LS Instruments | Radius of Gyration

Radius of Gyration

The optical Radius of Gyration $R_g$ is defined as:

$R_g^2 = \int r^2 g(r)dr$

Where $r$ is the distance from a reference point and $g(r)$ is the so-called pair distance function:

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

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:

$R^2_g = \frac{3}{5} R^2$

Spherical shell

Spherical shell with outer and inner radii $R_{\mathrm{o}}$ and $R_i$ , respectively:

$R^2_g = \frac{3}{5} \frac{ R^5_{\mathrm{o}} - R^5_{\mathrm{i}} }{ R^3_{\mathrm{o}} - R^3_{\mathrm{i}} }$

Cylinder

Cylinder with radius $R$ and length $h$ :

$R^2_g = \frac{ R^2}{2} + \frac{h^2}{12}$

Hollow Cylinder

Hollow cylinder of length $h$ with outer and inner radii $R_o$ and $R_i$ , respectively:

$R^2_g = \frac{ R_{\mathrm{o}}^2 + R_{\mathrm{i}}^2}{2} + \frac{h^2}{12}$

Ellipsoid

Ellipsoid with semiaxes a,b, and c:

$R^2_g = \frac{ a^2 + b^2 + c^2}{5}$

Disk

Flat disk with radius $R$

$R^2_g = \frac{ R^2 }{2}$

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