Nuclear Physics and Radioactive Decay Notes

Nuclear Particles and the Standard Model

• The Standard Model classifies elementary particles into three groups: quarks, leptons, and bosons.

• Quarks: Combine in groups of two or three to form hadrons.

  • Two quarks form Mesons.

  • Three quarks form Baryons.

• Leptons: Light particles (e.g., electrons and neutrinos) that do not form combinations.

• Bosons: Particles that carry fundamental forces.

  • Gluons carry the strong force.

  • Photons (e.g., gamma rays) carry the electromagnetic force.

• Sub-atomic particles important to everyday experience: protons, neutrons, electrons, and photons.

Modeling the Nucleus

• The nucleus can be modeled as a sphere of uniform density.

• The volume of the nucleus is proportional to the mass number, $A$ (total number of protons and neutrons).

• The radius $R$ of the nucleus is proportional to $A^{1/3}$.

• Formula for the radius of a nucleus: $R ≈ (1.2 × 10^{-15} m)A^{1/3}$

• Example: Gold-197 ($^{197}Au$) has $A = 197$.

  • $R ≈ (1.2 × 10^{-15} m)(197)^{1/3} ≈ 7.0 × 10^{-15} m$

• The radius of a gold atom is approximately $1.66 × 10^{-10} m$, which is much larger than its nucleus.

Nuclear Processes and the Standard Model

• Gamma rays interacting with a nucleus can create mass out of energy.

• The Standard Model organizes particles into quarks, leptons, and bosons, differentiated by their properties and interactions.

• Muon neutrinos are leptons with negligible mass and no charge.

• Known bosons are classified as Gauge (e.g., gluons, photons) or Scalar (e.g., Higgs boson).

• Antimatter particle interactions with their equivalent particles can result in:

  • Positron and electron annihilation.

  • Two massless gamma-ray photons.

  • $1.022 \, MeV$ of energy.

  • An electron/positron pair.

Nuclear Radius Calculations

• Calculate differences in nuclear radii using the formula $R ≈ (1.2 × 10^{-15} m)A^{1/3}$.

• Example: Comparing aluminum-27 ($<em>{13}^{27}Al$) and sodium-23 ($</em>{11}^{23}Na$) requires calculating radii for both and finding the difference.

• Atomic nuclei are significantly denser than materials like quartz. The density difference is around $1.0 × 10^{14}$.

Nuclear Forces and Binding Energy

• Protons and neutrons in the nucleus are attracted to each other by the strong force.

• The strong force dominates over other forces (weak, electromagnetic, gravity) within the nucleus.

• Energy is required to break the nucleus into separate particles; this energy is the binding energy.

• Binding energy is also the energy released when particles bind together to form a nucleus.

• Einstein’s mass-energy relationship ($E = mc^2$) implies that adding energy to separate nuclear particles increases the system's mass.

Binding Energy Calculation

• Binding energy can be calculated by comparing the mass of a nucleus to the mass of its separated particles.

• Formula: $B = (Z m<em>p + N m</em>n – m_{nucleus})c^2$

  • $B$ = binding energy

  • $Z$ = number of protons

  • $m_p$ = mass of a proton ($938.27 \, MeV/c^2$)

  • $N$ = number of neutrons

  • $m_n$ = mass of a neutron ($939.57 \, MeV/c^2$)

  • $m_{nucleus}$ = mass of the nucleus

  • $c$ = speed of light

• Example: Calculating the binding energy for carbon-12 ($^{12}C$).

  • $m_{nucleus} = 12.0000 \, u = 11,177.9 \, MeV/c^2$

  • $B = [(6 \, protons)(938.27 \, MeV/c^2) + (6 \, neutrons)(939.57 \, MeV/c^2) – (11,177.9 \, MeV/c^2)]c^2$

  • $B = (5629.62 + 5637.42 – 11,177.9) \, MeV = 89.1 \, MeV$

• Nuclear processes involve small mass changes relative to the total mass, requiring high precision in calculations.

Nuclear Decay and Mass-Energy Relationship

• Beta-plus decay involves the conversion of a proton into a neutron, a positron, and a neutrino.

• The total number of particles (nucleons) in the nucleus remains the same during beta decay.

• According to Einstein’s mass-energy relationship, processes that result in more mass at the end could include:

  • The joining of two bottom quarks.

  • An unstable nucleus undergoing beta-minus decay.

  • The interaction of two gamma rays.

Binding Energy per Nucleon

• The average binding energy per nucleon is calculated by dividing the total binding energy by the number of nucleons.

• Example: For lead-208 ($^{208}Pb$), if the mass of the nucleus is $193,752 \, MeV/c^2$, calculate the total binding energy first, then divide by 208.

Nuclear Fission and Fusion

• Nuclear Fission: A large nucleus splits into two smaller nuclei.

• Nuclear Fusion: Two small nuclei combine to form a larger nucleus.

• Energy release depends on the relative positions of reactant and product nuclei on the curve of nuclear binding energy.

• Energy is released if the average binding energy in the products is greater than in the reactants.

Energy Calculation in Nuclear Reactions

• Energy released can be calculated using the difference in mass between reactants and products ($E = mc^2$).

• Alternatively, energy released can be calculated using the average binding energy of reactants and products.

• Formula: $\Delta E = \text{(product binding energy)} – \text{(reactant binding energy)}$

• Example: Fission of plutonium-239 ($^{239}Pu$) into xenon-134 ($^{134}Xe$) and zirconium-103 ($^{103}Zr$) plus free neutrons.

  • $\Delta E = [(134 \, nucleons)(8.25 \, MeV/nucleon) + (103 \, nucleons)(8.35 \, MeV/nucleon)] – (239 \, nucleons)(7.56 \, MeV/nucleon)$

  • $\Delta E = [1105.50 \, MeV + 860.05 \, MeV] – 1806.84 \, MeV = 158.71 \, MeV ≈ 159 \, MeV$

Conservation Laws in Nuclear Processes

• In any nuclear process, the following are conserved:

  • Number of nucleons.

Fission Energy Release

• Fission releases significantly more energy per gram compared to chemical combustion (e.g., methane combustion).

• Calculate the energy released per gram in fission reactions using binding energies and compare to combustion energy.

Radioactive Decay

• Radioactive decay is a random process at the atomic level, but predictable for large samples.

• The decay curve has a long