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Reactivity

Normalized measure of how far a reactor is from criticality, used throughout reactor control and kinetics calculations.

Written by the Vorzep Team

NuclearReactor PhysicsKinetics

Reactivity CalculatorNuclear Reactor

ρ = ( keff − 1 ) / keff
ρ = reactivity  ·  keff = effective multiplication factor
⟹ Solve⟷ρ, keff
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Presets:
Reactivity (ρ)—
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ρ: —keff: —
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Reactivity Magnitude—
Low (< 0.01) Medium (0.01–0.05) High (0.05–0.1) Very High (> 0.1)
ρ = (keff − 1) / keff  ·  ρ < 0: subcritical, ρ = 0: critical, ρ > 0: supercritical
ρ = (k_eff - 1) / k_eff
Reactivity

Variables

SymbolQuantityUnit
ρReactivity
k_effEffective multiplication factor

What it means

Reactivity (ρ) is a dimensionless parameter that indicates how far a nuclear reactor is from criticality. It is defined as ρ = (k_eff − 1)/k_eff, where k_eff is the effective multiplication factor. When ρ = 0, the reactor is critical; ρ > 0 means supercritical (power increasing); ρ < 0 means subcritical (power decreasing). Reactivity is often expressed in units of "dollars" (where $1 = the amount of reactivity equal to the delayed neutron fraction) or in pcm (percent mille, i.e., 0.001 dollar). Reactivity changes are caused by control rod movement, temperature effects, fuel burnup, and poison buildup. Understanding reactivity is fundamental for reactor control, safety analysis, and reactivity feedback mechanisms. It is used in the point kinetics equations and in the design of reactivity control systems.

Worked example

Reactivity – Two Examples

Real‑World
Scenario: A reactor has k_eff = 1.001. The nuclear engineer calculates the reactivity to determine the amount of control rod insertion needed to return the reactor to criticality during a power manoeuvre.
ParameterValue
k_eff1.001
1ρ = (1.001−1)/1.001 = 0.001/1.001 = 9.99×10⁻⁴
Result 9.99×10⁻⁴ ✓ Positive reactivity
Scenario: A reactor is subcritical with k_eff = 0.995. The reactor operator calculates the negative reactivity to determine how far the reactor is from criticality and the shutdown margin for safe maintenance operations.
ParameterValue
k_eff0.995
1ρ = (0.995−1)/0.995 = −0.005/0.995 = −0.00503
Result −0.00503 ✓ Subcritical
Nuclear insight: Reactivity measures the deviation from criticality. Positive reactivity increases power, negative reactivity decreases power. Reactivity is zero at critical.

Common mistakes

  • Reactivity ρ: A measure of the deviation from criticality – dimensionless (often in pcm, 1 pcm = 10⁻⁵).
  • Formula: ρ = (k_eff − 1) / k_eff – not (k_eff − 1) without dividing.
  • Sign: Positive for supercritical, negative for subcritical, zero at critical.
  • Units: Often expressed in %Δk/k or pcm – be consistent.
  • Reactivity worth: Control rods, poisons, temperature coefficients are expressed in reactivity units.

Applications

Reactivity, ρ = (k_eff − 1)/k_eff, quantifies the deviation of a nuclear reactor from criticality. It is a dimensionless measure of the fractional change in neutron population per generation. Reactivity is used to express the effect of control rod movement, fuel burnup, temperature changes, and other perturbations. Reactor physicists and operators use reactivity to monitor the state of the core, to calculate the worth of control devices, and to predict the response to transients. It is central to the design of reactivity control systems and to the analysis of reactor accidents. By tracking reactivity, engineers can ensure that the reactor operates within safe margins and that all safety systems function as designed.

  • Reactor core monitoring and control
  • Calibration of control rods and measurement of rod worth
  • Analysis of temperature and void reactivity feedbacks
  • Design of safety systems for reactivity control
  • Assessment of core burnup and fuel cycle management