Statistics of polymer chains embedded on a dimond lattice. I. Exact results for the lattice version of the freely rotating chain

A polymer of finite length is embedded on a diamond lattice where the angle between adjacent monomers is cos^-1(-1/3)=109°. We show that the characteristic function C_n (k) can be calculated in closed form for an n link chain if all configurations (i.e., trans and gauche) are given equal weights. It is necessary to do a spherical average to get rid of the cubic symmetry artifically imposed by the diamond lattice. This represents the only model, other than the freely jointed chain, for which the characteristic function is known in closed form. This model may be regarded as the lattice version of the freely rotating chain. From the characteristic function we extract the first few moments 〈R²〉_n, 〈R⁴〉_n, and 〈R⁶〉 _n. We show that the results for the second and fourth moments are identical to those for the freely rotating chain, but the sixth moment is different. Various approximations to the probability distribution function W_n (R) [the Fourier transform of C_n (k)] are tested against the exact result.