Editing Radius of gyration (section)

==Molecular applications==

[[File:IUPAC definition for radius of gyration.png|thumb|right|550px|link=https://doi.org/10.1351/goldbook.R05121|IUPAC definition for radius of gyration]]

In [[polymer physics]], the radius of gyration is used to describe the dimensions of a [[polymer]] [[ideal chain|chain]].  The radius of gyration of an individual homopolymer with degree of polymerization N at a given time is defined as:<ref>{{Cite journal|last=Fixman|first=Marshall|date=1962|title=Radius of Gyration of Polymer Chains|journal=The Journal of Chemical Physics|volume=36|issue=2|pages=306–310|doi=10.1063/1.1732501|bibcode=1962JChPh..36..306F}}</ref> 
:<math>
  R_\mathrm{g}^2 \ \stackrel{\mathrm{def}}{=}\ 
  \frac{1}{N} \sum_{k=1}^{N} \left|\mathbf{r}_k - \mathbf{r}_\mathrm{mean} \right|^2
</math>

where <math>\mathbf{r}_\mathrm{mean}</math> is the [[mean]] position of the monomers.
As detailed below, the radius of gyration is also proportional to the root mean square distance between the monomers:

:<math>
  R_\mathrm{g}^2 \ \stackrel{\mathrm{def}}{=}\ 
  \frac{1}{2N^2} \sum_{i\ne j} \left| \mathbf{r}_i - \mathbf{r}_j \right|^2
</math>

As a third method, the radius of gyration can also be computed by summing the principal moments of the [[gyration tensor]].

Since the chain [[Chemical structure|conformations]] of a polymer sample are quasi infinite in number and constantly change over time, the "radius of gyration" discussed in polymer physics must usually be understood as a mean over all polymer molecules of the sample and over time.  That is, the radius of gyration which is measured as an ''average'' over time or [[Ensemble average|ensemble]]:

:<math>
  R_{\mathrm{g}}^2 \ \stackrel{\mathrm{def}}{=}\ 
  \frac{1}{N} \left\langle \sum_{k=1}^{N} \left| \mathbf{r}_k - \mathbf{r}_\mathrm{mean} \right|^2 \right\rangle
</math>

where the angular brackets <math>\langle \ldots \rangle</math> denote the [[ensemble average]].

An entropically governed polymer chain (i.e. in so called theta conditions) follows a random walk in three dimensions. The radius of gyration for this case is given by

:<math>R_\mathrm{g} = \frac{1}{\sqrt{6}\ } \ \sqrt{N}\ a</math>

Note that although <math>aN</math> represents the [[contour length]] of the polymer, <math>a</math> is strongly dependent of polymer stiffness and can vary over orders of magnitude. <math>N</math> is reduced accordingly.

One reason that the radius of gyration is an interesting property is that it can be determined experimentally with [[static light scattering]] as well as with [[Small-angle neutron scattering|small angle neutron-]] and [[Small-angle X-ray scattering|x-ray scattering]]. This allows theoretical polymer physicists to check their models against reality.
The [[hydrodynamic radius]] is numerically similar, and can be measured with [[Dynamic Light Scattering]] (DLS).

===Derivation of identity===
To show that the two definitions of <math>R_{\mathrm{g}}^{2}</math> are identical, 
we first multiply out the summand in the first definition:

:<math>
R_{\mathrm{g}}^{2} \ \stackrel{\mathrm{def}}{=}\   
\frac{1}{N} \sum_{k=1}^{N} \left( \mathbf{r}_{k} - \mathbf{r}_{\mathrm{mean}} \right)^{2} = 
\frac{1}{N} \sum_{k=1}^{N} \left[ \mathbf{r}_{k} \cdot \mathbf{r}_{k} + 
\mathbf{r}_{\mathrm{mean}} \cdot \mathbf{r}_{\mathrm{mean}} 
 - 2 \mathbf{r}_{k} \cdot \mathbf{r}_{\mathrm{mean}} \right]
</math>

Carrying out the summation over the last two terms and using the definition of <math>\mathbf{r}_{\mathrm{mean}}</math> gives the formula

:<math>
R_{\mathrm{g}}^{2} \ \stackrel{\mathrm{def}}{=}\   
-\mathbf{r}_{\mathrm{mean}} \cdot \mathbf{r}_{\mathrm{mean}} + 
\frac{1}{N} \sum_{k=1}^{N} \left( \mathbf{r}_{k} \cdot \mathbf{r}_{k} \right)
</math>

On the other hand, the second definition can be calculated in the same way as follows. 
:<math>
\begin{align}
R_{\mathrm{g}}^{2} \ &\stackrel{\mathrm{def}}{=}\ 
  \frac{1}{2N^2} \sum_{i,j} \left| \mathbf{r}_i - \mathbf{r}_j \right|^2 \\
&= \frac{1}{2N^2} \sum_{i,j} \left( \mathbf{r}_{i} \cdot \mathbf{r}_{i} - 2 \mathbf{r}_{i} \cdot \mathbf{r}_{j} + 
\mathbf{r}_{j} \cdot \mathbf{r}_{j} \right) \\
&= \frac{1}{2N^2} \left[ 
N \sum_{i} \left(\mathbf{r}_{i} \cdot \mathbf{r}_{I} \right) - 2 \sum_{i,j} \left(\mathbf{r}_{i} \cdot \mathbf{r}_{j} \right) + N \sum_{j} \left( \mathbf{r}_{j} \cdot \mathbf{r}_{j}\right) \right] \\
&= \frac{1}{N} \sum_{k}^{N} \left( \mathbf{r}_{k} \cdot \mathbf{r}_{k} \right)- \frac{1}{N^2} \sum_{i,j} \left(\mathbf{r}_{i} \cdot \mathbf{r}_{j} \right) \\
&= \frac{1}{N} \sum_{k}^{N} \left(\mathbf{r}_{k} \cdot \mathbf{r}_{k} \right)- \mathbf{r}_{\mathrm{mean}} \cdot \mathbf{r}_{\mathrm{mean}}
\end{align}
</math>
Thus, the two definitions are the same. 

The last transformation uses the relationship 

:<math>
\begin{align}
\frac{1}{N^2}\sum_{i,j} \left(\mathbf{r}_{i} \cdot \mathbf{r}_{j} \right) &= \frac{1}{N^2} \sum_{i} \mathbf{r}_{i} \cdot \left( \sum_{j} \mathbf{r}_{j} \right) \\
&= \frac{1}{N} \sum_{i} \mathbf{r}_{i}\cdot \mathbf{r}_{\mathrm{mean}} \\
&= \mathbf{r}_{\mathrm{mean}} \cdot \mathbf{r}_{\mathrm{mean}}.
\end{align}
</math>