Nuclear Fission Chain Reaction Simulator

The core physics of a sustained chain reaction is captured by the Six-Factor Formula, which accounts for all the ways neutrons are produced, lost, or utilized in a reactor core.

$\eta$ (neutrons per fission): Average number of neutrons released per fission event. Primarily depends on fuel enrichment.
$f$ (thermal utilization factor): Fraction of thermal neutrons absorbed by the fuel (not by control rods or moderator).
$p$ (resonance escape probability): Chance a neutron slows down (moderates) without being captured by U-238 at intermediate energies.
$\varepsilon$ (fast fission factor): Small increase in neutrons from fissions caused by fast neutrons before they slow down.
$P_{NL}$ (non-leakage probability): Chance that neutrons do not escape the reactor core entirely. Our simulator assumes an infinite core, so $P_{NL}= 1$.

The simulator uses this formula to calculate keff based on your settings. The resulting keff then determines the neutron population growth or decay over time, following a simple exponential model.

$N(t)$: Neutron population at time $t$.
$N_0$: Initial neutron count you set.
$n$: Number of fission generations over time $t$.
When $k_{eff}\gt 1$, $N(t)$ grows exponentially (supercritical). When $k_{eff}\lt 1$, it decays (subcritical). At $k_{eff} = 1$, it remains constant (critical).

Nuclear Power Generation: Commercial reactors like Pressurized Water Reactors (PWRs) are meticulously designed to operate at a steady-state critical condition (keff = 1). Operators constantly adjust control rods and boron concentration in the coolant to maintain this balance, responding to changes in power demand.

Research Reactors: These facilities, used for producing medical isotopes or for neutron scattering experiments, often require a very stable neutron flux. Engineers select specific moderator materials (like heavy water or beryllium) to maximize the neutron economy and achieve criticality with low-enriched fuel.

Nuclear Reactor Safety & Design: Before any reactor is built, CAE simulations using models like the six-factor formula are run thousands of times. They test different accident scenarios, like control rod insertion failures or coolant loss, to predict how keff would change and ensure shutdown systems are fail-safe.

Nuclear Propulsion & Space Exploration: Concepts for nuclear thermal rockets for deep-space missions rely on compact, high-power reactors. These designs push for very high-temperature moderators and highly enriched fuel to achieve a high keff in a small, lightweight package, a trade-off you can explore in the simulator.

First, you might wonder why neutrons don't immediately drop to zero even when you set the "Control Rod Absorption Rate" to its maximum in this simulator. This is because control rods absorb neutrons "probabilistically." For example, even with a 90% absorption rate, a neutron might, with some luck, slip through and contribute to the next generation. In a real reactor as well, even when control rods are inserted fully, neutrons decay exponentially and do not stop instantaneously.

Next, you might think that increasing the "Uranium Enrichment" will always lead to criticality, but this is not the case if the moderator or core size is inappropriate. For instance, even if you set enrichment to over 90% (weapons-grade), if you set the moderator to "None," fast neutrons are less likely to sustain a fission chain, and the system may remain subcritical. Conversely, a natural uranium reactor (0.7% enrichment) using "Light Water" as a moderator is theoretically not feasible. This is because light water absorbs too many neutrons; actual "Light Water Reactors" only become viable when enrichment is increased to 3–5%.

Finally, note that the "Neutrons per Generation" graph in the simulator shows a smooth curve. This is the result of averaging a stochastic process. In a real reactor, especially when neutron counts are low (e.g., during startup), the birth and death of neutrons involve probabilistic "fluctuations" and can sometimes show unexpected variations. For more detailed analysis using CAE, methods like the "Monte Carlo method," which tracks this stochastic behavior, become necessary.