The radius of gyration

An alternative measure of the size of a polymer chain is provided by its radius of gyration , which may be measured by light scattering experiments . It is defined by

$$R_{g}^{2}= \frac{1}{(N+1)}\sum_{i=0}^{N}\langle (\vec{R}_{i}-\vec{R}_{G})^{2}\rangle$$ (1.42)

$$\vec{R}_{G}=\frac{1}{(N+1)}\sum_{i=0}^{N}\vec{R}_{i}.$$ (1.43)

It measures the average squared distance to the centre of gravity  $\vec{R}_{G}$.

A little manipulation yields

 

$$\frac{1}{(N+1)} \sum_{i=0}^N \langle R_i^2 - 2 \vec{R}_i \cdot \vec{R}_G + R_G^2 \rangle$$ R_g² =

$$\frac{1}{(N+1)} \sum_{i=0}^N \langle R_i^2 \rangle - \frac{1}{(N+1)^2} \sum_{i=0}^N \sum_{j=0}^N \langle \vec{R}_i \cdot \vec{R}_j \rangle$$ =

$$\frac{1}{(N+1)^2} \sum_{i=0}^N \sum_{j=0}^N \langle R_i^2 - \vec{R}_i \cdot \vec{R}_j \rangle$$ =

$$\frac{1}{2(N+1)^2} \sum_{i=0}^N \sum_{j=0}^N \langle (\vec{R}_i-\vec{R} _j)^2 \rangle$$ =

$$\frac{1}{(N+1)^2} \sum_{i=0}^{N-1} \sum_{j=i+1}^N \langle (\vec{R}_i- \vec{R}_j)^2 \rangle .$$ = (1.44)

It is clear that expressions for R_g can easily be obtained using methods similar to the ones of the previous section.