# Static Light Scattering: The Form Factor

For homogeneous objects the form factor is expressed as

\(P(\mathbf{q}) = \left[\frac{\int_{V_p}dV\exp[-i\mathbf{q}\cdot\mathbf{r}]}{V_p}\right]^2 \)

For a sphere of radius \( R \) the formula above evaluates to

\( P(q) = \left[\frac{3}{(qR)^3}(\sin(qR)-qR\cos(qR))\right]^2 \)

Fig. 1 shows the form factor of a sphere according to the formula above. In agreement with what stated in the previous paragraph, we notice that for \( q R \ll 1\) the form factor attains a plateau at a value of 1, whereas as soon as \( q R \) becomes substantially larger than 1 the form factor is effected by the interparticle interference effects.

Figure 1: Sphere Form Factor.

A simpler general formula for the form factor can be obtained by introducing the so-called pair distance function, \( g(r) \) i.e. the probability to find two points belonging to the colloidal particle at a distance \( r \):

\( P(q) = \left[\int_0^\infty r^2 g(r)\sin(qr)/(qr)dr\right]^2 \)

By expanding in series the term \( \sin(qr)/(qr) \) we obtain the Guinier approximation for the form factor:

\(P(q) \simeq 1 - \frac{(q R_g)^2}{3}, \)

where the (optical) radius of gyration is defined as

\(R_g^2 = \int_0^\infty r^2 g(r)dr. \)

The importance of such approximation lies in the fact that it allows for the determination of a size parameter, namely \( R_g \) by performing a simple linear fit in the plot \( I_s \) vs. \(q^2\), the so-called Guinier plot. As an example Fig. 3 shows the approximation for a sphere for which \(R_g = \sqrt(3/5)R\).

Figure 2: Sphere Form Factor, Guinier Plot.

Learn more about **static light scattering** by following our online technology section step by step.

It is advisable to read the different sections in the suggested order if you want to understand all the details. For a quick reference you can also jump to each section individually.

**Rayleigh-Gans-Debye Scattering****Form Factor****Structure Factor****Scattering from Macromolecules****Excess Rayleigh Ratio****Radius of Gyration****Guinier Plot****Zimm Plot**

A very detailed source of information is the slide show **"Light Scattering Fundamentals"**.