Ideal chains pair distribution function

According to this result, the pair distribution function of an ideal chain is given by a general function go which depends on the dimensionless quantity... 

Sd is known as the Debye-structure function of an ideal chain, and a plot is shown in Fig. 2.9. In correspondence to the pair distribution function, the Debye-structure function can also be expressed in a reduced form, with v as general variable. Both the equations for the pair distribution function and for the scattering law indicate that all ideal chains are similar to each other, differing only in the length scale as expressed by Rq. 

It is possible to deduce from the scaling law Eq. (2.83) the fractal dimension of expanded chains, as well as characteristic properties of the pair distribution function and the structure function. The fractal dimension d follows from the same argument as applied above for the ideal chains, by estimating the average number of monomers included in a sphere of radius r which is now given by... 

More detailed considerations show that, as in the case of the ideal chains, the pair distribution function of expanded chains may also be presented in a reduced form... 

Equation (2.105) describes the pair distribution function for distances that lie within the chain interior, i.e., for r < Rp. Note the difference to the respective relation for ideal chains, g(r) oc r (Eq. (2.54)). 

The form factor of ideal polymer chain can be calculated explicitly making use of the Gaussian distribution of distances between each pair of the monomer units and is described by the Debye function ... 

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