• 20% of U.S. electricity from nuclear
• First plant in U.S. in 1958
• 2021 stats
  • 94 reactors, ~1 GW each (avg)
  • 56 plants
  • 28 states
  • Average age is 39 years
• Oldest plant is Nine Mile Point (NY), Dec. 1969, 52 years
• Newest is Vogtle Unit 4, 2023, Georgia

Equation for radiactive decay?

• Radioactive decay is governed by \(\frac{dN}{dt} = -N/\tau\), \(N(0) = N_0\).
• Here, \(\tau\) is the characteristic decay time.
• The analytic solution is

  \[N = N_0e^{-t/\tau}\]

• Relate this to the half-life: \(\frac{N}{N_0} = \frac{1}{2} = e^{-t_{1/2}/\tau}\rightarrow \tau = -t_{1/2}/\ln(1/2)\) \[\tau = t_{1/2}/\ln(2)\] \[t_{1/2} = \tau\ln(2)\]

  \[N = N_0\left(\frac{1}{2}\right)^{t/t_{1/2}}\]

• Neutron fate:
  • Fission
  • Absorbed (e.g., by U-238)
  • Escape
    • (bare n is unstable \(\rightarrow\) p+\(e^-\))
    • half life ~ 10 minutes.
• Chain reaction
  • 1 fission creates \(\ge\) 1 fission
• Energy from mass: \[\Delta m = m_\text{reactants} - m_\text{products}\] \[\Delta m = \frac{E}{c^2}\]
  • mass difference as kinetic energy of fission products \(\rightarrow\) slows \(\rightarrow\) heat.
• Neutrons + protons are conserved, electrons are not (formed).
• Energy released as more stable nuclei are formed (sounds familiar)

• Cross section is an area of interaction of a target particle
• May be bigger or smaller than the geometrical area
  • In this sense it’s like a probability of interation
• Units: 1 barn = 10\(^{-28}\) m\(^3\)
  • as in “you couldn’t hit the broad side of a barn”
• Depends on the nucleus and the neutron energy
  • Fast (prompt) neutrons are released during fusion (0.1 - 10 MeV).
  • These have low fission cross sections though, so they are slowed down to “thermal” by neutron interactions with a moderator.
  • Thermal neutrons (0.025 eV at 17 \(^o\)C, ~2.2 km/s=4900 mph)

• 40 times more common than silver, 500 times more common than gold.
• Found everywhere.
• Highest grade deposits in Canada

• Pressurized water reactor
• Pressurized water in contact with fuel
• Water transfers heat and moderates the neutrons
• Nonboiling (high P)
• Heat transfer to separate steam loop
• T: 290 \(\rightarrow\) 325 \(^oC\)
• P: 15 MPa (2250 psi)

• Boiling water reactor
• Steam is formed directly in the reactor (direct cycle)
  • No separate heat transfer loop or steam generators needed.
• Latent vs sensible heat \(\rightarrow\) less water flow.
• Water becomes radioactive \(\rightarrow\) turbine, pumps, pipes, etc. shielded.
• P: 7 MPA (900 psi)
• T: 290 \(^oC\)
• Larger unit for same energy vs. PWR; (BWR lower energy per volume)

• Advanced Gas Reactor
• CO\(_2\) cooled
  • non-reactive with fuel or moderator, non-absorbing
  • not pressurized
• Graphite moderated
• enriched U-235 (2.5-3.5%)
• T: 650 \(o^C\).
  • Higher efficiencies: 41%
• Indirect steam but heat transfer internal to the reactor vessel.