Nuclear powerEnergy

Nuclear reactor

A reactor's neutronics from first principles — six-group point kinetics with Doppler feedback and xenon-135: the prompt jump, the prompt-critical cliff, self-limiting power excursions, the post-shutdown iodine pit, load-follow, and SCRAM.

DjiniousLabModelling & simulation

A log-scale plot of reactor power after a reactivity insertion — running away without feedback, but self-limiting to a bounded level once Doppler temperature feedback is included

delayed-neutron kinetics: 6 groupsdelayed-neutron kinetics
Doppler-limited (vs 28 830× runaway): 2.4×Doppler-limited (vs 28 830× runaway)
xenon pit, 8 h post-shutdown: −4459 pcmxenon pit, 8 h post-shutdown
period at +0.2 $ rod worth: 37 speriod at +0.2 $ rod worth
decay heat (1 s → 1 h): 6 % → 1 %decay heat (1 s → 1 h)
design notebooks: 10design notebooks

Why a reactor is controllable — and why it stops itself.

A nuclear reactor lives on a knife-edge: the neutron population can double in a fraction of a second, yet operators steer it by hand. Two things make that possible — the delayed neutrons that slow the response to human time scales, and the temperature feedback that makes the core fight its own power rise. This program models both, from six-group point kinetics up through Doppler self-limiting, the xenon poison that can lock a reactor out of restart, load-follow control, and emergency shutdown.

Delayed neutrons buy the time.

If a reactor ran on prompt neutrons alone, a small reactivity insertion would blow the power up in milliseconds — uncontrollable. But a fraction of a percent of neutrons emerge seconds to minutes later, from decaying fission products, and those delayed neutrons set the pace: insert a little reactivity and the power climbs on a period of tens of seconds, slow enough to steer. The model captures the prompt jump (a +0.5 dollar step leaps power to 2.07×, matching theory), the inhour relation between reactivity and period, and the cliff at one dollar where the delayed neutrons stop mattering and the reactor goes prompt-critical.

A family of reactor-power responses to reactivity steps on a log scale, the slope steepening as the inserted reactivity grows — Reactor power after reactivity steps, on a log scale: each step settles onto an exponential climb whose period shortens as the inserted reactivity grows. This is the delayed-neutron-controlled response that makes a reactor steerable — until the insertion approaches one dollar.

The inhour curve — reactor period against inserted reactivity in dollars, the period collapsing as reactivity approaches one dollar — The inhour curve: how reactor period depends on inserted reactivity. A tenth of a dollar gives a comfortable 98-second period; half a dollar, 5.7 seconds; approach a full dollar and the period collapses toward zero — the prompt-critical cliff every reactor control system is built to stay away from.

The core that fights its own power rise.

Reactivity isn't fixed — it depends on temperature. As fuel heats, the Doppler broadening of absorption resonances captures more neutrons, pushing reactivity down. That negative feedback is what makes a power excursion self-limit instead of running away. The model shows it starkly: a +0.6 dollar insertion that explodes to nearly thirty-thousand times power with no feedback instead peaks at 2.4× and settles, once Doppler is in the loop. It's the same physics that makes a well-designed reactor inherently stable.

Reactor power after a reactivity insertion on a log scale: an exponential runaway without feedback versus a bounded, self-limited response with Doppler feedback — Doppler self-limiting: the same +0.6 dollar insertion runs away to ~30 000× power with no temperature feedback (red), but with Doppler feedback (green) the rising fuel temperature pushes reactivity back down and the excursion self-limits to a few times power. Inherent stability, drawn in one chart.

Xenon reactivity after a reactor shutdown, dipping further negative to a pit at about 8 hours before recovering over the next day — The iodine pit: after a shutdown, xenon-135 keeps building from decaying iodine even as its burn-off stops, so its negative reactivity deepens for about eight hours — to −4459 pcm here — before decaying away. A reactor that can't override that pit cannot restart until it passes; the transient that famously shapes operations.

Decay heat after a reactor SCRAM, falling from about 6 percent of full power at one second toward one percent after an hour — After SCRAM: even with the chain reaction stopped, fission-product decay keeps the core generating heat — about 6% of full power a second after trip, still 1% an hour later. Removing that decay heat is the safety problem that defines reactor shutdown cooling.

Every number is one you can re-run.

The sign-off notebook re-derives each requirement from the same point-kinetics model the program builds.

Result

Prompt jump (+0.5 $ step): 2.07×
Doppler-limited excursion: 2.4× peak
Xenon iodine-pit: −4459 pcm @ 8 h
Load-follow tracking error: 2.4 %
Requirements verified: 6 / 6

Requirement

Prompt jump (+0.5 $ step): theory 2.0
Doppler-limited excursion: bounded, not runaway
Xenon iodine-pit: post-shutdown
Load-follow tracking error: 100→60→100 %
Requirements verified: PASS

0-D neutronics, a teaching & controls model.

The reactor is 0-D point kinetics — the neutron population as a single number, with no spatial flux shape, so there are no spatial xenon oscillation modes (a noted follow-up). Thermal feedback is lumped, the xenon is single-node, and the cross-sections are illustrative. This is a teaching and control-design model, not a licensing-grade neutron-transport code. It is exactly the fidelity that builds intuition and supports control logic — the prompt jump, the inhour relation, the safety feedbacks, the xenon transient, load-follow and SCRAM — on reproducible numbers, before a spatial diffusion or Monte-Carlo transport solver.