Formula used
- Rayleigh (small particles): σ = (8π/3) k⁴ a⁶ |(m²−1)/(m²+2)|², with k = 2π nₘ/λ and m = nₚ/nₘ.
- Thomson (free electron): σT = (8π/3) re², using re = 2.81794×10⁻¹⁵ m.
- Geometric: σ ≈ Q π r², where Q is an efficiency factor.
- Nuclear size: R = r₀ A^(1/3) and σ ≈ πR².
- Transport extras: mean free path ℓ = 1/(nσ), optical depth τ = nσt, probability P = 1 − e^(−τ).
How to use this calculator
- Select the scattering model that matches your physical situation.
- Enter the required parameters shown for that model.
- Optionally add number density and path length for transport metrics.
- Press Estimate Cross Section to view results above the form.
- Use the CSV or PDF buttons to export the same inputs and outputs.
Example data table
| Scenario | Model | Key inputs | Estimated σ | Notes |
|---|---|---|---|---|
| Nanoparticle in water | Rayleigh | λ=532 nm, a=50 nm, nₚ=1.59, nₘ=1.33 | Order of 10⁻²⁴–10⁻²¹ m² | Strong size dependence: σ ∝ a⁶ |
| Low‑energy photons | Thomson | Electron target | 6.65×10⁻²⁹ m² | Energy corrections may matter at high energies |
| Dust grain | Geometric | r=1 µm, Q=1.0 | ~3.14×10⁻¹² m² | Good for large objects and simple estimates |
| Iron nucleus | Nuclear size | A=56, r₀=1.2 fm | ~0.53 barns | Only a size scale, not a reaction prediction |
Scattering cross section: practical estimates
1) Why cross sections matter
In transport problems, the cross section sets interaction likelihood. A larger σ means more frequent scattering, shorter mean free paths, and stronger attenuation through a material. Engineers use σ to forecast shielding performance, optical haze, detector rates, and collision-driven chemistry.
2) Common units and scale awareness
Cross sections are areas. This page reports m², cm², nm², and barns, where 1 barn equals 10⁻²⁸ m². Atomic and nuclear processes often lie near barns or millibarns, while micron-sized geometric targets can reach 10⁻¹² m² and above.
3) Rayleigh regime data trends
Rayleigh scattering applies when radius a is far smaller than wavelength λ. The estimate scales roughly as σ ∝ a⁶ and σ ∝ λ⁻⁴, explaining why blue light scatters more than red. Small size changes can dominate results, so measure particle radius carefully.
4) Thomson baseline reference value
The Thomson cross section is a standard benchmark for low-energy photons scattering on free electrons. It is σT ≈ 6.65×10⁻²⁹ m², about 0.665 barns. At higher photon energies, the Klein–Nishina formula reduces σ relative to this value.
5) Geometric limit and efficiency
When an object is comparable to or larger than λ, a quick estimate is σ ≈ Qπr². The efficiency factor Q captures diffraction and absorption effects, and often falls between 0 and 2 for simple targets. This model is useful for dust, droplets, and large inclusions.
6) Nuclear size benchmarking
For a crude nuclear scale, the radius model R ≈ r₀A^(1/3) with r₀ ≈ 1.2 fm gives σ ≈ πR². For example, A=56 yields a size-scale cross section near 0.5 barns. Real nuclear cross sections vary strongly with energy and reaction channel.
7) From σ to attenuation through matter
If number density n is known, the mean free path is ℓ = 1/(nσ). Typical n values are about 2.5×10²⁵ m⁻³ for air at room conditions and about 3.3×10²⁸ m⁻³ for liquid water. With thickness t, optical depth τ = nσt and interaction probability is P = 1 − e^(−τ).
8) Practical input guidance
Choose the simplest model matching your regime, then compare with a second model as a sanity check. If your particle is nanometer-scale and λ is visible light, Rayleigh is a good start. If r is micron-scale, geometric estimates often bracket the outcome. Always keep units consistent and review the notes shown in the result panel.
FAQs
1) What does “cross section” mean physically?
It is an effective area that represents the chance of an interaction. Larger cross sections imply more frequent scattering for the same target density and path length.
2) When should I use the Rayleigh option?
Use Rayleigh when particle radius is much smaller than the wavelength, commonly a/λ ≪ 1. The estimate is very sensitive to size because σ scales with a⁶.
3) Why does the calculator show barns?
Barns are convenient for nuclear and particle physics because many interaction areas are near 10⁻²⁸ m². They provide readable numbers compared with very small m² values.
4) What is a reasonable range for the efficiency factor Q?
For many simple scatterers, Q often falls between 0 and 2. Values outside that range can occur in specific resonant or absorbing cases, but they require justification.
5) How is mean free path computed here?
If you provide number density n, the calculator uses ℓ = 1/(nσ). This is a standard dilute-medium estimate that assumes independent scattering events.
6) What does the interaction probability represent?
It is the chance of at least one interaction over thickness t, computed as P = 1 − e^(−nσt). It is a Poisson-process result for random independent events.
7) Are these results accurate for all energies and materials?
No. These are first-pass estimates. Many real systems require energy dependence, angular distributions, multiple scattering, and material-specific effects beyond the simplified models used here.