Neutron Diffusion Length

Inputs

Choose what to solve for, then fill in the other two. All three quantities must use consistent length units — cm, m, mm, in or ft — the tool converts automatically. Scientific notation such as 1.97e-2 is accepted.

Results

Flux decay over the diffusion length

Pure exponential attenuation of the neutron flux in a homogeneous absorbing medium. After one diffusion length L the flux has dropped to 1/e ≈ 36.8 % of its starting value; after two L it is 1/e² ≈ 13.5 %; after three it is 1/e³ ≈ 5.0 %. The hatched markers show those fractions on the curve.

Point-source flux φ(r) vs distance

The full point-source solution φ(r) = S · e^−r/L / (4πD·r) plotted against distance on a logarithmic flux axis. Close to the source the 1/r geometric factor dominates; far from the source the exponential factor takes over and the curve steepens. The amber marker sits at r = L.

What is being calculated

Common moderators and their diffusion lengths

Values at thermal energies for pure moderators. Real reactor cores have lower L because fuel and structural materials add absorption. Illustrative values, not manufacturer data.

Diffusion data of common moderators

These are textbook values for thermal neutrons (most probable energy 0.0253 eV, "2200 m/s"). The Fermi age τ is the mean square distance travelled during slowing-down from fission energies to thermal.

Key relations and unit conversions

Formula

• L: diffusion length — the distance over which the neutron flux falls to 1/e (cm)
• D: diffusion coefficient of the medium, D = 1/(3Σ_tr) (cm)
• Σ_a: macroscopic absorption cross-section, Σ_a = N σ_a (cm⁻¹)
• Σ_tr: macroscopic transport cross-section (cm⁻¹), N number density, σ_a microscopic absorption cross-section

Example — light water. For H₂O at thermal energies, D ≈ 0.16 cm and Σ_a ≈ 0.0197 cm⁻¹. Then L = √(0.16 / 0.0197) ≈ 2.85 cm. That is the distance in which the thermal flux from a localized source falls to 1/e of its starting value — a very short distance, which is exactly why light-water reactors are so compact.

The diffusion length is the geometric mean of two very different scales. The transport mean free path λ_tr = 1/Σ_tr sets how far a neutron travels between scattering events; the absorption mean free path λ_a = 1/Σ_a sets how far it can go before being captured. The diffusion length combines them as L² = λ_tr λ_a / 3. For light water λ_a ≈ 50 cm but λ_tr ≈ 0.5 cm, so L is small: a neutron is absorbed long before it can diffuse far. For heavy water λ_a is enormous (Σ_a ≈ 3 × 10⁻⁵ cm⁻¹, λ_a ≈ 33 m), so L is huge — 170 cm — which is why D₂O-moderated reactors are physically large.

Physically, L is the answer to the question "how far does a neutron wander before it is absorbed?" More precisely, in the diffusion-theory picture where the flux obeys ∇²φ = φ/L², the flux decays as e^−r/L in a spherically symmetric medium. The diffusion length is therefore a decay constant, not a physical boundary: L is not where the flux stops, just where it has dropped to 37 %. After 5L the flux is below 1 %; after 10L it is negligible.

In a bare reactor, the diffusion length combines with the geometric buckling B² of the core shape to give the effective multiplication factor:

k_eff = k_∞ / (1 + L² B²)

The term L²B² is the leakage correction: it accounts for the fraction of neutrons that diffuse out of the core before they can be absorbed and cause another fission. Increasing the absorption (adding boron control rods, for example) reduces L, which increases L²B² only weakly, but it strongly reduces k_∞ and shuts the reactor down. Increasing the core size reduces B², which reduces L²B² and pushes k_eff toward k_∞ — the physical reason there is a minimum critical mass below which no chain reaction can be sustained.

Diffusion theory itself is an approximation: it assumes the flux varies slowly over a mean free path and that the medium is infinite and homogeneous, and it uses Fick's law for the neutron current. Near strong absorbers, boundaries and air gaps, the corrections of transport theory are needed. The diffusion length is nonetheless the central parameter of reactor core physics, and the value of L together with the Fermi age τ (which describes the slowing-down process) determines the migration area M² = L² + τ and hence the size of the core.

Advanced mode adds the diffusion area L², the buckling B², the migration area and length if the Fermi age τ is supplied, and the point-source flux φ(r) = S e^−r/L / (4πDr) if a source strength S and a distance r are given.

References

• Wikipedia, Neutron diffusion: the diffusion length, diffusion equation and point-source solution
• Wikipedia, Neutron moderator: moderator properties and typical diffusion lengths
• Lamarsh, J.R., Baratta, A.J. (2001). Introduction to Nuclear Engineering, 3rd ed., Prentice Hall — Chapters 5 and 6 for diffusion theory and reactor criticality.
• Duderstadt, J.J., Hamilton, L.J. (1976). Nuclear Reactor Analysis, Wiley — the standard reference for one- and two-group diffusion theory.
• NNDC, Brookhaven National Laboratory: cross-section data
• NIST SP 811, Guide for the Use of the SI: unit conversion factors

How to Use This Tool

1. Choose what to solve for: L, D, or Σ_a.
2. Pick a preset for the medium you have in mind, or type the two known values.
3. In Advanced, add the Fermi age τ to get the migration area and length, and a source strength S with a distance r to get the point-source flux.
4. Click Calculate.
5. Read the highlighted answer, then check the decay diagram and the point-source flux chart.

What This Tool Offers

• Solves L = √(D / Σ_a) for L, D or Σ_a
• Automatic length-unit conversion (cm, m, mm, in, ft) for D, Σ_a and L
• Diffusion area L², buckling B² and the k_eff leakage term L²B²
• Migration area and length M² = L² + τ when the Fermi age is supplied (Advanced)
• Point-source flux φ(r) = S e^−r/L / (4πDr) and 1/e, 1/e², 1/e³ reference distances
• Dynamic flux-decay diagram and a log-axis point-source flux chart
• Reference table of D, Σ_a, L, τ and M² for the common moderators
• Ready-made presets covering H₂O, D₂O, graphite, beryllium and reactor cores