# The Rayleigh-Gans-Debye (RGD) Scattering

Consider a colloidal particle suspension illuminated by laser light. If the particles have a different refractive index $$n$$ than the solvent they will scatter the laser light. But how much light is scattered in which direction? In order to understand this, we have to consider that light is an electromagnetic wave. The corresponding oscillating electric field of this wave deforms the electronic cloud of the atoms making up the particle. As a result, the oscillating electrons, emit ("scatter") electromagnetic radiation.

Each small volume hit by the laser light in our sample will scatter light according to the aforementioned mechanism, thus the intensity measured will be the sum of all such contributions. It is easily seen that, within this description, an homogeneous sample wouldn't scatter any light. Indeed for every small volume scattering a wave with a phase $$\varphi$$, on the direction connecting it to the detector, we can find another small volume at a distance such that its scattered light has a phase of $$\varphi + \pi (2n + 1)$$.

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This results in destructive interference. On the other hand, if the sample in the cuvette is not homogeneous, i.e. contains colloidal particles or polymer chains, this argument doesn't apply as the two volumes might have different properties in terms of dielectric constant, such as their polarizabilities. The two volume elements will scatter the light with the same phase but different amplitudes, thus interference will not be completely destructive.
One simple yet very useful theory describing the light scattering of colloidal samples is the so-called Rayleigh-Gans-Debye (RGD) theory. It's basic assumption is that light is not reflected at the medium-particle boundary nor is it attenuated within the particle. This is true if the following conditions are met \begin{gather*} \left| 1 - m \right| \ll 1 \\ \frac{2 \pi n_s}{\lambda} a \left| 1 - m \right| \ll 1, \end{gather*} here $$\lambda$$ is the wavelength of light $$m \equiv n_p/n_s$$ is the relative refractive index i.e. the ratio of the refractive index of the colloidal particles to that of the suspending medium, and $$a$$ is the characteristic size of the colloidal object.
Within this assumption each small portion of particle is subject to same electromagnetic radiation conveyed by the incident light and scatters light as if isolated from the rest of the particle. In a situation where light with intensity $$I_i$$ impinges on a sample containing a monodisperse colloidal suspension of number concentration $$c$$ the RGD assumption enables us to write the scattered intensity collected at the angle, $$\theta$$ at a distance $$R_0$$ from the sample as
\begin{gather*} I_s(\mathbf{q},c) = I_i \frac{f(\theta)}{R_0^2} c V_s \Delta \rho ^2 V_p^2 P(\mathbf{q}) S(\mathbf{q},c), \end{gather*} here $$V_p$$ and $$V_s$$ are the particle and scattering volumes, respectively, $$f(\theta)$$ is a geometrical factor accounting for the polarization of incident and collected light, $$P(\mathbf{q})$$ and $$S(\mathbf{q},c)$$ are factors accounting for intra- and inter-particle interference respectively, and the scattering length density difference $$\Delta \rho$$ reflects for the scattering ability of the sample and is expressed as \begin{gather*} \Delta \rho = \frac{3 \pi n_s^2}{\lambda^2} \frac{m^2 - 1}{m^2 + 2}. \end{gather*} In the formulas above the scattering vector $$\mathbf{q}$$, defined as the difference between the incident and scattered light wavevectors, accounts for the angle dependent phase difference of interfering scattered waves. Its modulus has the following form \begin{gather*} q = \frac{4 \pi n_s}{\lambda} \sin (\theta/2 ). \end{gather*} The modulus of the scattering vector has dimension of the inverse of a length. $$q^{-1}$$ physically represents the lenghtscale at which we are "looking" at the colloidal system. For example at suitably large scattering angle $$q^{-1}$$ will be smaller than the colloids correlation length (e.g. the particle size in non-interacting colloids) resulting in substantial interference, on the contrary at small angles the opposite situation will occur and the scattering intensity will reach a plateau.