# Rayleigh-Gans scattering

## RAYLEIGH-GANS SCATTERING

In a large number of important cases, the relative refractive index of particles (with respect to that of the ambient medium) is close to one. Such particles are called “optically soft,” and the corresponding approximation could be named that of optically soft particles. The known examples of optically soft particles are organic particles in natural water [see the books by Shifrin and Oliver (1988) and Mobley (1994)] and suspended cells considered by Lopatin and Sid’ko (1988). Optical properties of soft particles are also considered in recent monographs by Kokhanovsky (2004) and Sharma and Sommerford (2006). The condition |m - 1|<< 1 is not sufficient for radical simplification of the problem to the case of the so-called Rayleigh-Gans scattering. The following two conditions should be satisfied:

(1)

The first of these conditions results in a small reflection of the incident wave at the interface. The second condition corresponds to a small change of the phase and amplitude of an incident wave inside the particle. A detailed discussion of the physical sense of the approximate theory for not too large optically soft particles was given by van de Hulst (1957). It is important that the Rayleigh-Gans approximation can be employed to determine the characteristics of particles of complex shape.

In the particular case of a homogeneous spherical particle, the following expressions for the absorption efficiency factor and absorption cross section can be derived:

(2)

(3)

According to van de Hulst (1957), the scattering efficiency factor is given by

(4)

where

(5)

The scattering function for Rayleigh-Gans scattering is much more complex than that for Rayleigh scattering. An analysis of the expression

(6)

shows that asymmetry of scattering increases with the diffraction parameter, and one can find strongly forward scattering for large particles. The scattering function is not monotonic and the number of extremums is approximately equal to the particle diffraction parameter.

In the limit of x << 1, i.e., for Rayleigh particles, Eqs. (4) and (5) are radically simplified and the efficiency factor of scattering is given by

(7)

In the opposite case of large particles (x >> 1), one can also find a rather simple but quite different formula,

(8)

Note that solutions for ellipsoids and circular cylinders of finite length can be found in the book by van de Hulst (1957). Some other solutions based on the Rayleigh-Gans approximation and diverse applications of these solutions can be found in the books by Shifrin and Oliver (1988), Mobley (1994), and Lopatin and Sid’ko (1988), and also in papers by Wriedt (1998), Farias et al. (1995), Lednei et al. (2001), Lopatin et al. (2001), Shepelevich et al. (2001), Rysakov and Ston’ (2001), Posselt et al. (2002), and Galletto et al. (2005). A more detailed bibliography on this subject can be found in the recent monograph by Dombrovsky and Baillis (2010).

#### REFERENCES

Dombrovsky, L. A. and Baillis, D., Thermal Radiation in Disperse Systems: An Engineering Approach, Begell House, Redding, CT, and New York, 2010.

Farias, T. L., Carvalho, M. G., Köylü, Ü. Ö., and Faeth, G. M., Computational evaluation of approximate Rayleigh-Debye-Gans/fractal-aggregate theory for the absorption and scattering properties of soot, ASME J. Heat Transfer, vol. 117, no. 1, pp. 152-159, 1995.

Galletto, P., Lin, W., Mishchenko, M. I., and Borkovec, M., Light-scattering form factors of asymmetric particle dimers from heteroaggregation experiments, J. Chem. Phys., vol. 123, no. 6), 064709, 2005.

Kokhanovsky, A. A., Optics of Light Scattering Media: Problems and Solutions, 3rd ed., Praxis, Chichester, UK, 2004.

Lednei, M. F., Pinkevich, I. P., Reshetnyak, V. Yu., and Sluckin, T. J., Rayleigh-Gans theory of light scattering by liquid crystals filled with cylindrical particles, J. Molec. Liq., vol. 92, no. 1-2, pp. 139-146, 2001.

Lopatin, V. N. and Sid’ko, F. Ya., Introduction to Optics of Suspended Cells, Nauka, Novosibirsk (in Russian), 1988.

Lopatin, V. V., Priezzhev, A. V., and Fedoseev, V. V., Numerical simulation of light propagation and scattering in turbid biological media, Crit. Rev. Biomed. Eng., vol. 29, no. 3, pp. 400-419, 2001.

Mobley, C. D., Light and Water: Radiative Transfer in Natural Waters, Academic Press, New York, 1994.

Posselt, B., Farafonov, V. G., Il’in, V. B., and Prokopjeva, M. S., Light scattering by multi-layered ellipsoidal particles in the quasistatic approximation, Meas. Sci. Tech., vol. 13, no. 3, pp. 256-262, 2002,

Rysakov, W. and Ston’, M., Light scattering by spheroids, J. Quant. Spectr. Radiat. Transfer, vol. 69, no. 5, pp. 651-665, 2001.

Sharma, S. K. and Sommerford, D. J., Light Scattering by Optically Soft Particles: Theory and Applications: Praxis, Chichester, England, 2006.

Shepelevich, N. V., Prostakova, I. V., and Lopatin, V. N., Light-scattering by optically soft randomly oriented spheroids, J. Quant. Spectrosc. Radiat. Transfer, vol. 70, no. 4-6): 375-381, 2001.

Shifrin, K. S. and Oliver, D., Physical Optics of Ocean Water, AIP Press, Melville NY, 1988 (Transl. of orig. Russian book by K. S. Shifrin, Introduction to Optics of the Ocean, Hydrometeoizdat, Leningrad, 1983).

van de Hulst, H. C., Light Scattering by Small Particles, Wiley, Hoboken, NJ, 1957 (also Dover Publ., 1981).

Wriedt, T., A review of elastic light scattering theories, Part. Part. Syst. Charact., vol. 15, no. 2, pp. 67-74, 1998.