End-point distribution and structure function of polymers

The coefficients in the expansion of the structure function S(k) as small momentum transfer k can be related to universal ratios 〈r2m〉/〈r2〉m, and 〈r2n〉 = (l/N2)σNMn 〈(Rm−Rn)2n 〉. where Rm denotes the position of monomer m. N is the monomer number and the average is taken over the configurations of the polymer. We show that the moments 〈r3m〉 calculated using the end-point distribution PN(r) = AN(r/Nnr0)nexp[−(r/r0Nn)n]. where r is the correlation length exponent. σ = (1−v)−1, and AN a normalization constant give universal ratios in disagreement with Monte Carlo results in two and three dimensions.