Polarizability tensor of a Gaussian chain

For the Zimm model, the structure factor can be obtained in the same way if the linearization approximation is used. The result for the 0 

Let Xap(r, N) be the average polarizability of a polymer chain consisting of N units with its end-to-end vector being fixed at r. By symmetry, ar ,jg(r, N) is written as 

The second condition is written as follows. Suppose that the chain is divided into two parts each consisting of Ni and N2 units, then 

Comparison of the coefficients of the power of TV indicates that this equality is satisfied only when 

See general review, Stoclmayer, W. H., Dynamics of chain molecules, in Mokcular Fluids (eds R. Balian and G. Weill). Gordon Breach, New York (1976). 

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