Nuclear Physics and Radioactive Decay Notes

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.

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.

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.

Calculate differences in nuclear radii using the formula R ≈ (1.2 × 10^{-15} m)A^{1/3}.
Example: Comparing aluminum-27 ({13}^{27}Al) and sodium-23 ({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}.

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 can be calculated by comparing the mass of a nucleus to the mass of its separated particles.
Formula: B = (Z mp + N mn – 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.

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.

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: 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 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

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 is a random process at the atomic level, but predictable for large samples.
The decay curve has a long