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BLD-NUCL84-02 · Lesson 4 · 28 clock hours · theory

Reactor Kinetics: Prompt and Delayed Neutrons, Reactor Period

Objective. Students will distinguish prompt from delayed neutrons, explain why delayed neutrons make reactor control possible, and calculate approximate reactor period from reactivity insertion using the inhour equation in simplified form.

⚠ Pending professional review

This lesson, its worked examples and its practice questions has not yet been verified by a licensed professional engineer or master tradesperson or by a certification or approval body. It is a study draft and may contain errors. Do not use it as instructions for work on a job site or on live equipment. Always follow your instructor, your supervising professional, and current official references and regulations.

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Reactor Kinetics: Prompt and Delayed Neutrons, Reactor Period

Why Neutron Timing Matters

A fission chain reaction depends on neutrons produced by fission causing new fissions in turn. The speed at which the neutron population grows or shrinks determines how quickly reactor power changes. Understanding when those neutrons appear after fission is the key to understanding why a nuclear reactor can be controlled by mechanical systems at all.

Prompt Neutrons

The vast majority of neutrons released during fission appear almost instantaneously—within roughly 10⁻¹⁴ seconds of the fission event. These are called prompt neutrons. For U-235 fission, roughly 2.43 neutrons are released per fission on average, and about 99.35% of them are prompt. Because they appear so fast, if the chain reaction depended only on prompt neutrons, the time between successive neutron generations—the prompt neutron lifetime in a thermal reactor—would be on the order of microseconds to a fraction of a millisecond. A reactor that reacts to control-rod movement in microseconds cannot be managed by any mechanical or even electronic control system in a safe, deliberate way.

Delayed Neutrons

The remaining roughly 0.65% of neutrons from U-235 fission are delayed neutrons. They are not released directly by the fissioning nucleus; instead, certain fission product nuclei (called delayed neutron precursors) are left in highly unstable states and undergo beta decay. It is the daughter nucleus of that decay that emits a neutron. This two-step process takes time—from a fraction of a second up to roughly 55 seconds, depending on the precursor group.

Delayed neutrons are organized into six groups (Keepin's six-group model for U-235), each characterized by a fractional yield and a decay constant. The total delayed neutron fraction, β (beta), for U-235 is approximately 0.0065 (0.65%). Although small in absolute number, this fraction is enormously important.

Why Delayed Neutrons Make Control Possible

Because the chain reaction requires all neutrons—prompt and delayed—to sustain itself at a steady power level, the effective neutron generation time is extended from microseconds to on the order of tenths of a second. That slowed effective response time falls within the range that mechanical control rods and operator actions can manage safely.

The critical boundary to understand is prompt criticality: the condition in which the reactor could sustain a chain reaction on prompt neutrons alone, without needing the delayed fraction. Prompt criticality occurs when reactivity (ρ) equals or exceeds β. Operating at or beyond prompt criticality must always be avoided—power would rise on a microsecond timescale, faster than any safety system can respond.

Reactivity and the Inhour Equation

Reactivity (ρ) measures how far a reactor is from exactly critical. It is defined as ρ = (k − 1) / k, where k is the effective multiplication factor. When ρ = 0 the reactor is exactly critical; positive ρ means supercritical (rising power); negative ρ means subcritical (falling power). Reactivity is often expressed in units of dollars ($) where 1 dollar = β, or in percent mille (pcm) where 1 pcm = 10⁻⁵.

The reactor period (T) is the time in seconds for reactor power to change by a factor of e (≈ 2.718). It is the single most important kinetic quantity for operators. A short positive period means rapidly rising power; a negative period means falling power.

The exact relationship between ρ and T is given by the inhour equation. For small positive reactivities well below prompt criticality, a simplified (one-delayed-group) approximation is often used in training:

T ≈ β / (ρ · λ)

where λ is the effective delayed neutron decay constant (approximately 0.08 s⁻¹ for U-235 as a rough single-group value) and ρ is in the same units as β (both as fractions).

This approximation is valid only for small reactivities (ρ << β). Near prompt criticality, the full inhour equation must be used.

Operational Safety Implications

Reading the Semi-Log Power Plot

On semi-log paper, power rising at a constant reactor period appears as a straight line with a positive slope. A longer period gives a gentler slope; a very short period gives a steep slope. Falling power (negative period) slopes downward. Recognizing these curves at a glance is an essential operator skill.

Key terms10

Prompt neutronsNeutrons emitted within ~10⁻¹⁴ seconds of a fission event, comprising roughly 99.35% of all neutrons from U-235 fission.
Delayed neutronsNeutrons emitted seconds to tens of seconds after fission, released by the beta decay of fission product precursor nuclei; ~0.65% of U-235 fission neutrons.
Delayed neutron fraction (β)The fraction of all fission neutrons that are delayed; approximately 0.0065 for U-235. It sets the boundary for prompt criticality.
Prompt criticalityThe condition in which reactivity equals or exceeds β, meaning the reactor could sustain a chain reaction on prompt neutrons alone; a dangerous condition to be avoided in all operations.
Reactivity (ρ)A measure of the departure from exactly critical, defined as (k − 1)/k; positive values indicate supercritical, negative values subcritical, and zero exactly critical.
Reactor period (T)The time in seconds for reactor power to change by a factor of e (~2.718); a short positive period indicates rapidly rising power.
Inhour equationThe mathematical relationship that gives reactor period as a function of reactivity and delayed neutron group parameters; simplified forms apply only for small reactivities well below prompt criticality.
Dollar ($)A unit of reactivity equal to one delayed neutron fraction (β); reactivity of exactly 1 dollar equals prompt criticality.
Delayed neutron precursorA fission product nucleus that undergoes beta decay and whose daughter then emits a delayed neutron; grouped into six families (Keepin model) for U-235.
SCRAMEmergency rapid shutdown of a reactor achieved by quickly inserting all control rods; initiated automatically or manually when protective limits are exceeded.

Worked examples4

1. A reactor operating on U-235 fuel has a small positive reactivity insertion of ρ = 0.001 (as a fraction). Using the simplified one-group inhour approximation T ≈ β / (ρ · λ), with β = 0.0065 and λ = 0.08 s⁻¹, calculate the reactor period and determine whether a period alarm would be expected.
  1. Step 1 – Identify given values: ρ = 0.001, β = 0.0065, λ = 0.08 s⁻¹.
  2. Step 2 – Apply the simplified inhour approximation: T = β / (ρ × λ).
  3. Step 3 – Substitute values: T = 0.0065 / (0.001 × 0.08) = 0.0065 / 0.00008.
  4. Step 4 – Calculate: T = 81.25 seconds.
  5. Step 5 – Classify: A period of ~81 seconds is a slow, controlled rise. Depending on the plant's period alarm setpoint (commonly around 18–60 s), this may or may not trigger an alarm, but it represents a manageable, stable transient well below prompt criticality (ρ << β).
Answer: T ≈ 81 seconds; the reactor power rises slowly and the transient is manageable. Reactivity is far below the prompt critical threshold (ρ = β = 0.0065).
2. Using the same simplified formula (T ≈ β / (ρ · λ)), calculate the reactor period for a reactivity insertion of ρ = 0.005, with β = 0.0065 and λ = 0.08 s⁻¹. Compare ρ to β and state the operational concern.
  1. Step 1 – Identify given values: ρ = 0.005, β = 0.0065, λ = 0.08 s⁻¹.
  2. Step 2 – Apply the formula: T = β / (ρ × λ).
  3. Step 3 – Substitute: T = 0.0065 / (0.005 × 0.08) = 0.0065 / 0.0004.
  4. Step 4 – Calculate: T = 16.25 seconds.
  5. Step 5 – Compare to β: ρ = 0.005 is about 77% of β (0.0065). The reactor is not yet prompt critical but is approaching that boundary. In dollars: ρ = 0.005 / 0.0065 ≈ 0.77 dollars.
  6. Step 6 – Classify: A period of ~16 seconds would trigger period alarms on virtually all plants. Immediate corrective action (control rod insertion) is required. The simplified formula is reaching the edge of its valid range at this reactivity level.
Answer: T ≈ 16 seconds; ~0.77 dollars of reactivity. Period alarm expected; power rising rapidly. Immediate rod insertion required. Reactor is approaching but has not yet reached prompt criticality.
3. A negative reactivity of ρ = −0.002 is inserted into a U-235 reactor (e.g., by inserting a control rod). Using T ≈ β / (ρ · λ) with β = 0.0065 and λ = 0.08 s⁻¹, calculate the reactor period and describe the power behavior.
  1. Step 1 – Identify given values: ρ = −0.002, β = 0.0065, λ = 0.08 s⁻¹.
  2. Step 2 – Apply the formula: T = β / (ρ × λ).
  3. Step 3 – Substitute: T = 0.0065 / (−0.002 × 0.08) = 0.0065 / (−0.00016).
  4. Step 4 – Calculate: T = −40.6 seconds.
  5. Step 5 – Interpret sign: A negative period means the power is decreasing by a factor of e every ~40.6 seconds. On a semi-log plot this appears as a downward-sloping straight line.
  6. Step 6 – Classify: This is a controlled, intentional power reduction—normal operational behavior when shutting down or reducing power.
Answer: T ≈ −41 seconds; reactor power is falling by a factor of e every 41 seconds. This represents a controlled, stable power decrease.
4. Express a reactivity of ρ = 0.0065 in dollars and state the significance of that value.
  1. Step 1 – Recall the definition: 1 dollar = β = 0.0065 for U-235.
  2. Step 2 – Convert: ρ (dollars) = ρ (fraction) / β = 0.0065 / 0.0065 = 1.00 dollar.
  3. Step 3 – Apply the simplified inhour formula to see what period this gives: T = β / (ρ × λ) = 0.0065 / (0.0065 × 0.08) = 0.0065 / 0.00052 ≈ 12.5 seconds. NOTE: At exactly 1 dollar the simplified one-group formula breaks down and the full inhour equation is needed; the actual period approaches zero (or becomes undefined) as ρ → β because prompt neutrons alone sustain criticality.
  4. Step 4 – State significance: ρ = 1 dollar is the prompt criticality boundary. At this point the reactor is critical on prompt neutrons alone, and power would rise on a microsecond timescale—far beyond the ability of mechanical control systems or operators to manage safely. This condition must never be reached during normal or abnormal operations.
Answer: ρ = 0.0065 = 1.00 dollar = prompt criticality threshold. This is the absolute operational boundary; exceeding it produces an uncontrollable power excursion.

The math in this lesson2

Learners compute reactor period with T = β/(ρλ) from given reactivity and decay constant.
A-SSE.1F-LE.1
Brush up in the math halls at newSchoolWorks.com:
In class: Lecture with timeline diagrams: students annotate a fission event timeline showing prompt neutron emission (~10⁻¹⁴ s) versus delayed neutron emission (up to ~55 s), then explain in writing why a reactor controlled only by prompt neutrons would be uncontrollable at normal reactivity levels. · Delayed neutron group table exercise: using a standard six-group delayed neutron data table for U-235, students calculate the weighted mean delayed neutron fraction (β) and compare to the prompt criticality threshold, reinforcing the concept of 'prompt critical' as an operational boundary to avoid. · Reactor period calculation practice: students use the simplified inhour approximation to compute reactor period for three given reactivity insertions (small positive, near-prompt-critical, and small negative), plot power vs. time on semi-log paper, and identify which case represents an unacceptable transient.
Materials: Delayed neutron group data table (U-235, Keepin values), semi-log graph paper, scientific calculators, Lamarsh & Baratta textbook, whiteboard
Check: Problem set of five reactor period calculations; students must correctly compute period and classify reactor behavior (stable, period alarm, prompt critical approach) for each. Pass mark 80%.

Practice check8

Answer each question, then check it. Nothing is recorded; this is your own practice.

NRC Non-Licensed Operator Prep / Reactor Theory
1. What is the primary reason delayed neutrons make reactor control possible?
NRC Reactor Operator (RO) License / Reactor Kinetics
2. For U-235 fuel, the total delayed neutron fraction (β) is approximately:
NRC Reactor Operator (RO) License / Reactor Kinetics
3. Prompt criticality is defined as the condition in which:
NRC Reactor Operator (RO) License / Reactor Kinetics
4. Using the simplified inhour approximation T ≈ β / (ρ · λ) with β = 0.0065 and λ = 0.08 s⁻¹, what is the reactor period for a reactivity insertion of ρ = 0.002?
NUCP Uniform Curriculum (NEI) / Reactor Theory
5. On a semi-log power-versus-time plot, a reactor operating with a constant negative reactor period would appear as:
NRC Reactor Operator (RO) License / Operational Safety
6. A reactor period of 15 seconds with a positive sign most likely indicates:
NRC Non-Licensed Operator Prep / Reactor Physics
7. Which of the following best describes a delayed neutron precursor?
OSHA 10 / Safety & Health
8. During maintenance on a control rod drive mechanism, which safety practice is mandatory before any work begins?
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