Average density of chain atoms at points along the contour of freely jointed chains

The density of chain atoms in the volume element dV centered on atom i has been obtained for short and long freely jointed (FJ) chains. The Gaussian distribution function is used for long chains, and Treloar's corrections to this distribution are imposed for short chains. For long FJ chains, the density of chain atoms is a maximum when atom i is located midway along the contour of the chain, and it is a minimum when atom i is any of the three atoms at either end of the chain. The ratio of these densities is 2. Short FJ chains can exhibit qualitatively different behavior, with the site of the maximum density shifting from the middle to the end of the chain when the number of bonds decreases from 8 to 7. This shift is imposed by the strong deviations from the Gaussian distribution when n is small.