Rayleigh–Gans approximation, also known as Rayleigh–Gans–Debye approximation and Rayleigh–Gans–Born approximation, is an approximate solution to

light scattering Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...

by optically soft particles. Optical softness implies that the relative

refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, o ...

of particle is close to that of the surrounding medium. The approximation holds for particles of arbitrary shape that are relatively small but can be larger than

Rayleigh scattering Rayleigh scattering ( ), named after the 19th-century British physicist Lord Rayleigh (John William Strutt), is the predominantly elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the ...

limits. The theory was derived by

Lord Rayleigh John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. Amo ...

in 1881 and was applied to homogeneous spheres, spherical shells, radially inhomogeneous spheres and infinite cylinders.

Peter Debye Peter Joseph William Debye (; ; March 24, 1884 – November 2, 1966) was a Dutch-American physicist and physical chemistry, physical chemist, and List of Nobel laureates in Chemistry, Nobel laureate in Chemistry. Biography Early life Born Petr ...

has contributed to the theory in 1881. The theory for homogeneous sphere was rederived by

Richard Gans Richard is a male given name. It originates, via Old French, from Old Frankish and is a compound of the words descending from Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'stro ...

in 1925. The approximation is analogous to

Born approximation Generally in scattering theory and in particular in quantum mechanics, the Born approximation consists of taking the incident field in place of the total field as the driving field at each point in the scatterer. The Born approximation is named a ...

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...

Theory

The validity conditions for the approximation can be denoted as: :

, n-1,  \ll 1

kd, n-1,  \ll 1

k

is the wavevector of the light (

k=\frac

), whereas

d

refers to the linear dimension of the particle.

n

is the

complex refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...

of the particle. The first condition allows for a simplification in expressing the material polarizability in the derivation below. The second condition is a statement of the

, that is, that the incident field is not greatly altered within one particle so that each volume element is considered to be illuminated by an intensity and phase determined only by its position relative to the incident wave, unaffected by scattering from other volume elements. The particle is divided into small volume elements, which are treated as independent Rayleigh scatterers. For an inbound light with s polarization, the

scattering amplitude In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process. :

dS_1(\theta, \phi)=i \frac k^3 \left( \frac \right) e^ dV

where

\delta

denotes the phase difference due to each individual element, and the fraction in parenthesis is the electric

polarizability Polarizability usually refers to the tendency of matter, when subjected to an electric field, to acquire an electric dipole moment in proportion to that applied field. It is a property of all matter, considering that matter is made up of elementar ...

as found from the refractive index using the

Clausius–Mossotti relation The Clausius–Mossotti relation expresses the dielectric constant (relative permittivity, ''ε''r) of a material in terms of the atomic polarizability, α, of the material's constituent atoms and/or molecules, or a homogeneous mixture thereof. It ...

. Under the condition ''(n-1) << 1'', this factor can be approximated as ''2(n-1)/3''. The phases

\delta

affecting the scattering from each volume element are dependent only on their positions with respect to the incoming wave and the scattering direction. Integrating, the scattering amplitude function thus obtains: :

S_1(\theta, \phi)  \approx    \frack^3(n-1) \int e^ dV

in which only the final integral, which describes the

interfering Interference is the act of interfering, invading, or poaching. Interference may also refer to: Communications * Interference (communication), anything which alters, modifies, or disrupts a message * Adjacent-channel interference, caused by extra ...

phases contributing to the scattering direction (θ, φ), remains to be solved according to the particular geometry of the scatterer. Calling ''V'' the entire volume of the scattering object, over which this integration is performed, one can write that scattering parameter for scattering with the electric field polarization normal to the plane of incidence (s polarization) as :

S_1=\frack^3(n-1) V R(\theta, \phi)

and for polarization ''in'' the plane of incidence (p polarization) as :

S_2=\frack^3(n-1) V R(\theta, \phi)cos \theta

where

R(\theta, \phi)

denotes the "form factor" of the scatterer: :

R(\theta, \phi)=\frac \int e^ dV

In order to only find intensities we can define ''P'' as the squared magnitude of the form factor: :

P(\theta, \phi)=\left( \frac \right) \left,  \int e^ dV \^2

Then the scattered radiation intensity, relative to the intensity of the incident wave, for each polarization can be written as: :

I_1/I_0 = \left(\frac \right) (n-1)^2 P(\theta, \phi)

I_2/I_0 = \left(\frac \right) (n-1)^2 P(\theta, \phi)cos^2 \theta

where ''r'' is the distance from the scatterer to the observation point. Per the

optical theorem In physics, the optical theorem is a general law of wave scattering theory, which relates the forward scattering amplitude to the total cross section of the scatterer. It is usually written in the form :\sigma_\mathrm=\frac~\mathrm\,f(0), where ...

absorption Absorption may refer to: Chemistry and biology *Absorption (biology), digestion **Absorption (small intestine) *Absorption (chemistry), diffusion of particles of gas or liquid into liquid or solid materials * Absorption (skin), a route by which ...

cross section is given as: :

C_=2kV \mathbb(n)

which is independent of the polarization.

Applications

Rayleigh–Gans approximation has been applied on the calculation of the optical cross sections of

fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as il ...

aggregates. The theory was also applied to

anisotropic Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's phys ...

spheres for nanostructured polycristalline alumina and

turbidity Turbidity is the cloudiness or haziness of a fluid caused by large numbers of individual particles that are generally invisible to the naked eye, similar to smoke in air. The measurement of turbidity is a key test of water quality. Fluids ...

calculations on biological structures such as lipid vesicles and

bacteria Bacteria (; singular: bacterium) are ubiquitous, mostly free-living organisms often consisting of one biological cell. They constitute a large domain of prokaryotic microorganisms. Typically a few micrometres in length, bacteria were am ...

. A

nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...

Rayleigh−Gans−Debye model was used to investigate

second-harmonic generation Second-harmonic generation (SHG, also called frequency doubling) is a nonlinear optical process in which two photons with the same frequency interact with a nonlinear material, are "combined", and generate a new photon with twice the energy of ...

malachite green Malachite green is an organic compound that is used as a dyestuff and controversially as an antimicrobial in aquaculture. Malachite green is traditionally used as a dye for materials such as silk, leather, and paper. Despite its name the dye is ...

molecules

adsorbed Adsorption is the adhesion of atoms, ions or molecules from a gas, liquid or dissolved solid to a surface. This process creates a film of the ''adsorbate'' on the surface of the ''adsorbent''. This process differs from absorption, in which ...

polystyrene Polystyrene (PS) is a synthetic polymer made from monomers of the aromatic hydrocarbon styrene. Polystyrene can be solid or foamed. General-purpose polystyrene is clear, hard, and brittle. It is an inexpensive resin per unit weight. It is ...

particles.

References

{{DEFAULTSORT:Rayleigh-Gans approximation Scattering, absorption and radiative transfer (optics) Radio frequency propagation X-ray scattering

Theory

Applications

See also

References