# 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.

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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.

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".