# Light Scattering by a dispersion of spheres

Despite the fact that it is just an ideal case almost impossible to reproduce in real world, the knowledge of light interaction with a suspension of perfect spheres is very useful to have an idea of the behaviour of the light interacting in a real complex medium.

### One sphere

The problem of light scattering by a sphere has been solved analytically by Gustav Mie. His calculation gives the efficiency factor Q, the anisotropy factor g and the scattering diagram P(θ) with respect to sphere size and the material index, for both polarizations. Q is very small for particles << λ and converges to 2 for particles >> λ (λ is the wavelength). Also, for what concerns the anisotropy factor g, it is close to 0 (isotropic scattering) if the particle size is small with respect to λ and it gets closer to 1 (forward scattering) when the particle size is much bigger than λ.

A good way to calculate all this is to use the code provided by the appendice of the Bohren and Huffman book. Fortran code is available (Thomas Robitaille) here. We have adapted this code to java here Figure: scattering diagrams for a spherical particle of Titane dioxide in water, unpolarized, wavelength 650nm, for 3 cases of size: 0.5µm (blue), 1µm (purple) and 2µm (red)

### A suspension of spheres: the mean free path l and the transport length l*

For a suspension of mono-disperse spherical particles, the mean free path, that is the typical length recovered by a photon without being scattered, is noted l and can be calculated by the formula : l=2d/(3 Φ Q) with Φ the volume fraction of spheres.

The interaction with light by a concentrated suspension is better modelized by the parameter l* called transport length. Basically, it is the depth of penetration of light in the suspension or in other words the distance needed to randomized the direction of a photon injected in the suspension. l* tells about the opacity of a material. l* can be calculated by the formula  l*=l/(1-g). With g the anisotropy coefficient of the light scattering by one sphere mentioned before. In the other way, one see that, knowing l*, it is possible to calculate the particle size. Of course, at the condition one knows the volume fraction and the particle refractive index. This is the case for example when one formulates an oil in water emulsion and wants to have the oil dropplets size.

### Conclusion

The optical parameters corresponding to light scattering can be exactly calculated for the case of a monodisperse suspension of sphere. Hence this theoretical case is the first one to approximate to a real life case in order to make estimation, for example the mean size of a polydisperse suspension of particles

### References

1. C.F. Bohren, D.R. Huffman, Absorption and Scattering of Light by Small Particles (WileyVCH,Berlin, 1983)
2. Particle Sizing and Characterization, Edition: ACS Symposium Series, Vol. 881, Chapter: 3, Publisher: American Chemical Society, pp.45-60 (get pdf)
3. Multiple light scattering in random systems: Analysis of the backscattering spot image. The European Physical Journal Applied Physics 6(01):81 - 87 · April 1999 DOI: 10.1051