It is well known that optical properties of some small particles can be predicted on the basis of the so-called Rayleigh theory developed by Lord Rayleigh long before the general theory of scattering (Lord Rayleigh, 1871). The limiting case of Rayleigh scattering takes place when two conditions are satisfied,

The first of these conditions is true for particles of very small radius in comparison to the wavelength. The second condition gives a limitation for the optical properties of particle substance.

Consider optical properties of two-layer spherical particles in the Rayleigh region,

In this case, Mie theory relations are significantly simplified, since it is sufficient to
take into account only the first coefficient a_{1} corresponding to the dipole
scattering. This corresponds to the uniform internal field in the particle. In the
Rayleigh region, the scattering function is symmetric (μ = 0). For randomly
polarized incident radiation, we have the following expression for the scattering
function:

and polarization degree is given by the formula

so that scattered radiation is totally polarized in direction μ = 0 (σ = π/2). The Rayleigh scattering function (3) is shown in Fig. 1. This scattering function does not depend on particle size. As a result, it is impossible to determine the size of Rayleigh particles from the measurements of the scattering angular dependence.

**Figure 1. Rayleigh scattering function for randomly polarized radiation**

The efficiency factors of absorption and scattering for two-layer spheres are

where

Equations (5) and (6) enable us to find the condition when the core-mantled Rayleigh particles are invisible (Qa = Qs = 0),

There are some combinations of the parameters m', m'', and δ that satisfy this condition.

For homogeneous particles, Eqs. (5) become

The radiation energy absorbed by Rayleigh particles is characterized by the
above-introduced absorption cross section a = πa^{2}Qa. For absorbing materials
(κ > 0), the value of a is proportional to volumes of the particle V and inversely
proportional to the wavelength,

One can see from Eq. (8) that scattering by Rayleigh particles is much less than absorption, and the scattering cross section is inversely proportional to the fourth power of the wavelength,

It is important that the absorption coefficient of a small volume containing Rayleigh particles is determined by the volume fraction of the particles and does not depend on the particle size distribution. This follows from Eq. (9) for the absorption cross section.

The calculations show that the Rayleigh approximation is usually inapplicable at diffraction parameter x > 0.3 (Naumenko et al., 1970; Kerker et al., 1978; Ku and Felske, 1984). To extend the range of applicability of the small-particle approach, one can generalize the Rayleigh approximation by taking into account some partial waves of higher order and corresponding terms of the Mie series as was done by Penndorf (1962). Following Dombrovsky (1996) and Dombrovsky and Baillis (2010), consider an extension of the Penndorf approximation. For homogeneous particles at small diffraction parameter, one can use the following expansions for efficiency factors of extinction and scattering:

A similar equation for the asymmetry factor of scattering can be written as follows:

In a monograph by van de Hulst (1957), the following expressions for the Mie coefficients at small diffraction parameter were given:

More complete expansions have the form (Dombrovsky, 1996; Dombrovsky and Baillis, 2010)

where

It was shown by Penndorf (1962) that even terms with x^{10} may be important to
extend the range of the expansion applicability. Substituting the expressions for the
Mie coefficients in Eq. (11), one can obtain the analytical expressions for the
efficiency factors of extinction and scattering. The same, but in more general form,
has been done by Penndorf (1962), and the resulting equations were used by Selamet
and Arpaci (1989) in calculations of radiative properties of soot. Particularly, it was
shown that a rather high accuracy of calculations can be reached up to diffraction
parameter x = 0.8. Selamet and Arpaci (1989) did not consider any characteristics of
scattering anisotropy. Therefore, the series of equations derived is not complete. One
can substitute the approximate expressions for the Mie coefficients in Eq. (12)
to develop, after transformations, a relation for the asymmetry factor of
scattering. But it is simpler to use Eqs. (11), (12), (14), and (15) immediately. A
comparison to exact Mie calculations presented in the book by Dombrovsky (1996)
show that the transport efficiency factor of scattering Qs^{tr} for spherical soot
particles in the near-infrared spectral range can be determined with good
accuracy. At the wavelength λ = 1 μm (i.e., for not so large n and κ), one
can obtain reliable results, even for the value of diffraction parameter x =
1.

The analytical solutions for absorption and scattering of radiation in the Rayleigh region can be obtained not only for spheres, but also for particles of more complex shape. It is sufficient to determine the dipolar polarizability of the particle in the uniform electric field. The results for ellipsoids and finite cylinders can be found in books by van de Hulst (1957) and Bohren and Huffman (1983).

Strictly speaking, the infinite cylinder cannot be considered as a Rayleigh particle even at a very small diffraction parameter calculated for the cylinder radius. But the case of normal incidence gives the solution that can be treated as the Rayleigh limit. For thin homogeneous cylinders, the efficiency factors of extinction and scattering are given by

The most interesting parameter is the absorption efficiency factor for randomly
polarized incident radiation Qa = (Qa^{E} + Qa^{H})/2,

Two terms of this expression correspond to the incident radiation components with
different orientations of the polarization plane. As in all cases of Rayleigh scattering,
the absorption cross section C_{a} is proportional to the particle volume and inversely
proportional to the wavelength.

An analysis of the internal radiation field in a spherical Rayleigh particle shows that this field is uniform. It is similar to the case of an electrostatic field in the particle formed by an electric dipole. Moreover, the particle polarizability appears to be the same as that in Rayleigh theory. This observation can be used to obtain characteristics of absorption and scattering of electromagnetic radiation by Rayleigh particles of relatively complex shapes. It is sufficient to determine the polarizability tensor of these particles by solving the electrostatic problem. The examples of such a solution can be found in monographs by van de Hulst (1957) and Bohren and Huffman (1983).

#### REFERENCES

Bohren, C. F. and Huffman, D. R., Absorption and Scattering of Light by Small Particles, Wiley, Hoboken, NJ, 1983.

Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, Begell House, Redding, CT, and New York, 1996.

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

Kerker, M., Scheiner, P., and Cooke, D. D., The range of validity of the Rayleigh and Thomson limits for Lorenz-Mie scattering, J. Opt. Soc. Am., vol. 68, no. 1, pp. 135-137, 1978.

Ku, J. C. and Felske, J. D., The range of validity of the Rayleigh limit for computing Mie scattering and extinction efficiencies, J. Quant. Spectr. Radiat. Transfer, vol. 31, no. 6, pp. 569-574, 1984.

Lord Rayleigh, On the light from the sky, its polarization and colour, Philos. Mag., vol. 41, pp. 107-120, 274-279, 1871 (reprinted in “Scientific Papers by Lord Rayleigh,” vol. 1, no. 8, pp. 1869-1881, Dover Publ., New York, 1964).

Naumenko, E. K., Ivanov, A. P., and Prishivalko, A. P., Applicability range of small particles approximation in calculations of light extinction and scattering coefficients, J. Appl. Spectr., vol. 13, no. 5, pp. 898-903, 1970.

Penndorf, R. B., Scattering and extinction coefficients for small absorbing and nonabsorbing aerosols, J. Opt. Soc. Am., vol. 52, no. 8, pp. 896-904, 1962.

Selamet, A. and Arpaci, V. S., Rayleigh limit--Penndorf extension, Int. J. Heat Mass Transfer, vol. 32, no. 10, pp. 1809-20, 1989.

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