. 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

31. 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 > 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

35. 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

36. 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
λ 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 t=0

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- λt
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

53. 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 doesn’t change Z, N, or A
The emitted photon has an energy of hƒ equal to DE between the two nuclear energy levels

60. 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 < 15 x 106 K) take place through the proton-proton cycle
In stars with hotter cores (T > 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