Particle-size distributions: independent scattering#

PyMieSim can average the response of homogeneous spheres over a number-based particle-size distribution, using discrete weights or quadrature for continuous distributions.

Warning

This feature uses the non-interacting (independent-scattering) approximation only. Each particle responds to the prescribed incident illumination in isolation. It excludes electromagnetic coupling between particles, multiple scattering, positional correlations, and interference between fields scattered by different particles.

It does not predict the response of a dense suspension, an aggregate, or an optically thick sample. Low concentration alone does not guarantee that multiple scattering is negligible: propagation distance also matters. PyMieSim does not infer or verify whether your sample meets these assumptions.

For a better approximation when particle correlations matter, refer to PackLab’s correlation-based dependent-scattering calculations. These incorporate interparticle correlations; they are not a general solver for full electromagnetic multiple scattering.

Averaging uses the independent-scattering approximation without a mode argument. Results record this assumption in their metadata, including when converted to typed results or dataframes.

Discrete sizes#

Supply particle counts or number fractions at each diameter:

from PyMieSim import (
    Measure,
    Experiment, ParticleSizeDistribution, PlaneWaveSet,
    PolarizationSet, SphereSet, ureg,
)

distribution = ParticleSizeDistribution(
    diameters=[100, 200, 400] * ureg.nanometer,
    number_weights=[1, 2, 1],
)
source = PlaneWaveSet(
    wavelength=[450, 550, 650] * ureg.nanometer,
    polarization=PolarizationSet(angles=0 * ureg.degree),
    amplitude=[1] * ureg.volt / ureg.meter,
)
spheres = SphereSet(
    diameter=distribution.diameters,
    material=[1.5],
    medium=[1.0],
)
experiment = Experiment(scatterer_set=spheres, source_set=source)
result = experiment.average_size_distribution(
    distribution, Measure.CSCA, Measure.CEXT, Measure.G,
    as_result=True,
)
scattering = result["Csca"].to("nanometer ** 2")
table = scattering.as_labeled_array().as_dataframe()

For these three wavelengths, each typed result has shape (3,) after removing the diameter axis. Cross sections carry area units and g is dimensionless. The exported table contains one averaged response per wavelength, not a separate response for every diameter.

Weights are normalized to sum to one. They must be number weights, not volume fractions, mass fractions, or intensity-weighted size estimates. For histogram measurements, use the number of particles in each bin. Samples of a probability density require integration weights (including bin widths); supplying only density values generally gives the wrong average.

The experiment diameter nodes must match distribution.diameters in the same order. The method removes only the diameter axis and optionally singleton axes; wavelength, material, medium, and detector sweeps remain separate. It applies the same size distribution at every remaining grid point. It does not represent a joint distribution of size and material. Sequential parameter sets, cylinders, and core-shell particles are currently rejected.

Lognormal sizes#

A lognormal number distribution can be constructed with:

distribution = ParticleSizeDistribution.lognormal(
    median_diameter=200 * ureg.nanometer,
    geometric_std=1.3,
    sampling=32,
)

Build the SphereSet using these new distribution.diameters before averaging. The median is the number median diameter, and geometric_std is dimensionless: the standard deviation of log diameter is its natural logarithm. A geometric standard deviation of one produces a single diameter.

Gauss-Hermite quadrature samples the full lognormal distribution without an explicit cutoff. Nodes and weights are integration points, not histogram bins. Increase sampling and compare the optical results to check convergence; broad distributions and sharp Mie resonances can require more points. Tail nodes can extend far beyond an intended particle-size range and increase computation cost; use a discrete distribution for explicitly bounded populations.

More distribution families#

All constructors return the same ParticleSizeDistribution type, so use distribution.diameters in the SphereSet and call experiment.average_size_distribution(distribution, Measure.CSCA) as above. All weights represent particle number. The non-interacting approximation and PackLab guidance at the top of this page apply to every family.

Available constructors#
Constructor	Parameters	Interpretation
`monodisperse`	`diameter`	All particles have one diameter.
`uniform`	`minimum_diameter`, `maximum_diameter`, `sampling=32`	Constant number density per unit diameter within the bounds.
`lognormal`	`median_diameter`, `geometric_std`, `sampling=32`	Normal distribution of log diameter.
`truncated_normal`	`mean_diameter`, `standard_deviation`, required positive bounds, `sampling=64`	Normal density conditioned on the bounded interval.
`triangular`	`minimum_diameter`, `mode_diameter`, `maximum_diameter`, `sampling=16`	Linear density on either side of a peak.
`mixture`	`distributions`, `number_weights`	Combine populations by their particle counts or number fractions.

For example:

single_size = ParticleSizeDistribution.monodisperse(diameter=200 * ureg.nanometer)
uniform = ParticleSizeDistribution.uniform(
    minimum_diameter=100 * ureg.nanometer,
    maximum_diameter=300 * ureg.nanometer,
)
normal = ParticleSizeDistribution.truncated_normal(
    mean_diameter=200 * ureg.nanometer,
    standard_deviation=40 * ureg.nanometer,
    minimum_diameter=100 * ureg.nanometer,
    maximum_diameter=300 * ureg.nanometer,
)
triangular = ParticleSizeDistribution.triangular(
    minimum_diameter=100 * ureg.nanometer,
    mode_diameter=180 * ureg.nanometer,
    maximum_diameter=300 * ureg.nanometer,
)
bimodal = ParticleSizeDistribution.mixture(
    distributions=[single_size, uniform],
    number_weights=[3, 1],
)

The normal parameters describe the untruncated distribution. Conditioning on the specified bounds generally changes its actual mean and standard deviation. Bounds are explicit so negative particle sizes are never generated. Use monodisperse when the width is zero.

Uniform and truncated-normal constructors use Gauss-Legendre quadrature. Triangular distributions integrate each nonempty side of the peak separately: sampling is the point count per side, so an interior peak produces twice that many nodes. An endpoint peak is supported. Always compare optical results at increasing sampling, particularly for a normal distribution on an interval much wider than its standard deviation or when Mie resonances are sharp.

Mixtures preserve each component’s quadrature nodes. In the example, 75% of particles belong to single_size and 25% to uniform, regardless of how many nodes either component has. Duplicate diameters are retained. These are mixtures of sizes, not a joint distribution of size and composition.

What the average means#

Let \(w_i\) be normalized number fractions, \(d_i\) diameters, and \(A_i=\pi d_i^2/4\) projected areas. The supported reductions are:

\[\overline{C} = \sum_i w_i C_i, \qquad Q_{\mathrm{ensemble}} = \frac{\sum_i w_i C_i}{\sum_i w_i A_i}, \qquad g_{\mathrm{ensemble}} = \frac{\sum_i w_i C_{\mathrm{sca},i} g_i} {\sum_i w_i C_{\mathrm{sca},i}}.\]

Cross sections Csca, Cext, Cabs, Cback, Cforward, and Cpr are means per particle, in area units. Their corresponding Q measures divide by mean projected area; they are not arithmetic number averages of individual efficiencies. Asymmetry g is weighted by scattered power through Csca; it is undefined when the mean scattering cross section is zero, in which case the method raises ValueError.

coupling is the number-averaged single-particle detector power, in watts, for identical illumination and detector geometry. Even for a coherent mode detector, this averages individual powers, never complex amplitudes. There is no integration over particle positions, illumination gradients, or relative optical phases. It is not the total detected power of a suspension. Complex multipole coefficients, Qratio, and Cratio are not supported.

A sample’s particle count, concentration, and path length are not supplied or inferred. Multiplying a mean cross section by number density gives an independent-scattering coefficient; predicting transmitted or detected light through a sample requires a suitable propagation model and its own validity assessment.

The result’s attrs (or metadata for typed results) records the approximation, assumptions, weighting rule, diameter nodes, and number fractions. These records survive unit conversion and labeled-array/dataframe export.

For the cross-section integration and scattering-weighted asymmetry conventions, see Geer et al. (2021), Section 2.

API#

class ParticleSizeDistribution#

Bases: pybind11_object

Discrete particle diameters and normalized number fractions.

Optical averaging uses the non-interacting, independent-scattering approximation. Refer to PackLab when particle correlations matter.

property diameters#: Diameter nodes as a Pint quantity in meters.

static lognormal(median_diameter: object, geometric_std: object, sampling: object = 32) → PyMieSim.distributions.ParticleSizeDistribution#

static mixture(distributions: collections.abc.Iterable, number_weights: object) → PyMieSim.distributions.ParticleSizeDistribution#: Combine distributions using component number fractions.

static monodisperse(diameter: object) → PyMieSim.distributions.ParticleSizeDistribution#: Create a distribution containing one diameter.

property number_fractions#: Normalized particle-number fractions as a NumPy array.

static triangular(minimum_diameter: object, mode_diameter: object, maximum_diameter: object, sampling: object = 16) → PyMieSim.distributions.ParticleSizeDistribution#

static truncated_normal(mean_diameter: object, standard_deviation: object, *, minimum_diameter: object, maximum_diameter: object, sampling: object = 64) → PyMieSim.distributions.ParticleSizeDistribution#

static uniform(minimum_diameter: object, maximum_diameter: object, sampling: object = 32) → PyMieSim.distributions.ParticleSizeDistribution#

Experiment.average_size_distribution(distribution, *measures, drop_unique_level=True, as_result=False)#

Average sphere sizes in the non-interacting approximation only.

Parameters:

distribution (ParticleSizeDistribution) – Number-based distribution with nodes matching the SphereSet diameter axis, in the same order. Other parameter axes remain independent.
measures (str | Measure) – Cross sections (Csca, Cext, Cabs, Cback, Cforward, Cpr), corresponding efficiencies, g, or coupling. Amplitudes and ratios are unsupported.
drop_unique_level (bool) – Remove remaining parameter axes containing only one value.
as_result (bool) – Return named typed results instead of a LabeledArray. Both retain the approximation and weighting rules in their metadata.

Return type:

LabeledArray | SimulationResult | SimulationResults

Notes

This uses the non-interacting (independent-scattering) approximation: particles respond in isolation to the prescribed illumination. It excludes interparticle electromagnetic coupling, multiple scattering, positional correlations, and interference between different particles. For a better approximation when particle correlations matter, refer to PackLab’s correlation-based dependent-scattering calculations: https://martinpdes.github.io/PackLab/docs/latest/scattering.html. PackLab is not a general solver for full electromagnetic multiple scattering.

Cross sections and detector powers are number averages per particle, assuming the same illumination for each particle. Efficiencies are mean cross section divided by mean projected area; g is weighted by scattering cross section. These are not predictions of a dense or multiply scattering sample, nor sums of coherent particle fields.