Modeling the effect of particle size distribution on spectra for optically large particles
Authors: Penttilä Antti , Martikainen Julia , Muinonen Karri 
Published: September 2022

&lt;p&gt;&lt;strong&gt;Background:&lt;/strong&gt; Spectral observations in visual and near-infrared wavelengths are used to infer their surface material composition. Similar laboratory measurements of meteorites and various minerals and are used to link the known composition in laboratory measurements to asteroid composition.&lt;/p&gt;
&lt;p&gt;However, many parameters in observations and measurements can introduce effects on the resulting spectra. With regolith-type powdered grains packed to a slab, the observing angles, particle size distribution, particle shapes, and space-weathering state of the material can all alter the spectra. Here, we try to model the particle size effects so that at least this effect could be understood when comparing different spectra.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Size effects in spectra:&lt;/strong&gt; One can find spectral measurements of the same material but with varying size fractions in spectral databases such as RELAB (https://pds-speclib.rsl.wustl.edu/) and SSHADE (https://www.sshade.eu/). The typical spectral effects of different particle size are that the smaller particles are generally brighter, have redder slope, and deeper (absolute) absorption band depths. However, it is also possible to find counterexamples of different behavior.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Trend from light-scattering theory:&lt;/strong&gt; If we limit to optically large particles (several times larger than the wavelength) and assume that different particle sizes are simply scaled versions of one particle, the particle size effects in spectra can be understood as interplay between surface reflections and absorption in the volume. If we assume that the surfaces are flat in the wavelength-scale, the surface reflections follow Fresnel reflections. For light that is refracted from the surface into the homogeneous material, the Beer-Lambert absorption gives the attenuation in the volume.&lt;/p&gt;
&lt;p&gt;The Fresnel-type reflections do not absorb and are also only weakly dependent on the imaginary part &lt;em&gt;k&lt;/em&gt; of the complex refractive index of the material, &lt;em&gt;m&lt;/em&gt;=&lt;em&gt;n&lt;/em&gt;+i&lt;em&gt;k&lt;/em&gt;. The Beer-Lambert absorption, on the other hand, is only dependent on &lt;em&gt;k&lt;/em&gt; and the distance traveled in the material, &lt;em&gt;d&lt;/em&gt;. If the same volume is divided into smaller particles, typical &lt;em&gt;d&lt;/em&gt; in the material between air-surface or surface-air interfaces and reflections or refractions from these is decreasing. Since absorption is taking place only in the volume, the increased surface reflections are increasing the brightness. This is the simple mechanism behind increasing brightness with decreasing particle size. At the same time, the exponent-term in the Beer-Lambert absorption affects so that similar change in &lt;em&gt;d&lt;/em&gt; in the particle for a bright material has larger effect than the same &lt;em&gt;d&lt;/em&gt; in darker material. This means that the absorption band depths are also increasing, in absolute units, when particle size is decreasing.&lt;/p&gt;
&lt;p&gt;The possible increase of red slope with decreasing particle size in spectra cannot be explained with optically large particles. The longer wavelengths are &amp;#8216;seeing&amp;#8217; the distances inside the particles shorter, but the effect is quite small for particles that are several times larger than the wavelength. However, if the sample also includes wavelength-scale particles, these are relatively more abundant in smaller size fractions and can introduce a red slope. This is because the scattering efficiency of particles increase when they reach wavelength-scale whereas for optically large particles it is constant.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Numerical modeling:&lt;/strong&gt; We will verify and quantify the abovementioned theoretical trends with numerical simulations. We use a two-fold scheme where we first simulate the single-scattering properties (single-scattering albedo and the phase function) of individual random particle shapes, and then simulate the multiple-scattering effect in a packed slab of these particles.&lt;/p&gt;
&lt;p&gt;We model the particles with a Voronoi cell particle model, see Fig. 1. In this model, a large volume is first filled with random seed points. Then, a 3d Voronoi division is done on the points, resulting the volume to be divided into Voronoi cells having flat surfaces and sharp corners. Finally, single cells are extracted from the volume, except the cells close to the edge of the volume to avoid edge-effects.&lt;/p&gt;
&lt;p&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;
&lt;p&gt;The light-scattering properties of single Voronoi particles are computed using a geometric-optics ray-tracing code SIRIS (https://bitbucket.org/planetarysystemresearch/siris4-framework/). These computations are run for 200 realizations of the particle model and changing the size parameter (physical size divided by the wavelength) and the real and imaginary parts of the refractive index. The results are averaged over the particle shape realizations.&lt;/p&gt;
&lt;p&gt;The single-particle properties are fed into the Monte Carlo radiative transfer code RT-CB (https://bitbucket.org/planetarysystemresearch/rtcb_public/) where the amount of reflected light from a large slab containing the single particles is computed. This gives us the brightness as a function of particle size, wavelength, and real and imaginary parts of the refractive index.&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;Preliminary results:&lt;/strong&gt; We have a grid of size parameters and refractive indices computed, and the results are consistent with what we expected from the simple theory of the changing ratio of surface reflections to volume absorption. We can, for example, plug in a typical behavior of the imaginary part of olivine as a function of wavelength, keep the real part of the refractive index constant for simplicity (it does not change much in visual/near-infrared wavelengths), and plot the spectra of different-sized olivine particles, see Fig. 2.&lt;/p&gt;
&lt;p&gt;&lt;img src=&quot;&quot; alt=&quot;&quot; /&gt;&lt;/p&gt;
&lt;p class=&quot;Sectiontext&quot;&gt;&lt;strong&gt;Conclusions and future work:&lt;/strong&gt; Currently, we are increasing the parameter grid to include both smaller size parameters and smaller imaginary &lt;em&gt;k&lt;/em&gt; values. From the results we have, it seems that we can fit an analytical function to the results with quite small prediction errors. If successful, this analytical model can be used to model the size effect on the reflectance spectra of particulate materials if the particles are clearly larger than the wavelength. If we can model, we can also remove the effect if we know the particle sizes and receive size-standardized spectra that can be better compared between the samples.&lt;/p&gt;
&lt;p class=&quot;Sectiontext&quot;&gt;As a side product, we can use this data and the model to invert the imaginary part of the refractive index for materials from which we have reflectance spectra measurements and the particle size.&lt;/p&gt;