Polymer chains in a d-dimensional space

Let us consider a continuous polymer chain and let r(n) be the position vector of a point on the chain. The coordinate n of the point along the chain varies from — N/2 to + N/2, and it will be assumed that this quantity is proportional to the number of links. The form factor is then defined by 

For large values of q, this expression simplifies and, neglecting the contributions of the chain ends, we may write 

A critical chain, like the Brownian chain or the Kuhnian chain, is characterized by a size exponent v corresponding to the dimensionality D — 1/v. In this case, the order of magnitude of the distances r(n) — (0) contributing to H(q) is given by the length x defined by 

Now we can see that (E.22) is very analogous to (E.20) the form function of a random chain of dimension D, embedded in a d-dimensional space, has the same asymptotic dependence on to q, as a solid body of dimension D, oriented at random in a d-dimensional space (with d D). 

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