⚛️Nuclear Physics Unit 3 – Nuclear Models and Stability

Key Concepts and Terminology

• Atomic nucleus consists of protons and neutrons (nucleons) held together by the strong nuclear force
• Isotopes are atoms with the same number of protons but different numbers of neutrons
  • Isotopes of an element have the same chemical properties but different physical properties (mass, radioactivity)
• Mass defect is the difference between the mass of an atomic nucleus and the sum of the masses of its constituent protons and neutrons
• Binding energy is the energy required to disassemble a nucleus into its constituent protons and neutrons
  • Binding energy per nucleon is a measure of nuclear stability
• Magic numbers are specific numbers of protons or neutrons (2, 8, 20, 28, 50, 82, 126) that result in increased nuclear stability
• Radioactive decay is the spontaneous emission of particles or energy from an unstable atomic nucleus
  • Types of radioactive decay include alpha decay, beta decay, and gamma decay

Historical Development of Nuclear Models

• Early models of the atom (Thomson's plum pudding model, Rutherford's nuclear model) laid the groundwork for understanding atomic structure
• Discovery of the neutron by James Chadwick in 1932 led to the development of more accurate nuclear models
• Liquid drop model proposed by George Gamow in 1935 treated the nucleus as a drop of incompressible nuclear fluid
• Shell model developed by Maria Goeppert Mayer and J. Hans D. Jensen in 1949 introduced the concept of nucleons occupying discrete energy levels
• Collective model proposed by Aage Bohr and Ben Mottelson in 1953 described nuclei as having both individual particle and collective behaviors
• Interacting boson model developed by Akito Arima and Francesco Iachello in 1974 treated nucleons as pairs of interacting bosons
• Advancements in experimental techniques (particle accelerators, detectors) have enabled more precise measurements and refinements of nuclear models

The Liquid Drop Model

• Models the atomic nucleus as a drop of incompressible nuclear fluid
• Accounts for the spherical shape of most nuclei and the existence of nuclear fission
• Binding energy is analogous to the surface tension of a liquid drop
  • Binding energy increases with the volume of the nucleus (number of nucleons) but decreases with the surface area
• Coulomb repulsion between protons contributes to the instability of large nuclei
• Predicts the semi-empirical mass formula, which estimates the binding energy and mass of a nucleus based on its number of protons and neutrons
  • $BE = a_VA - a_SA^{2/3} - a_CZ(Z-1)A^{-1/3} - a_A(A-2Z)^2/A + \delta(A,Z)$
• Limitations include the inability to explain magic numbers and the existence of non-spherical nuclei

The Shell Model

• Describes the atomic nucleus in terms of energy levels (shells) occupied by individual nucleons
• Nucleons fill energy levels in a manner analogous to electrons in the atomic shell model
  • Pauli exclusion principle applies, with each energy level having a specific capacity for protons and neutrons
• Magic numbers correspond to completely filled energy levels, resulting in increased nuclear stability
• Explains the existence of isotopes with even numbers of protons and neutrons being more stable than those with odd numbers
• Predicts the existence of nuclear isomers, which are excited states of atomic nuclei with relatively long half-lives
• Limitations include the inability to fully account for the properties of deformed nuclei and the need for a large number of adjustable parameters

Other Nuclear Models

• Fermi gas model treats nucleons as a gas of non-interacting fermions, useful for understanding the gross properties of nuclear matter
• Interacting boson model describes nuclei in terms of pairs of interacting bosons (protons and neutrons) and helps explain collective nuclear properties
• Cluster models treat nuclei as composed of clusters of nucleons (e.g., alpha particles) and are useful for describing light nuclei
• Ab initio models aim to describe nuclei from first principles using quantum chromodynamics, the fundamental theory of strong interactions
  • Computationally intensive and currently limited to light nuclei
• Density functional theory models use energy density functionals to describe nuclear properties and have been successful in predicting the properties of heavy nuclei
• Each model has its strengths and limitations, and a combination of models is often used to describe different aspects of nuclear structure and behavior

Nuclear Stability and Binding Energy

• Nuclear stability refers to the ability of a nucleus to remain intact and not undergo radioactive decay
• Binding energy is a measure of nuclear stability, with higher binding energy per nucleon indicating greater stability
  • Iron-56 has the highest binding energy per nucleon and is the most stable nucleus
• Nuclei with magic numbers of protons or neutrons (2, 8, 20, 28, 50, 82, 126) have increased stability due to completely filled energy levels
• Nuclei with even numbers of protons and neutrons tend to be more stable than those with odd numbers
  • Even-even nuclei are the most stable, followed by odd-even, even-odd, and odd-odd nuclei
• Neutron-to-proton ratio affects nuclear stability, with heavier nuclei requiring more neutrons to remain stable
  • Beta decay occurs when the neutron-to-proton ratio is too high or too low for a given nucleus
• Liquid drop model predicts the general trend of binding energy as a function of mass number, with deviations due to shell effects and pairing

Radioactive Decay and Half-Life

• Radioactive decay is the spontaneous emission of particles or energy from an unstable atomic nucleus
• Types of radioactive decay include alpha decay (emission of alpha particles), beta decay (emission of electrons or positrons), and gamma decay (emission of high-energy photons)
  • Alpha decay typically occurs in heavy nuclei, while beta decay occurs when the neutron-to-proton ratio is too high or too low
• Decay rate is the number of decays per unit time and is proportional to the number of unstable nuclei present
  • Decay rate follows first-order kinetics, with the half-life being the time required for half of the original unstable nuclei to decay
• Half-life is a characteristic property of each radioactive isotope and can range from fractions of a second to billions of years
  • Carbon-14 has a half-life of 5,730 years and is used for radiocarbon dating of organic materials
• Decay chains occur when the product of a radioactive decay is also radioactive, leading to a series of decays until a stable nucleus is reached
  • Uranium-238 decay chain includes thorium-234, protactinium-234, and other radioactive isotopes before reaching stable lead-206

Applications and Real-World Relevance

• Radioisotopes are used in various applications, including medical imaging and therapy, industrial process control, and scientific research
  • Technetium-99m is widely used in nuclear medicine for diagnostic imaging
  • Cobalt-60 is used in radiation therapy for cancer treatment
• Nuclear power plants rely on controlled nuclear fission reactions to generate electricity
  • Uranium-235 is the most common fuel used in nuclear reactors
• Radiocarbon dating uses the decay of carbon-14 to determine the age of organic materials up to ~50,000 years old
  • Used in archaeology, paleontology, and earth sciences to date artifacts and geological samples
• Nuclear weapons rely on uncontrolled fission (atomic bombs) or fusion (hydrogen bombs) reactions to release massive amounts of energy
  • International treaties and non-proliferation efforts aim to limit the spread of nuclear weapons
• Study of nuclear structure and reactions helps in understanding the origin of elements in the universe (nucleosynthesis)
  • Big Bang nucleosynthesis produced hydrogen, helium, and trace amounts of lithium
  • Stellar nucleosynthesis in stars and supernovae is responsible for the production of heavier elements