Macroscopic Cross Section Calculator

Inputs

Select an input method and provide interaction data.

Advanced unit handling Mean free path Transmission

Input mode

Use density mode for solids, liquids, and gases.

Interaction type

Label only; computation uses your σ value.

Microscopic cross section, σ

1 barn = 10^-24 cm².

Density, ρ

Used to estimate number density from composition.

Molar mass, A

g/mol

For compounds, use average molar mass.

Porosity (optional)

fraction

Effective density becomes ρ·(1−φ).

Packing factor (optional)

0–1

Useful for powders and packed beds.

Shield thickness, x (optional)

Computes uncollided transmission exp(−Σx).

Formula used

The macroscopic cross section links microscopic interaction probability to bulk material behavior.

Macroscopic cross section: Σ = N · σ, where Σ is in 1/length, N is number density, and σ is microscopic cross section.
Number density from material data: N = ρ · N_A / A, with density ρ, Avogadro constant N_A, and molar mass A.
Mean free path: λ = 1/Σ, the average travel distance between interactions.
Uncollided transmission: I/I₀ = exp(−Σx) for thickness x.

How to use this calculator

Select Density + molar mass or Direct number density.
Enter the microscopic cross section σ and its unit.
Provide ρ and A, or enter N directly.
Optionally add porosity or packing factor for effective density.
Optionally enter thickness to estimate I/I0 transmission.
Press Calculate to view results above the form.

Example data table

Material	ρ (g/cm³)	A (g/mol)	σ (barn)	Σ (1/cm)	λ (cm)
Iron (Fe), example σ	7.87	55.845	2.5	0.211	4.74
Water (H₂O), example σ	1.00	18.015	0.66	0.0221	45.2
Concrete, example σ	2.30	100.0	1.2	0.0166	60.2

These rows are illustrative and use example σ values. Replace σ with your energy-dependent data for accurate shielding or reactor calculations.

Macroscopic Cross Section Guide

1) What the macroscopic cross section represents

The macroscopic cross section, Σ, expresses how strongly a bulk material interacts with a particle beam. It is an interaction probability per unit path length, typically reported in 1/cm or 1/m. Larger values imply more frequent interactions inside the material.

2) Core relation used by the calculator

The standard relation is Σ = N σ, where N is number density (atoms per cm³) and σ is microscopic cross section (cm², often given in barns). Remember 1 barn = 10^-24 cm².

3) Number density from density and molar mass

If you know density ρ and molar mass M, number density is N = (ρ / M) N_A, with N_A as Avogadro’s constant. For typical solids, N is often on the order of 10²² atoms/cm³. For aluminum (ρ = 2.70 g/cm³, M = 26.98 g/mol), N ≈ 6.0×10²² atoms/cm³; with σ = 1 barn, Σ ≈ 0.060 1/cm.

4) Unit handling and conversions

Inputs can be entered in barns, cm², or m² for σ. Density can be g/cm³ or kg/m³, and molar mass can be g/mol or kg/mol. The calculator converts to consistent internal units and reports Σ in both 1/cm and 1/m.

5) Total, scattering, and absorption components

When separate microscopic values are available, compute Σ_s = Nσ_s and Σ_a = Nσ_a, then combine them as Σ_t = Σ_s + Σ_a. This separation is useful for shielding and detector design.

6) Mean free path and attenuation intuition

The mean free path is λ = 1/Σ. If Σ = 0.50 1/cm, then λ = 2.0 cm, so interactions occur within a few centimeters on average. A simple transmission model is e^-Σx for thickness x.

7) Mixtures and compounds

For mixtures, a common approximation is a weighted sum: Σ = Σ_i N_iσ_i. If you only have an effective microscopic value, estimate N from the bulk density and an effective molar mass, then apply Σ = Nσ.

8) Practical checks and typical ranges

Microscopic cross sections can range from <1 to hundreds of barns depending on energy and isotope. If your Σ looks unrealistic, re-check unit selections (barns vs m²) and confirm density and molar mass are for the same material composition.

FAQs

1) What is the difference between microscopic and macroscopic cross section?

Microscopic cross section (σ) describes a single nucleus interaction area, usually in barns. Macroscopic cross section (Σ) scales that by number density, giving an interaction probability per unit distance in bulk material.

2) Why does the calculator ask for density and molar mass?

Density and molar mass determine the number density N of atoms in the material. Once N is known, the macroscopic cross section follows directly from Σ = Nσ.

3) Which unit should I use for σ?

Use barns when you have nuclear data tables, or cm²/m² if your source provides SI units. The tool converts units internally, so consistent selection is more important than the unit choice itself.

4) What does a higher Σ mean physically?

A higher Σ means interactions occur more frequently as the particle travels through the material. This typically corresponds to stronger attenuation and a shorter mean free path λ = 1/Σ.

5) Can I compute scattering and absorption separately?

Yes. If you have σ_s and σ_a, compute Σ_s = Nσ_s and Σ_a = Nσ_a. Add them for the total Σ_t.

6) How do I estimate Σ for a compound?

Ideally, sum contributions from each element using their own number densities and microscopic values. If only an effective microscopic value is available, use the compound density and effective molar mass to estimate N, then apply Σ = Nσ.

7) Why do my results look too large or too small?

Most issues come from unit mismatch: barns vs m², or kg/m³ vs g/cm³. Also confirm molar mass units and that density corresponds to the same material state and composition as your cross section source.