Formula Used
Mean free path is the average distance a neutron travels between interactions.
- λ = 1 / Σ
- Σ = N · σt (for a single target species)
- N = (ρ / M) · NA · f (density pathway)
Transmission through a slab of thickness x is T = e−x/λ, and interaction probability is 1 − T.
How to Use This Calculator
- Select an input method: macroscopic Σ, number density N, or density ρ with molar mass M.
- If using N or ρ, enter σ values. Provide σt directly or use σs+σa+σf.
- Optionally add thickness x to estimate transmission and interaction probability.
- Optionally add distance d to estimate expected collisions (d/λ).
- Click Calculate to see results above the form.
- Use the export buttons to save results as CSV or PDF.
Example Data Table
| Material (illustrative) | Method | Inputs | Computed λ (approx.) |
|---|---|---|---|
| Light water | Macroscopic | Σ = 0.20 1/cm | 5.0 cm |
| Iron | N + σ | N = 8.5×1022 1/cm³, σt = 2.5 barn | 4.7 cm |
| Graphite | ρ + M + σ | ρ = 1.7 g/cm³, M = 12 g/mol, σt = 4.8 barn | 3.1 cm |
Examples are simplified for demonstration. Use reference data for precise work.
Neutron Mean Free Path Guide
1) What mean free path represents
Mean free path (λ) is the average distance a neutron travels before an interaction such as scattering, absorption, or fission. Smaller λ means more frequent events and stronger attenuation; larger λ means longer travel between interactions in the same material.
2) Macroscopic and microscopic viewpoints
The calculator uses λ = 1/Σ, where Σ is the macroscopic cross section in 1/length. If you begin with microscopic data, Σ comes from Σ = N·σt, using number density N and microscopic total cross section σt. Many references report σ in barns (1 barn = 10−28 m²).
3) Estimating number density
When N is unknown, estimate it from density and molar mass: N = (ρ/M)·NA·f. For many solids, N is typically around 1028 atoms/m³ (about 1022 atoms/cm³). The fraction f scales the target species in mixtures or enriched materials.
4) Energy dependence and moderation
Cross sections vary strongly with neutron energy. A “thermal” σ may differ greatly from a “fast” σ, so λ can change by orders of magnitude. In shielding or reactor problems, moderation shifts energy along the path, meaning λ is not constant unless the spectrum is controlled.
5) Building a practical σt
If a single total value is unavailable, a practical approximation is σt = σs + σa + σf for the dominant processes. This tool accepts σt directly or component values. Keep the same units and ensure each component represents the same energy range.
6) Turning λ into transmission
For a slab thickness x, the uncollided transmission is T = e−x/λ, and the interaction probability is 1 − T. These expressions are most useful for narrow-beam, first-pass estimates. In thick shields, scattered neutrons can add build-up beyond this simple model.
7) Sensitivity and uncertainty
Because λ is the inverse of Σ, a 10% change in Σ produces about a 10% change in λ. Uncertainty usually comes from σ selection (energy, temperature, isotopes) rather than arithmetic. Document your σ source, density basis, and fraction assumptions for reproducible results.
8) Where mean free path is used
Mean free path supports reactor physics (collision spacing and moderation scales), shielding comparisons (attenuation trends), detector design (interaction probability in a converter), and materials screening. Treat the output as a clear intermediate step that links nuclear data to geometry and thickness decisions.
FAQs
1) What is the difference between Σ and σ?
σ is a microscopic cross section per nucleus (area). Σ is the macroscopic cross section per path length (1/length) for a bulk material. They are related by Σ = N·σ.
2) Which σ should I use for this calculator?
Use the total interaction cross section for your neutron energy range, or provide components (scattering, absorption, fission) that sum to a practical σt. Be consistent with units and data source conditions.
3) Why does the calculator ask for a fraction f?
f scales the number density of the target species within a mixture or isotopic composition. For a pure element, use 1. For a component in a compound or alloy, use an appropriate fraction estimate.
4) Is transmission T = e−x/λ always accurate?
It predicts uncollided attenuation in an idealized, uniform slab. In real shielding, scattered neutrons can contribute additional flux (build-up). Use it as a first-order estimate and refine with transport methods when needed.
5) What units are supported?
You can enter Σ in 1/cm or 1/m, number density in 1/cm³ or 1/m³, density in g/cm³ or kg/m³, and cross sections in barns, cm², or m². Outputs include meters and centimeters.
6) How do I estimate N if I only know density?
Use the density pathway: provide ρ and molar mass M, then the calculator uses N = (ρ/M)·NA·f. This is a standard conversion from bulk density to atomic number density.
7) What does “expected collisions” mean?
It reports d/λ, the average number of interactions expected over a straight-line distance d in a homogeneous medium. It is a mean value; the actual number of interactions varies around that mean.