WHY a sphere? A sphere has the least surface area per unit volume → least leakage → smallest critical mass. (Bare 235U sphere ≈ 52 kg; with a neutron reflector, far less.)
Recall Quick self-test
Q: Two identical subcritical half-spheres are slammed together. What happens to k?
A: Combining halves makes one ball with lower surface-to-volume ratio → less leakage → k jumps above 1 → supercritical (this is the "gun-type" bomb principle).
Splitting of a heavy nucleus into lighter fragments + 2–3 neutrons + ~200 MeV energy.
Why does fission release energy?
Fragments have higher binding-energy-per-nucleon than the parent (closer to the iron peak); the gain is released as kinetic energy.
Energy released per U-235 fission?
~200 MeV.
Formula linking mass defect to energy (with unit shortcut)?
E=Δmc2; 1u→931.5MeV.
Define multiplication factor k.
Ratio of fission-causing neutrons in one generation to the previous; governs whether chain grows (k>1), holds (k=1), or dies (k<1).
What does k=1 mean physically?
Critical: steady self-sustaining chain — the reactor operating point.
Why does a chain reaction need a minimum (critical) mass?
Production scales with volume (r3), leakage with surface (r2); only above a minimum size do enough neutrons stay in for k≥1.
Why a sphere for minimum critical mass?
Minimum surface area per volume → least neutron leakage.
Role of moderator?
Slows fast neutrons to thermal speeds (elastic collisions with light nuclei) so U-235 captures them efficiently.
Role of control rods?
Absorb excess neutrons (Cd, B) to keep k=1.
Thermal vs fast reactor — one key difference?
Thermal uses slowed (moderated) neutrons + low enrichment; fast uses unmoderated fast neutrons + high enrichment, can breed fuel.
What does a breeder reactor do?
Converts fertile 238U into fissile 239Pu using fast neutrons, making more fuel than consumed.
Reaction chain U-238 → Pu-239?
238U + n → 239U →(β−) 239Np →(β−) 239Pu.
Recall Feynman: explain to a 12-year-old
Imagine a room packed with mousetraps, each loaded with two ping-pong balls. Throw in ONE ball. It springs a trap, which flings two more balls, which spring two more traps... soon the whole room explodes with bouncing balls. That's a chain reaction. Each uranium atom is a loaded mousetrap; a neutron is the ping-pong ball. If the room is too small, most balls fly out the open door before hitting a trap, and it fizzles — you need a big enough pile (critical mass). In a power plant, we put up a few catchers (control rods) to grab exactly the extra balls, so it stays a calm, steady snapping — never an explosion.
Dekho, fission ka idea simple hai: 235U jaisa bhaari nucleus already tension mein rehta hai — 92 protons ek doosre ko repel kar rahe hain. Ek neutron maaro, nucleus wobble karke do tukdo mein toot jaata hai aur saath mein 2–3 fresh neutrons nikalta hai plus bahut saari energy (~200 MeV). Energy aati kahan se hai? Binding energy per nucleon ka graph iron ke paas peak karta hai; heavy nucleus uss peak ke neeche right side pe hai, toot ke products peak ke kareeb chale jaate hain — yani zyada tightly bound, aur bachi hui "tightness" energy ban ke nikalti hai.
Ab chain reaction: agar har fission se nikle neutrons aage aur fission cause karein, toh avalanche ban jaata hai. Iska scorekeeper hai k (multiplication factor). k<1 matlab reaction mar jaayega (subcritical), k=1 matlab steady chalega (reactor ka sweet spot), k>1 matlab exponential — bomb. Yaad rakho: 2-3 neutrons nikalne ka matlab k>1 nahi hota, kyunki bahut saare neutrons leak ho jaate hain ya bina fission ke absorb ho jaate hain.
Critical mass ka funda surface-vs-volume hai. Neutron banta hai volume mein (r3), lekin leak hota hai surface se (r2). Lump jitna chhota, surface/volume utna bada, utna zyada leakage — isliye ek minimum mass chahiye taaki enough neutrons andar ruke aur k=1 ho. Sphere best hai kyunki uska surface sabse kam hota hai per volume.
Reactor mein hum k ko exactly 1 pe pakad ke rakhte hain. Moderator (paani, graphite) fast neutron ko slow karta hai taaki U-235 use easily pakad ke fission kare — yeh hai thermal reactor. Control rods (Cd, B) extra neutrons ko soak karte hain. Fast/breeder reactor mein moderator nahi hota — fast neutrons use karke fertile 238U ko fissile 239Pu mein convert kar deta hai, yani naya fuel "breed" karta hai. Mnemonic: "Mom Cooks Cool Food" = Moderator, Control rods, Coolant, Fuel.
