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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

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


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

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

A little manipulation yields

 
Rg2 = $\displaystyle \frac{1}{(N+1)} \sum_{i=0}^N \langle R_i^2 - 2 \vec{R}_i \cdot
\vec{R}_G + R_G^2 \rangle$  
  = $\displaystyle \frac{1}{(N+1)} \sum_{i=0}^N \langle R_i^2 \rangle - \frac{1}{(N+1)^2}
\sum_{i=0}^N \sum_{j=0}^N \langle \vec{R}_i \cdot \vec{R}_j \rangle$  
  = $\displaystyle \frac{1}{(N+1)^2} \sum_{i=0}^N \sum_{j=0}^N \langle R_i^2 - \vec{R}_i
\cdot \vec{R}_j \rangle$  
  = $\displaystyle \frac{1}{2(N+1)^2} \sum_{i=0}^N \sum_{j=0}^N \langle (\vec{R}_i-\vec{R}
_j)^2 \rangle$  
  = $\displaystyle \frac{1}{(N+1)^2} \sum_{i=0}^{N-1} \sum_{j=i+1}^N \langle (\vec{R}_i-
\vec{R}_j)^2 \rangle .$ (1.44)

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


next up previous contents index
Next: Some results for polyethylene Up: The Rotational Isomeric State Previous: The mean square end-to-end
W.J. Briels