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One of the important observable that can be measured is called the radius of gyration, s. It is essentially a measure of the ‘size’ of a polymer chain
with the following mathematical definition:
Note that the quantity is usually describes in its square form as it is easier (and faster) to compute in this form. The Ri is the coordinate position for Bead labelled i. The N is the chain length or number of linkages. There are N+1 beads in a chain of length N.
In words, s is the average distance of all beads from their common centre
of gravity, Rc, or a point where a chain will be in a balanced position when placed on top of an imaginary tip:
The angular brackets is a mathematical notation of an average over many samples (chain patterns). In scientific jargon it is a thermodynamic averaging of the statistical sample. If you are familiar with statistics, the quantity s is similar to a deviation. The accuracy of s depends on the number of samples we collect.
The more sample we collect, the more accurate s can be obtained. Note that there is always an uncertainty associate with any thermodynamic average quantities since we are not able to sample all possible chain structures. It is meaningless to simply describe a value of s for a particular chain structure as it is not a true representative of the overall chain size.
There is another average quantity called the end-to-end distance, or the distance between two terminal beads. It is much simpler and quicker to compute. However, it is only interesting purely from theoretical point of view as it is difficult to measure experimentally. On the other hand, s is an
important quantity and can be measured in experiment.
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