Rayleigh–Gans approximation

Rayleigh–Gans approximation, also known as Rayleigh–Gans–Debye approximation[1] and Rayleigh–Gans–Born approximation,[2] is an approximate solution to light scattering by optically soft particles. Optical softness implies that the relative refractive index 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 limits.[1]

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 has contributed to the theory in 1881. The theory for homogeneous sphere was rederived by Richard Gans in 1925. The approximation is analogous to Born approximation in quantum mechanics.[3]

Theory

The validity conditions for the approximation can be denoted as:

${\displaystyle |n-1|\ll 1}$

${\displaystyle kd|n-1|\ll 1}$

${\textstyle k}$ is the wavevector of the light (${\textstyle k={\frac {2\pi }{\lambda }}}$), whereas ${\textstyle d}$ refers to the linear dimension of the particle. ${\displaystyle n}$ is the complex refractive index of the particle. The former condition implies that reflection from the particle interface should be minimum and the latter indicates that the incident light should not undergo appreciable change in phase or amplitude after impacting the particle.[1]

The particle is divided into small volume elements, which are treated as independent Rayleigh scatterers. For an inbound light with perpendicular polarization, scattering amplitude function for each element is given as:[3]

${\displaystyle dS_{1}(\theta )=i{\frac {3}{4\pi }}k^{3}\left({\frac {n^{2}-1}{n^{2}+2}}\right)e^{i\delta }dV}$

where ${\displaystyle \delta }$ denotes the phase difference due to each individual element.[3] The derivation is based on Clausius–Mossotti relation for polarizability.[4] The fundamental approximation in Rayleigh–Gans theory dictates that the particle dimensions and relative refractive index are not too large; as a result, phase ${\displaystyle \delta }$ for each element is dependent only on its location. In the far-field, scattering amplitude function thus becomes:

${\displaystyle S_{1}(\theta )={\frac {i}{2\pi }}k^{3}(m-1)\int e^{i\delta }dV}$

which indicates that the function is dependent on the interference of each wave component.[3] Also considering particle volume ${\textstyle V}$, the corresponding S-matrix elements can be written as:[5]

${\displaystyle S_{1}={\frac {i}{2\pi }}k^{3}(m-1)VR(\theta ,\phi )}$

${\displaystyle S_{2}={\frac {i}{2\pi }}k^{3}(m-1)VR(\theta ,\phi )cos\theta }$

where ${\textstyle R(\theta ,\phi )}$ denotes the form factor:[5]

${\displaystyle R(\theta ,\phi )={\frac {1}{V}}\int {}{}e^{i\delta }dV}$

For scattered radiation intensity, form factor can alternatively be defined as ${\textstyle P(\theta )}$:[3]

${\displaystyle P(\theta )=\left({\frac {1}{V^{2}}}\right)\left|\int e^{i\delta }dV\right|^{2}}$

The scattered radiation intensity for both polarization are given as:[3]

${\displaystyle I_{1}=\left({\frac {k^{4}V^{2}}{4\pi ^{2}r^{2}}}\right)(m-1)^{2}P(\theta )}$

${\displaystyle I_{2}=\left({\frac {k^{4}V^{2}}{4\pi ^{2}r^{2}}}\right)(m-1)^{2}P(\theta )cos^{2}\theta }$

Per the optical theorem, absorption cross section is given as:

${\displaystyle C_{abs}=2kV\mathbb {Im} (m)}$

which is independent of the polarization.[1]

Applications

Rayleigh–Gans approximation has been applied on the calculation of the optical cross sections of fractal aggregates.[6] The theory was also applied to anisotropic spheres for nanostructured polycristalline alumina and turbidity calculations on biological structures such as lipid vesicles[7] and bacteria.[8]

A nonlinear Rayleigh−Gans−Debye model was used to investigate second-harmonic generation in malachite green molecules adsorbed on polystyrene particles.[9]

See also

• Mie scattering
• Anomalous diffraction theory
• Discrete dipole approximation
• Gans theory

References

1. Bohren, C. F.; Huffmann, D. R. (2010). Absorption and scattering of light by small particles. New York: Wiley-Interscience. ISBN 978-3-527-40664-7.
2. Turner, Leaf (1973). "Rayleigh-Gans-Born Light Scattering by Ensembles of Randomly Oriented Anisotropic Particles". Applied Optics. 12 (5): 1085–1090. Bibcode:1973ApOpt..12.1085T. doi:10.1364/AO.12.001085. PMID 20125471.
3. Kerker, Milton (1969). Loebl, Ernest M. (ed.). The Scattering of Light and Other Electromagnetic Radiation. New York: Academic Press. ISBN 9780124045507.
4. Leinonen, Jussi; Kneifel, Stefan; Hogan, Robin J. (12 June 2017). "Evaluation of the Rayleigh–Gans approximation for microwave scattering by rimed snowflakes". Advances in Remote Sensing of Rainfall and Snowfall. 144: 77–88. doi:10.1002/qj.3093.
5. van de Hulst, H. C. (1957). Light scattering by small particles. New York: John Wiley and Sons. ISBN 9780486139753.
6. Farias, T. L.; Köylü, Ü. Ö.; Carvalho, M. G. (1996). "Range of validity of the Rayleigh–Debye–Gans theory for optics of fractal aggregates". Applied Optics. 35 (33): 6560–6567. Bibcode:1996ApOpt..35.6560F. doi:10.1364/AO.35.006560. PMID 21127680.
7. Chong, C.S.; Colbow, Konrad (17 June 1976). "Light scattering and turbidity measurements on lipid vesicles". Biochimica et Biophysica Acta (BBA) - Biomembranes. 436 (2): 260–282. doi:10.1016/0005-2736(76)90192-9. PMID 1276217.
8. Koch, Arthur L. (19 August 1961). "Some calculations on the turbidity of mitochondria and bacteria". Biochimica et Biophysica Acta. 51 (3): 429–441. doi:10.1016/0006-3002(61)90599-6. PMID 14457538.
9. Jen, Shih-Hui; Dai, Hai-Lung; Gonella, Grazia (18 February 2010). "The Effect of Particle Size in Second Harmonic Generation from the Surface of Spherical Colloidal Particles. II: The Nonlinear Rayleigh−Gans−Debye Model". The Journal of Physical Chemistry C. 114 (10): 4302–4308. doi:10.1021/jp910144c.