Static Light Scattering: The Form Factor


For homogeneous objects the form factor is expressed as \begin{gather*} P(\mathbf{q}) = \left[\frac{\int_{V_p}dV\exp[-i\mathbf{q}\cdot\mathbf{r}]}{V_p}\right]^2 \end{gather*} For a sphere of radius \( R \) the formula above evaluates to \begin{gather*} P(q) = \left[\frac{3}{(qR)^3}(\sin(qR)-qR\cos(qR))\right]^2 \end{gather*}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 \( r \): \begin{gather*} P(q) = \left[\int_0^\infty r^2 g(r)\sin(qr)/(qr)dr\right]^2 \end{gather*} By expanding in series the term \( \sin(qr)/(qr) \) we obtain the Guinier approximation for the form factor: \begin{gather*} P(q) \simeq 1 - \frac{(q R_g)^2}{3}, \end{gather*} where the (optical) radius of gyration is defined as \begin{gather*} R_g^2 = \int_0^\infty r^2 g(r)dr. \end{gather*} 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.

 

 

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