Radius of Gyration

 
 

The optical Radius of Gyration \(R_g\) is defined as: 

 

\begin{gather*} R_g^2 = \int r^2 g(r)dr \end{gather*} Where \(r\) is the distance from a reference point and \(g(r)\) is the so-called pair distance function:

 

\begin{gather} g(r) = r^2 \int \Delta \rho(\mathbf{r}')\Delta \rho(\mathbf{r} - \mathbf{r}') d^3 \mathbf{r}'\end{gather}

where \(\Delta \rho(r)\) is the scattering length density. It's worth noting that \(g(r)\) contains information about shape and size of the particle and/or macromolecule, to each particle shape corresponds a  well defined \(g(r)\).

 

The radius of gyration can be obtained from static light scattering by performing either a Guinier or Zimm Plot.

 
 
 


 
 

Relation of the Radius of Gyration to the geometry of homogeneous objects:

Sphere

For a sphere the radius of gyration is related to the spheres geometric radius, \(R\) in the following way:\begin{gather} R^2_g = \frac{3}{5} R^2\end{gather}.

 

Spherical shell

Spherical shell with outer and inner radii \(R_{\mathrm{o}}\) and \(R_{\mathrm{i}}\), respectively:\begin{gather} R^2_g = \frac{3}{5} \frac{ R^5_{\mathrm{o}} - R^5_{\mathrm{i}} }{ R^3_{\mathrm{o}} - R^3_{\mathrm{i}} } \end{gather} .

 

Cylinder

Cylinder with radius \(R\) and length \(h\):\begin{gather}R^2_g = \frac{ R^2}{2} + \frac{h^2}{12} \end{gather} .

 

Hollow Cylinder

Hollow cylinder of length \(h\) with outer and inner radii \(R_{\mathrm{o}}\) and \(R_{\mathrm{i}}\), respectively:\begin{gather}R^2_g = \frac{ R_{\mathrm{o}}^2 + R_{\mathrm{i}}^2}{2} + \frac{h^2}{12} \end{gather} .

 

Ellipsoid

Ellipsoid with semiaxes a,b, and c:\begin{gather}R^2_g = \frac{ a^2 + b^2 + c^2}{5}  \end{gather} .

 

Disk

Flat disk with radius \(R\):\begin{gather}R^2_g = \frac{ R^2 }{2}  \end{gather} .

 
 


 
 

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".