Excess Reactivity Calculator
Convert between the multiplication factor and reactivity, express it in Δk/k, %Δk/k, pcm, mk, dollars or cents — with a reactivity-versus-k chart, a reactor-state gauge, βeff presets, and sourced reference tables.
k above 1 → positive ρ
A supercritical core has k > 1 and positive reactivity: each neutron generation is larger than the last, so power rises unless control is inserted.
k below 1 → negative ρ
A subcritical core has k < 1 and negative reactivity: the neutron population dies away, which is the state shutdown margin is measured in.
ρ ≈ k − 1 near criticality
For k close to 1, dividing by k changes the answer only slightly, so ρ ≈ k − 1. The gap widens for larger departures — at k = 1.10, ρ is 9,091 pcm rather than 10,000.
Dollars and βeff
Reactivity measured in dollars is ρ divided by βeff. At 1 $ or more a reactor is prompt critical and no longer relies on delayed neutrons. Because βeff shifts with fuel and burnup, the same pcm value can mean very different dollars.
How to Use This Calculator
The reactivity relation links the multiplication factor k and the reactivity ρ. Pick which one to solve, then enter the other.
1. Choose what to solve for
Pick ρ (from k) or k (from ρ) — the tool applies ρ = (k − 1)/k or its inverse k = 1/(1 − ρ).
2. Pick the reactivity unit
Show ρ in Δk/k, %Δk/k, pcm, mk, dollars or cents — the field converts automatically.
3. Set βeff
Choose a preset delayed neutron fraction or type your own. It only affects dollars, cents and the prompt-critical check.
4. Enter k or ρ, or load an example
Use a core-state preset, drag the slider, type a value, or load one of the published worked examples.
5. Read the result, steps, and charts
The state badge shows subcritical, critical, supercritical or prompt critical; the chart plots ρ against k with your point marked.
6. Go Advanced, convert, or copy
Advanced mode adds sensitivity, the reactor-state gauge and every unit at once; the converters below handle any reactivity unit.
What This Tool Offers
Built around the definition of reactivity, ρ = (k − 1)/k, the quantity reactor physicists use to describe how far a core sits from criticality.
Solve either direction
Go from k to reactivity, or from a reactivity back to the multiplication factor that produces it.
Six reactivity units
Δk/k, %Δk/k, pcm, mk, dollars and cents, with live conversion in the main field and in a standalone converter.
Reactivity-versus-k chart
The full ρ(k) curve with subcritical and supercritical regions shaded, the prompt-critical line, and your point marked.
Reactor-state gauge
A dollar-scaled gauge showing where the core sits: subcritical, delayed supercritical, or prompt critical.
βeff presets & examples
Delayed neutron fractions for U-235, U-233, Pu-239, U-238 and light-water-reactor cycle extremes, plus published worked examples.
Conversion & reference tables
k-to-pcm and pcm-to-k lookup tables, unit factors, reactor-state definitions, and sourced β values.
Conversion Tables & References
Live converters, followed by lookup tables and sourced reference data for reactivity units and delayed neutron fractions.
Reactivity unit converter
k ↔ ρ converter
Reactivity Unit Conversion Factors
| Unit | Equals (Δk/k) | Notes |
|---|---|---|
| 1 Δk/k | 1 | Dimensionless base unit |
| 1 %Δk/k | 0.01 | = 1,000 pcm |
| 1 pcm (per cent mille) | 10⁻⁵ | = 0.001 %Δk/k |
| 1 mk (milli-k) | 10⁻³ | = 100 pcm; usage varies by facility |
| 1 $ (dollar) | βeff | e.g. 650 pcm when βeff = 0.0065 |
| 1 ¢ (cent) | βeff / 100 | = 1/100 of a dollar |
k → Reactivity Lookup
| k | ρ (Δk/k) | ρ (%Δk/k) | ρ (pcm) | State |
|---|---|---|---|---|
| 0.900 | −0.11111 | −11.111 | −11,111 | Subcritical |
| 0.950 | −0.05263 | −5.263 | −5,263 | Subcritical |
| 0.980 | −0.02041 | −2.041 | −2,041 | Subcritical |
| 0.990 | −0.01010 | −1.010 | −1,010 | Subcritical |
| 0.995 | −0.00503 | −0.503 | −503 | Subcritical |
| 1.000 | 0 | 0 | 0 | Critical |
| 1.001 | 0.000999 | 0.0999 | 99.9 | Supercritical |
| 1.002 | 0.001996 | 0.1996 | 199.6 | Supercritical |
| 1.005 | 0.004975 | 0.4975 | 497.5 | Supercritical |
| 1.010 | 0.009901 | 0.9901 | 990.1 | Supercritical |
| 1.020 | 0.019608 | 1.9608 | 1,960.8 | Supercritical |
| 1.050 | 0.047619 | 4.7619 | 4,761.9 | Supercritical |
| 1.100 | 0.090909 | 9.0909 | 9,090.9 | Supercritical |
Reactivity → k Lookup
| ρ (pcm) | ρ (%Δk/k) | k = 1/(1 − ρ) |
|---|---|---|
| −5,000 | −5.0 | 0.952381 |
| −1,000 | −1.0 | 0.990099 |
| −500 | −0.5 | 0.995025 |
| 0 | 0 | 1.000000 |
| 100 | 0.1 | 1.001001 |
| 500 | 0.5 | 1.005025 |
| 650 (≈ 1 $ at βeff = 0.0065) | 0.65 | 1.006543 |
| 1,000 | 1.0 | 1.010101 |
| 5,000 | 5.0 | 1.052632 |
Delayed Neutron Fractions (β)
The delayed neutron fraction depends on the fissioning isotope and neutron energy, so it is not a universal constant.
| Fuel / case | β | 1 $ ≈ (pcm) |
|---|---|---|
| U-235, thermal fission | ≈ 0.0064 – 0.0067 (0.0065 often used) | ≈ 650 |
| U-233, thermal fission | 0.0026 | 260 |
| Pu-239, thermal fission | 0.0021 | 210 |
| U-238, fast fission | 0.0148 | 1,480 |
| LWR effective β, start of cycle | ≈ 0.007 | ≈ 700 |
| LWR effective β, end of cycle | ≈ 0.005 | ≈ 500 |
| Measured βeff, a highly enriched uranium fast reactor | 0.00645 (± 3.0%) | ≈ 645 |
Sources: Wikipedia — Delayed neutron (U-235, Pu-239, U-238); ScienceDirect Topics (U-233, typical U-235 value); nuclear-power.com (LWR cycle values); arXiv 1510.06276 (measured fast-reactor value). Values differ slightly between references.
Reactor States
| Condition | k | ρ | Meaning |
|---|---|---|---|
| Subcritical | < 1 | < 0 | Chain reaction dies away |
| Critical | = 1 | = 0 | Steady neutron population |
| Supercritical (delayed) | > 1 | 0 < ρ < βeff | Power rises, paced by delayed neutrons (controllable) |
| Prompt critical | > 1 | ρ ≥ βeff (≥ 1 $) | Chain reaction sustained by prompt neutrons alone; power rises extremely fast |
Published Worked Examples
Load any of these with the example buttons above.
| Example | ρ | βeff | Published | Calculator result |
|---|---|---|---|---|
| Control-rod withdrawal error, U-235 reactor | 0.004 Δk/k | 0.0065 | ≈ $0.61 | $0.615 (k = 1.004016) |
| Highly supercritical reactor | 0.007 Δk/k | 0.007 | 1 $ → prompt critical | $1.000 (k = 1.007049) |
| 500 pcm at start of LWR cycle | 500 pcm | 0.007 | — | $0.714 (calculated) |
| 500 pcm at end of LWR cycle | 500 pcm | 0.005 | — | $1.000 (calculated) |
Sources: Lecture 16 notes (delayed neutrons and the inhour equation); Wikipedia — Dollar (reactivity). The two LWR rows apply the β values from nuclear-power.com and are calculated here, not quoted: the same 500 pcm is far less than a dollar at start of cycle but equals one dollar at end of cycle.
Sources & Further Reading
| Source | What it covers |
|---|---|
| nuclear-power.com — Reactivity | Definition of reactivity in terms of k and common units for power reactors |
| nuclear-power.com — pcm | Why pcm and %Δk/k are preferred for power reactors |
| Wikipedia — Dollar (reactivity) | Dollars, cents and prompt criticality |
| Wikipedia — Delayed neutron | β by isotope and effective β |
| Lecture 16 notes | Reactivity in dollars and the inhour equation |
What Is Excess Reactivity?
The multiplication factor k tells you how the neutron population changes from one generation to the next. Reactivity re-expresses that as a fractional departure from criticality: how far above (positive) or below (negative) the steady state the core sits. Excess reactivity is the positive reactivity a core has beyond exactly critical — the built-in margin that must later be offset by control rods, burnable absorbers or soluble poison so the reactor can run steadily and make up for fuel burnup, temperature and fission-product buildup.
Reactivity Formula
The defining relationship is:
Rearranged to recover the multiplication factor from a known reactivity:
Reactivity in dollars normalizes by the delayed neutron fraction:
Reactivity Variables
| Symbol | Meaning | Notes |
|---|---|---|
| ρ | Reactivity (excess reactivity when positive) | Dimensionless; expressed as Δk/k, %Δk/k, pcm, mk, $ or ¢ |
| k | Effective multiplication factor | < 1 subcritical, = 1 critical, > 1 supercritical |
| βeff | Effective delayed neutron fraction | Needed only for dollars and cents; varies with fuel and burnup |
How k Affects Reactivity
Reactivity rises with k but not in a straight line: because ρ divides by k, the curve flattens as k grows. Close to criticality the two are almost interchangeable (ρ ≈ k − 1), which is why small reactivities are conveniently quoted in pcm. Reactor operators also care about the unit: pcm and %Δk/k are common for power reactors, while dollars express how close the core is to prompt criticality — but the value of a dollar changes with the fuel, since βeff falls as plutonium builds up during burnup.
Reactivity Calculation Examples
ρ = (1.005 − 1) / 1.005 = 0.005 / 1.005
ρ = 0.004975 Δk/k = 497.5 pcm = 0.4975 %Δk/k
With βeff = 0.0065 → ρ = 0.004975 / 0.0065 = $0.765
k = 1 / (1 − 0.005) = 1 / 0.995
k = 1.005025
ρ ($) = 0.004 / 0.0065
ρ = ≈ $0.62 — below one dollar, so still delayed-neutron-controlled
Frequently Asked Questions
Dividing by k measures the change relative to the new generation's neutron population, which makes reactivity behave consistently in the reactor kinetics equations. For k close to 1 the two are nearly identical; the difference only shows up for larger departures from criticality.
A pcm ("per cent mille") is one hundred-thousandth of Δk/k, so 1 pcm = 10⁻⁵ and 1,000 pcm = 1 %Δk/k. It is one of the most common reactivity units for power reactors, together with %Δk/k.
A dollar is the reactivity equal to the effective delayed neutron fraction, βeff. At one dollar or more, a reactor is prompt critical: the chain reaction is sustained by prompt neutrons alone, so power can rise far faster than delayed-neutron-based control can respond. A cent is one hundredth of a dollar.
Because βeff changes as fuel burns. In light-water reactors it drops from roughly 0.007 at the start of a cycle to about 0.005 at the end, as plutonium-239 (with a delayed fraction near 0.0021) contributes more of the fissions. The same pcm value therefore corresponds to different dollars over the cycle — see the 500 pcm examples above.
Yes. Negative reactivity means k < 1: the core is subcritical and the neutron population decays. Shutdown margin is the amount of negative reactivity available when control is fully inserted. The calculator accepts any reactivity below 1 Δk/k.
No. The reactor period depends on the delayed-neutron groups and the neutron lifetime, and is found from the inhour equation or full kinetics — not from ρ = (k − 1)/k alone. The calculator gives the reactivity and how it compares with βeff, not the transient response.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| ρ | Excess Reactivity | dimensionless |
| k | Multiplication Factor | dimensionless |
What it means
Excess reactivity indicates how much control is needed to keep the reactor critical. It must be managed to ensure safe operation.
Worked example
Excess Reactivity (ρ = (k − 1) / k)
Reactor Physics| Parameter | Value |
|---|---|
| Effective Multiplication Factor (k) | 1.06 |
| Excess Reactivity (ρ = (k − 1) / k) | 0.0566 (or 5.66 % Δk/k) |
End‑of‑Cycle Excess Reactivity (ρ = (k − 1) / k)
Reactor Physics| Parameter | Value |
|---|---|
| Effective Multiplication Factor (k) | 1.02 |
| Excess Reactivity (ρ = (k − 1) / k) | 0.0196 (1.96 % Δk/k) |
Research Reactor Excess Reactivity (ρ = (k − 1) / k)
Reactor Physics| Parameter | Value |
|---|---|
| Effective Multiplication Factor (k) | 1.005 |
| Excess Reactivity (ρ = (k − 1) / k) | 0.00498 (0.498 % Δk/k) |
Common mistakes
- Confusing reactivity ρ with reactivity worth: ρ is the state of the reactor; worth is the change caused by a control device.
- Using absolute k instead of (k‑1)/k: Some use ρ = k‑1, which is the reactivity in dollars, but the standard definition divides by k.
- Ignoring sign conventions: Positive ρ means supercritical, negative ρ means subcritical; reversing the sign leads to wrong control actions.
Applications
- Reactivity meters: Used in instrumentation to display the reactor’s current state.
- Fuel burnup management: Tracks how excess reactivity decreases over the cycle to schedule refuelling.
- Safety limits: Ensures that the reactor never exceeds the maximum allowed excess reactivity.
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