# Rayleigh and Mie Scattering

**DOI:**https://doi.org/10.1007/978-1-4419-8071-7_218

## Synonyms

## Definitions

*Rayleigh scattering* refers primarily to the elastic scattering of light from atomic and molecular particles whose diameter is less than about one-tenth the wavelength of the incident light.

*Rayleigh line* refers to the unshifted central peak observed in the spectroscopic analysis of scattered light.

*Mie scattering* refers primarily to the elastic scattering of light from atomic and molecular particles whose diameter is larger than about the wavelength of the incident light.

*Thomson scattering* is elastic scattering of light from free electrons.

*Raman scattering* is inelastic scattering of light from objects whereby the scattered photon has a lower (Raman Stokes scattering) or higher (Raman anti-Stokes scattering) energy than the incident photon.

## Introduction

From ancient times, people have gazed up at the sky in daylight and asked the perennial question “Why is the sky blue?” [1]. Other similar and related questions are “Why is the night sky black?,” “Why are sunrises and sunsets red?,” and “Why are the clouds white?” Rayleigh [2, 3, 4, 5] and Mie scattering [6] lie behind the long-sought answers to all such questions about the colors seen in the sky.

It is also noteworthy that Rayleigh contributed widely to scattering theory in eight different categories; for an overview of these different contributions, see Reference [8]. However, this entry is limited to a description of classical Rayleigh scattering of light from single small objects such as molecules and atoms. Thus, it is ignoring coherence effects that arise in solids, liquids, and gases at atmospheric pressure or even free electrons where it is known as Thomson scattering.

## Classical Description

*λ*, of the incident light. Rayleigh considered a plane wave incident on a dielectric sphere of radius r and of relative permittivity (dielectric constant)

*ε*, as illustrated in Fig. 3 [10]. The probability that the sphere scatters light at angle

*θ*is proportional to the differential scattering cross section, dσ(

*θ*)/dΩ, which is defined as the ratio of the power scattered into the solid angle dΩ between

*θ*and d

*θ*(see Fig. 3) to the incident power per unit area. For unpolarized incident light, the differential cross section becomes [10]

*ε*

_{0}is the relative permittivity and n

_{0}the refractive index of the medium surrounding the sphere of refractive index n. Integrating this equation over the entire solid angle yields the total cross section:

*I*

_{R}of the Rayleigh scattered light is given by [11]

*I*is the intensity of the unpolarized incident light and D is the distance between the particle and the observer.

^{2}

*θ*) term in Eq. 3 contains the angular dependence of the scattering and indicates that Rayleigh scattering intensity at 90° is one-half that in the forward or backward directions.

For particle sizes larger than *λ*, Mie scattering predominates [6, 12] and for particles that are much larger than *λ*, a third type of atmospheric scattering, known as nonselective scattering, occurs [13]. A description of this last type of scattering, which can be considered as comprising a combination of Mie scattering, absorption, and multiple scattering, is outside the scope of this entry. Nonselective scattering is not wavelength dependent and is the primary cause of haze in the lower atmosphere. Water droplets and large dust particles can cause this type of scattering.

*σ*

_{M}, can be expressed in the form of an infinite series as [10]

_{i}and b

_{i}are expressed in terms of spherical Hankel functions and spherical Bessel functions of the first kind; these functions are dependent on the magnetic permeabilities of the sphere and surrounding medium and the values of parameters m = n/n

_{0}and x = 2πn

_{0}r/

*λ*, which is termed the size parameter.

*σ*/πr

^{2}. This figure demonstrates that for small spheres with mx ≪1, (x = 2πn

_{0}r/

*λ*and m = n/n

_{0}= 1.59/1.33), the scattering efficiencies of Rayleigh and Mie scattering are very nearly the same, whereas for larger spheres with high mx values, the behavior is completely different. The Mie scattering efficiency departs from the Rayleigh

*λ*

^{−4}behavior and approaches a limiting value of Q = 2, as a consequence of the “extinction paradox” related to equal geometrical and diffraction contributions to the cross section [10]. The diffraction contribution is most readily discerned at a distance from the sphere and is strongly peaked in the forward direction. Thus Mie scattering produces a scattering pattern like an antenna lobe with a forward lobe that becomes more intense and sharper with increasing particle size. For larger particle sizes, as Fig. 6 shows, Mie scattering is not strongly wavelength dependent. This is the reason that such scattering from water droplets in clouds, mist, or fog produces white light, as can be seen in Fig. 4 in the lower part of the sky behind the lion. Likewise, due to its larger forward lobe, Mie scattering produces the whitish glare that is seen surrounding the sun when large particulate matter is present in the air.

## Quantum Description

*ħω*(

*ω*is the angular frequency and

*ħ*=

*h*/2π, where

*h*is Planck’s constant) of the incident light beam is destroyed and a quantum of scattered light of energy

*ħω*

_{s}is created. When

*ω*

_{s}=

*ω*, the process is called elastic scattering or Rayleigh scattering in the nomenclature of this field of research, which is where the use of terminology leads to naming confusions for different scattering processes [7]. When

*ω*

_{s}≠

*ω*, the process is called inelastic scattering or Raman scattering; the energy difference thus created is accommodated by the atom or molecule that is involved in the scattering process. Energy may be transferred to, or taken from, the atom or molecule via an electronic transition, in which case it is call Raman Stokes, or Raman anti-Stokes, scattering. These three different electronic energy transition processes are illustrated in Fig. 7. Electronic transitions can occur between any states but usually arise from or terminate at the ground state or thermally populated higher states, and the excited states are usually virtual states, that is, they are not stationary states of the system. However, if the energy of the incident or scattered light is close to or at a stationary energy state, electronic resonant scattering occurs.

*σ*/dΩ, can be written as [15]

**ε**and

**ε**

_{s}are the unit polarization vectors for the incident and scattered light, and

**D**is the sum of the electron coordinates and is proportional to the atomic-dipole moment. Evaluating the matrix elements that determine

**D**is not trivial and has only been performed for simple systems such as atomic hydrogen [16].

*ω*is much larger than an atomic excitation frequency

*ω*

_{i}, but much less than mc

^{2}/

*ħ*where m is the electron mass. In this high-frequency limit (corresponding to Thomson scattering), the cross section reduces simply to [15]

_{e}= e

^{2}/4πε

_{0}mc

^{2}is the classical electron radius. Interestingly, this result from the quantum theory for the elastic scattering cross section in the high-frequency limit is identical to that obtained from the classical theory for Thomson scattering [15].

*ω*is much smaller than all atomic excitation frequencies

*ω*

_{i}, in which case Eq. 5 becomes [15]

_{R}= me

^{4}/32π

^{2}ε

_{0}

^{2}ħ

^{2}is the ground state binding energy of hydrogen.

_{1i}are the appropriate electric-dipole matrix elements. Note that in the frequency region below ω

_{R}, there are several strong electronic transition resonances evident (only those up to the fifth excited state were calculated by Gavrila), separated by zeros in the differential cross section. Such resonances could not be predicted from the classical model used by Rayleigh. The inclusion of radiative damping in the theory removes the infinite values from the cross section and the accompanying zeroes evident in Fig. 8. Note also that at high frequencies, the cross section value approaches that of Thomson scattering, as given by Eq. 6 with Z = 1, and eventually reaches it when ω ≫ ω

_{i}.

## Representative Scattering Media

Here we review Rayleigh and Mie scattering processes at work in different media of interest, including solids, liquids, and gases. Some examples of practical applications of Rayleigh and Mie scattering in these representative media are also provided.

In fluids, conventional Rayleigh scattering is most commonly observed. Although proof of Rayleigh’s law in gases was obtained quite early on [2, 17], at just a few isolated wavelengths, it was not until later that similar information was obtained over a wide wavelength range. In 1973, Stone [18] reported on measurements of the Rayleigh scattering from CCl_{4} and C_{2}Cl_{4} as a function of wavelength between 600 and 1,060 nm by placing the liquid sample in a hollow fused-quartz fiber and measuring the light scattered by the liquid through the fiber wall. Stone determined that the scattering loss rate was 25 dB/km for CCl_{4} and 68 dB/km for C_{2}Cl_{4} at 632.8 nm and that the scattering loss rate followed a *λ*^{−4} dependence over the entire spectral range.

*α*

_{R}, given by [20, 21]

*β*

_{T}the isothermal compressibility, and T is a fictitious temperature at which the density fluctuations are “frozen” into the glass (~1,500 K for fused silica). Note that as the contribution of Rayleigh scattering to the attenuation coefficient scales with the inverse fourth power of the wavelength, the Rayleigh scattering losses predominate at shorter wavelengths. A more unusual example is the observation of recoilless Rayleigh scattering by atoms in solids. This has been observed, for example, by employing equipment designed for Mossbauer effect studies (i.e., photon sources and analyzers with extreme selectivity in energy) to observe Rayleigh scattering in Pt, Al, graphite, and paraffin using an incident light energy of 23.8 keV [22].

*λ*

^{−4}Rayleigh-type decay of the scattering coefficient with increasing wavelength [23]. Measurements of light scattering spectra from rutile structure fluorides using circularly polarized incident light have revealed Rayleigh optical activity. This occurs when the incident light is tilted away from the crystal

*c*-axis direction of the tetragonal crystal structure [24]. Rayleigh optical activity is observed experimentally through a difference in the intensity of the Rayleigh scattering when excited with right and left circularly polarized incident light. The Rayleigh circular intensity differential (CID), Δ

_{α}, is defined by

_{α}

^{R}and I

_{α}

^{L}are the scattered light intensities with

*α*polarization due to right and left circularly polarized incident light, respectively. The dependence of the CID spectrum on small rotations of the crystal about the laboratory y-axis was measured in optically transparent MgF

_{2}and in crown glass for reference purposes. For the case of the crystal

*c*(

*a*) axis nearly aligned along the direction of the monochromatic incident(scattered) laser light, the strong dependence of the Rayleigh CID spectrum on the crystal angle is shown in Fig. 11. Most importantly for correct identification purposes, the CID changes sign as the crystal is rotated through the true

*c*-axis alignment at

*θ*

_{e}= −0.7°. Notably, no CID is observed for small rotations of the crystal about the orientation where the incident light and scattered light propagate perpendicular to the optic (

*c*) axis, nor is any CID observed from the crown glass. These results indicate that the Rayleigh scattering CID observed in the MgF

_{2}crystal is due to depolarization effects associated with its birefringence and not polarization-dependent scattering cross sections, as confirmed theoretically [24].

Mie scattering is also found everywhere in nature: in the lower atmosphere, as noted above, in fluids like milk and latex paint, and even in biological tissue. In the latter case, Mie theory has been applied to determine if scattered light from appropriately treated tissue can be used to diagnose cancerous from healthy cells [25, 26]. Mie scattering is used in particle size determination for particles in non-absorbing media [27], in the determination of the oil concentration in polluted water [28], in parasitology [29], and in the design of metamaterials [30].

In summary, it is evident that Rayleigh scattering and Mie scattering are ubiquitous, being found in the everyday and colorful optical wonders that surround us.

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