Inputs
Enter values and press Calculate. The form uses three columns on large screens, two on tablets, and one on phones.
Example Data Table
Sample values for visible light and small particles. Use these to test the calculator.
| λ₀ | a | nₚ | nₘ | N (1/m³) | L | θ | R |
|---|---|---|---|---|---|---|---|
| 450 nm | 30 nm | 1.50 | 1.00 | 1.0e20 | 1 cm | 90° | 1 m |
| 550 nm | 50 nm | 1.50 | 1.33 | 5.0e19 | 5 mm | 60° | 50 cm |
| 650 nm | 20 nm | 1.45 | 1.00 | 2.0e20 | 2 cm | 120° | 1 m |
Formula Used
The intensity estimate assumes single scattering and far-field observation. For larger particles, consider Mie scattering methods.
How to Use This Calculator
- Enter the vacuum wavelength and particle radius using convenient units.
- Provide refractive indices for the particle and surrounding medium.
- Set number density and path length to compute attenuation and transmission.
- Choose scattering angle and observation distance for angular intensity output.
- Press Calculate to show results above the form. Use CSV or PDF downloads as needed.
Rayleigh Scattering Guide
1) When Rayleigh scattering is valid
Rayleigh theory assumes particles are much smaller than the wavelength in the surrounding medium. The calculator reports the size parameter x = 2πa/λmedium. Values near or below ~0.3 are typically Rayleigh-like; larger x signals a shift toward Mie behavior.
2) The strong wavelength trend
The cross section scales as σ ∝ 1/λ4, so shorter wavelengths scatter more. For the same particle and indices, 450 nm scatters about 4.3× more than 650 nm, which contributes to blue skies.
3) Sensitivity to particle size
Rayleigh scattering is extremely sensitive to radius because σ ∝ a6. Doubling radius increases σ by 26 = 64. A change from 20 nm to 50 nm increases σ by (50/20)6 ≈ 244.
4) Role of refractive contrast
The material contrast enters through the factor ((m²−1)/(m²+2))² where m = np/nm. With np = 1.50 and nm = 1.00, the factor is about 0.086. If the same particle sits in water (nm ≈ 1.33), contrast drops and scattering typically decreases.
5) From single-particle σ to bulk attenuation
In a suspension or haze, the scattering coefficient is α = Nσ, where N is number density. Over path length L, optical depth is τ = αL and transmission is T = e−τ. If τ = 0.1 the beam retains ~90% intensity; if τ = 1 it retains ~37%, often seen as strong haze in many conditions.
6) Angular scattering and observed brightness
The Rayleigh phase function used here is proportional to 1 + cos²θ, giving comparable forward and backward scattering and a minimum at 90°. The displayed dσ/dΩ converts σ into a per-steradian measure, and Is estimates the brightness at distance R using the expected 1/R² falloff.
7) Practical measurement choices
When designing a lab setup, keep R consistent when comparing samples because Is changes with distance. Use moderate N and short L to remain in the single-scattering regime; once multiple scattering dominates, the exponential transmission and simple angular pattern break down.
8) Limits and extensions
This calculator assumes real refractive indices and spherical particles. Strongly absorbing materials require complex indices, and irregular shapes can deviate from the Rayleigh pattern. If x is not small, or if you need full spectral behavior, a Mie solver or measured scattering data is the better choice.
FAQs
1) What does the size parameter x mean?
x = 2πa/λmedium measures how “small” the particle is compared with the wavelength. Values below about 0.3 are typically Rayleigh-like; larger x suggests increasing Mie effects and reduced accuracy for Rayleigh formulas.
2) Should I enter wavelength in vacuum or in the medium?
Enter the vacuum wavelength λ₀. The calculator converts internally using λmedium = λ₀/nm and uses k = 2πnm/λ₀, which is consistent for a homogeneous surrounding medium.
3) Can the calculator model absorbing particles?
Not fully. It assumes real refractive indices and does not include absorption or complex polarizability. For strongly absorbing particles, results can be biased; use a model that supports complex n and separates scattering from absorption.
4) What is number density N?
N is the number of particles per cubic meter. It converts single-particle σ into a bulk coefficient α = Nσ. If you want single-particle outputs only, set N to zero and focus on σ and dσ/dΩ.
5) Why does transmission sometimes stay near 1?
If τ = αL is very small, then T = e−τ ≈ 1 − τ. Low N, small a, long λ, or short L can make τ tiny, so the change in transmission may be too small to notice.
6) How should I interpret the scattered intensity Is?
Is is an estimate of intensity at distance R for a single scattering event: Is ≈ I₀(dσ/dΩ)/R². It is most useful for relative comparisons across θ, λ, and a under the same geometry.
7) When should I use a different model?
Use a different model when x is not small, when particles are not spherical, when absorption matters, or when refractive index varies strongly with wavelength. In these cases, Mie theory or measured phase functions provide better accuracy.