MP

PHY 1020 – Chapter 5: Chain Reactions, Nuclear Reactors, and Atomic Bombs

Chain Reactions – Fundamental Concepts

  • Definition: A chain reaction is a sequence of events in which one reaction initiates additional reactions, leading to a self-amplifying process.
  • Mathematical growth law for binary chain reactions:
    • Each event produces 2 new events → number of events after N steps is 2^N (law of doubling).
    • Growth is exponential, not linear.
  • Physical requirements:
    • A trigger event (e.g., neutron striking a fissile nucleus).
    • Sufficient propagation medium (enough nuclei, cells, people, etc.).
    • No dominant damping mechanism (losses, removal, exhaustion of resources).

Biological Analogies: Fetal Development & Cancer

  • Human embryo/fetus as a biological chain reaction:
    • One fertilized egg divides via mitosis → 2 → 4 → 8 … 2^N cells.
    • Total cells in an adult ≈ 10^{11}.
    • Solve 2^N = 10^{11} → N = \log_2 10^{11} \approx 11\times3.32 \approx 36.52 → about 37 cell-division days (idealized, ignores differentiation & apoptosis).
  • Cancer = “cells gone wild”:
    • Uncontrolled mitotic chain reaction when regulatory mechanisms (immune surveillance, programmed cell death) fail.
    • Younger/smaller bodies usually eliminate defective cells faster; loss of control → tumors.
    • Illustrates why damping processes are vital to prevent runaway growth.

Population, Viral & Rumor Spread

  • Human population growth shows near-exponential phases, but environmental carrying capacity limits sustainment (resource depletion, disease, social factors).
  • Other self-propagating systems:
    • Biological/computer viruses.
    • Rumors & urban legends in social networks.
    • Electron avalanche in lightning discharges.

Nuclear Fission Chain Reactions and Bomb Design

  • Fissile isotopes: ^{235}\text{U} and ^{239}\text{Pu} naturally emit neutrons.
  • Mechanism:
    1. Neutron absorbed → nucleus fissions → releases kinetic energy + \approx 2{-}3 fast neutrons.
    2. If >1 neutron on average induces new fission, reaction grows exponentially.
  • Critical mass = minimum mass/geometry/density of fissile material needed so that each generation produces ≥1 subsequent fission.
    • Depends on density, shape, purity (isotope ratio), neutron absorbers, and reflectors.
    • Typical bare-sphere values (no reflector):
    – Uranium-235: \approx 15\,\text{kg}.
    – Plutonium-239: \approx 5\,\text{kg} (higher neutron yield & lower critical mass).
  • Two classic bomb architectures:
    1. Gun-type ("Little Boy") – Fires a hollow ^{235}\text{U} projectile into a ^{235}\text{U} target → super-critical mass; yield ≈ 15 kt.
    2. Implosion-type ("Fat Man") – Symmetric high-explosive lenses compress ^{239}\text{Pu} core → higher density lowers critical mass; yield ≈ 20 kt.

Thermonuclear (Fusion) Weapons

  • Stage-1 fission device acts as trigger; Stage-2 fusion (deuterium + tritium or Li-6) yields vastly higher energy.
  • “Hydrogen bomb” yields in megaton range; Hiroshima/Nagasaki become negligible by comparison.
  • Example: USSR “Tsar Bomba” 50 Mt.

Effects Comparison: Hiroshima vs Tsar Bomba

  • Hiroshima (15 kt):
    • Fatalities ≈ 29,140; injuries ≈ 30,360.
    • Destructive radius ≈ 1.91 km.
  • Tsar Bomba (50 Mt) modeled over Jacksonville, FL:
    • Fatalities ≈ 633,650; injuries ≈ 347,040.
    • Destructive radius ≈ 60 km.
  • Demonstrates cubic scaling of blast volume with yield.

Uranium Enrichment & Manhattan Project

  • Natural uranium (~99.3 % ^{238}\text{U}, 0.7 % ^{235}\text{U}) too dilute for bombs/reactors.
  • Converted to uranium hexafluoride (UF6) for gaseous diffusion or centrifuge separation.
  • WWII K-25 plant (Oak Ridge, TN) produced highly enriched uranium shipped to Los Alamos for bomb assembly.
  • Different enrichment grades:
    • Reactors: ~3–5 % ^{235}\text{U} ("low enriched").
    • Weapons: >90 % ^{235}\text{U} ("highly enriched").
  • Time scale matters: bombs must assemble super-critical mass faster than neutron generation; reactors purposely keep chain reaction slow and controlled.

Nuclear Reactors – Principles & Components

  • Definition: Device sustaining a controlled chain reaction; heat → boil water → mechanical/electric power.
  • Key distinction from bombs:
    • Reactors use thermal (slow) neutrons; bombs rely on fast neutrons.
    • Excess reactivity controlled with moderators, control rods, coolant.
  • Major parts:
    • Fuel rods (usually \text{UO}_2 pellets, ~3–5 % ^{235}\text{U}).
    • Moderator (light water, heavy water, graphite) reduces neutron speed via elastic collisions.
    • Control rods (boron, cadmium, hafnium) absorb neutrons to throttle reaction.
    • Coolant loop extracts heat; produces steam for turbines.
  • Cherenkov radiation: blue glow when charged particles (β electrons) outrun local light speed in water.
  • Breeder reactors: Design converts fertile ^{238}\text{U} or ^{232}\text{Th} into fissile ^{239}\text{Pu} or ^{233}\text{U}; can generate more fuel than consumed.
  • Generation IV concepts (e.g., lead-cooled fast reactors) rely on passive convection cooling for added safety.

Reactor Accidents & Lessons

  1. Three Mile Island (1979, USA):
    • Cost-cutting → loss of secondary loop flow → primary loop overheated → partial core melt.
    • Containment held; minimal radiation release.
    • Lesson: importance of redundant cooling & transparent regulation.
  2. Chernobyl (1986, USSR):
    • Deliberate power-down test; low water flow + positive void coefficient.
    • Graphite-tipped control rods displaced neutron-absorbing water, spiked reactivity.
    • Steam explosion & graphite fire → wide radioactive release.
    • Lesson: negative feedback & robust containment non-negotiable.
  3. Fukushima Daiichi (2011, Japan):
    • 9.0 earthquake + tsunami disabled grid and backup diesel generators (placed behind seawall).
    • Loss of coolant → hydrogen buildup → four building explosions; significant release.
    • Different failure mode vs. Chernobyl (no prompt criticality, but cooling failure).

Nuclear Waste Management – Yucca Mountain

  • High-level waste remains radioactive 10³–10⁵ years.
  • Proposed deep-geologic repository: Yucca Mountain, Nevada.
    • ~300 m below surface, above water table.
    • Access ramps & tunnels isolate canisters.
  • Political, ethical, and logistical challenges: long-term stewardship, intergenerational equity, security.

Fusion Power – Stellar, Tokamak, Laser & Muon Catalysis

  • Fusion = combining light nuclei (e.g., D + T) → heavier nucleus + energy (via mass defect).
  • Occurs naturally in stars:
    • Hydrogen burning in sun → helium + photons → supports stellar radiation pressure.
    • Larger stars produce heavier elements up to Fe, supernovae for elements beyond.
  • Tokamak magnetic confinement:
    • Donut-shaped torus; toroidal + poloidal magnetic fields confine high-temperature (~10⁸ K) plasma.
    • ITER (France) largest ongoing experiment.
  • Inertial confinement (laser) fusion:
    • Symmetric laser pulses compress pellet to high density/temperature before it disassembles.
    • National Ignition Facility (NIF) achieved > kilojoule-scale target energies.
  • Muon-catalyzed (cold) fusion:
    • Muon replaces electron, shrinking D-T molecule → quantum tunneling allows fusion at low temp.
    • Challenge: each muon must catalyze many fusions (>200) to offset production energy; ~1 % of muons “stick” to α, halting cycle → not yet practical.

Chapter Takeaways

  • Chain reactions underpin phenomena from fetal growth to nuclear weapons.
  • Exponential doubling 2^N yields rapid escalation; control mechanisms differentiate benign from catastrophic outcomes.
  • Fission bombs need super-critical mass & microsecond assembly; critical mass ≈15 kg ^{235}\text{U}, 5 kg ^{239}\text{Pu}.
  • Thermonuclear bombs dwarf fission devices using staged D-T fusion.
  • Reactors harness slow-neutron fission for power; cannot explode like bombs but pose meltdown & waste challenges.
  • Accidents (TMI, Chernobyl, Fukushima) illustrate diverse failure paths; robust safety culture essential.
  • Future solutions: breeder & Gen-IV designs, effective waste repositories, and ultimately controlled fusion to emulate the sun’s power with minimal long-lived waste.