Point kinetics equations

Point kinetics equations (also called the point reactor kinetics equations) are a set of differential equations used in nuclear reactor dynamics to describe how the neutron population of a reactor changes with time, ignoring spatial variations. In the point kinetics model, the neutron flux is assumed to have a fixed spatial shape, so the equations track only the time-dependent neutron population and the concentrations of the delayed-neutron precursors. The equations are used to analyze reactor startups, shutdowns, and accidental power transients.

Derivation

The point kinetics model is obtained from the time-dependent neutron diffusion equation. The neutron flux is assumed to remain proportional to a fixed spatial shape during a transient, so that it can be written as the product of a shape function and a time-dependent amplitude (the fundamental mode assumption). Removing the spatial dependence in this way reduces the problem to a system of ordinary differential equations in time alone.

Standard form

The standard form of the point kinetics equations uses six groups of delayed-neutron precursors:

${\frac {dn}{dt}}={\frac {\rho (t)-\beta }{\Lambda }}n(t)+\sum _{i=1}^{6}\lambda _{i}C_{i}(t),$

${\frac {dC_{i}}{dt}}={\frac {\beta _{i}}{\Lambda }}n(t)-\lambda _{i}C_{i}(t)\qquad (i=1,\dots ,6).$

where $n(t)$ is the neutron population (proportional to reactor power), $\rho (t)$ is the reactivity, $\Lambda$ is the prompt neutron generation time, $\beta$ is the total delayed neutron fraction, $\beta _{i}$ and $\lambda _{i}$ are the delayed neutron fraction and decay constant of the $i$ -th precursor group, and $C_{i}(t)$ is the concentration of that group.

The effective delayed neutron fraction $\beta _{\mathrm {eff} }$ is usually slightly larger than $\beta$ in thermal reactors and depends on the reactor size and neutron spectrum; it is the quantity that sets the reactivity thresholds in the state classification described in the role of delayed neutrons section.

The delayed neutrons are traditionally represented by six groups whose yields and decay constants are empirical parameters obtained from least-squares fits to experimental measurements. For uranium-235 fission the total delayed neutron fraction is $\beta \approx 0.0065.$

Six-group delayed neutron data for uranium-235 fission
Group	Half-life $T_{1/2}$ (s)	Relative yield $\beta _{i}/\beta$
1	54.51	0.038
2	21.84	0.213
3	6.00	0.188
4	2.23	0.407
5	0.496	0.128
6	0.179	0.026

The group decay constants follow from $\lambda _{i}=\ln 2/T_{1/2}$ and range from about 0.013 s⁻¹ (55-second group) to about 3.9 s⁻¹ (0.18-second group).

Role of delayed neutrons

The prompt neutron lifetime in a thermal reactor is on the order of 10⁻⁴ seconds, and is shorter (around 10⁻⁸–10⁻⁶ s) in fast reactors. The generation time is not itself the reactor's power-growth time: the reactor period depends on the reactivity and the reactor's kinetic parameters, and for a small positive reactivity it is approximately the mean generation time, including the effect of the delayed neutrons, divided by the reactivity. If only prompt neutrons were present, even a very small reactivity insertion would give a very short period: for example, with a prompt neutron lifetime of 2×10⁻⁵ s, a 10 pcm insertion would give a period of only 0.2 s, and the power would rise by a factor of $e^{50}$ in 10 s. Delayed neutrons lengthen the mean generation time, to about 0.085 s in that same example; for the same 10 pcm insertion the period is then about 850 s, slow enough for control systems such as control rods to respond.

Depending on where the reactivity $\rho$ lies relative to the effective delayed neutron fraction $\beta _{\mathrm {eff} }$ , the following regimes are distinguished:

Subcritical ( $\rho <0$ ): the chain reaction is not self-sustaining; without a neutron source the population decays in time, while with an external or internal neutron source present the reactor is in the state called subcritical multiplication.
Delayed critical ( $\rho =0$ ): prompt and delayed neutrons together sustain the chain reaction, so the population is constant in time at equilibrium.
Delayed supercritical ( $0<\rho <\beta _{\mathrm {eff} }$ ): the population grows, with delayed neutrons essential to sustaining the chain reaction. This allows slower, controllable power changes, although the growth can become dangerously rapid as $\rho$ approaches $\beta _{\mathrm {eff} }$ .
Prompt critical ( $\rho =\beta _{\mathrm {eff} }$ ): prompt neutrons alone exactly balance neutron losses, making the chain reaction self-sustaining without delayed neutrons.
Prompt supercritical ( $\rho >\beta _{\mathrm {eff} }$ ): prompt neutrons alone support a growing chain reaction, and the neutron number rises on the prompt-neutron timescale, allowing very rapid power increases.

The distinction between the last two states follows from the prompt-neutron balance. Power reactors are designed for operation under conditions in which delayed neutrons make the chain reaction controllable, and safety systems prevent prompt criticality; a specially designed pulsed test reactor is an exception, since it is operated super prompt critical for a short fraction of each cycle.

Approximate solutions

Prompt jump approximation

For a step insertion of reactivity $\rho <\beta$ from a critical state, the power first changes sharply on the prompt timescale (the precursors cannot respond that quickly) and then follows a slower asymptotic period. The prompt jump approximation assumes the initial change is instantaneous and gives the ratio of the power just after to just before the insertion as

${\frac {n(0^{+})}{n(0^{-})}}={\frac {\beta }{\beta -\rho }}.$

This approximation is usually valid for moderate insertions, for example $\rho <\beta /2.$

One delayed group

For one effective delayed group with an average decay constant $\lambda$ , precursor concentration $C$ , and a constant step reactivity $\rho _{0}$ , the point kinetics equations in terms of the instantaneous power $P(t)$ simplify to

${\frac {dP}{dt}}={\frac {\rho _{0}-\beta }{\Lambda }}P(t)+\lambda C(t),$

${\frac {dC}{dt}}={\frac {\beta }{\Lambda }}P(t)-\lambda C(t),$

with the general solutions

$P(t)=P_{1}e^{s_{1}t}+P_{2}e^{s_{2}t},$

$C(t)=C_{1}e^{s_{1}t}+C_{2}e^{s_{2}t},$

where the exponents are, approximately (for $\lambda \Lambda /\beta \ll 1$ and $|\rho _{0}|\ll \beta$ ),

$s_{1}={\frac {\lambda \rho _{0}}{\beta -\rho _{0}}},$

$s_{2}=-({\frac {\beta -\rho _{0}}{\Lambda }}).$

The time constant expressing the more slowly varying asymptotic behavior, $T=1/s_{1}$ , is sometimes referred to as the stable reactor period.

Reactor period and the inhour equation

Once the fast transients have died out, the power varies as a single exponential characterized by the stable reactor period $T.$ The relation between a step reactivity $\rho$ and the resulting period is given by the inhour equation:

$\rho ={\frac {\Lambda }{T}}+\sum _{i=1}^{6}{\frac {\beta _{i}}{1+\lambda _{i}T}}.$

In terms of the reciprocal period $s=1/T$ this is a seventh-order polynomial: for a given reactivity it has seven roots, of which the dominant (rightmost) root gives the stable reactor period.

Numerical solution

The prompt timescale (on the order of 10⁻⁴ s) is orders of magnitude shorter than the delayed-neutron precursor half-lives (up to about 55 s for the longest-lived group), which makes the system difficult to integrate with finite-difference schemes. The prompt jump approximation is frequently used in numerical studies of the point kinetics equations precisely because it eliminates this very short time scale.

Applications

The point kinetics equations are used by engineers to study the kinetic behavior of a reactor: to analyze reactor startups, shutdowns, and accidental power transients (including the response to reactivity changes, such as a control-rod step), and to serve as a benchmark for testing the numerical methods used in more detailed space-time kinetics calculations.

References

1 2 3 4 5 6 7 8 9 10 11 Duderstadt, James J.; Hamilton, Louis J. (1976). Nuclear Reactor Analysis. New York: Wiley. ISBN 978-0-471-22363-4.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 "Point Kinetics Equations". Nuclear Power. Retrieved 2026-10-09.
1 2 3 4 5 6 7 8 Yoshiaki Oka; Katsuo Suzuki, eds. (2013). Nuclear Reactor Kinetics and Plant Control. An Advanced Course in Nuclear Engineering. Tokyo: Springer. doi:10.1007/978-4-431-54195-0. ISBN 978-4-431-54194-3.
↑ Bell, George I.; Glasstone, Samuel (1970). Nuclear Reactor Theory. New York: Van Nostrand Reinhold. ISBN 978-0-442-20684-0.
1 2 3 4 Zohuri, Bahman (2019). Neutronic Analysis For Nuclear Reactor Systems (2nd ed.). Cham: Springer. doi:10.1007/978-3-030-04906-5. ISBN 978-3-030-04906-5.
1 2 Hetrick, David L. (1993). Dynamics of Nuclear Reactors. La Grange Park, Ill.: American Nuclear Society. ISBN 0-894-48453-2.
↑ Lewis, E. E.; Miller, W. F. Jr. (1984). Computational Methods of Neutron Transport. New York: Wiley-Interscience. ISBN 0-471-09245-2.

[duderstadt-1] 1 2 3 4 5 6 7 8 9 10 11 Duderstadt, James J.; Hamilton, Louis J. (1976). Nuclear Reactor Analysis. New York: Wiley. ISBN 978-0-471-22363-4.

[npcom-2] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 "Point Kinetics Equations". Nuclear Power. Retrieved 2026-10-09.

[okasuzuki-3] 1 2 3 4 5 6 7 8 Yoshiaki Oka; Katsuo Suzuki, eds. (2013). Nuclear Reactor Kinetics and Plant Control. An Advanced Course in Nuclear Engineering. Tokyo: Springer. doi:10.1007/978-4-431-54195-0. ISBN 978-4-431-54194-3.

[bell_glasstone1970-4] Bell, George I.; Glasstone, Samuel (1970). Nuclear Reactor Theory. New York: Van Nostrand Reinhold. ISBN 978-0-442-20684-0.

[zohuri-5] 1 2 3 4 Zohuri, Bahman (2019). Neutronic Analysis For Nuclear Reactor Systems (2nd ed.). Cham: Springer. doi:10.1007/978-3-030-04906-5. ISBN 978-3-030-04906-5.

[hetrick-6] 1 2 Hetrick, David L. (1993). Dynamics of Nuclear Reactors. La Grange Park, Ill.: American Nuclear Society. ISBN 0-894-48453-2.

[lewismler-7] Lewis, E. E.; Miller, W. F. Jr. (1984). Computational Methods of Neutron Transport. New York: Wiley-Interscience. ISBN 0-471-09245-2.