PHY 1020 – Chapter 4: Nuclei and Radioactivity

Elements and Atoms

• Heat is the sensation produced by microscopic motion (translation, vibration, rotation) of molecules and atoms.
  • In physics, the amount of heat present is quantified by temperature.
  • Kelvin is the absolute scale; absolute zero is 0\,\text{K} (all molecular motion theoretically ceases).
  • Common laboratory/­daily scales: \text{^\circ C} and \text{^\circ F}.
• Elements are distinct kinds of atoms, each defined uniquely by the number of protons (atomic number Z).
  • General atomic structure: nucleus (protons p^+ and neutrons n^0) + surrounding electrons e^-.

Isotopes

• Isotopes = atoms of the same element (same Z) but different mass number A (due to varying neutrons).
  • Example: Hydrogen family
  • ^1_1\text{H} (protium) – 0 neutrons.
  • ^2_1\text{H} (deuterium) – 1 neutron.
  • ^3_1\text{H} (tritium) – 2 neutrons; radioactive, beta-emitter.
• Stability landscape
  • Many neutron-rich or neutron-poor isotopes are unstable ➜ undergo radioactive decay to reach a more stable n/p ratio.
  • Chart-of-nuclides (p.4) categorises known nuclides, their half-lives, and dominant decay modes (α, β±, EC, γ, p, n, spontaneous fission).
  • Key concept: “Valley of stability” where binding energy per nucleon is maximised.

Fusion (Building Bigger Nuclei)

• Definition: combining two light nuclei into a heavier one, releasing energy if final binding energy per nucleon increases.
  • Stellar fusion pathway (proton–proton chain, CNO cycle) converts ^1\text{H} to ^4\text{He}, then progressively to heavier elements.
  • Sun = natural fusion reactor; needs ≥ \sim 10^{26}\,\text{kg} mass to reach required core pressure/temperature.
  • Jupiter possesses similar composition to Sun but lacks sufficient mass ➜ no sustained fusion.
• Fusion significance
  • Source of almost all naturally occurring elements heavier than helium (“stellar nucleosynthesis”).
  • Drives stellar luminosity, hence life-sustaining energy on Earth.

Radioactivity (Breaking Down Bigger Nuclei)

• Radioactive decay = spontaneous transformation of an unstable nucleus into another nucleus plus radiation.
• Major decay modes
  1. Alpha (α) decay: ^AZX \rightarrow ^{A-4}{Z-2}Y + ^4_2\text{He}^{2+}
  • Emits a helium nucleus (2 p, 2 n); large mass ➜ low penetration.
  1. Beta minus (β⁻): neutron → proton + electron + antineutrino.
  • n \rightarrow p + e^- + \bar{\nu}_e.
  1. Beta plus (β⁺ / positron emission) or Electron Capture (EC): proton → neutron.
  • p \rightarrow n + e^+ + \nue (β⁺) or p + e^- \rightarrow n + \nue (EC).
  1. Gamma (γ): de-excitation of nucleus emits high-energy photon; no change in A or Z.
  2. Spontaneous/­induced fission: heavy nucleus splits into two medium nuclei + neutrons + energy.
• Energy release in decay quantified by Q-value: Q = (m{initial}-m{final})c^2; positive Q = energetically allowed.

Radiation Types & Penetration Depth

• α: stopped by a sheet of paper or epidermis; harmful if ingested/­inhaled.
• β: penetrates paper, stopped by few mm Al or plastic.
• γ / X-ray: electromagnetic; requires thick concrete/­lead and distance.
• Neutron: deeply penetrating; best shield = hydrogen-rich (water, polyethylene) + concrete.
• Schematic penetration hierarchy (thin → thick): \alpha < \beta < \gamma \;\text{or}\; n.

Cosmic Radiation

• Continual flux of high-energy charged particles (mainly protons, α’s, electrons) from Sun + distant astrophysical sources.
• Planetary defenses: Earth’s magnetic field deflects, and atmosphere absorbs cascades.
• Flying at high altitude or space travel significantly increases dose.

Free Neutrons

• Produced in nuclear reactions & cosmic-ray spallation.
• Mean lifetime ≈ 880\,\text{s} \;(\approx 15\,\text{min}) before β⁻ decay: n \rightarrow p + e^- + \bar{\nu}_e.

Measuring Radiation Dose

• Dose quantifies biological effect: rem (old US) or Sievert (SI).
  • Conversion: 1\,\text{rem} = 1000\,\text{mrem} = 0.01\,\text{Sv}.
• Recommended annual occupational limit: 5000\,\text{mrem} = 0.05\,\text{Sv} = 50\,\text{mSv}.
• Typical exposures
  • Natural background: \sim 300\,\text{mrem/yr} (cosmic, terrestrial, internal K-40, radon).
  • Five commercial flights: \approx 25\,\text{mrem}.
  • Chest CT: \sim 1000\,\text{mrem}.
• Acute thresholds
  • Radiation sickness onset: \sim 100\,\text{rem} in a short time.
  • LD50 (≈50 % lethal) single dose: \sim 500\,\text{rem}.

Radiation & Cancer — Linear No-Threshold (LNT) Hypothesis

• Assumes risk ∝ dose with no safe threshold.
  • Statistic: every additional 2.5\,\text{rem} increases lifetime fatal-cancer probability by 1/1000.
• Medical imaging dominates anthropogenic ionizing exposure: X-ray, CT, mammography, PET.
• Case studies
  1. Chernobyl (1986)
  • 134 workers received 70{,}000–1{,}340{,}000\,\text{mrem} → 28 acute deaths.
  • Global integrated dose ≈ 6.0\times10^7\,\text{rem} ➜ LNT predicts 24{,}000 extra cancers.
  1. Hiroshima survivors
  • 52{,}000 individuals, mean 20\,\text{rem} each → total 1.04\times10^6\,\text{rem}.
  • LNT predicts 416 cancers (0.8 %), observed ≈2 % – discrepancy hints at complexities (dose rate, quality, biology).
  1. Denver, CO
  • Elevated natural background by 0.1\,\text{rem/yr} for 2.4\times10^6 people ⇒ 1.2\times10^7\,\text{rem} total.
  • LNT projects 4800 additional cancers, yet Denver’s cancer incidence is lower than US average → indicates LNT oversimplification or confounding lifestyle factors.

Half-Life (t₁⁄₂)

• Definition: time required for half of a given radioactive sample to decay.
• Exponential law: remaining fraction N(t)=N0\; 2^{-t/t{1/2}}.
• Misconception clarified: decay is probabilistic; each succeeding half-life reduces the current amount by ½, not eliminates remainder.
• Example with ^{14}\text{C}: starting 100 kg ➜ after one t₁⁄₂ (5730 yr) 50 kg, after two 25 kg, after three 12.5 kg, etc.

Fission

• Occurs spontaneously (e.g.
  ^{238}\text{U}) or when a fissile nucleus (e.g.
  ^{235}\text{U}, ^{239}\text{Pu}) absorbs a neutron.
• Releases ≈ 200\,\text{MeV} per event as
  • kinetic energy of fragments,
  • prompt γ-rays,
  • additional neutrons (→ chain reaction).
• Example decay chain (simplified) for ^{238}\text{U} ➜ sequential α, β decays through ^{234}\text{Th}, ^{230}\text{Th}, ^{226}\text{Ra}, ^{222}\text{Rn}, …, ^{206}\text{Pb} (stable).
  • Half-lives range from 0.000164\,\text{s} (Po-214) to 4.47\times10^{9}\,\text{yr} (U-238).
• Neutron-induced fission foundation for reactors & nuclear weapons; controlled chain reaction demands moderation, geometry, neutron absorbers.

Practical Applications of Radioactivity

• Radioisotope Thermoelectric Generators (RTGs)
  • Convert heat from α-decay (often ^{238}\text{Pu}) into electricity via thermoelectric effect.
  • New Horizons probe: 11\,\text{kg} Pu-238 producing 600\,\text{W/kg} → 6.6\,\text{kW} heat; ~7 % efficiency ⇒ \approx 460\,\text{W} electric (enough for eight 60 W bulbs).
  • Longevity determined by Pu-238 t₁⁄₂ = 87.7\,\text{yr}.
• Smoke Detectors
  • Utilize ^{241}\text{Am} (α-emitter) to ionize air, enabling small standing current.
  • Smoke particles disrupt ion flow ➜ circuit senses drop ➜ alarm.
• Radiometric Dating
  1. Carbon-14
  • Living organisms incorporate ^{14}\text{C}; activity ~12 decays / min per gram of carbon.
  • After death, no new intake; activity declines via t₁⁄₂ = 5730\,\text{yr}.
  • Age calculation: n\;\text{half-lives}=\log_2(\text{initial rate}/\text{measured rate}).
    • Example: 3 dpm measured ⇒ n=\log_2(12/3)=2 ⇒ age \approx 11{,}460\,\text{yr}.
  • Effective dating window ≈ \le 60{,}000\,\text{yr} (beyond that, activity too low).
  1. Potassium-40
  • Natural abundances: ^{39}\text{K} (93.26 %), ^{41}\text{K} (6.73 %), ^{40}\text{K} (0.0117 %).
  • ^{40}\text{K} t₁⁄₂ = 1.248\times10^{9}\,\text{yr}.
    • 89.1 % β⁻ to ^{40}\text{Ca}, 10.9 % β⁺/EC to ^{40}\text{Ar}.
  • Widely used to date geological samples (volcanic rock) in 10^6–10^9\,\text{yr} range.

Radiation Penetration Illustration

  Radiation Type   Typical Shield
  ---------------  --------------
  Alpha (α)        sheet of paper / skin
  Beta  (β)        few mm Al / plastic
  Gamma (γ)        many cm Pb or tens cm concrete
  Neutron (n)      water, polyethylene + concrete

Chapter 4 — Core Takeaways

• Every element is characterised by proton count; isotopes vary neutron count ➜ impacts stability.
• Fusion builds heavier nuclei in stars; fission or decay can break them apart, both releasing vast nuclear energy.
• Radiation manifests as particles (α, β, n) or photons (γ, X-ray); penetrating power and biological hazard vary.
• Dose metrics (rem/Sv) guide safety; biological risk approximated by LNT but real-world data show complexities.
• Half-life governs radioactive decay kinetics; fundamental to dating techniques and waste management.
• Nuclear science yields practical technologies—from electrical power in deep-space probes to everyday smoke alarms.