Question: Converting DLS I(x) to N(x) and V(x) using Mie theory

[AlexShalby]

Hello @scottprahl,

First, thank you for creating and maintaining the miepython library – it's an excellent tool.

Context:
We are working with Dynamic Light Scattering (DLS) data and have obtained the intensity-weighted size distribution, I(x).

Goal:
We need to convert this I(x) into:

• Number-weighted distribution N(x)
• Volume-weighted distribution V(x)

Question:
We understand that Mie theory provides the scattering intensity for particles of a given size. Could you please advise on the correct way to use miepython to perform this conversion?

Specifically:

1. What is the mathematical relationship or workflow to transform I(x) → N(x) and I(x) → V(x) using Mie scattering functions?
2. Are there any existing functions in miepython or examples that demonstrate this conversion?
3. If not built-in, what would be the recommended approach using the outputs from miepython?

We would greatly appreciate any guidance, code snippets, or references you could share on this topic.

Thank you for your time and for this valuable library.

[scottprahl]

Dynamic Light Scattering (DLS) instruments commonly report a scattering intensity–weighted particle size distribution, typically written as $I(d)$, where $d$ is the particle diameter.

1. Mathematical Relationships

The relationship between distributions depends on the brightness per particle, $B(d)$, that is the scattered intensity of a single particle of diameter d under the same geometry.

The DLS intensity distribution is proportional to:

$$ I(d) \propto N(d) B(d) $$

Rearranging gives the unnormalized number distribution:

$$ N(d) \propto \frac{I(d)}{B(d)} $$

After normalization:

$$ \tilde{N}(d)=\frac{I(d)/B(d)}{\sum_j I(d_j)/B(d_j)} $$

For spherical particles, volume is proportional to d³, so:

$$ V(d) \propto N(d) d^3 $$

Normalized gives:

$$ \tilde{V}(d)= \frac{N(d) d^3}{\sum_j N(d_j) d_j^3} $$

2. Computing Single-Particle Brightness

Most modern instruments measure intensity at a specific angle (commonly 90° or 173°, especially for backscatter DLS). In that case:

$$ B(d) = I_{\text{single}}(d, \theta_{\text{detector}}) $$

and can be computed with miepython :

Ipar, Iper = miepython.intensities(m, d, lambda0, mu)
B = 0.5 * (Ipar + Iper)  # unpolarized light

3. Example Python Workflow

import numpy as np
import miepython as mie

# Input: measured intensity-weighted distribution
d = np.array([...])            # diameters (must match DLS binning)
I_intensity = np.array([...])  # intensity-weighted values

# Experimental parameters
m = 1.59 + 0.001j              # refractive index (particle / medium)
lambda0 = 633.0                # nm
theta_det = 173.0              # scattering angle (degrees)

# Compute per-particle brightness
mu = np.cos(np.deg2rad(theta_det))
Ipar, Iper = mie.intensities(m, d, lambda0, mu)
B = 0.5 * (Ipar + Iper)  # unpolarized brightness

# Convert intensity → number distribution
eps = 1e-30
N_raw = I_intensity / np.maximum(B, eps)
N_number = N_raw / np.sum(N_raw)

# Convert number → volume distribution
V_raw = N_raw * d**3
V_volume = V_raw / np.sum(V_raw)

4. Practical Notes

• In the Rayleigh scattering regime, intensity scales approximately as d⁶, so converting to N(d) will amplify noise for small particles. Smoothing or regularization may help.

• Accurate complex refractive index values at the measurement wavelength are essential.

• This analysis assumes that all your particles are spherical and that there is no aggregation.