Rayleigh–Gans approximation
Approximate solution for light scattering by optically soft particles.
The Rayleigh–Gans approximation, also referred to as the Rayleigh–Gans–Debye or Rayleigh–Gans–Born approximation, provides a way to estimate how light scatters from particles that are optically soft—meaning their refractive index is very similar to that of the surrounding medium. This approach works for particles of any shape, as long as they are relatively small, though they may be larger than what the Rayleigh scattering limit allows.
Lord Rayleigh first derived the theory in 1881, applying it to homogeneous spheres, spherical shells, radially inhomogeneous spheres, and infinite cylinders. Peter Debye also contributed to the theory in the same year. Richard Gans later rederived the theory for homogeneous spheres in 1925. The approximation is analogous to the Born approximation used in quantum mechanics.
The validity of the approximation depends on two conditions. The first involves the wavevector of the light and the particle’s linear dimension, while the second relates to the complex refractive index. The first condition simplifies how the material’s polarizability is expressed. The second condition, a statement of the Born approximation, means the incident field is not significantly altered within a single particle, so each volume element is illuminated by an intensity and phase determined solely by its position relative to the incoming wave, unaffected by scattering from other volume elements.
In the theory, the particle is divided into small volume elements, each treated as an independent Rayleigh scatterer. For incoming light with s polarization, the scattering amplitude from each element depends on a phase difference and the electric polarizability, which is derived from the refractive index using the Clausius–Mossotti relation. When the refractive index difference is small, this polarizability factor simplifies. The phases affecting scattering from each element depend only on their positions relative to the incoming wave and the scattering direction. Integrating over the entire volume gives the scattering amplitude function, where only the integral describing the interfering phases for a given scattering direction remains to be solved based on the particle’s shape. The form factor of the scatterer is defined from this integral, and its squared magnitude is used to find the scattered intensity for both s and p polarizations.
- Named after
- Lord Rayleigh, Richard Gans, Peter Debye
- Derived by
- Lord Rayleigh in 1881
- Applied to
- homogeneous spheres, spherical shells, radially inhomogeneous spheres, infinite cylinders
- Rederived for homogeneous sphere by
- Richard Gans in 1925
- Contributed by
- Peter Debye in 1881
- Analogous to
- Born approximation in quantum mechanics
Lore & Background
The theory was derived by Lord Rayleigh in 1881 and was applied to homogeneous spheres, spherical shells, radially inhomogeneous spheres and infinite cylinders. Peter Debye contributed to the theory in 1881. The theory for homogeneous sphere was rederived by Richard Gans in 1925. The approximation is analogous to the Born approximation in quantum mechanics.
The validity conditions require that |n-1| << 1 and that k a |n-1| << 1, where k is the wavevector of the light, a is the linear dimension of the particle, and n is the complex refractive index. The first condition allows simplification in expressing material polarizability; the second is a statement of the Born approximation, meaning the incident field is not greatly altered within one particle so that each volume element is illuminated by an intensity and phase determined only by its position relative to the incident wave.
The particle is divided into small volume elements treated as independent Rayleigh scatterers. The scattering amplitude is obtained by integrating over the volume, with a form factor describing the interfering phases. The scattered intensity for s and p polarizations is expressed in terms of the squared magnitude of the form factor. The absorption cross section, given by the optical theorem, is independent of polarization.
Reader's Guide
The Rayleigh–Gans approximation has been applied to the calculation of the optical cross sections of fractal aggregates. It has also been applied to anisotropic spheres for nanostructured polycrystalline alumina and to turbidity calculations on biological structures such as lipid vesicles and bacteria. A nonlinear Rayleigh–Gans–Debye model was used to investigate second-harmonic generation in malachite green molecules adsorbed on polystyrene particles. The approximation provides a bridge between Rayleigh scattering and Mie scattering for optically soft particles, allowing treatment of particles larger than the Rayleigh limit while retaining a relatively simple analytical form. Its connection to the Born approximation in quantum mechanics gives it a firm theoretical foundation, and its applicability to arbitrary shapes makes it versatile for modeling scattering from complex structures in both physical and biological contexts.
Did You Know?
- The approximation is also known as the Rayleigh–Gans–Debye approximation and Rayleigh–Gans–Born approximation.
- It holds for particles of arbitrary shape that are relatively small but can be larger than Rayleigh scattering limits.
- The theory was derived by Lord Rayleigh in 1881 and rederived for homogeneous spheres by Richard Gans in 1925.
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