Formula & Calculator
Delayed Neutron Fraction
The delayed neutron fraction is the sum of the fractions of delayed neutrons from each precursor group. Delayed neutrons are emitted with half‑lives ranging from milliseconds to minutes, and they are essential for reactor control. The total β is typically about 0.0065 for thermal fission of U‑235. This parameter appears in the point kinetics equations and determines the reactivity worth of control rods and the safety margins.
Delayed NeutronsReactor KineticsNuclear Fission
Delayed Neutron Fraction
Nuclear Engineering · Reactor Physics
β = Σ βi
β =
Σ
βi
·
β = total delayed fraction ·
βi = individual group fractions
Delayed neutron fraction β is the fraction of fission neutrons that are emitted
from fission products with a measurable delay (milliseconds to minutes). It is the sum of
individual precursor group fractions βi.
For thermal fission of U-235, β ≈ 0.0065 (650 pcm).
For fast fission, β is slightly lower. This parameter is critical for reactor control,
kinetics analysis, and safety assessments.
Presets:
—
—
β unit:
Solve for:
| Group | βi | Contribution |
|---|
Target: β
Calculation Steps
Ready| Step | Operation | Value |
|---|---|---|
| Enter values and press Calculate | ||
Delayed Neutron Reference
Typical β values for different fuels
| Fuel / Condition | β (fraction) | β (pcm) |
|---|
β = Σ βi · U-235 thermal: β ≈ 0.0065 (650 pcm) · Delayed neutrons enable reactor control
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| β | Total Delayed Neutron Fraction | dimensionless |
| β_i | Fraction for Group i | dimensionless |
What it means
The delayed neutron fraction determines the amount of reactivity that can be inserted before the reactor becomes prompt critical.
Worked example
Delayed Neutron Fraction (β = Σ βi)
Reactor Physics
Scenario: A reactor physicist analyses the delayed neutron fraction for a PWR core. Delayed neutrons are emitted from fission product precursors with characteristic half‑lives. The total delayed neutron fraction (β) is the sum of the fractions from each precursor group (six groups for U‑235). This parameter determines the reactor's response to reactivity changes and is essential for control rod calibration and safety analysis.
| Group | βi (×10⁻⁴) | Half‑Life (s) |
|---|---|---|
| Group 1 | 2.1 | 55.7 |
| Group 2 | 14.0 | 22.7 |
| Group 3 | 12.6 | 6.22 |
| Group 4 | 25.0 | 2.30 |
| Group 5 | 7.4 | 0.610 |
| Group 6 | 2.7 | 0.230 |
| Total Delayed Neutron Fraction (β = Σ βi) | ||
| β = (2.1 + 14.0 + 12.6 + 25.0 + 7.4 + 2.7) × 10⁻⁴ = 0.0065 | ||
1Obtain the delayed neutron data for the fissile isotope (U‑235 or Pu‑239) from nuclear data libraries (ENDF, JENDL).
2List the fractions (βi) for each of the six precursor groups (with their respective half‑lives).
3Sum all the group fractions to obtain the total delayed neutron fraction (β).
Total Delayed Neutron Fraction
β = 0.0065
For U‑235 thermal fission, β = 0.0065 (650 pcm). This is a fundamental parameter that determines reactor control and safety margins.
Common mistakes
- Adding fractions directly as percentages: β is a fraction (e.g., 0.0065); incorrectly writing it as 0.65% in pcm calculations causes errors.
- Ignoring energy dependence: β varies slightly with neutron energy and fuel composition; using a single constant for all conditions is an approximation.
- Confusing β with βeff: βeff accounts for the importance of delayed neutrons and differs from the physical β.
Applications
- Reactor licensing: Used to demonstrate that the reactor has sufficient delayed neutron fraction for safe control.
- Fuel cycle studies: Evaluates how changing fuel (e.g., breeding plutonium) affects the delayed neutron fraction.
- Transient analysis: Essential for analysing reactivity insertion accidents (e.g., rod ejection).