# 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.

**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"**.