Fission Reactor Criticality

Katie Nath
March 24, 2026

Submitted as coursework for PH241, Stanford University, Winter 2026

Introduction

Fig. 1: Enrichment ξ versus volume fraction of fuel Ω required for criticality with light water moderation, as generated by Eq. (1) with the paramgers in Table 1. (Image source: F. Nath)

The condition for reactor criticality can be understood most directly as a balance between neutron production and neutron loss. For the reactor to be critical, the two must be exactly equal.

Rather than working with a full neutron transport equation that can account for reflection and loss of neutrons at the reactor core surfaces, we shall consider a simplified limit in which the reactor core fills space and has a fraction Ω of fuel and a fraction 1 - Ω of moderator, in our case light water. Then the neutron n is uniform, and we need only consider its time dependence.

If dn/dt = 0, meaning that the neutron denstiy is constant, the reactor is said to be critical. [1] This is the condition required for steady power producion. If dn/dt < 0 the reactor is said to be subcritical, meaning that its neutron population is dying away. It then cannot sustain a chain cration. If dn/dt > 0 the reactor is said to be supercrtiical, meaning that the chain reaction is accelerating. This latter is required for powering up the reactor, but it must be done delicately so that the reactor does not run away.

As a practial matter, the reactor core is designed to be slightly supercritical, and then is brought back to criticality by the action of control rods. These absorb neutrons in an amount that can be varied by moving the rods mechanically up or down. But the rods only absorb neutrons, so for them to work properly it is necessary for the reactor to be at least critical when the rods are fully withdrawn.

This latter is a condition on the enrichment ξ of the fuel. This means that fraction ξ of the Uranium atoms are fissile U-235, the remaining 1 - ξ being non-fissile U-238. For a given value of Ω we wish to find the minimum value of ξ required to make the reactor just critical with the control rods fully withdrawn. Real reactor fuel must have an enrichment slightly above this value.

Growth Equation

Ths time dependence of the neutron density is given approximately by

Ω n_U v ξ σ_f²³⁵ ν [g₀ Ω + (1 − Ω)] - Ω n_U v [ξ (σ_a²³⁵ + σ_f²³⁵) + (1 − ξ) σ_a²³⁸] - (1 − Ω) n_H v σ_a^H

(1)

with parameters defined as in Table 1. [2,3] Setting this to zero we obtain for the criticality condition

Ω σ_a²³⁵ + (1 − Ω) (n_H/n_U) σ_a^H

σ_f²³⁵ ν [g₀ Ω + (1 − Ω)] - (σ_a²³⁵ + σ_f²³⁵) + σ_a²³⁸

(2)

This is plotted in Fig. 1

Symbol	Meaning	Microscopic Data at T = 300°K
Symbol	Meaning	Pure U-235	Pure U-238	Moderator (Hydrogen)
n_U,H	Atomic Density	4.83 × 10²² cm^-3	4.83 × 10²² cm^-3	6.68 × 10²² cm^-3
v	Thermal Neutron Average Speed	2.72 × 10⁵ cm sec^-1	2.72 × 10⁵ cm sec^-1	2.72 × 10⁵ cm sec^-1
σ_a	Absorption Cross Section	9.8 × 10^-23 cm²	2.71 × 10^-24 cm²	3.32 × 10^-25 cm²
σ_f	Fission Cross Section	5.80 × 10^-22 cm^-2	(Negligible)	N/A
ν	Average Neutrons Per Fission	2.43	N/A	N/A
Ω	Fuel Volume Fraction	1	1	0
g₀	Fraction of Neutrons That Make It to Thermalization	0.5	-	-
ξ	Fraction of U Atoms That are U₂₃₅	1	0	-

Table 1: Microscopic parameters at T = 273°K used in evaluating the neutron growth equation. The listed number densities and cross sections determine the balance of fission production and absorption losses. [2,3]

Thermal Neutron Loss and Growth Rates

We can separate the neutron balance into a loss rate (absorption) and a growth rate (net fission production). Each term is written as a probability per unit time so that the criticality condition can be stated as a balance between measurable microscopic quantities.

The thermal neutron absorption (loss) rate is

Loss rate

n_UΩ v [ ξ (σ_a²³⁵ + σ_f²³⁵) + (1 − ξ) σ_a²³⁸ ] + (1 - ω) n_H v σ_a^H

(3)

The first term represents absorption in the fuel. The factor Ω is the probability that the neutron is in the fuel at a given time. A fraction ξ of uranium nuclei are U-235 and a fraction (1 − ξ) are U-238, so the absorption contributions are weighted accordingly. The second term represents absorption in the moderator (hydrogen), with probability (1 − Ω). The units check is explicit: number densities n have units of cm⁻³, neutron speed v has units of cm s⁻¹, and microscopic cross sections σ have units of cm². Thus each product n v σ has units of s⁻¹, as required for a rate.

The thermal neutron production (growth) rate is

Growth rate

Ωξ n_U v σ_f²³⁵ ν [g₀Ω + (1 − Ω)]

(4)

It is due entirely to U-235 fissions. The factor ν accounts for the net number of neutrons released per fission after one neutron is consumed to induce the event. The bracketed factor [g₀ Ω + (1 - Ω)] represents the survival probability during thermalization. If the neutron thermalizes while in the fuel (probability Ω) it survives with probability g₀. If it thermalizes in the moderator (probability 1 - Ω) the survival probability is taken as unity.

The parameter g₀ is determined by requiring consistency with the observed properties uranium metal. Unmoderated uranium becomes critical only when enriched to approximately ξ = 0.06. Setting Ω = 1 and ξ = 0.06 in Eq. (2) and solving for g₀ yields g₀ ≈ 0.5. Thus g₀ is a fit parameter anchored to experimental criticality data.

Atomic Density and Thermal Velocity

The natural Uranium atomic density n_U is given in terms of the measured mass density ρ, Avogadro's number N_A and Uranium molar mass M_U by

n_U

ρ_U N_A

M_U

19.1 g cm^-3 × 6.022 × 10²³ atoms mole^-1

238 gm mole^-1

4.83 × 10²² atoms cm^-3

(5)

For water we have similarly

n_H

2 ×

ρ_H₂O N_A

M_H₂O

2 atoms molecule^-1 × 1 g cm^-3 × 6.022 × 10²³ molecles mole^-1

18 gm mole^-1

6.68 × 10²² atoms cm^-3

(6)

The average thermal speed is given in terms of Boltzmann's constant k_B and the neutron mass m_n by

(

3 k_B T

m_n

)^1/2

(

3 × 1.380 × 10^-16 erg °K^-1 × 300 °K

1.674 × 10^-24 g

)^1/2

2.72 × 10⁵ cm sec^-1

(7)

Conclusion

Reactor criticality is fundamentally a problem of neutron balance. By expressing the time rate of change of the neutron population in terms of microscopic cross sections, number densities, enrichment fraction, and moderation probability, the condition for criticality emerges as the requirement dN/dt = 0. This balance equates neutron production from U-235 fission with absorption losses in U-235, U-238, and the moderator.

The minimum of the criticality curve occurs at ξ ≈ 0.04 - 0.06, indicating that several percent enrichment in U-235 is required for criticality in light water systems, because absorption in U-238 and hydrogen reduces neutron population. Increasing enrichment strengthens the positive fission term and restores the balance required for sustained chain reaction. The minimum in the enrichmentmoderation curve reflects optimal moderation, where thermalization is sufficient to enhance fission without excessive absorption losses.

The competition between fission production and absorptive loss, quantified through measurable cross sections and number densities, determines whether a reactor is subcritical, critical, or supercritical.

© Katie Nath. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

References

[1] G. I. Bell and S. Glasstone, Nuclear Reactor Theory (Krieger Pub. Co. 1970).

[2] S. Glasstone and A. Sesonske, Nuclear Reactor Engineering, 2nd ed. (Van Nostrand Reinhold,1963).

[3] G. C. Hanna et al., "Revision of Values for the 2200 m/s Neutron Constants for Four Fissile Nuclides," Atom. Energy Rev. 7, 3 (1969).